GIFT OF 
 MICHAEL REESE 
 
THE 
 
 THEORY OF SOUND 
 
THE 
 
 THEOEY OF SOUND 
 
 J er - 
 
 BY 
 
 JOHN WILLIAM STRUTT, BARON RAYLEIGH, Sc.D., F.R.S. 
 
 / I 
 
 HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE. 
 
 IN TWO VOLUMES 
 
 VOLUME II. 
 
 SECOND EDITION REVISED AND ENLARGED 
 
 Ionian 
 
 MACMILLAN AND CO., LTD. 
 
 NEW YORK: MACMILLAN & CO. 
 1896 
 
 [All Rights reserved.] 
 
^\ 
 
 First Edition printed 1878. 
 Second Edition revised and enlarged 1896. 
 
 CAMBRIDGE: PRINTED BY j. AND c. F. CLAY, 
 
 AT THE UNIVERSITY PRESS. 
 
PREFACE. 
 
 THE appearance of this second and concluding volume has 
 been delayed by pressure of other work that could not well 
 be postponed. As in Vol. i. the additions down to 348 are 
 indicated by square brackets, or by letters following the number 
 of the section. From that point onwards the matter is new 
 with the exception of 381, which appeared in the first edition 
 as 348. 
 
 The additions to Chapter XIX. deal with aerial vibrations in 
 narrow tubes where the influence of viscosity and heat conduction 
 are important, and with certain phenomena of the second order 
 dependent upon viscosity. Chapter xx. is devoted to capillary 
 vibrations, and the explanation thereby of many beautiful obser- 
 vations due to Savart and other physicists. The sensitiveness of 
 flames and smoke jets, a very interesting department of acoustics, is 
 considered in Chapter XXL, and an attempt is made to lay the 
 foundations of a theoretical treatment by the solution of problems 
 respecting the stability, or otherwise, of stratified fluid motion. 
 371, 372 deal with "bird-calls," investigated by Sondhauss, and 
 with aeolian tones. In Chapter xxil. a slight sketch is given of 
 the theory of the vibrations of elastic solids, especially as regards 
 the propagation of plane waves, and the disturbance due to a 
 harmonic force operative at one point of an infinite solid. The 
 important problems of the vibrations of plates, cylinders and 
 spheres, are perhaps best dealt with in works devoted specially to 
 the theory of elasticity. 
 
 The concluding chapter on the facts and theories of audition 
 could not well have been omitted, but it has entailed labour out of 
 
Vlll PREFACE. 
 
 proportion to the results. A large part of our knowledge upon 
 this subject is due to Helmholtz, but most of the workers who 
 have since published their researches entertain divergent views, in 
 some cases, it would seem, without recognizing how fundamental 
 their objections really are. And on several points the observations 
 recorded by well qualified observers are so discrepant that no satis- 
 factory conclusion can be drawn at the present time. The future 
 may possibly shew that the differences are more nominal than real. 
 In any case I would desire to impress upon the student of this 
 part of our subject the importance of studying Helmholtz's views 
 at first hand. In such a book as the present an imperfect outline 
 of them is all that can be attempted. Only one thoroughly 
 familiar with the Tonempfindungen is in a position to appreciate 
 many of the observations and criticisms of subsequent writers. 
 
 TERLING PLACE, WITHAM. 
 February, 1896 
 
CONTENTS. 
 
 CHAPTER XI. 
 
 236254 
 
 Aerial vibrations. Equality of pressure in all directions. Equations of 
 motion. Equation of continuity. Special form for incompressible fluid. 
 Motion in two dimensions. Stream function. Symmetry about an axis. 
 Velocity-potential. Lagrange's theorem. Stokes' proof. Physical in- 
 terpretation. Thomson's investigation. Circulation. Equation of con- 
 tinuity in terms of velocity-potential. Expression in polar co-ordinates. 
 Motion of incompressible fluid in simply connected spaces is determined 
 by boundary conditions. Extension to multiply connected spaces. Sphere 
 of irrotationally moving fluid suddenly solidified would have no rotation. 
 Irrotational motion has the least possible energy. Analogy with theories 
 of heat and electricity. Equation of pressure. General equation for 
 sonorous motion. Motion in one dimension. Positive and negative pro- 
 gressive waves. Relation between velocity and condensation. Har- 
 monic type. Energy propagated. Half the energy is potential, and 
 half kinetic. Newton's calculation of velocity of sound. Laplace's cor- 
 rection. Expression of velocity in terms of ratio of specific heats. 
 Experiment of Clement and Desormes. Rankine's calculation from 
 Joule's equivalent. Possible effect of radiation. Stokes' investigation. 
 Rapid stifling of the sound. It appears that communication of heat has 
 no sensible effect in practice. Velocity dependent upon temperature. 
 Variation of pitch of organ-pipes. Velocity of sound in water. Exact 
 differential equation for plane waves. Application to waves of theory 
 of steady motion. Only on one supposition as to the law connecting 
 pressure and density can a wave maintain its form without the assist- 
 ance of an impressed force. Explanation of change of type. Poisson's 
 equation. Relation between velocity and condensation in a progressive 
 wave of finite amplitude. Difficulty of ultimate discontinuity. Earn- 
 shaw's integrals. Riemann's investigation. Limited initial disturbance. 
 [Phenomena of second order. Repulsion of resonators. Rotatory force 
 upon a suspended disc due to vibrations. Striations in Kundt's tubes. 
 Konig's theory.] Experimental determinations of the velocity of sound. 
 
CONTENTS. 
 
 CHAPTER XII. 
 255266 
 
 Vibrations in tubes. General form for simple harmonic type. Nodes and 
 loops. Condition for an open end. In stationary vibrations there must 
 be nodes at intervals of |X. Keflection of pulses at closed and open ends. 
 Problem in compound vibrations. Vibration in a tube due to external 
 sources. Both ends open. Progressive wave due to disturbance at open 
 end. Motion originating in the tube itself. Forced vibration of piston. 
 Kundt's experiments. Summary of results. Vibrations of the column 
 of air in an organ-pipe. Relation of length of wave to length of pipe. 
 Overtones. Frequency of an organ-pipe depends upon the gas. Com- 
 parison of velocities of sound in various gases. Examination of 
 vibrating column of air by membrane and sand. By Konig's flames. 
 Curved pipes. Branched pipes. Conditions to be satisfied at the 
 junctions of connected pipes. Variable section. Approximate calcula- 
 tion of pitch for pipes of variable section. Influence of variation of 
 section on progressive waves. Variation of density. 
 
 CHAPTER XIII. 
 267 272 a 
 
 Aerial vibrations in a rectangular chamber. Cubical box. Resonance of 
 rooms. Rectangular tube. Composition of two equal trains of waves. 
 Reflection by a rigid plane wall. [Nodes and loops.] Green's investiga- 
 tion of reflection and refraction of plane waves at a plane surface. Law 
 of sines. Case of air and water. Both media gaseous. Fresnel's ex- 
 pression. Reflection at surface of air and hydrogen. Reflection from 
 warm air. Tyndall's experiments. Total reflection. Reflection from a 
 plate of finite thickness. [Reflection from a corrugated surface. Case 
 where the second medium is impenetrable.] 
 
 CHAPTER XIV. 
 273-295 
 
 Arbitrary initial disturbance in an unlimited atmosphere. Poisson's solu- 
 tion. Verification. Limited initial disturbance. Case of two dimen- 
 sions. Deduction of solution for a disturbance continually renewed. 
 Sources of sound. Harmonic type. Verification of solution. Sources 
 distributed over a surface. Infinite plane wall. Sheet of double 
 sources. Waves in three dimensions, symmetrical about a point. Har- 
 monic type. A condensed or rarefied wave cannot exist alone. Con- 
 tinuity through pole. Initial circumstances. Velocity-potential of a 
 given source. Calculation of energy emitted. Speaking trumpet. 
 Theory of conical tubes. Position of nodes. Composition of vibrations 
 from two simple sources of like pitch. Interference of sounds from 
 electrically maintained tuning forks. Points of silence. Existence 
 
CONTENTS. XI 
 
 PAGE 
 
 often to be inferred from considerations of symmetry. Case of bell. 
 Experimental methods. Mayer's experiment. Sound shadows. Aperture 
 in plane screen. Fresnel's zones. General explanation of shadows. 
 Oblique screen. Conditions of approximately complete reflection. 
 Diverging Waves. Variation of intensity. Foci. Eeflection from 
 curved surfaces. Elliptical and parabolic reflectors. Fermat's prin- 
 ciple. Whispering galleries. Observations in St Paul's cathedral. 
 Probable explanation. Kesonance in buildings. Atmospheric refrac- 
 tion of sound. Convective equilibrium of temperature. Differential 
 equation to path of ray. Eefraction of sound by wind. Stokes' 
 explanation. Law of refraction. Total reflection from wind overhead. 
 In the case of refraction by wind the course of a sound ray is not 
 reversible. Observations by Eeynolds. Tyndall's observations on fog 
 signals. Law of divergence of sound. Speaking trumpet. Diffraction 
 of sound through a small aperture in an infinite screen. [Experiments 
 on diffraction. Circular grating. Shadow of circular disc. ] Extension 
 of Green's theorem to velocity -potentials. Helmholtz's theorem of reci- 
 procity. Application to double sources. Variation of total energy 
 within a closed space. 
 
 CHAPTER XV. 
 296302 . . .149 
 
 Secondary waves due to a variation in the medium. Eelative importance of 
 secondary waves depends upon the wave-length. A region of altered 
 compressibility acts like a simple source, a region of altered density like 
 a double source. Law of inverse fourth powers inferred by method of 
 dimensions. Explanation of harmonic echos. Alteration of character 
 of compound sound. Secondary sources due to excessive amplitude. 
 Alteration of pitch by relative motion of source and recipient. Experi- 
 mental illustrations of Doppler's principle. Motion of a simple source. 
 Vibrations in a rectangular chamber due to internal sources. Simple 
 source situated in an unlimited tube. ^nergv_emitted. Comparison 
 with conical tube. Further discussion of the motion. Calculation of 
 the reaction of the air on a vibrating circular plate, whose plane is com- 
 pleted by a fixed flange. Equation of motion for the plate. Case of 
 coincidence of natural and forced periods. 
 
 CHAPTER XVI. 
 303 322 & . 170 
 
 Jd 
 
 Theory of resonators. Eesonator composed of a piston and air reservoir. 
 Potential energy of compression. Periodic time. In a large class of air 
 resonators the compression is sensibly uniform throughout the reservoir, 
 and the kinetic energy is sensibly confined to the neighbourhood of the 
 air passages. Expression of kinetic energy of motion through passages 
 in terms of electrical conductivity. Calculation of natural pitch. Case 
 of several channels. Superior and inferior limits to conductivity of 
 
Xll CONTENTS. 
 
 PAGE 
 
 channels. Simple apertures. Elliptic aperture. Comparison with cir- 
 cular aperture of equal area. In many cases a calculation based on area 
 only is sufficient. Superior and inferior limits to the conductivity of 
 necks. Correction to length of passage on account of open end. Con- 
 ductivity of passages bounded by nearly cylindrical surfaces of revolution. 
 Comparison of calculated and observed pitch. Multiple resonance. 
 Calculation of periods for double resonator. Communication of energy 
 to external atmosphere. Kate of dissipation. Numerical example. 
 Forced vibrations due to an external source. Helmholtz's theory of 
 open pipes. Correction to length. Eate of dissipation. Influence of 
 flange. Experimental methods of determining the pitch of resonators. 
 Discussion of motion originating within an open pipe. Motion due to 
 external sources. Effect of enlargement at a closed end. Absorption of 
 sound by resonators. Quincke's tubes. Operation of a resonator close 
 to a source of sound. Keinforcement of sound by resonators. Ideal 
 resonator. Operation of a resonator close to a double source. Savart's 
 experiment. Two or more resonators. Question of formation of jets 
 during sonorous motion. [Free vibrations initiated. Influence of wind 
 upon pitch of organ-pipes. Maintaining power of wind. Overtones. 
 Mutual influence of neighbouring organ -pipes. Whistling. Maintenance 
 of vibrations by heat. Trevelyan's rocker. Communication of heat and 
 aerial vibrations. Singing flames. Sondhauss' observations. Sounds 
 discovered by Kijke and Bosscha. Helmholtz's theory of reed 
 instruments.] 
 
 CHAPTER XVII. 
 323335 236 
 
 Applications of Laplace's functions to acoustical problems. General solution 
 involving the term of the w th order. Expression for radial velocity. Di- 
 vergent waves. Origin at a spherical surface. The formation of sonorous 
 waves requires in general a certain area of moving surface ; otherwise the 
 mechanical conditions are satisfied by a local transference of air without 
 appreciable condensation or rarefaction. Stokes' discussion of the effect 
 of lateral motion. Leslie's experiment. Calculation of numerical results. 
 The term of zero order is usually deficient when the sound originates in 
 the vibration of a solid body. Eeaction of the surrounding air on a 
 rigid vibrating sphere. Increase of effective inertia. When the sphere 
 is small in comparison with the wave-length, there is but little commu- 
 nication of energy. Vibration of an ellipsoid. Multiple sources. In 
 cases of symmetry Laplace's functions reduce to Legendre's functions. 
 [Table of zonal harmonics.] Calculation of the energy emitted from a 
 vibrating spherical surface. Case when the disturbance is limited to a 
 small part of the spherical surface. Numerical results. Effect of a 
 small sphere situated close to a source of sound. Analytical trans- 
 formations. Case of continuity through pole. Analytical expressions 
 for the velocity-potential. Expression in terms of Bessel's functions of 
 fractional order. Particular cases. Vibrations of gas confined within a 
 rigid spherical envelope. Radial vibrations. Diametral vibrations. 
 Vibrations expressed by a Laplace's function of the second order. Table 
 
CONTENTS. Xlll 
 
 PAGE 
 
 of wave-lengths. Relative pitch of various tones. General motion ex- 
 pressible by simple vibrations. Case of uniform initial velocity. Vibra- 
 tions of gas included between concentric spherical surfaces. Spherical 
 sheet of gas. Investigation of the disturbance produced when plane 
 waves of sound impinge upon a spherical obstacle. Expansion of the 
 velocity-potential of plane waves. Sphere fixed and rigid. Intensity of 
 secondary waves. Primary waves originating in a source at a finite 
 distance. Symmetrical expression for secondary waves. Case of a 
 gaseous obstacle. Equal compressibilities. 
 
 CHAPTER XVIII. 
 336343 285 
 
 Problem of a spherical layer of air. Expansion of velocity-potential in 
 Fourier's series. Differential equation satisfied by each term. Ex- 
 pressed in terms of fj, and of v. Solution for the case of symmetry. 
 Condition to be satisfied when the poles are not sources. Reduction 
 to Legendre's functions. Conjugate property. Transition from sphe- 
 rical to plane layer. Bessel's function of zero order. Spherical 
 layer bounded by parallels of latitude. Solution for spherical layer 
 bounded by small circle. Particular cases soluble by Legendre's func- 
 tions. General problem for unsymmetrical motion. Transition to 
 two dimensions. Complete solution for entire sphere in terms of 
 Laplace's functions. Expansion of an arbitrary function. Formula 
 of derivation. Corresponding formula in Bessel's functions for two 
 dimensions. Independent investigation of plane problem. Transverse 
 vibrations in a cylindrical envelope. Case of uniform initial velocity. 
 Sector bounded by radial walls. Application to water waves. Vibra- 
 tions, not necessarily transverse, within a circular cylinder with plane 
 ends. Complete solution of differential equation without restriction 
 as to absence of polar source. Formula of derivation. Expression of 
 velocity-potential by descending semi-convergent series. Case of purely 
 divergent wave. Stokes' application to vibrating strings. Importance 
 of sounding-boards. Prevention of lateral motion. Velocity-potential 
 of a linear source. Significance of retardation of \. Problem of 
 plane waves impinging upon a cylindrical obstacle. Fixed and rigid 
 cylinder. Mathematically analogous problem relating to the transverse 
 vibrations of an elastic solid. Application to theory of light. Tyndall's 
 experiments shewing the smallness of the obstruction to sound offered 
 by fabrics, whose pores are open. [Reflection from series of equidistant 
 and parallel sheets.] 
 
 CHAPTER XIX. 
 344352 .312 
 
 Fluid Friction. Nature of viscosity. Coefficient of viscosity. Independent 
 of the density of the gas. Maxwell's experiments. Comparison of 
 equations of viscous motion with those applicable to an elastic solid. 
 Assumption that a motion of uniform dilatation or contraction is not 
 opposed by viscous forces. Stokes' expression for dissipation function. 
 
XIV CONTENTS. 
 
 PAGE 
 
 Application to theory of plane waves. Gradual decay of harmonic 
 waves maintained at the origin. To a first approximation the velocity 
 of propagation is unaffected by viscosity. Numerical calculation of 
 coefficient of decay. The effect of viscosity at atmospheric pressure is 
 sensible for very high notes only. A hiss becomes inaudible at a mode- 
 rate distance from its source. In rarefied air the effect of viscosity is 
 much increased. Transverse vibrations due to viscosity. Application 
 to calculate effects of viscosity on vibrations in narrow tubes. Helm- 
 holtz's and Kirchhoff's results. [Kirchhoff's investigation. Plane 
 waves. Symmetry round an axis. Viscosity small.] Observations of 
 Schneebeli and Seebeck. [Exceedingly small tubes. Porous wall. Ee- 
 sonance of buildings. Dvorak's observation on circulation due to vibra- 
 tion in Kundt's tubes. Theoretical investigation.] 
 
 CHAPTER XX. 
 353364 343 
 
 [Waves moving under gravity and cohesion. Kelvin's formula. Minimum 
 velocity of propagation. Numerical values for water. Capillary tension 
 determined by method of ripples. Values for clean and greasy water. 
 Faraday's crispations. They have a period double that of the support. 
 Lissajous' phenomenon. Standing waves on running water. Scott 
 Russell's wave pattern. Equilibrium of liquid cylinder. Potential 
 energy of small deformation. Plateau's theorem. Kinetic energy. 
 Frequency equation. Experiments of Bidone and Magnus. Transverse 
 vibrations. Application to determine T for a recently formed surface. 
 Instability. A maximum when \ = 4-51x2a. Numerical estimates. 
 Application of theory to jet. Savart's laws. Plateau's theory. Experi- 
 ments on vibrations of low frequency. Influence of overtones. Bell's 
 experiments. Collisions between drops. Influence of electricity. Obli- 
 que jets. Vibrations of detached drops. Theoretical calculation. 
 Stability due to cohesion may be balanced by instability due to electri- 
 fication. Instability of highly viscous threads, leading to a different 
 law of resolution.] 
 
 CHAPTER XXI. 
 365372 376 
 
 [Plane vortex-sheet. Gravity and capillarity. Infinite thickness. Equal and 
 opposite velocities. Tendency of viscosity. General equation for small 
 disturbance of stratified motion. Case of stability. Layers of uniform 
 vorticity. Fixed walls. Stability and instability. Various cases of 
 infinitely extended fluid. Infinities occurring when n + kU=0. Sensi- 
 tive flames. Early observations thereon. Is the manner of break-down 
 varicose or sinuous? Nodes and loops. Places of maximum action 
 are loops. Dependence upon azimuth of sound. Prejudicial effect of 
 obstructions in the supply pipes. Various explanations. Periodic view 
 of disintegrating smoke-jets. Jets of liquid in liquid. Influence of 
 
CONTENTS. XV 
 
 PAGE 
 
 viscosity. Warm water. Mixture of water and alcohol. Bell's experi- 
 ment. Bird-calls. Sondhauss' laws regulating pitch. Notes examined 
 by flames. Aeolian tones. Strouhal's observations. Aeolian harp 
 vibrates transversely to direction of wind. Dimensional formula.] 
 
 CHAPTER XXII. 
 373381 415 
 
 [Vibrations of solid bodies. General equations. Plane waves dilatational 
 and distortional. Stationary waves. Initial disturbance limited to a 
 finite region. Theory of Poisson and Stokes. Waves from a single 
 centre. Secondary waves dispersed from a small obstacle. Linear 
 source. Linear obstacle. Complete solution for periodic force opera- 
 tive at a single point of an infinite solid. Comparison with Stokes and 
 Hertz. Eeflection of plane waves at perpendicular incidence.] Principle 
 of dynamical similarity. Theory of ships and models. Application of 
 principle of similarity to elastic plates. 
 
 CHAPTER XXIII. 
 382397 432 
 
 [Facts and theories of audition. Eange of pitch over which the ear is capable 
 of perceiving sound. Estimation of pitch. Preyer's observations. 
 Amplitude necessary for audibility. Estimate of Toepler and Boltzmann. 
 Author's observations by whistle and tuning-forks. Binaural audition. 
 Location of sounds. Ohm's law of audition. Necessary exceptions. 
 Two simple vibrations of nearly the same pitch. Bosanquet's observa- 
 tions. Mayer's observation that a grave sound may overwhelm an 
 acute sound, but not vice versa. Effect of fatigue. How best to hear 
 overtones. Helmholtz's theory of audition. Degree of damping of 
 vibrators internal to the ear. Helmholtz's estimate. Mayer's results. 
 How many impulses are required to delimit pitch? Kohlrausch's 
 results. Beats of overtones. Consonant intervals mainly defined thereby. 
 Combination-tones. According to Helmholtz, due to a failure of super- 
 position. In some cases combination-tones exist outside the ear. 
 Difference-tone on harmonium. Helmholtz's theory. Summation- 
 tones. The difficulty in hearing them perhaps explicable by Mayer's 
 observation. Are powerful generators necessary for audibility of 
 difference-tones? Can beats pass into a difference-tone? Periodic 
 changes of suitable pitch are not always recognised as tones. The 
 difference-tone involves a vibration of definite amplitude and phase. 
 Audible difference- tones from inaudible generators. Consonant intervals 
 of pure tones. Helmholtz's views. Delimitation of the Fifth by diffe- 
 rential tones of the second order. Order of magnitude of various 
 differential tones. When the Octave is added, the first differential tone 
 suffices to delimit the Fifth. Does the ear appreciate phase-differences? 
 Helmholtz's observations upon forks. Evidence of mistuned consonances. 
 Lord Kelvin finds the beats of imperfect harmonics perceptible even 
 
XVI CONTENTS. 
 
 PAGE 
 
 when the sounds are faint. Konig's observations and theories. Beat- 
 tones. The wave-siren. Quality of musical sounds as dependent upon 
 upper partials. Willis' theory of vowel sounds. Artificial imitations. 
 Helmholtz's form of the theory. No real inconsistency. Kelative 
 pitch characteristic, versus fixed pitch characteristic. Auerbach's re- 
 sults. Evidence of phonograph. Hermann's conclusions. His analysis 
 of A. Comparison of results by various writers. In Lloyd's view 
 double resonance is fundamental. Is the prime tone present ? Helm- 
 holtz's imitation of vowels by forks. Hermann's experiment. Whispered 
 vowels.] 
 
 NOTE TO 86' . . . . . 479 
 APPENDIX TO CH. V. 1 480 
 
 On the vibrations of compound systems when the amplitudes are not 
 infinitely small. 
 
 NOTE TO 273 2 486 
 
 APPENDIX A. ( 307) 2 487 
 
 On the correction for an open end. 
 
 . 
 
 INDEX OF AUTHORS . . * 492 
 
 INDEX OF SUBJECTS . . ,, 496 
 
 1 Appears now for the first time. 
 
 2 Appeared in the First Edition. 
 
 ERRATA. 
 
 Vol. i. p. 407, footnote. Add reference to Chree, Caml). Phil. Trans., Vol. xiv. 
 p. 250, 1887. 
 
 Vol. ii. p. 46, for A. Konig read W. Konig. 
 
 Vol. ii. p. 236, footnote. Add reference to Gray and Mathews Bessel's Func- 
 tions, Macmillan, 1895. 
 

 CHAPTER XL 
 
 AERIAL VIBRATIONS. 
 
 236. SINCE the atmosphere is the almost universal vehicle of 
 sound, the investigation of the vibrations of a gaseous medium 
 has always been considered the peculiar problem of Physical 
 Acoustics ; but in all, except a few specially simple questions, 
 chiefly relating to the propagation of sound in one dimension, the 
 mathematical difficulties are such that progress has been very 
 slow. Even when a theoretical result is obtained, it often happens 
 that it cannot be submitted to the test of experiment, in default 
 of accurate methods of measuring the intensity of vibrations. In 
 some parts of the subject all that we can do is to solve those 
 problems whose mathematical conditions are sufficiently simple to 
 admit of solution, and to trust to them and to general principles 
 not to leave us quite in- the dark with respect to other questions 
 in which we may be interested. 
 
 In the present chapter we shall regard fluids as perfect, that is 
 to say, we shall assume that the mutual action between any two 
 portions separated by an ideal surface is normal to that surface. 
 Hereafter we shall say something about fluid friction ; but, in 
 general, acoustical phenomena are not materially disturbed by 
 such deviation from perfect fluidity as exists in the case of air 
 and other gases. 
 
 The equality of pressure in all directions about a given point . 
 
 is a necessary consequence of perfect fluidity, whether there be 
 
 rest or motion, as is proved by considering the equilibrium of a 
 
 small tetrahedron under the operation of the fluid pressures, the 
 
 R. II. 1 
 
2 EQUATIONS OF FLUID MOTION. [236. 
 
 impressed forces, and the reactions against acceleration. In the 
 limit, when the tetrahedron is taken indefinitely small, the fluid 
 pressures on its sides become paramount, and equilibrium requires 
 that their whole magnitudes be proportional to the areas of the 
 faces over which they act. The pressure at the point x, y, z will 
 be denoted by p. 
 
 237. If pXdV, pYdV, pZdV, denote the impressed forces 
 acting on the element of mass pdV, the equation of equilibrium 
 is 
 
 dp = p(Xdx + Ydy + Zdz), 
 
 where dp denotes -the variation of pressure corresponding to 
 changes dx, dy, dz in the co-ordinates of the point at which the 
 pressure is estimated. This equation is readily established by 
 considering the equilibrium of a small cylinder with flat ends, the 
 projections of whose axis on those of co-ordinates are respectively 
 dx, dy } dz. To obtain the equations of motion we have, in accord- 
 ance with D'Alembert's Principle, merely to replace X, &c. by 
 X Du/Dt, &c., where DujDt, &c. denote the accelerations of the 
 particle of fluid considered. Thus 
 
 - (x D 
 di f \r TX 
 
 dp_ 
 
 dx 
 dy 
 
 In hydrodynamical investigations it is usual to express the veloci- 
 ties of the fluid u, v, w in terms of x, y, z and t. They then 
 denote the velocities of the particle, whichever it may be, that at 
 the time t is found at the point x, y, z. After a small interval of 
 time dt, a new particle has reached x, y } z\ du/dt . dt expresses 
 the excess of its velocity over that of the first particle, while 
 Du/Dt . dt on the other hand expresses the change in the velocity 
 of the original particle in the same time, or the change of velocity 
 at a point, which is not fixed in space, but moves with the fluid. 
 To this notation we shall adhere. In the change contemplated in 
 d/dt, the position in space (determined by the values of x, y, z) is 
 retained invariable, while in DjDt it is a certain particle of the 
 
237.] EQUATION OF CONTINUITY. 3 
 
 fluid on which attention is fixed. The relation between the two 
 kinds of differentiation with respect to time is expressed by 
 D d d d d 
 
 -T^L = ^I + U -J- + V J- + W J- ( 2 )> 
 
 Dt dt dx dy dz 
 
 and must be clearly conceived, though in a large class of impor- 
 tant problems with which we shall be occupied in the sequel, the 
 distinction practically disappears. Whenever the motion is very 
 small, the terms ud/dx, &c. diminish in relative importance, and 
 ultimately D/Dt^d/dt. 
 
 238. We have further to express the condition that there is 
 no creation or annihilation of matter in the interior of the fluid. 
 If a, ft, 7 be the edges of a small rectangular parallelepiped 
 parallel to the axes of co-ordinates, the quantity of matter which 
 passes out of the included space in time dt in excess of that which 
 enters is 
 
 |<^) ^) + <^)i 
 
 ( dx dy dz } 
 
 and this must be equal to the actual loss sustained, or 
 
 - 
 
 Hence 
 
 dp d(pu) d(pv) d(pw)_ 
 dt* dx dy dz 
 
 the so-called equation of continuity. When p is constant (with 
 respect to both time and space), the equation assumes the simple 
 form 
 
 ^ + J + ^ = (2). 
 
 dx dy dz 
 
 In problems connected with sound, the velocities and the varia- 
 tion of density are usually treated as small quantities. Putting 
 p p (1 + s), where s, called the condensation, is small, and neg- 
 lecting the products u ds/dx, &c., we find 
 
 ds du dv dw , . 
 
 JT T j h i r -7- = U lo). 
 
 dt dx dy dz 
 
 In special cases these equations take even simpler forms. In 
 the case of an incompressible fluid whose motion is entirely 
 parallel to the plane of xy, 
 
 du dv 
 
 -j- + -T- = (4), 
 
 dx dy 
 
 12 
 
4 STREAM-FUNCTION. [238. 
 
 from which we infer that the expression udy vdx is a perfect 
 differential. Calling it dty, we have as the equivalent of (4) 
 
 difr dty 
 
 U = -J L , fl = -_x ................... (6)1 
 
 dy dx 
 
 where -^ is a function of the co-ordinates which so far is perfectly 
 arbitrary. The function ^ is called the s?'0a??i-function, since the 
 motion of the fluid is everywhere in the direction of the curves 
 ^ = constant. When the motion is steady, that is, always the 
 same at the same point of space, the curves ty = constant mark 
 out a system of pipes or channels in which the fluid may be sup- 
 posed to flow. Analytically, the substitution of one function -^ 
 for the two functions u and v is often a step of great consequence. 
 
 Another case of importance is when there is symmetry round 
 an axis, for example, that of x. Everything is then expressible in 
 terms of x and r, where r = V(2/ 2 + &\ an d the motion takes place 
 in planes passing through the axis of symmetry. If the velocities 
 respectively parallel and perpendicular to the axis of symmetry be 
 u and q, the equation of continuity is 
 
 which, as before, is equivalent to 
 
 ty being the stream-function. 
 
 239. In almost all the cases with which we shall have to 
 deal, the hydrodynamical equations undergo a remarkable sim- 
 plification in virtue of a proposition first enunciated by Lagrange. 
 If for any part of a fluid mass udx + vdy + wdz be at one moment 
 a perfect differential d(f>, it will remain so for all subsequent 
 time. In particular, if a fluid be originally at rest, and be then 
 set in motion by conservative forces and pressures transmitted 
 from the exterior, the quantities 
 
 dv dw dw du du dv 
 dz ~~ dy ' dx dz ' dy dx' 
 
 (which we shall denote by f , ij, f) can never depart from zero. 
 
239.] LAGRANGE'S THEOREM. 5 
 
 We assume that p is a function of j), and we shall write for 
 brevity 
 
 The equations of motion obtained from (1), (2), 237, are 
 
 d v du du du du /ON 
 
 -=- X jiu-j v -y w -T- ............ (2), 
 
 dx at dx dy dz 
 
 with two others of the same form relating to y and z. By 
 
 hypothesis, 
 
 dX = dY. 
 dy dx ' 
 
 so that by differentiating the first of the above equations with 
 respect to y and the second with respect to x, and subtracting, 
 we eliminate OT and the impressed forces, obtaining equations 
 which may be put into the form 
 
 D du dv du dv 
 
 with two others of the same form giving D/Dt, 
 
 In the case of an incompressible fluid, we may substitute for 
 du/dx + dv/dy its equivalent dw/dz, and thus obtain 
 
 du . dv dw p //1X 
 
 ' + 6 * ................. (4) ' 
 
 which are the equations used by Helmholtz as the foundation 
 of his theorems respecting vortices. 
 
 If the motion be continuous, the coefficients of f, 77, f in 
 the above equations are all finite. Let L denote their greatest 
 numerical value, and II the sum of the numerical values of f , 77, f . 
 By hypothesis, O is initially zero; the question is whether in 
 the course of time it can become finite. The preceding equa- 
 tions shew that it cannot; for its rate of increase for a given 
 particle is at any time less than 3Zfl, all the quantities con- 
 cerned being positive. Now even if its rate of increase were 
 as great as 3ZH, H would never become finite, as appears from 
 the solution of the equation 
 
6 LAGRANGE'S THEOREM. [239. 
 
 A fortiori in the actual case, H cannot depart from zero, and 
 the same must be true of ( , 77, f. 
 
 It is worth notice that this conclusion would not be disturbed 
 by the presence of frictional forces acting on each particle pro- 
 portional to its velocity, as may be seen by substituting X K u, 
 Y KV, ZKW, for X, Y, Z in (2) 1 . But it is otherwise with 
 the frictional forces which actually exist in fluids, and are de- 
 pendent on the relative velocities of their parts. 
 
 The first satisfactory demonstration of the important pro- 
 position now under discussion was given by Cauchy; but that 
 sketched above is due to Stokes 2 . It is not sufficient merely to 
 shew that if, and whenever, f, 77, f vanish, their differential 
 coefficients Dg/Dt, &c. vanish also, though this is a point that is 
 often overlooked. When a body falls from rest under the action 
 of gravity, s oc s* ; but it does not follow that s never becomes 
 finite. To justify that conclusion it would be necessary to prove 
 that s vanishes in the limit, not merely to the first order, but 
 to all orders of the small quantity t\ which, of course, cannot 
 be done in the case of a falling body. If, however, the equation 
 had been s oc s, all the differential coefficients of s with respect 
 to t would vanish with t, if s did so, and then it might be in- 
 ferred legitimately that s could never vary from zero. 
 
 By a theorem due to Stokes, the moments of momentum about 
 the axes of co-ordinates of any infinitesimal spherical portion 
 of fluid are equal to f, 77, f, multiplied by the moment of 
 inertia of the mass ; and thus these quantities may be regarded 
 as the component rotatory velocities of the fluid at the point to 
 which they refer. 
 
 If ?> *7> ? vanish throughout a space occupied by moving 
 fluid, any small spherical portion of the fluid if suddenly solidified 
 would retain only a motion of translation. A proof of this 
 proposition in a generalised form will be given a little later. 
 Lagrange's theorem thus consists in the assertion that particles 
 of fluid at any time destitute of rotation can never acquire it. 
 
 1 By introducing such forces and neglecting the terms dependent on inertia, we 
 should obtain equations applicable to the motion of electricity through uniform 
 conductors. 
 
 2 Cambridge Trans. Vol. vm. p. 307, 1845. B. A. Report on Hydrodynamics, 
 
 1847. 
 
240.] ROTATORY VELOCITIES. 7 
 
 240. A somewhat different mode of investigation has been 
 adopted by Thomson, which affords a highly instructive view 
 of the whole subject 1 . 
 
 By the fundamental equations 
 
 _ , Du 7 Dv 7 Dw , 
 
 d-n = Xdx + Ydy + Zdz -~- dx - ^ dy - -^7 dz. 
 
 Now Xdx+ Ydy + Zdz = dR, if the forces be conservative, 
 
 and 
 
 Du 7 Dv 1 Dw , 
 
 _ dc +_dy + _ < fe 
 
 D Ddx Ddy Ddz 
 
 = _ (udx + vd y +wdz) - U _ r - V ^- W - m , 
 
 in which 
 
 Ddx 7 Dx 7 
 -el-^^&c, 
 
 Thus, if U' 2 = u z 4- v 2 + w 2 , we have 
 
 dvr = dR jc- (udx +vdy + wdz) + %dU* ......... (1), 
 
 or -(udx + vdy + wdz) = d(R + %U*-vT) ...... (2). 
 
 Integrating this equation along any finite arc PiP 2 , moving 
 with the fluid, we have 
 
 -^ {(udx + vdy + wdz} = (R + ^U*- w) 2 - (R + i U 2 - ts\. . .(3), 
 
 in which suffixes denote the values of the bracketed function 
 at the points P 2 and P l respectively. If the arc be a complete 
 circuit, 
 
 y^7 / (udx + vdy + wdz) = Q ............... (4) ; 
 
 or, in words, 
 
 The line-integral of the tangential component velocity round 
 any closed curve of a moving fluid remains constant throughout all 
 time. 
 
 The line-integral in question is appropriately called the circu- 
 lation, and the proposition may be stated : 
 
 The circulation in any closed line moving with the fluid re- 
 mains constant. 
 
 1 Vortex Motion. Edinburgh Transactions, 1869. 
 
8 CIRCULATION. [240. 
 
 In a state of rest the circulation is of course zero, so that, 
 if a fluid be set in motion by pressures transmitted from the 
 outside or by conservative forces, the circulation along any closed 
 line must ever remain zero, which requires that udx + vdy + wdz 
 be a complete differential. 
 
 But it does not follow conversely that in irrotational motion 
 there can never be circulation, unless it be known that </> is single- 
 valued ; for otherwise fd<> need not vanish round a closed circuit. 
 In such a case all that can be said is that there is no circu- 
 lation round any closed curve capable of being contracted to 
 a point without passing out of space occupied by irrotationally 
 moving fluid, or more generally, that the circulation is the same 
 in all mutually reconcilable closed curves. Two curves are said 
 to be reconcilable, when one can be obtained from the other 
 by continuous deformation, without passing out of the irrota- 
 tionally moving fluid. 
 
 Within an oval space, such as that included by an ellipsoid, all 
 circuits are reconcilable, and therefore if a mass of fluid of that 
 form move irrotationally, there can be no circulation along any 
 closed curve drawn within it. Such spaces are called simply- 
 connected. But in an annular space like that bounded by the 
 surface of an anchor ring, a closed curve going round the ring is 
 not continuously reducible to a point, and therefore there may be 
 circulation along it, even although the motion be irrotational 
 throughout the whole volume included. But the circulation is 
 zero for every closed curve which does not pass round the ring, and 
 has the same constant value for all those that do. 
 
 [In the above theorems " circulation " is defined without 
 reference to mass. If the fluid be of uniform density, the momen- 
 tum reckoned round a closed circuit is proportional to circulation, 
 but in the case of a compressible fluid a distinction must be 
 drawn. The existence of a velocity-potential does not then imply 
 evanescence of the integral momentum reckoned round a closed 
 circuit.] 
 
 241. When udx + vdy + wdz is an exact differential d(j), the 
 velocity in any direction is expressed by the corresponding rate 
 of change of <, which is called the velocity-potential, and 
 
 du dv dw 
 dx dy dz 
 
241.] VELOCITY-POTENTIAL. 
 
 may be replaced by 
 
 dty &$ 
 
 dx* + df dz* ' 
 
 If 8 denote any closed surface, the rate of flow outwards across the 
 element dS is expressed by dS . dcf)/dn, where d(f>/dn is the rate of 
 variation of <f> in proceeding outwards along the normal. In the 
 case of constant density, the total loss of fluid in time dt is thus 
 
 the integration ranging over the whole surface of 8. If the space 
 8 be full both at the beginning and at the end of the time dt, 
 the loss must vanish ; and thus 
 
 d(D ,y -. ,-, x 
 
 -, Gwo = U ( I ). 
 
 dn 
 
 The application of this equation to the element dxdydz gives for 
 the equation of continuity of an incompressible fluid 
 
 <) (2) 
 
 dx* df dz* 
 
 or, as it is generally written, 
 
 V 2 </, = (3); 
 
 when it is desired to work with polar co-ordinates, the trans- 
 formed equation is more readily obtained directly by applying (1) 
 to the corresponding element of volume, than by transforming (2) 
 in accordance with the analytical rules for effecting changes in the 
 independent variables. 
 
 Thus, if we take polar co-ordinates in the plane xy, so that 
 
 x = r cos 0, y r sin 6, 
 we find 
 
 or, if we take polar co-ordinates in space, 
 
 x r sin 6 cos &>, y = r sin 6 sin &>, z = r cos 6, 
 
 2d(f) 1 d ( _. 0d<\ 1 ^ 2 < 
 
 d$J"*Vsii 
 
 Simpler forms are assumed in special cases, such, for example, as 
 that of symmetry round z in (5). 
 
10 PROPERTY OF IRROTATIONAL MOTION. [241. 
 
 When the fluid is compressible, and the motion such that the 
 squares of small quantities may be neglected, the equation of con- 
 tinuity is by (3), 238, 
 
 where any form of V 2 ( may be used that may be most convenient 
 for the problem in hand. 
 
 242. The irrotational motion of incompressible fluid within 
 any simply-connected closed space S is completely determined by 
 the normal velocities over the surface of S. If S be a material 
 envelope, it is evident that an arbitrary normal velocity may be im- 
 pressed upon its surface, which normal velocity must be shared 
 by the fluid immediately in contact, provided that the whole 
 volume inclosed remain unaltered. If the fluid be previously at 
 rest, it can acquire no molecular rotation under the operation of 
 the fluid pressures, which shews that it must be possible to de- 
 termine a function $, such that V 2 < = throughout the space 
 inclosed by S, while over the surface d(f>/dn has a prescribed value, 
 limited only by the condition 
 
 d8=Q (1) - 
 
 An analytical proof of this important proposition is indicated 
 in Thomson and Tait's Natural Philosophy, 317. 
 
 There is no difficulty in proving that but one solution of the 
 problem is possible. By Green's theorem, if V 2 < = 0, 
 
 the integration on the left-hand side ranging over the volume, 
 and on the right over the surface of S. Now if </> and <f> + A< 
 be two functions, satisfying Laplace's equation, and giving pre- 
 scribed surface- values of d(f)/dn, their difference A< is a function 
 also satisfying Laplace's equation, and making d&(j>/dn vanish 
 over the surface of S. Under these circumstances the double 
 integral in (2) vanishes, and we infer that at every point of S 
 d&cfr/dx, dA<j>/dy, d^jdz must be equal to zero. In other words 
 A</> 'must be constant, and the two motions identical. As a par- 
 ticular case, there can be no motion of the irrotational kind 
 
242.] MULTIPLY-CONNECTED SPACES. 11 
 
 within the volume S, independently of a motion of the surface. 
 The restriction to simply-connected spaces is rendered necessary 
 by the failure of Green's theorem, which, as was first pointed 
 out by Helmholtz, is otherwise possible. 
 
 When the space S is multiply- connected, the irrotational 
 motion is still determinate, if besides the normal velocity at 
 every point of S there be given the values of the constant 
 circulations in all the possible irreconcilable circuits. For a 
 complete discussion of this question we must refer to Thomson's 
 original memoir, and content ourselves here with the case of a 
 doubly-connected space, which will suffice for illustration. 
 
 Let A BCD be an endless tube within which fluid moves 
 irrotationally. For this motion there must exist a velocity-poten- 
 tial, whose differential coefficients, 
 expressing, as they do, the com- 
 ponent velocities, are necessarily 
 single-valued, but which need not 
 itself be single-valued. The simplest 
 way of attacking the difficulty pre- 
 sented by the ambiguity of (f>, is to 
 conceive a barrier AB taken across 
 the ring, so as to close the passage. 
 The space ABCDBAEF is then 
 simply continuous, and Green's theo- 
 rem applies to it without modifica- 
 
 tion, if allowance be made for a possible finite difference in the 
 value of (/> on the two sides of the barrier. This difference, if it 
 exist, is necessarily the same at all points of AB, and in the 
 hydrodynamical application expresses the circulation round the 
 ring. 
 
 In applying the equation 
 
 we have to calculate the double integral over the two faces of 
 the barrier as well as over the original surface of the ring. Now 
 
 since -=4 has the same value on the two sides, 
 dn 
 
 |J</> ^ dS (over two faces of AB) = (Jg 
 
12 MULTIPLY-CONNECTED SPACES. [242. 
 
 if K denote the constant difference of </>. Thus, if K vanish, 
 or there be no circulation round the ring, we infer, just as for 
 a simply-connected space, that <f> is completely determined by 
 the surface- values of d^/dn. If there be circulation, < is still 
 determined, if the amount of the circulation be given. For, 
 if </> and </> + A$ be two functions satisfying Laplace's equation 
 and giving the same amount of circulation and the same normal 
 velocities at S, their difference A^> also satisfies Laplace's equa- 
 tion and the condition that there shall be neither circulation 
 nor normal velocities over S. But, as we have just seen, under 
 these circumstances A</> vanishes at every point. 
 
 Although in a doubly-connected space irrotational motion 
 is possible independently of surface normal velocities, yet such 
 a motion cannot be generated by conservative forces nor by 
 motions imposed (at any previous time) on the bounding surface, 
 for we have proved that if the fluid be originally at rest, there 
 can never be circulation along any closed curve. Hence, for 
 multiply-connected as well as simply-connected spaces, if a fluid 
 be set in motion by arbitrary deformation of the boundary, the 
 whole mass comes to rest so soon as the motion of the boundary 
 ceases. 
 
 If in a fluid moving without circulation all the fluid outside 
 a reentrant tube-like surface of uniform section become instan- 
 taneously solid, then also at the same moment all the fluid 
 within the tube comes to rest. This mechanical interpretation, 
 however unpractical, will help the student to understand more 
 clearly what is meant by a fluid having no circulation, and it 
 leads to an extension of Stokes' theorem with respect to mole- 
 cular rotation. For, if all the fluid (moving subject to a 
 velocity-potential) outside a spherical cavity of any radius be- 
 come suddenly solid, the fluid inside the cavity can retain no 
 motion. Or, as we may also state it, any spherical portion of 
 an irrotationally moving [incompressible] fluid becoming suddenly 
 solid would possess only a motion of translation, without rotation 1 . 
 
 A similar proposition will apply to a cylinder disc, or cylinder 
 with flat ends, in the case of fluid moving irrotationally in two 
 dimensions only. 
 
 1 Thomson on Vortex Motion, loc. cit. 
 
242.] ANALOGY WITH HEAT AND ELECTRICITY. 13 
 
 The motion of an incompressible fluid which has been once 
 at rest partakes of the remarkable property ( 79) common to that 
 of all systems which are set in motion with prescribed velocities, 
 namely, that the energy is the least possible. If any other 
 motion be proposed satisfying the equation of continuity and 
 the boundary conditions, its energy is necessarily greater than 
 that of the motion which would be generated from rest 1 . 
 
 243. The fact that the irrotational motion of incompressible 
 fluid depends upon a velocity-potential satisfying Laplace's 
 equation, is the foundation of a far-reaching analogy between 
 the motion of such a fluid, and that of electricity or heat in 
 a uniform conductor, which it is often of great service to bear 
 in mind. The same may be said of the connection between 
 all the branches of Physics which depend mathematically on 
 a potential, for it often happens that the analogous theorems 
 are far from equally obvious. For example, the analytical 
 theorem that, if V 2 > = 0, 
 
 over a closed surface, is most readily suggested by the fluid 
 interpretation, but once obtained may be interpreted for electric 
 or magnetic forces. 
 
 Again, in the theory of the conduction of heat or electricity, 
 it is obvious that there can be no steady motion in the interior 
 of S, without transmission across some part of the bounding 
 surface, but this, when interpreted for incompressible fluids, gives 
 an important and rather recondite law. 
 
 244. When a velocity-potential exists, the equation to deter- 
 mine the pressure may be put into a simpler form. We have from 
 (1), 240, 
 
 dv**dIl-^d<l> + $dU* ................... (1), 
 
 whence by integration 
 
 1 [The reader who wishes to pursue the study of general hydrodynamics is 
 referred to the treatises of Lamb and Basset.] 
 
14 EQUATION OF PRESSURE. [244. 
 
 Now 
 so that 
 
 which is the form ordinarily given. 
 
 If p be constant, I is replaced, of course, by . 
 
 The relation between p and <f> in the case of impulsive motion 
 from rest may be deduced from (2) by integration. We see that 
 
 IP dt = <f> ultimately. 
 
 The same conclusion may be arrived at by a direct application of 
 mechanical principles to the circumstances of impulsive motion. 
 
 If p = /cp, equation (2) takes the form 
 
 R-&-W (3). 
 
 If the motion be such that the component velocities are always the 
 same at the same point of space, it is called steady, and <f> becomes 
 independent of the time. The equation of pressure is then 
 
 d j = *-W W, 
 
 or in the case when there are no impressed forces, 
 
 5 n 1 772 /K\ 
 
 - = O -$u (5). 
 
 In most acoustical applications of (2), the velocities and condensa- 
 tion are small, and then we may neglect the term J 7 2 , and sub- 
 
 stitute for I , if Sp denote the small variable part of p ; thus 
 PQ J P 
 
 PQ J P 
 
 *P-R- d( t> 
 Po' dt 
 which with 
 
 are the equations by means of which the small vibrations of an 
 elastic fluid are to be investigated. 
 
244.] PLANE WAVES. 15 
 
 If a 2 = dp I dp, so that $p = afp^s, (6) becomes 
 
 * = -R-s ..... ....................... (8). 
 
 and we get on elimination of s, 
 
 245. The simplest kind of wave-motion is that in which the 
 excursions of every particle are parallel to a fixed line, and are the 
 same in all planes perpendicular to that line. Let us therefore 
 (assuming that E = 0) suppose that <f> is a function of x (and t) 
 only. Our equation (9) 244 becomes 
 
 
 the same as that already considered in the chapter on Strings. 
 We there found that the general solution is 
 
 $=f(a;-at) + F(a; + at) .................. (2), 
 
 representing the propagation of independent waves in the positive 
 and negative directions with the common velocity a. 
 
 Within such limits as allow the application of the approximate 
 equation (1), the velocity of sound is entirely independent of the 
 form of the wave, being, for example, the same for simple waves 
 
 2?r , x 
 
 9 = A cos (x at), 
 
 A> 
 
 whatever the wave-length may be. The condition satisfied by the 
 positive wave, and therefore by the initial disturbance if a posi- 
 tive wave alone be generated, is 
 
 or by (8) 244 
 
 u-as = Q ............................ (3). 
 
 Similarly, for a negative wave 
 
 u + as = Q ............................. (4). 
 
 Whatever the initial disturbance may be (and u and s are both 
 arbitrary), it can always be divided into two parts, satisfying 
 respectively (3) and (4), which are propagated undisturbed. In 
 
16 PLANE PROGRESSIVE WAVES. [245. 
 
 each component wave the direction of propagation is the same as 
 that of the motion of the condensed parts of the fluid. 
 
 The rate at which energy is transmitted across unit of area of 
 a plane parallel to the front of a progressive wave may be re- 
 garded as the mechanical measure of the intensity of the radiation. 
 In the case of a simple wave, for which 
 
 2-7T 
 
 < = A cos (x at) ..................... (5), 
 
 A, 
 
 the velocity f of the particle at x (equal to dty/dx) is given by 
 
 G). O__ 
 
 Z = -^Asm^(x-at) .................. (6), 
 
 A, X 
 
 and the displacement f is given by 
 
 f jcoe?^(*-aO .................... (7). 
 
 The pressure p=p + Sp ) where by (6) 244 
 
 TT 
 Sp = p Q aA sin (x at) ............... (8). 
 
 A, A, 
 
 Hence, if W denote the work transmitted across unit area of the 
 plane x in time t, 
 
 dW . 
 
 -w=(p + 
 
 Sp) f = -|-p a ( -- ) A 2 + periodic terms. 
 V A, / 
 
 If the integration with respect to time extend over any number of 
 complete periods, or practically whenever its range is sufficiently 
 long, the periodic terms may be omitted, and we may take 
 
 A* ..................... (9); 
 
 or by (3) and (6), if f now denote the maximum value of the 
 velocity and s the maximum value of the condensation, 
 
 TT=ip pa*=i/v^ .................. (10). 
 
 Thus the work consumed in generating waves of harmonic type 
 is the same as would be required to give the maximum velocity f 
 to the whole mass of air through which the waves extend 1 . 
 
 1 The earliest statement of the principle embodied in equation (10) that I have 
 met with is in a paper by Sir W. Thomson, " On the possible density of the 
 luminiferous medium, and on the mechanical value of a cubic mile of sun-light." 
 Phil. Mag. ix. p. 36. 1855. 
 
245.] ENERGY OF PLANE WAVES. 17 
 
 In terms of the maximum excursion f by (7) and (9) 
 
 where r(=\/a) is the periodic time. In a given medium the 
 mechanical measure of the intensity is proportional to the square 
 of the amplitude directly, and to the square of the periodic time 
 inversely. The reader, however, must be on his guard against 
 supposing that the mechanical measure of intensity of undulations 
 of different wave lengths is a proper measure of the loudness of 
 the corresponding sounds, as perceived by the ear. 
 
 In any plane progressive wave, whether the type be harmonic 
 or not, the whole energy is equally divided between the potential 
 and kinetic forms. Perhaps the simplest road to this result is 
 to consider the formation of positive and negative waves from an 
 initial disturbance, whose energy is wholly potential 2 . The total 
 energies of the two derived progressive waves are evidently equal, 
 and make up together the energy of the original disturbance. 
 Moreover, in each progressive wave the condensation (or rare- 
 faction) is one -half of that which existed at the corresponding 
 point initially, so that the potential energy of each progressive 
 wave is one-quarter of that of the original disturbance. Since, as 
 we have just seen, the whole energy is one-half of the same 
 quantity, it follows that in a progressive wave of any type one- 
 half of the energy is potential and one-half is kinetic. 
 
 The same conclusion may also be drawn from the general 
 expressions for the potential and kinetic energies and the relations 
 between velocity and condensation expressed in (3) and (4). 
 The potential energy of the element of volume dV is the work 
 that would be gained during the expansion of the corresponding 
 quantity of gas from its actual to its normal volume, the expansion 
 being opposed throughout by the normal pressure p Q . At any 
 stage of the expansion, when the condensation is s', the effective 
 pressure &p is by 244 a 2 p s', which pressure has to be multiplied 
 by the corresponding increment of volume dV.ds'. The whole 
 work gained during the expansion from dV to dV (I +s) is 
 therefore a 2 p dV '. j ' Q s' 'ds' or ^a~p Q dV.s. The general expressions 
 for the potential and kinetic energies are accordingly 
 
 1 Bosanquet, Phil. Mag. XLV. p. 173. 1873. 
 
 2 Phil. Mag. (5) i. p. 260. 1876. 
 
 R. II. 2 
 
18 NEWTON'S INVESTIGATION. [245. 
 
 potential energy = J a 2 p MJs 2 d V ............... (12), 
 
 kinetic energy = ^pQ\\\u-dV 
 
 and these are equal in the case of plane progressive waves for 
 which 
 
 u = as. 
 
 If the plane progressive waves be of harmonic type, u and s 
 at any moment of time are circular functions of one of the space 
 co-ordinates (#), and therefore the mean value of their squares 
 is one-half of the maximum value. Hence the total energy of 
 the waves is equal to the kinetic energy of the whole mass of 
 air concerned, moving with the maximum velocity to be found in 
 the waves, or to the potential energy of the same mass of air 
 when condensed to the maximum density of the waves. 
 
 [It may be worthy of notice that when terms of the second 
 order are retained, a purely periodic value of u does not correspond 
 to a purely periodic motion. The quantity of fluid which passes 
 unit of area at point x in time dt is pudt, or p (l +s)udt If u 
 be periodic, fudt = 0, but jsudt may be finite. Thus in a positive 
 progressive wave 
 
 fsudt = afs z dt, 
 
 and there is a transference of fluid in the direction of wave 
 propagation.] 
 
 246. The first theoretical investigation of the velocity of 
 sound was made by Newton, who assumed that the relation be- 
 tween pressure and density was that formulated in Boyle's law. If 
 we assume p /cp, we see that the velocity of sound is expressed 
 by \//c, or *Jp -7- Vp, in which the dimensions of p (= force + area) 
 are [M] [L]~ l [T]~' 2 , and those of p ( mass -r volume) are [M] [L]~ 3 . 
 Newton expressed the result in terms of the ' height of ike homo- 
 geneous atmosphere, 1 defined by the equation 
 
 9ph=p ............................... (1), 
 
 where p and p refer to the pressure and the density at the earth's 
 surface. The velocity of sound is thus ^(gh), or the velocity which 
 would be acquired by a body falling freely under the action of 
 gravity through half the height of the homogeneous atmosphere. 
 
 To obtain a numerical result we require to know a pair of 
 simultaneous values of p and p. 
 
246.] LAPLACE'S CORRECTION. 19 
 
 [It is found by experiment 1 that at Cent, under the pressure 
 due (at Paris) to 760 mm. of mercury at the density of dry air 
 is "0012933 gms. per cubic centimetre. If we assume as the 
 density of mercury at 13'5953 2 , and #=980-939, we have 
 in c.G.S. measure 
 
 p = 760 x 13-5953 x 980'939, p = "0012933, 
 whence a = J(p/p) = 27994'5 ; 
 
 so that the velocity of sound at would be 279'945 metres per 
 second, falling short of the result of direct observation by about a 
 sixth part.] 
 
 Newton's investigation established that the velocity of sound 
 should be independent of the amplitude of the vibration, and also 
 of the pitch, but the discrepancy between his calculated value 
 (published in 1687) and the experimental value was not explained 
 until Laplace pointed out that the use of Boyle's law involved 
 the assumption that in the condensations and rarefactions ac- 
 companying sound the temperature remains constant, in contra- 
 diction to the known fact that, when air is suddenly compressed, 
 its temperature rises. The laws of Boyle and Charles supply only 
 one relation between the three quantities, pressure, volume, 
 temperature, of a gas, viz. 
 
 (2), 
 
 where the temperature is measured from the zero of the gas 
 thermometer ; and therefore without some auxiliary assumption it 
 is impossible to specify the connection between p and v (or p). 
 Laplace considered that the condensations and rarefactions con- 
 cerned in the propagation of sound take place with such rapidity 
 that the heat and cold produced have not time to pass away, and 
 that -therefore the relation between volume and pressure is sensibly 
 the same as if the air were confined in an absolutely non-con- 
 ducting vessel. Under these circumstances the change of pressure 
 corresponding to a given condensation or rarefaction is greater 
 than on the hypothesis of constant temperature, and the velocity 
 of sound is accordingly increased. 
 
 1 On the Densities of the Principal Gases, Proc. Roy. Soc. vol. LIII. p. 147, 
 1893. 
 
 2 Volkmann, Wied. Ann. vol. xm. p. 221, 1881. 
 
 22 
 
20 LAPLACE'S CORRECTION. [246. 
 
 In equation (2) let v denote the volume and p the pressure of 
 the unit of mass, and let 6 be expressed in centigrade degrees 
 reckoned from the absolute zero 1 . The condition of the gas (if 
 uniform) is denned by any two of the three quantities p, v, 6, and 
 the third may be expressed in terms of them. The relation 
 between the simultaneous variations of the three quantities is 
 
 dO dp dv 
 
 T~7 H 'T 
 
 In order to effect the change specified by dp and dv, it is 
 in general necessary to communicate heat to the gas. Calling 
 the necessary quantity of heat dQ, we may write 
 
 Suppose now (a) that dp 0. Equations (3) and (4) give 
 
 v 
 
 ^ (p const.) =(~p!z> 
 where -^ (_p const.) expresses the specific heat of the gas under a 
 constant pressure. This being denoted by K P , we have 
 
 V dv/ 6 
 
 Again, suppose (6) that dv = 0. We find in a similar manner 
 that, if K V denote the specific heat under a constant volume, 
 
 In order to obtain the relation between dp and dv when 
 there is no communication of heat, we have only to put dQ = 0. 
 Thus 
 
 or, on substituting for the differential coefficients of Q their values 
 in terms of K V , K P , 
 
 dv dp ^ 
 
 - ........................ 7 - 
 
 Since v l/p, dv/v dp/p ; 
 
 so that a * = ^=P?P = P ,. <(8) 
 
 dp p K V p ' 
 
 1 On the ordinary centigrade scale the absolute zero is about - 273. 
 
246.] EXPERIMENT OF CLEMENT AND DESORMES. 21 
 
 if, as usual, the ratio of the specific heats be denoted by 7. 
 Laplace's value of the velocity of sound is therefore greater than 
 Newton's in the ratio of \/7 : 1- 
 
 By integration of (8), we obtain for the relation between 
 p and p, on the supposition of no communication of heat, 
 
 ................ ........... (9)', 
 
 where p , p are two simultaneous values. Under the same 
 circumstances the relation between pressure and temperature is 
 by (3) 
 
 The magnitude of 7 cannot be determined with accuracy by direct 
 experiment, but an approximate value may be obtained by a 
 method of which the following is the principle. Air is compressed 
 into a reservoir capable of being put into communication with 
 the external atmosphere by opening a wide valve. At first the 
 temperature of the compressed air is raised, but after a time 
 the superfluous heat passes away and the whole mass assumes 
 the temperature of the atmosphere . Let the pressure (measured 
 by a manometer) be p. The valve is now opened for as short 
 a time as is sufficient to permit the equilibrium of pressure to 
 be completely established, that is, until the internal pressure 
 has become equal to that of the atmosphere P. If the experiment 
 be properly arranged, this operation is so quick that the air in the 
 vessel has not sufficient time to receive heat from the sides, and 
 therefore expands nearly according to the law expressed in (9). 
 Its temperature 6 at the moment the operation is complete 
 is therefore determined by 
 
 HlX 1 (11) - 
 
 The enclosed air is next allowed to absorb heat until it has 
 regained the atmospheric temperature , and its pressure (p f ) is 
 then observed. During the last change the volume is constant, 
 and therefore the relation between pressure and temperature 
 gives 
 
 -P 6 
 
 / = <B) 
 
 1 It is here assumed that y is constant. This equation appears to have been 
 given first by Poisson. 
 
22 RATIO OF SPECIFIC HEATS. [246. 
 
 so that by elimination of 
 
 whence 
 
 log 2) - logp 
 
 By experiments of this nature Clement and Desormes de- 
 termined 7 = T3492 ; but the method is obviously not susceptible 
 of any great accuracy. The value of 7 required to reconcile 
 the calculated and observed velocities of sound is 1/408, of the 
 substantial correctness of which there can be little doubt. 
 
 We are not, however, dependent on the phenomena of sound 
 for our knowledge of the magnitude of 7. The value of K P 
 the specific heat at constant pressure has been determined 
 experimentally by Regnault ; and although on account of in- 
 herent difficulties the experimental method 1 may fail to yield 
 a satisfactory result for K V , the information sought for may be 
 obtained indirectly by means of a relation between the two 
 specific heats, brought to light by the modern science of Thermo- 
 dynamics. 
 
 If from the equations 
 
 dQ_ ( dv dp] 
 
 > 
 d0_ dv dp 
 
 /j "r 
 
 6 v p 
 we eliminate dp, there results 
 
 dQ = (tc p - K V )? V 4-tc v d0 ............... (15). 
 
 Let us suppose that dQ 0, or that there is no communication 
 of heat. It is known that the heat developed during the com- 
 pression of an approximately perfect gas, such as air, is almost 
 exactly the thermal equivalent of the work done in compressing 
 it. This important principle was assumed by Mayer in his 
 celebrated memoir on the dynamical theory of heat, though 
 on grounds which can hardly be considered adequate. However 
 that may be, the principle itself is very nearly true, as has since 
 been proved by the experiments of Joule and Thomson. 
 
 If we measure heat in dynamical units, Mayer's principle may 
 be expressed /c v dBpdv on the understanding that there is 
 
 1 [See, however, Joly, Phil. Trans, vol. CLXXXII.A, 1891.] 
 
246.] RANKINE'S CALCULATION. 23 
 
 no communication of heat. Comparing this with (15), we see 
 that 
 
 ic p -* v =R (16), 
 
 and therefore 
 
 .(17). 
 
 The value of pv in gravitation measure (gramme, centimetre) 
 is 1033 -r -001293, at Cent, so that 
 
 J033 
 
 ~ -001293 x272-8~5' 
 
 By Regnault's experiments the specific heat of air is '2379 
 of that of water ; and in order to raise a gramme of water one 
 degree Cent., 42350 gramme-centimetres of work must be done 
 on it. Hence with the same units as for R, 
 
 Kp = -2379 x 42350. 
 
 Calculating from these data, we find 7 = 1*410, agreeing almost 
 exactly with the value deduced from the velocity of sound. This 
 investigation is due to Rankine, who employed in it 1850 to 
 calculate the specific heat of air, taking Joule's equivalent 
 and the observed velocity of sound as data. In this way he 
 anticipated the result of Regnault's experiments, which were 
 not published until 1853. 
 
 247. Laplace's theory has often been the subject of mis- 
 apprehension among students, and a stumblingblock to those 
 remarkable persons, called by De Morgan ' paradoxers.' But there 
 can be no reasonable doubt that, antecedently to all calculation, 
 the hypothesis of no communication of heat is greatly to be 
 preferred to the equally special hypothesis of constant temperature. 
 There would be a real difficulty if the velocity of sound were 
 not decidedly in excess of Newton's value, and the wonder is 
 rather that the cause of the excess remained so long undiscovered. 
 
 The only question which can possibly be considered open, 
 is whether a small part of the heat and cold developed may not 
 escape by conduction or radiation before producing its full effect. 
 Everything must depend on the rapidity of the alternations. 
 Below a certain limit of slowness, the heat in excess, or defect, 
 would have time to adjust itself, and the temperature would 
 remain sensibly constant. In this case the relation between 
 
24 STOKES' INVESTIGATION [247. 
 
 & 
 
 pressure and density would be that which leads to Newton's value 
 
 of the velocity of sound. On the other hand, above a certain 
 limit of quickness, the gas would behave as if confined in a 
 non-conducting vessel, as supposed in Laplace's theory. Now 
 although the circumstances of the actual problem are better 
 represented by the latter than by the former supposition, there 
 may still (it may be said) be a sensible deviation from the law of 
 pressure and density involved in Laplace's theory, entailing a 
 somewhat slower velocity of propagation of sound. This question 
 has been carefully discussed by Stokes in a paper published 
 in 185 1 1 , of which the following is an outline. 
 
 The mechanical equations for the small motion of air are 
 dp du 
 
 dx=-nt^ ......................... < 1} ' 
 
 with the equation of continuity 
 
 ds du dv dw 
 
 -j- + -J-+ j- + -j-=0 ..................... (2). 
 
 dt dx dy dz 
 
 The temperature is supposed to be uniform except in so far as 
 it is disturbed by the vibrations themselves, so that if 6 denote 
 the excess of temperature, 
 
 (3). 
 
 The effect of a small sudden condensation s is to produce an 
 elevation of temperature, which may be denoted by 0s. Let 
 dQ be the quantity of heat entering the element of volume in 
 time dt, measured by the rise of temperature that it would 
 produce, if there were no condensation. Then (the distinction 
 between D/Dt and d/dt being neglected) 
 
 M_ ds dQ 
 dt~^dt + dt ' 
 
 dQ/dt being a function of 6 and its differential coefficients with 
 respect to space, dependent on the special character of the 
 dissipation. Two extreme cases may be mentioned ; the first 
 when the tendency to equalisation of temperature is . due to 
 conduction, the second when the operating cause is radiation, 
 and the transparency of the medium such that radiant heat is 
 
 1 Phil. Mag. (4) i. 305. 
 
247.] OF EFFECT OF RADIATION. 25 
 
 not sensibly absorbed within a distance of several wave-lengths. 
 In the former case dQ/dt oc V 2 0, and in the latter, which is that 
 selected by Stokes for analytical investigation, dQ/dt oc ( 0), 
 Newton's law of radiation being assumed as a sufficient approxi- 
 mation to the truth. We have then 
 
 dd ds 
 
 In the case of plane waves, to which we shall confine our 
 attention, v and w vanish, while u, p, s, are functions of x (and t) 
 only. Eliminating p and u between (1), (2) and (3), we find 
 
 d 2 s /d^s d 2 0\ 
 .. i i Q . \ 
 
 from which and (5) we get 
 
 'd \ d-s f d \ d^ 
 
 if y be written (in the same sense as before) for 1 + ct{3. 
 
 If the vibrations be harmonic, we may suppose that s varies 
 as e int , and the equation becomes 
 
 ...(7). 
 
 da? K q + ijn 
 
 Let the coefficient of s in (7) be put into the form 
 where 
 
 ^.Jf + Kl 
 
 ^ <f + ry2 ?l 2 
 
 and 
 
 2^ = tan-^ - tan-^ l = tan- ^ ...... (9). 
 
 q q 7 
 
 Equation (7) is then satisfied by terms of the form 
 
 g?fx (cos fy i sin \jt) x 
 
 but (fju being positive, and i/r less than ^TT) if we wish for the 
 expression of the wave travelling in the positive direction, we 
 must take the lower sign. Discarding the imaginary part, we 
 find as the appropriate solution 
 
 s = Ae-* s[ll * x cos (nt - /j, cos ty x) ............ (10). 
 
26 THE AMPLITUDE IS MORE [247. 
 
 The first thing to be noticed is that the sound cannot be 
 propagated to a distance unless sin ty be insensible. 
 
 The velocity of propagation ( V) is 
 
 V = n ft' 1 sec ty (H)> 
 
 which, when sin -^ is insensible, reduces to 
 
 F=?i A t- 1 (12). 
 
 Now from (9) we see that ty cannot be insensible, unless 
 q/n is either very great, or very small. On the first supposition 
 from (11), or directly from (7), we have approximately, F=V* 
 (Newton) ; and on the second, V V(*7) (Laplace), as ought 
 evidently to be the case, when the meaning of q in (5) is con- 
 sidered. What we now learn is that, if q and n were comparable, 
 the effect would be not merely a deviation of V from either of 
 the limiting values, but a rapid stifling of the sound, which we 
 know does not take place in nature. 
 
 Of this theoretical result we may convince ourselves, as 
 Stokes explains, without the use of analysis. Imagine a mass 
 of air to be confined within a closed cylinder, in which a piston 
 is worked with a reciprocating motion. If the period of the 
 motion be very long, the temperature of the air remains nearly 
 constant, the heat developed by compression having time to 
 escape by conduction or radiation. Under these circumstances 
 the pressure is a function of volume, and whatever work has 
 to be expended in producing a given compression is refunded 
 when the piston passes through the same position in the reverse 
 direction; no work is consumed in the long run. Next suppose 
 that the motion is so rapid that there is no time for the heat 
 and cold developed by the condensations and rarefactions to 
 escape. The pressure is still a function of volume, and no work 
 is dissipated. The only difference is that now the variations 
 of pressure are more considerable than before in comparison 
 with the variations of volume. We see how it is that both on 
 Newton's and on Laplace's hypothesis the waves travel without 
 dissipation, though with different velocities. 
 
 But in intermediate cases, when the motion of the piston 
 is neither so slow that the temperature remains constant nor 
 so quick that the heat has no time to adjust itself, the result 
 is different. The work expended in producing a small condensa- 
 
247.] INFLUENCED THAN THE VELOCITY. 27 
 
 tion is no longer completely refunded during the corresponding 
 rarefaction on account of the diminished temperature, part of 
 the heat developed by the compression having in the meantime 
 escaped. In fact the passage of heat by conduction or radiation 
 from a warmer to a finitely colder body always involves dissipa- 
 tion, a principle which occupies a fundamental position in the 
 science of Thermodynamics. In order therefore to maintain the 
 motion of the piston, energy must be supplied from without, 
 and if there be only a limited store to be drawn from, the motion 
 must ultimately subside. 
 
 Another point to be noticed is that, if q and n were com- 
 parable, V would depend upon n t viz. on the pitch of the sound, 
 a state of things which from experiment we have no reason to 
 suspect. On the contrary the evidence of observation goes to 
 prove that there is no such connection. 
 
 From (10) we see that the falling off in the intensity, esti- 
 mated per wave-length, is a maximum with tan i/r, or ^ ; and 
 by (9) >Jr is a maximum when q : n = \Ay. In this case 
 
 p = W tf-i 7 -i } 2^ = tan- 1 7* - tan" 1 7-* (13), 
 
 whence, if we take 7 = T36, 2^ = 8 47'. 
 
 Calculating from these data, we find that for each wave- 
 length of advance, the amplitude of the vibration would be 
 diminished in the ratio '6172. 
 
 To take a numerical example, let 
 
 T = sfe f a second, X = wave-length = 44 inches [112cm.]. 
 
 In 20 yards [1828 cm.] the intensity would be diminished in 
 the ratio of about 7 millions to one. 
 
 Corresponding to this, 
 
 g = 219S (14). 
 
 If the value of q were actually that just written, sounds of 
 the pitch in question would be very rapidly stifled. We there- 
 fore infer that q is in fact either much greater or else much less. 
 But even so large a value as 2000 is utterly inadmissible, as 
 we may convince ourselves by considering the significance of 
 equation (5). 
 
28 EFFECT OF CONDUCTION. [247. 
 
 Suppose that by a rigid envelope transparent to radiant heat, 
 the volume of a small mass of gas were maintained constant, 
 then the equation to determine its thermal condition at any 
 time is 
 
 whence 6 = Ae~ qt ........................... (15), 
 
 where A denotes the initial excess of temperature, proving that 
 after a time l/q the excess of temperature would fall to less than 
 half its original value. To suppose that this could happen in a 
 two thousandth of a second of time would be in contradiction to 
 the most superficial observation. 
 
 We are therefore justified in assuming that q is very small 
 in comparison with n, and our equations then become ap- 
 proximately 
 
 (16). 
 
 The effects of a small radiation of heat are to be sought for 
 rather in a damping of the vibration than in an altered velocity of 
 propagation. 
 
 Stokes calculates that if 7 = 1-414, F=1100, the ratio (N : 1) 
 in which the intensity is diminished in passing over a distance x, 
 is given by Iog 10 N = '0001 156 qx in foot-second measure. Although 
 we are not able to make precise measurements of the intensity of 
 sound, yet the fact that audible vibrations can be propagated for 
 many miles excludes any such value of q as could appreciably 
 affect the velocity of transmission. 
 
 Neither is it possible to attribute to the air such a conducting 
 power as could materially disturb the application of Laplace's 
 theory. In order to trace the effects of conduction, we have only 
 to replace q in (5) by q'd*/dx?. Assuming as a particular 
 solution 
 
 s = Ae i(nt+mx) , 
 
 we find m*inicy = in 3 + q'n-m? icq'm*, 
 
247.] VELOCITY DEPENDENT UPON TEMPERATURE. 29 
 
 whence, if q' be relatively small, 
 
 n /, 7 1 q'n .\ 
 m = -- (1- Y I i) .................. (17). 
 
 V(*y) V 7 2*7 / 
 
 Thus the solution in real quantities is 
 
 ...... (18), 
 
 7 
 
 leaving the velocity of propagation to this order of approximation 
 still equal to ^(tcy). 
 
 From (18) it appears that the first effect of conduction, as 
 of radiation, is on the amplitude rather than on the velocity of 
 propagation. In truth the conducting power of gases is so feeble r 
 and in the case of audible sounds at any rate the time during 
 which conduction can take place is so short, that disturbance from 
 this cause is not to be looked for. 
 
 In the preceding discussions the waves are supposed to be 
 propagated in an open space. When the air is confined within 
 a tube, whose diameter is small in comparison with the wave- 
 length, the conditions of the problem are altered, at least in the 
 case of conduction. What we have to say on this head will, 
 however, come more conveniently in another place. 
 
 248. From the expression *J(py/p), we see that in the same 
 gas the velocity of sound is independent of the density, because if 
 the temperature be constant, p varies as p (p = Rpd). On the 
 other hand the velocity of sound is proportional to the square 
 root of the absolute temperature, so that if a be its value at 
 Cent. 
 
 / w 
 
 a = a y 1 + 273 
 
 where the temperature is measured in the ordinary manner from 
 the freezing point of water. 
 
 The most conspicuous effect of the dependence of the velocity 
 of sound on temperature is the variability of the pitch of organ 
 pipes. We shall see in the following chapters that the period 
 of the note of a flue organ-pipe is the time occupied by a pulse 
 in running over a distance which is a definite multiple of the 
 length of the pipe, and therefore varies inversely as the velocity 
 of propagation. The inconvenience arising from this alteration 
 
30 VELOCITY OF SOUND IN WATER. [248. 
 
 of pitch is aggravated by the fact that the reed pipes are not 
 similarly affected ; so that a change of temperature puts an organ 
 out of tune with itself. 
 
 Prof. Mayer 1 has proposed to make the connection between 
 temperature and wave-length the foundation of a pyrometric 
 method, but I am not aware whether the experiment has ever 
 been carried out. 
 
 The correctness of (1) as regards air at the temperatures of 
 and 100 has been verified experimentally by Kundt. See 260. 
 
 In different gases at given temperature and pressure a is 
 inversely proportional to the square roots of the densities, at least 
 if 7 be constant 2 . For the non-condensable gases 7 does not 
 sensibly vary from its value for air. [Thus in the case of hydrogen 
 the velocity is greater than for air in the ratio 
 
 V(1'2933) : V008993), 
 or 3-792 : 1.] 
 
 The velocity of sound is not entirely independent of the 
 degree of dry ness of the air, since at a given pressure moist air 
 is somewhat lighter than dry air. It is calculated that at 50 F. 
 [10 C.], air saturated with moisture would propagate sound 
 between 2 and 3 feet per second faster than if it were perfectly 
 dry. [1 foot = 30'5 cm.] 
 
 The formula a 2 = dp/dp may be applied to calculate the velocity 
 of sound in liquids, or, if that be known, to infer conversely the 
 coefficient of compressibility. In the case of water it is found by 
 experiment that the compression per atmosphere is '0000457. 
 Thus, if dp = 1033 x 981 in absolute C.G.s. units, 
 
 dp = -0000457, since p=l. 
 Hence a = 1489 metres per second, 
 
 which does not differ much from the observed value (1435). 
 
 249. In the preceding sections the theory of plane waves 
 has been derived from the general equations of motion. We 
 
 1 On an Acoustic Pyrometer. Phil. Mag. XLV. p. 18, 1873. 
 
 - According to the kinetic theory of gases, the velocity of sound is determined 
 solely by, and is proportional to, the mean velocity of the molecules. Preston, 
 Phil Mag. (5) IIL p. 441, 1877. [See also Waterston (1846), Phil. Trans, vol. 
 CLXXXIII. A, p. 1, 1892.] 
 
249.] EXACT DIFFERENTIAL EQUATION. 31 
 
 now proceed to an independent investigation in which the motion 
 is expressed in terms of the actual position of the layers of air 
 instead of by means of the velocity-potential, whose aid is no 
 longer necessary inasmuch as in one dimension there can be no 
 question of molecular rotation. 
 
 If y } y + dy/dx.dx, define the actual positions at time t of 
 neighbouring layers of air whose equilibrium positions are defined 
 by x and x + dx, the density p of the included slice is given by 
 
 : Po-1 = ^- (1) 
 
 dx 
 
 whence by (9) 246, 
 
 > 
 
 the expansions and condensations being supposed to take place 
 according to the adiabatic law. The mass of unit of area of 
 the slice is p Q dx, and the corresponding moving force is 
 
 dpjdx . dx, 
 giving for the equation of motion 
 
 Between (2) and (3) p is to be eliminated. Thus, 
 
 (dx) ~d^^^ dx 2 W' 
 
 Equation (4) is an exact equation defining the actual abscissa 
 y in terms of the equilibrium abscissa x and the time. If the 
 motion be assumed to be small, we may replace (dy/dx)y +l , which 
 occurs as the coefficient of the small quantity d 2 y/dtf, by its 
 approximate value unity ; and (4) then becomes 
 
 T//2 r~ 7/^2 (.*/ 
 
 Uii D() (JLnfj 
 
 the ordinary approximate equation. 
 
 If the expansion be isothermal, as in -Newton's theory, the 
 equations corresponding to (4) and (5) are obtained by merely 
 putting 7=1. 
 
 Whatever may be the relation between p and p, depending on 
 
32 WAVES OF PERMANENT TYPE. [249. 
 
 the constitution of the medium, the equation of motion is by 
 
 (1) and (3) 
 
 * 
 
 ^daej dt 2 dp dx~" 
 
 from which p, occurring in dp/dp, is to be eliminated by means of 
 the relation between p and dyjdx expressed in (1). 
 
 250. In the preceding investigations of aerial waves we 
 have supposed that the air is at rest except in so far as it is 
 disturbed by the vibrations of sound, but we are of course at 
 liberty to attribute to the whole mass of air concerned any 
 common motion. If we suppose that the air is moving in the 
 direction contrary to that of the waves and with the same actual 
 velocity, the wave form, if permanent, is stationary in space, 
 and the motion is steady. In the present section we will consider 
 the problem under this aspect, as it is important to obtain all 
 possible clearness in our views on the mechanics of wave propaga- 
 tion. 
 
 If u , p , p denote respectively the velocity, pressure, and 
 density of the fluid in its undisturbed state, and if u, p, p be the 
 corresponding quantities at a point in the wave, we have for the 
 equation of continuity 
 
 and by (5) 244 for the equation of energy 
 
 .--' - 
 
 Eliminating u, we get 
 
 JpoP P*"' 
 
 determining the law of pressure under which alone it is possible 
 for a stationary wave to maintain itself in fluid moving with 
 velocity u . From (3) 
 
 (4), 
 
 I: 
 
 dp V 
 
 U 
 
 2 
 
 or p = constant - - ..................... (5). 
 
 Since the relation between the pressure and the density of 
 actual gases is not that expressed in (5), we conclude that a self- 
 maintaining stationary aerial wave is an impossibility, whatever 
 
250.] WAVES OF PERMANENT TYPE. 33 
 
 may be the velocity u of the general current, or in other words that 
 a wave cannot be propagated relatively to the undisturbed parts 
 of the gas without undergoing an alteration of type. Nevertheless, 
 when the changes of density concerned are small, (5) may be 
 satisfied approximately ; and we see from (4) that the velocity of 
 stream necessary to keep the wave stationary is given by 
 
 which is the same as the velocity of the wave estimated relatively 
 to the fluid. 
 
 This method of regarding the subject shews, perhaps more 
 clearly than any other, the nature of the relation between velocity 
 and condensation 245 (3), (4). In a stationary wave-form a loss 
 of velocity accompanies an augmented density according to the 
 principle of energy, and therefore the fluid composing the con- 
 densed parts of a wave moves forward more slowly than the 
 undisturbed portions. Relatively to the fluid therefore the 
 motion of the condensed parts is in the same direction as that in 
 which the waves are propagated. 
 
 When the relation between pressure and density is other than 
 that expressed in (5), a stationary wave can be maintained only 
 by the aid of an impressed force. By (1) and (2) 237 we have, 
 on the supposition that the motion is steady, 
 
 Y _ du I dp /(n 
 
 A U ~j ~\ ~j \ ' / ? 
 
 cix p ax 
 
 while the relation between u and p is given by (1). If we suppose 
 that p = a?p, (7) becomes 
 
 X = (u 2 - a 2 } d^8 u /3\ 
 
 (IcG * ' 
 
 shewing that an impressed force is necessary at every place where 
 u is variable and unequal to a. 
 
 251. The reason of the change of type which ensues when a 
 wave is left to itself is not difficult to understand. From the 
 ordinary theory we know that an infinitely small disturbance is 
 propagated with a certain velocity a, which velocity is relative 
 to the parts of the medium undisturbed by the wave. Let us 
 consider now the case of a wave so long that the variation 
 R. u. 
 
34 SUPERPOSITION OF PARTICLE VELOCITY. [251. 
 
 velocity and density are insensible for a considerable distance 
 along it, and at a place where the velocity (u) is finite let us 
 imagine a small secondary wave to be superposed. The velocity 
 with which the secondary wave is propagated through the 
 medium is a, but on account of the local motion of the medium 
 itself the whole velocity of advance is a + u, and depends upon 
 the part of the long wave at which the small wave is placed. 
 What has been said of a secondary wave applies also to the parts 
 of the long wave itself, and thus we see that after a time t the 
 place, where a certain velocity u is to be found, is in advance of 
 its original position by a distance equal, not to at, but to (a + u) t : 
 or, as we may express it, u is propagated with a velocity a + u. 
 In symbolical notation u =f{x (a + u) t}, where/ is an arbitrary 
 function, an equation first obtained by Poisson 1 . 
 
 From the argument just employed it might appear at first 
 sight that alteration of type was a necessary incident in the 
 progress of a wave, independently of any particular supposition as 
 to the relation between pressure and density, and yet it was 
 proved in 250 that in the case of one particular law of pressure 
 there would be no alteration of type. We have, however, tacitly 
 assumed in the present section that a is constant, which is tanta- 
 mount to a restriction to Boyle's law. Under any other law of 
 pressure ^(dp/dp) is a function of p, and therefore, as we shall see 
 presently, of u. In the case of the law expressed in (5) 250, the 
 relation between u and p for a progressive wave is such that 
 ^ (dp/dp) + u is constant, as much advance being lost by slower 
 propagation due to augmented density as is gained by superposi- 
 tion of the velocity u. 
 
 So far as the constitution of the medium itself is concerned 
 there is nothing to prevent our ascribing arbitrary values to both 
 u and p, but in a progressive wave a relation between these two 
 quantities must be satisfied. We know already ( 245) that this 
 is the case when the disturbance is small, and the following 
 argument will not only shew that such a relation is to be expected 
 in cases where the square of the motion must be retained, but 
 will even define the form of the relation. 
 
 Whatever may be the law of pressure, the velocity of propaga- 
 tion of small disturbances is by 245 equal to ^/(dp/dp), and in 
 
 1 MSmoire sur la Th6orie du Son. Journal de Vecole poly technique, t. vii. 
 p. 319. 1808. 
 
251.] RELATION BETWEEN VELOCITY AND DENSITY. 35 
 
 a positive progressive wave the relation between velocity and 
 condensation is 
 
 If this relation be violated at any point, a wave will emerge, 
 travelling in the negative direction. Let us now picture to our- 
 selves the case of a positive progressive wave in which the changes 
 of velocity and density are very gradual but become important by 
 accumulation, and let us inquire what conditions must be satisfied 
 in order to prevent the formation of a negative wave. It is clear 
 that the answer to the question whether, or not, a negative wave 
 will be generated at any point will depend upon the state of 
 things in the immediate neighbourhood of the point, and not upon 
 the state of things at a distance from it, and will therefore be 
 determined by the criterion applicable to small disturbances. In 
 applying this criterion we are to consider the velocities and 
 condensations, not absolutely, but relatively to those prevailing in 
 the neighbouring parts of the medium, so that the form of (1) 
 proper for the present purpose is 
 
 f (2); 
 
 /* / //7/M\ x7^ 
 
 whence u 
 
 which is the relation between u and p necessary for a positive 
 progressive wave. Equation (2) was obtained analytically by 
 Earnshaw 1 . 
 
 In the case of Boyle's law, *J(dp/dp) is constant, and the rela- 
 tion between velocity and density, given first, I believe, by 
 Helmholtz 2 , is 
 
 u = alog (4), 
 
 Po 
 
 if p be the density corresponding to u 0. 
 
 In this case Poisson's integral allows us to form a definite idea 
 of the change of type accompanying the earlier stages of the 
 progress of the wave, and it finally leads us to a difficulty which 
 has not as yet been surmounted 3 . If we draw a curve to represent 
 
 1 Phil. Trans. 1859, p. 146. 
 
 2 Fortschritte der Physik, iv. p. 106. 1852. 
 
 3 Stokes, " On a difficulty in the Theory of Sound." Phil. Mag. Nov. 1848. 
 
 32 
 
36 ULTIMATE DISCONTINUITY. [251. 
 
 the distribution of velocity, taking x for abscissa and u for 
 ordinate, we may find the corresponding curve after the lapse of 
 time t by the following construction. Through any point on the 
 original curve draw a straight line in the positive direction parallel 
 to x, and of length equal to (a + u) t, or, as we are concerned with 
 the shape of the curve only, equal to u t. The locus of the ends of 
 these lines is the velocity curve after a time t. 
 
 But this law of derivation cannot hold good indefinitely. The 
 crests of the velocity curve gain continually on the troughs and 
 must at last overtake them. After this the curve would indicate 
 two values of u for one value of as, ceasing to represent anything 
 that could actually take place. In fact we are not at liberty to 
 push the application of the integral beyond the point at which the 
 velocity becomes discontinuous, or the velocity curve has a vertical 
 tangent. In order to find when this happens let us take two 
 neighbouring points on any part of the curve which slopes down- 
 wards in the positive direction, and inquire after what time this 
 part of the curve becomes vertical. If the difference of abscissae 
 be dx, the hinder point will overtake the forward point in the 
 time dx-r-(du). Thus the motion, as determined by Poisson's 
 equation, becomes discontinuous after a time equal to the reci- 
 procal, taken positively, of the greatest negative value of dujdx. 
 
 For example, let us suppose that 
 
 2_ 
 
 u = U cos [x (a + u) t}, 
 A, 
 
 where U is the greatest initial velocity. When t = 0, the greatest 
 negative value of du/dx is 2?r?7/X; so that discontinuity will 
 commence at the time t = \/2,7rU. 
 
 When discontinuity sets in, a state of things exists to which 
 the usual differential equations are inapplicable ; and the subse- 
 quent progress of the motion has not been determined. It is 
 probable, as suggested by Stokes, that some sort of reflection would 
 ensue. In regard to this matter we must be careful to keep 
 purely mathematical questions distinct from physical ones. In 
 practice we have to do with spherical waves, whose divergency 
 may of itself be sufficient to hold in check the tendency to 
 discontinuity. In actual gases too it is certain that before dis- 
 continuity could enter, the law of pressure would begin to change 
 its form, and the influence of viscosity could no longer be neglected. 
 But these considerations have nothing to do with the mathematical 
 
251.] EARNSHAW'S INVESTIGATION. 37 
 
 problem of determining what would happen to waves of finite 
 amplitude in a medium, free from viscosity, whose pressure is 
 under all circumstances exactly proportional to its density ; and 
 this problem has not been solved. 
 
 It is worthy of remark that, although we may of course conceive 
 a wave of finite disturbance to exist at any moment, there is a 
 limit to the duration of its previous independent existence. By 
 drawing lines in the negative instead of in the positive direction 
 we may trace the history of the velocity curve ; and we see that 
 as we push our inquiry further and further into past time the 
 forward slopes become easier and the backward slopes steeper. 
 At a time, equal to the greatest positive value of dx/du, antecedent 
 to that at which the curve is first contemplated, the velocity 
 would be discontinuous. 
 
 252. The complete integration of the exact equations (4) and 
 (6) 249 in the case of a progressive wave was first effected by 
 Earnshaw 1 . Finding reason for thinking that in a sound wave 
 the equation 
 
 (1) 
 
 dt dx 
 
 must always be satisfied, he observed that the result of differen- 
 tiating (1) with respect to t, viz. 
 
 da? " 
 
 can by means of the arbitrary function F be made to coincide 
 with any dynamical equation in which the ratio of d*y/dt 2 and 
 d^yjdx* is expressed in terms of dyjdx. The form of the function 
 F being thus determined, the solution may be completed by the 
 usual process applicable to such cases 2 . 
 
 Writing for brevity a in place of dyjdx, we have 
 
 and the integral is to be found by eliminating a between the 
 equations 
 
 0= x + F'(a)t + '( 
 a being equal to p /p, and <f> being an arbitrary function. 
 
 1 Proceedings of the Royal Society, Jan. 6, 1859. Phil. Trans. 1860, p. 133. 
 
 2 Boole's Differential Equations, Ch. xiv. 
 
38 EARNSHAW'S INVESTIGATION. [252. 
 
 If p = a-p, the exact equation (6 249) is 
 
 dx 
 by comparison of which with (2) we see that 
 
 * = ^ ........................... (5), 
 
 or on integration 
 
 F(ct) = Caloga ........................ (6), 
 
 as might also have been inferred from (4) 251. The constant C 
 vanishes, if F(a.\ viz. u, vanish when a 1, or p = /o ; otherwise 
 it represents a velocity of the medium as a whole, having nothing 
 to do with the wave as such. For a positive progressive wave the 
 lower signs in the ambiguities are to be used. Thus in place of 
 (3), we have 
 
 and u = a log a = a log ..................... (8). 
 
 po 
 
 If we subtract the second of equations (7) from the first, we get 
 y at + at log a = < (a) a </>' (a), 
 
 from which by (8) we see that y-(a + u)t is an arbitrary function 
 of a, or of u. Conversely therefore u is an arbitrary function of 
 y (a + u) t, and we may write 
 
 u=f{y-(a + u)t} ..................... (9). 
 
 Equation (9) is Poisson's integral, considered in the preceding 
 section, where the symbol x has the same meaning as here 
 attaches to y. 
 
 253. The problem of plane waves of finite amplitude attracted 
 also the attention of Riemann, whose memoir was communicated 
 to the Royal Society of Gottingen on the 28th of November, 1859 \ 
 Riemann's investigation is founded on the general hydrodynamical 
 equations investigated in 237, 238, and is not restricted to any 
 particular law of pressure. In order, however, not unduly to 
 
 1 Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. 
 Gottingen, Abhandlungen, t. vui. 1860. See also an excellent abstract in the 
 Fortschritte der Physik, xv. p. 123. [Reference may be made also to a paper by 
 C. V. Burton, Phil. Mag. xxxv. p. 317, 1893.] 
 
253.] RIEMANN'S EQUATIONS. 39 
 
 extend the discussion of this part of our subject, already perhaps 
 treated at greater length than its acoustical importance would 
 warrant, we shall here confine ourselves to the case of Boyle's law 
 of pressure. 
 
 Applying equations (1), (2) of 237 and (1) of 238 to the 
 circumstances of the present problem, we get 
 
 du du , d log p 
 
 + u = -a? &r ..................... QV 
 
 dt dx dx 
 
 dlogp d\ogp_ du 
 dt dx dx" 
 
 If we multiply (2) by + a, and afterwards add it to (1), we 
 obtain 
 
 dP .dP dQ dQ ,. 
 
 I <*+>! -dt=- (u -^lTx ...... (3) ' 
 
 where P = alogp + u, Q = alogp u ............ (4). 
 
 Thus dP = {dx-(u + d)dt} .................. (5), 
 
 (6). 
 
 These equations are more general than Poisson's and Earnshaw's 
 in that they are not limited to the case of a single positive, or 
 negative, progressive wave. From (5) we learn that whatever 
 may be the value of P corresponding to the point x and the time 
 t, the same value of P corresponds to the point x + (u 4- a) dt at 
 the time t + dt; and in the same way from (6) we see that Q 
 remains unchanged when x and t acquire the increments (u a) dt 
 and dt respectively. If P and Q be given at a certain instant of 
 time as functions of x, and the representative curves be drawn, we 
 may deduce the corresponding value of u by (4), and thus, as in 
 251, construct the curves representing the values of P and Q 
 after the small interval of time dt, from which the new values 
 of u arid p in their turn become known, and the process can be 
 repeated. 
 
 The element of the fluid, to which the values of P and Q at 
 any moment belong, is itself moving with the velocity u, so that 
 the velocities of P and Q relatively to the element are numerically 
 the same, and equal to a, that of P being in the positive direction 
 and that of Q in the negative direction. 
 
40 LIMITED INITIAL DISTURBANCE. [253. 
 
 We are now in a position to trace the consequences of an 
 initial disturbance which is confined to a finite portion of the 
 medium, e.g. between x a and x = $, outside which the medium 
 is at rest and at its normal density, so that the values of P and Q 
 are alog/o . Each value of P propagates itself in turn to the ele- 
 ments of fluid which lie in front of it, and each value of Q to those 
 that lie behind it. The hinder limit of the region in which P is 
 variable, viz. the place where P first attains the constant value 
 alogpo, comes into contact first with the variable values of Q, and 
 moves accordingly with a variable 1 velocity. At a definite time, 
 requiring for its determination a solution of the differential equa- 
 tions, the hinder (left hand) limit of the region through which P 
 varies, meets the hinder (right hand) limit of the region through 
 which Q varies, after which the two regions separate themselves, 
 and include between them a portion of fluid in its equilibrium 
 condition, as appears from the fact that the values of P and Q are 
 both alog p . In the positive wave Q has the constant value 
 a log p , so that u = a log (p/po), as in (4) 251 ; in the negative wave 
 P has the same constant value, giving as the relation between u 
 and p, u a log (p/po). Since in each progressive wave, when 
 isolated, a law prevails connecting the quantities u and p, we see 
 that in the positive wave du vanishes with dP, and in the negative 
 wave du vanishes with dQ. Thus from (5) we learn that in a 
 positive progressive wave du vanishes, if the increments of x and 
 t be such as to satisfy the equation dx (u + a)dt = 0, from which 
 Poisson's integral immediately follows. 
 
 It would lead us too far to follow out the analytical develop- 
 ment of Riemann's method, for which the reader must be referred 
 to the original memoir ; but it would be improper to pass over in 
 silence an error on the subject of discontinuous motion into which 
 Riemann and other writers have fallen. It has been held that a 
 state of motion is possible in which the fluid is divided into two 
 parts by a surface of discontinuity propagating itself with constant 
 velocity, all the fluid on one side of the surface of discontinuity 
 being in one uniform condition as to density and velocity, and on 
 the other side in a second uniform condition in the same respects. 
 Now, if this motion were possible, a motion of the same kind 
 in which the surface of discontinuity is at rest would also be 
 
 1 At this point an error seems to have crept into Riemann's work, which is 
 corrected in the abstract of the Fortecliritte d'r PJn/siJ:. 
 
253.] POISSON'S INTEGRAL. 41 
 
 possible, as we may see by supposing a velocity equal and 
 opposite to that with which the surface of discontinuity at first 
 moves, to be impressed upon the whole mass of fluid. In order to 
 find the relations that must subsist between the velocity and 
 density on the one side (u lt p^ and the velocity and density on the 
 other side (w 2 , pz)> we notice in the first place that by the principle 
 of conservation of matter p 2 u 2 = p^. Again, if we consider the 
 momentum of a slice bounded by parallel planes and including the 
 surface of discontinuity, we see that the momentum leaving the 
 slice in the unit of time is for each unit of area (/3 2 w 2 = /3 1 Wi)w 2 , 
 while the momentum entering it is p^i*. The difference of mo- 
 mentum must be balanced by the pressures acting at the boundaries 
 of the slice, so that 
 
 pi Ui (u, - MJ) =_>! -p 2 = a? (/>! - p z ), 
 whence 
 
 The motion thus determined is, however, not possible ; it satisfies 
 indeed the conditions of mass and momentum, but it violates the 
 condition of energy ( 244) expressed by the equation 
 
 Jwa'-iw^a 3 logpj-a 8 Iogp 2 ................ (8). 
 
 This argument has been already given in another form in 250, 
 which would alone justify us in rejecting the assumed motion, since 
 it appears that no steady motion is possible except under the law of 
 density there determined. From equation (8) of that section we 
 can find what impressed forces would be necessary to maintain the 
 motion defined by (7). It appears that the force X, though con- 
 fined to the place of discontinuity, is made up of two parts of 
 opposite signs, since by (7) u passes through the value a. The 
 whole moving force, viz. ^Xp dx, vanishes, and this explains how 
 it is that the condition relating to momentum is satisfied by (7), 
 though the force X be ignored altogether. 
 
 253 a. Among the phenomena of the second order which 
 admit of a ready explanation, a prominent place must be assigned 
 to the repulsion of resonators discovered independently by 
 Dvorak 1 and Mayer 2 . These observers found that an air resonator 
 of any kind (Ch. xvi.) when exposed to a powerful source 
 
 1 Pogg. Ann. CLVII. p. 42, 1876 ; Wied. Ann. in. p. 328, 1878. 
 
 2 Phil. Mag. vol. vi. p. 225, 1878. 
 
42 REPULSION OF RESONATORS. [253 a. 
 
 of sound experiences a force directed inwards from the mouth, 
 somewhat after the manner of a rocket. A combination of 
 four light resonators, mounted anemometer fashion upon a steel 
 point, may be caused to revolve continuously. 
 
 If there be no impressed forces, equation (2) 244 gives 
 
 Distinguishing the values of the quantities at two points of space 
 by suffixes, we may write 
 
 /A A. \ _L 1 TT 1 TT 2 /9\ 
 
 T&1 tT = -J- {<p (p-^) ~r TjrL/o 2^1 \*J* 
 
 This equation holds good at every instant. Integrating it over a 
 long range of time we obtain as applicable to every case of 
 fluid motion in which the flow between the two points does 
 not continually increase 
 
 / jj. r j*. irr7"'-/-7f IfTTs/Jf ( e )\ 
 
 C7j (Jili J w Qiliu := o J U o tliL "~~ i)] U i (JLL \*-^/' 
 
 The first point (with suffix 0) is now to be chosen at such a 
 distance that the variation of pressure and the velocity are 
 there insensible. Accordingly 
 
 This equation is true wherever the second point be taken. If it 
 be in the interior of a resonator, or at a corner where three fixed 
 walls meet, U l 0, and therefore 
 
 /(OTJ w ) dt= (5), 
 
 or the mean value of w in the interior is the same as at a distance 
 outside. 
 
 By (9) 246, if the expansions and contractions be adiabatic, 
 p x pv ; and -a =|) ( v-D/v. Thus 
 
 '-i)/y 
 
 If in (6) we suppose that the difference between p^ and p 
 is comparatively small, we may expand the function there contained 
 by the binomial theorem. The approximate result may be 
 expressed 
 
 ............... ( 7), 
 
 po 
 
 shewing that the mean value of (p 1 p^) is positive, or in other 
 
2 5 3 a. ] ATTRACTIONS . 4 3 
 
 words that the mean pressure in the resonator is in excess of the 
 atmospheric pressure 1 . The resonator therefore tends to move as 
 if impelled by a force acting normally over the area of its 
 aperture and directed inwards. 
 
 The experiment may be made (after Dvorak) with a 
 Helmholtz resonator by connecting the nipple with a horizontal 
 and not too narrow glass tube in which moves a piston of ether. 
 When a fork of suitable pitch, e.g. 256 or 512, is vigorously 
 excited and presented to the mouth of the resonator, the movement 
 of the ether shews an augmentation of pressure, while the similar 
 presentation of the non-vibrating fork is without effect. 
 
 If to the first order of small quantities 
 
 (p-po)/Po = P cos nt (8), 
 
 its mean value correct to the second order is P 2 /4>y, in which for air 
 and the principal gases 7 = 1*4. 
 
 If the expansions and contractions be supposed to take place 
 isothermally, the corresponding result is arrived at by putting 
 7=1 in (7). 
 
 253 b. In 253 a the effect to be explained is intimately 
 connected with the compressibility of the fluid which occupies the 
 interior of the resonator. In the class of phenomena now to be 
 considered the compressibility of the fluid is of secondary import- 
 ance, and the leading features of the explanation may be given 
 upon the supposition that the fluid retains a constant density 
 throughout. 
 
 If p be constant, (4) 253 a may be written 
 
 ffa-pJdt^-frfUSdt (1), 
 
 shewing that the mean pressure at a place where there is 
 motion is less than in the undisturbed parts of the fluid a 
 theorem due to Kelvin 2 , and applied by him to the explanation of 
 the attractions observed by Guthrie and other experimenters. 
 Thus a vibrating tuning-fork, presented to a delicately suspended 
 rectangle of paper, appears to exercise an attraction, the mean 
 value of U- being greater on the face exposed to the fork than 
 upon the back. 
 
 1 Phil. Mag. vol. vi. p. 270, 1878. 
 
 2 Proc. Roy. Soc. vol. xix. p. 271, 1887. 
 
44 A ROTATORY FORCE OPERATIVE [253 6. 
 
 In the above experiment the action depends upon the prox- 
 imity of the source of disturbance. When the flow of fluid, 
 whether steady or alternating, is uniform over a large region, the 
 effect upon an obstacle introduced therein is a question of shape. 
 In the case of a sphere there is manifestly no tendency to turn ; 
 and since the flow is symmetrical on the up-stream and down- 
 stream sides, the mean pressures given by (1) balance one another. 
 Accordingly a sphere experiences neither force nor couple. It is 
 otherwise when the form of the body is elongated or flattened. 
 That a flat obstacle tends to turn its flat side to the stream 1 may 
 be inferred from the general character of Fig. 54 a. 
 
 the lines of flow round it. The pressures 
 at the various points of the surface EC 
 (Fig. 54 a) depend upon the velocities of 
 the fluid there obtaining. The full 
 pressure due to the complete stoppage of 
 the stream is to be found at two points, 
 where the current divides. It is pretty evident that upon the up- 
 stream side this lies (P) on AB, and upon the down-stream side 
 upon AC at the corresponding point Q. The resultant of the 
 pressures thus tends to turn AB so as to face the stream. 
 
 When the obstacle is in the form of an ellipsoid, the mathe- 
 matical calculation of the forces can be effected ; but it must 
 suffice here to refer to the particular case of a thin circular disc, 
 whose normal makes an angle 6 with the direction of the un- 
 disturbed stream. It may be proved 2 that the moment M of the 
 couple tending to diminish has the value given by 
 
 M=%pa s W*sin'20 (2), 
 
 a. being the radius of the disc and W the velocity of the stream. 
 If the stream be alternating instead of steady, we have merely to 
 employ the mean value of TF 2 , as appears from (1). 
 
 The observation that a delicately suspended disc sets itself 
 across the direction of alternating currents of air originated in the 
 attempt to explain certain anomalies in the behaviour of a 
 magnetometer mirror 3 . In illustration, "a small disc of paper, 
 about the size of a sixpence, was hung by a fine silk fibre across 
 
 1 Thomson and Tait's Natural Philosophy, 336, 1867. 
 
 2 W. Konig, Wied. Ann. t. XLIII. p. 51, 1891. 
 
 3 Proc. Roy. Soc. vol. xxxn. p. 110, 1881. 
 
253 6.] UPON A SUSPENDED DISC. 45 
 
 the mouth of a resonator of pitch 128. When a sound of this 
 pitch is excited in the neighbourhood, there is a powerful rush of 
 air into and out of the resonator, and the disc sets itself promptly 
 across the passage. A fork of pitch 128 may be held near the 
 resonator, but it is better to use a second resonator at a little 
 distance in order to avoid any possible disturbance due to the 
 neighbourhood of the vibrating prongs. The experiment, though 
 rather less striking, was also successful with forks and resonators 
 of pitch 256." 
 
 Upon this principle an instrument may be constructed for 
 measuring the intensities of aerial vibrations of selected pitch 1 . 
 A tube, measuring three quarters of a wave length, is open at one 
 end and at the other is closed air-tight by a plate of glass. At 
 one quarter of a wave length's distance from the closed end 
 is hung by a silk fibre a light mirror with attached magnet, such 
 as is used for reflecting galvanometers. In its undisturbed 
 condition the plane of the mirror makes an angle of 45 with the 
 axis of the tube. At the side is provided a glass window, 
 through which light, entering along the axis and reflected by the 
 mirror, is able to escape from the tube and to form a suitable 
 image upon a divided scale. The tube as a whole acts as a 
 resonator, and the alternating currents at the loop ( 255) deflect 
 the mirror through an angle which is read in the usual manner. 
 
 In an instrument constructed by Boys 2 the sensitiveness 
 is exalted to an extraordinary degree. This is effected partly 
 by the use of a very light mirror with suspension of quartz fibre, 
 and partly by the adoption of double resonance. The large 
 resonator is a heavy brass tube of about 10 cm. diameter, closed 
 at one end, and of such length as to resound to e r . The mirror is 
 hung in a short lateral tube forming a communication between 
 the large resonator and a small glass bulb of suitable capacity. 
 The external vibrations may be regarded as magnified first by the 
 large resonator and then again by the small one, so that the 
 mirror is affected by powerful alternating currents of air. The 
 selection of pitch is so definite that there is hardly any response 
 to sounds which are a semi-tone too high or too low. 
 
 Perhaps the most striking of all the effects of alternating 
 aerial currents is the rib-like structure assumed by cork filings in 
 
 1 Phil. Mag. vol. xiv. p. 186, 1882. 
 
 2 Nature, vol. XLII. p. 604, 1890. 
 
46 
 
 EXPLANATION OF THE 
 
 [253 6. 
 
 Fig. 54 I. 
 
 Fig. 54 c. 
 
 Kundt's experiment 260. Close observation, while the vibrations 
 are in progress, shews that the filings are disposed in thin laminae 
 transverse to the tube and extending upwards to a certain distance 
 from the bottom. The effect is a maximum at the loops, and 
 disappears in the neighbourhood of the nodes. When the vibra- 
 tions stop, the laminae necessarily fall, and in so doing lose much 
 of their sharpness, but they remain visible as transverse streaks. 
 
 The explanation of this peculiar behaviour has been given by 
 A. Konig 1 . We have seen that a single spherical obstacle 
 experiences no force from an alternating current. But this 
 condition of things is disturbed by the presence of a neighbour. 
 Consider for simplicity the case of two spheres at a moderate 
 distance apart, and so situated that the line of centres is either 
 parallel to the stream, Fig. 54 b, or 
 perpendicular to it, Fig. 54 c. It is 
 easy to recognise that the velocity 
 between the spheres will be less in 
 the first case and greater in the 
 second than on the averted hemi- 
 spheres. Since the pressure increases 
 as the velocity diminishes, it follows 
 
 that in the first position the spheres will repel one another, 
 and that in the second position they will attract one another. 
 The result of these forces between neighbours is plainly a 
 tendency to aggregate in laminae. The case may be contrasted 
 with that of iron filings in a magnetic field, whose direction 
 is parallel to that of the aerial current. There is then attraction 
 in the first position and repulsion in the second, and the result is 
 a tendency to aggregate in filaments. 
 
 On the foundation of the analysis of Kirchhoff, Konig has 
 
 T 
 
 
 O 
 
 O 
 
 calculated the forces operative in the case 
 of two spheres which are not too close 
 together. If a ly a 2 be the radii of the 
 spheres, r their distance asunder, 6 the 
 angle between the line of centres and the 
 direction of the current taken as axis of 
 z (Fig. 54 d), W the velocity of the current, 
 then the components of force upon the 
 sphere B in the direction of z and of x 
 
 Fig. 54 d. 
 
 O 
 
 1 Wied. Ann. t. XLII. pp. 353, 549, 1891. 
 
2536.] STRIATIONS IN KUKDT's TUBES. 47 
 
 drawn perpendicular to z in the plane containing z and the line 
 of centres, are given by 
 
 - W "- cos 0(3-5 coM) .......... ..(3), 
 
 * W \in(> (1 -.5 cos**) ............ (4), 
 
 the third component F vanishing by virtue of the symmetry. In 
 the case of Fig. 54 b = 0, and there is repulsion equal to 
 QTTpafajW'i/r 4 ' ; in the case of Fig. 54 c = JTT, and the force is an 
 attraction 37rpa 1 s a, 2 s W' 2 /r 4 '. In oblique positions the direction of the 
 force does not coincide with the line of centres. 
 
 If the spheres be rigidly connected, the forces upon the system 
 reduce to a couple (tending to increase 0) of moment given by 
 
 - Z sin e + X cos = * W " s i n 20 ..... (5). 
 
 3 V 
 
 When the current is alternating, we are to take the mean 
 value of W 2 in (3), (4), (5). 
 
 254. The exact experimental determination of the velocity 
 of sound is a matter of greater difficulty than might have been 
 expected. Observations in the open air are liable to errors from 
 the effects of wind, and from uncertainty with respect to the 
 exact condition of the atmosphere as to temperature and dryness. 
 On the other hand when sound is propagated through air con- 
 tained in pipes, disturbance arises from friction and from transfer 
 of heat; and, although no great errors from these sources are 
 to be feared in the case of tubes of considerable diameter, such 
 as some of those employed by Kegnault, it is difficult to feel 
 sure that the ideal plane waves of theory are nearly enough 
 realized. 
 
 The following Table 1 contains a list of the principal experi- 
 mental determinations which have been made hitherto. 
 
 Names of Observers. Velocity of Sound at 
 
 Cent, in Metres. 
 
 Academie des Sciences (1738) ........... , ......... 332 
 
 Benzenberg (1811) ................................. 
 
 Goldingham (1821) ................................. 33M 
 
 Bureau des Longitudes (1822) .................. 330'6 
 
 Moll and van Beek ................................. 332'2 
 
 1 Bosanquet, Phil. Mag. April, 1877. 
 
48 VELOCITY OF SOUND. [254. 
 
 Names of Observers. Velocity of Sound at 
 
 Cent, in Metres. 
 
 Stampfer and Myrback 332*4 
 
 Bravais and Martins (1844) 332'4 
 
 Wertheim 331'6 
 
 Stone (1871) 332'4 
 
 LeRoux 330-7 
 
 Regnault 330'7 
 
 In Stone's experiments 1 the course over which the sound 
 was timed commenced at a distance of 640 feet from the source, 
 so that any errors arising from excessive disturbance were to 
 a great extent avoided. 
 
 A method has been proposed by Bosscha 2 for determining 
 the velocity of sound without the use of great distances. It 
 depends upon the precision with which the ear is able to decide 
 whether short ricks are simultaneous, or not. In Konig's 3 form of 
 the experiment, two small electro-magnetic counters are controlled 
 by a fork-interrupter ( 64), whose period is one-tenth of a second, 
 and give synchronous ticks of the same period. When the 
 counters are close together the audible ticks coincide, but as one 
 counter is gradually removed from the ear, the two series of ticks 
 fall asunder. When the difference of distances is about 34 metres, 
 coincidence again takes place, proving that 34 metres is about 
 the distance traversed by sound in a tenth part of a second. 
 
 [On the basis of experiments made in pipes Violle and Vautier 4 
 give 33110 as applicable in free air. The result includes a cor- 
 rection, amounting to 0'68, which is of a more or less theoretical 
 character, representing the presumed influence of the pipe (0'7 m in 
 diameter).] 
 
 1 Phil. Trans. 1872, p. 1. 2 Pogg. Ann. xcn. 486. 1854. 
 
 3 Pogg. Ann. cxvin. 610. 1863. 4 Ann. de Chim. t. xix.; 1890. 
 
CHAPTER XII. 
 
 VIBRATIONS IN TUBES. 
 
 255. WE have already ( 245) considered the solution of our 
 fundamental equation, when the velocity-potential, in an unlimited 
 fluid, is a function of one space co-ordinate only. In the absence 
 of friction no change would be caused by the introduction of any 
 number of fixed cylindrical surfaces, whose generating lines are 
 parallel to the co-ordinate in question ; for even when the surfaces 
 are absent the fluid has no tendency to move across them. If one 
 of the cylindrical surfaces be closed (in respect to its transverse 
 section), we have the important problem of the axial motion of air 
 within a cylindrical pipe, which, when once the mechanical condi- 
 tions at the ends are given, is independent of anything that may 
 happen outside the pipe. 
 
 Considering a simple harmonic vibration, we know ( 245) 
 that, if </> varies as e int , 
 
 where 
 
 * = ??-? ................................. (2). 
 
 X a 
 
 The solution may be written in two forms 
 
 (f> (A cos kx + B sin kx) e int } 
 
 <t> = (Ae ik *+Be- ik *)e int }" 
 
 of which finally only the real parts will be retained. The first 
 form will be most convenient when the vibration is stationary, or 
 
 R. II. 4 
 
50 HARMONIC WAVES IN ONE DIMENSION. [255. 
 
 nearly so, and the second when the motion reduces itself to a 
 positive, or negative, progressive undulation. The constants A 
 and B in the symbolical solution may be complex, and thus the 
 final expression in terms of real quantities will involve four arbi- 
 trary constants. If we wish to use real quantities throughout, we 
 
 must take 
 
 <f> = (A cos kx + B sin kx) cos nt 
 
 + (C cos kx 4- D sin kx) sin nt ................ (4), 
 
 but the analytical work would generally be longer. When no 
 ambiguity can arise, we shall sometimes for the sake of brevity 
 drop, or restore, the factor involving the time without express 
 mention. Equations such as (1) are of course equally true whether 
 the factor be understood or not. 
 
 Taking the first form in (3), we have 
 
 * 
 <j> = A cos kx + B sin kx \ 
 
 -~- = kA sin lex + kB cos kx | 
 dx ) 
 
 If there be any point at which either <f) or dfyjdx is permanently 
 zero, the ratio A : B must be real, and then the vibration is sta- 
 tionary, that is, the same in phase at all points simultaneously. 
 
 Let us suppose that there is a node at the origin. Then when 
 x 0, dfyjdx vanishes, the condition of which is B = 0. Thus 
 
 ......... (6), 
 
 from which, if we substitute Pe iB for A, and throw away the 
 imaginary part, 
 
 <= P cos kx cos (nt + 0) 
 
 - = -kPsinkxcos(nt+0) 
 
 From these equations we learn that d<f>jdx vanishes wherever 
 sin kx = ; that is, that besides the origin there are nodes at the 
 points x = m\, m being any positive or negative integer. At any 
 of these places infinitely thin rigid plane barriers normal to x 
 might be stretched across the tube without in any way alter- 
 ing the motion. Midway between each pair of consecutive nodes 
 there is a loop, or place of no pressure variation, since Sp = p$ 
 (6) 244. At any of these loops a communication with the 
 
255.] NODES AND LOOPS. 51 
 
 external atmosphere might be opened, without causing any disturb- 
 ance of the motion from air passing in or out. The loops are the 
 places of maximum velocity, and the nodes those of maximum 
 pressure variation. At intervals of X everything is exactly re- 
 peated. 
 
 If there be a node at x = I, as well as at the origin, sin kl = 0, 
 or X = 2//m, where m is a positive integer. The gravest tone 
 which can be sounded by air contained in a doubly closed pipe 
 of length I is therefore that which has a wave-length equal to 2Z. 
 This statement, it will be observed, holds good whatever be the 
 gas with which the pipe is filled; but the frequency, or the place 
 of the tone in the musical scale, depends also on the nature of 
 the particular gas. The periodic time is given by 
 
 T-- (8). 
 
 a a 
 
 The other tones possible for a doubly closed pipe have periods 
 which are submultiples of that of the gravest tone, and the whole 
 system forms a harmonic scale. 
 
 Let us now suppose, without stopping for the moment to in- 
 quire how such a condition of things can be secured, that there is 
 a loop instead of a node at the point x = l. Equation (6) gives 
 cos kl = 0, whence X = 4>l H- (2m + 1), where m is zero or a positive 
 integer. In this case the gravest tone has a wave-length equal 
 to four times the length of the pipe reckoned from the node to 
 the loop, and the other tones form with it a harmonic scale, from 
 which, however, all the members of even order are missing. 
 
 256. By means of a rigid barrier there is no difficulty in 
 securing a node at any desired point of a tube, but the condition 
 for a loop, i.e. that under no circumstances shall the pressure vary, 
 can only be realized approximately. In most cases the variation 
 of pressure at any point of a pipe may be made small by allowing 
 a free communication with the external air. Thus Euler and 
 Lagrange assumed constancy of pressure as the condition to be 
 satisfied at the end of an open pipe. We shall afterwards return 
 to the problem of the open pipe, and investigate by a rigorous 
 process the conditions to be satisfied at the end. For our im- 
 mediate purpose it will be sufficient to know, what is indeed 
 tolerably obvious, that the open end of a pipe may be treated as 
 
 42 
 
52 CONDITION FOR AN OPEN END. [256. 
 
 a loop, if the diameter of the pipe be neglected in comparison 
 with the wave-length, provided the external pressure in the neigh- 
 bourhood of the open end be not itself variable from some cause 
 independent of the motion within the pipe. When there is an 
 independent source of sound, the pressure at the end of the pipe 
 is the same as it would be in the same place, if the pipe were 
 away. The impediment to securing the fulfilment of the condition 
 for a loop at any desired point lies in the inertia of the machinery 
 required to sustain the pressure. For theoretical purposes we may 
 overlook this difficulty, and imagine a massless piston backed by 
 a compressed spring also without mass. The assumption of a 
 loop at an open end of a pipe is tantamount to neglecting the 
 inertia of the outside air. 
 
 We have seen that, if a node exist at any point of a pipe, 
 there must be a series, ranged at equal intervals JX, that midway 
 between each pair of consecutive nodes there must be a loop, and 
 that the whole vibration must be stationary. The same conclusion 
 follows if there be at any point a loop ; but it may perfectly well 
 happen that there are neither nodes nor loops, as for example in 
 the case when the motion reduces to a positive or negative pro- 
 gressive wave. In stationary vibration there is no transference of 
 energy along the tube in either direction, for energy cannot pass 
 a node or a loop. 
 
 257. The relations between the lengths of an open or closed 
 pipe and the wave-lengths of the included column of air may also 
 be investigated by following the motion of a pulse, by which is 
 understood a wave confined within narrow limits and composed 
 of uniformly condensed or rarefied fluid. In looking at the matter 
 from this point of view it is necessary to take into account care- 
 fully the circumstances under which the various reflections take 
 place. Let us first suppose that a condensed pulse travels in the 
 positive direction towards a barrier fixed across the tube. Since 
 the energy contained in the wave cannot escape from the tube, 
 there must be a reflected wave, and that this reflected wave is 
 also a wave of condensation appears from the fact that there is no 
 loss of fluid. The same conclusion may be arrived at in another 
 way. The effect of the barrier may be imitated by the introduc- 
 tion of a similar and equidistant wave of condensation moving in 
 the negative direction. Since the two waves are both condensed 
 and are propagated in contrary directions, the velocities of the 
 
257.] REFLEXION AT AN OPEN END. 53 
 
 fluid composing them are equal and opposite, and therefore neu- 
 tralise one another when the waves are superposed. 
 
 If the progress of the negative reflected wave be interrupted 
 by a second barrier, a similar reflection takes place, and the wave, 
 still remaining condensed, regains its positive character. When a 
 distance has been travelled equal to twice the length of the pipe, 
 the original state of things is completely restored, and the same 
 cycle of events repeats itself indefinitely. We learn therefore that 
 the period within a doubly closed pipe is the time occupied by a 
 pulse in travelling twice the length of the pipe. 
 
 The case of an open end is somewhat different. The supple- 
 mentary negative wave necessary to imitate the effect of the open 
 end must evidently be a wave of rarefaction capable of neutralising 
 the positive pressure of the condensed primary wave, and thus in 
 the act of reflection a wave changes its character from condensed 
 to rarefied, or from rarefied to condensed. Another way of con- 
 sidering the matter is to observe that in a positive condensed 
 pulse the momentum of the motion is forwards, and in the absence 
 of the necessary forces cannot be changed by the reflection. But 
 forward motion in the reflected negative wave is indissolubly 
 connected with the rarefied condition. 
 
 When both ends of a tube are open, a pulse travelling back- 
 wards and forwards within it is completely restored to its original 
 state after traversing twice the length of the tube, suffering in the 
 process two reflections, and thus the relation between length and 
 period is the same as in the case of a tube, whose ends are both 
 closed ; but when one end of a tube is open and the other closed, 
 a double passage is not sufficient to close the cycle of changes. 
 The original condensed or rarefied character cannot be recovered 
 until after two reflections from the open end, and accordingly in 
 the case contemplated the period is the time required by the pulse 
 to travel over four times the length of the pipe. 
 
 258. After the full discussion of the corresponding problems 
 in the chapter on Strings, it will not be necessary to say much on 
 the compound vibrations of columns of air. As a simple example 
 we may take the case of a pipe open at one end and closed at the 
 other, which is suddenly brought to rest at the time t 0, after 
 being for some time in motion with a uniform velocity parallel to 
 its- length. The initial state of the contained air is then one of 
 
54 PROBLEM. [258. 
 
 uniform velocity u parallel to x, and of freedom from compression 
 and rarefaction. If we suppose that the origin is at the closed 
 end, the general solution is by (7) 255, 
 
 <f> = (A! cos ??j + B l sin n^) cos k^x 
 + (As cos nj + B 2 sin n 2 t) cos h& 
 
 + .-.'. ............ ." ..... "- ....... "... ............. ax 
 
 where k r = (r J) jr/l, n r = ak r , and A , B l , A> 2 , R 2 . . . are arbitrary 
 constants. 
 
 Since <f> is to be zero initially for all values of x, the coeffi- 
 cients B must vanish ; the coefficients A are to be determined by 
 the condition that for all values of x between and I, 
 
 2 k r A r sin & r & 1 = w ..................... (2), 
 
 where the summation extends to all integral values of r from 
 1 to oo . The determination of the coefficients A from (2) is 
 effected in the usual way. Multiplying by smkrscdw, and inte- 
 grating from to I, we get 
 
 ^ = - ........................... (3). 
 
 The complete solution is therefore 
 
 2tt 0x ? r=ao coskr 
 
 *=-T 2 ~i ny- 008 ""* ............... w- 
 
 259. In the case of a tube stopped at the origin and open at 
 x = I, let <f) = cos nt be the value of the potential at the open end 
 due to an external source of sound. Determining P and 6 in 
 equation (7) 255, we find 
 
 cos for 
 9 = -- ,-. cos?* ........................ (1). 
 
 coskl 
 
 It appears that the vibration within the tube is a minimum, 
 when coakl = + 1, that is when I is a multiple of ^\, in which case 
 there is a node at x = l. When I is an odd multiple of JX, cosH 
 vanishes, and then according to (1) the motion would become 
 infinite. In this case the supposition that the pressure at the 
 open end is independent of what happens within the tube breaks 
 down ; and we can only infer that the vibration is very large, in 
 
259.J FORCED VIBRATION. 55 
 
 consequence of the isochronism. Since there is a node at x = 0, 
 there must be a loop when x is an odd multiple of \\, and we 
 conclude that in the case of isochronism the variation of pressure 
 at the open end of the tube due to the external cause is exactly 
 neutralised by the variation of pressure due to the motion within 
 the tube itself. If there were really at the open end a variation 
 of pressure on the whole, the motion must increase without limit 
 in the absence of dissipative forces. 
 
 If we suppose that the origin is a loop instead of a node, the 
 solution is 
 
 sin kx 
 
 where <f> = cos nt is the given value of <f> at the open end x = I. 
 In this case the expression becomes infinite, when Id mir, or 
 
 We will next consider the case of a tube, whose ends are both 
 open and exposed to disturbances of the same period, making < 
 equal to He int , Ke int respectively. Unless the disturbances at the 
 ends are in the same phase, one at least of the coefficients H, K 
 must be complex. 
 
 Taking the first form in (3) 255, we have as the general 
 expression for <f> 
 
 </> = e int (A cos kx + B sin kx). 
 
 If we take the origin in the middle of the tube, and assume that 
 the values He int , Ke int correspond respectively to x = I, x=l, 
 we get to determine A and B, 
 
 H = A cos kl + B sin kl, 
 K = A cos kl B sin kl, 
 whence 
 
 A _^ + ^ P_^~-^ /o\ 
 
 ~2cos&/' 2sin&r' 
 
 giving 
 
 _ pint Hsmk(l + x) + K*\ak(l-x) , 
 
 * = sin -2kt 
 
 This result might also be deduced from (2), if we consider that 
 the required motion arises from the superposition of the motion, 
 which is due to the disturbance He int calculated on the hypothesis 
 that the other end x l is a loop, on the motion, which is 
 due to Ke int on the hypothesis that the end x l is a loop. 
 
56 BOTH ENDS OPEN. [259. 
 
 The vibration expressed by (4) cannot be stationary, unless the 
 ratio H : K be real, that is unless the disturbances at the ends be 
 in similar, or in opposite, phases. Hence, except in the cases 
 reserved, there is no loop anywhere, and therefore no place at 
 which a branch tube can be connected along which sound will not 
 be propagated 1 . 
 
 At the middle of the tube, for which x = 0, 
 
 shewing that the variation of pressure (proportional to </>) vanishes 
 if ^T+jBT = 0, that is, if the disturbances at the ends be equal and 
 in opposite phases. Unless this condition be satisfied, the expres- 
 sion becomes infinite when Zl = J (2m + 1) X. 
 
 At a point distant \ from the middle of the tube the 
 expression for <f> is 
 
 vanishing when HK, that is, when the disturbances at the ends 
 are equal and in the same phase. In general <f> becomes infinite, 
 when sin kl = 0, or 21 = m\. 
 
 If at one end of an unlimited tube there be a variation of 
 pressure due to an external source, a train of progressive waves 
 will be propagated inwards from that end. Thus, if the length 
 along the tube measured from the open end be y, the velocity- 
 potential is expressed by < = cos (nt ny/a), corresponding to 
 </> = cos nt at y = ; so that, if the cause of the disturbance within 
 the tube be the passage of a train of progressive waves across the 
 open end, the intensity within the tube will be the same as in the 
 space outside. It must not be forgotten that the diameter of the 
 tube is supposed to be infinitely small in comparison with the 
 length of a wave. 
 
 1 An arrangement of this kind has been proposed by Prof. Mayer (Phil. Mag. 
 XLV. p. 90, 1873) for comparing the intensities of sources of sound of the same 
 pitch. Each end of the tube is exposed to the action of one of the sources to be 
 compared, and the distances are adjusted until the amplitudes of the vibrations 
 denoted by H and K are equal. The branch tube is led to a manometric capsule 
 ( 262), and the method assumes that by varying the point of junction the disturb- 
 ance of the flame can be stopped. From the discussion in the text it appears that 
 this assumption is not theoretically correct. 
 
259.] FORCED VIBRATION OF, PISTON. 
 
 Let us next suppose that the source of the motion is within the 
 tube itself, due for example to the inexorable motion of a piston 
 at the origin 1 . The constants in (5) 255 are to be determined 
 by the conditions that when x 0, d^/dx = cos nt (say), and that, 
 when x = I, <f> = 0. Thus k A = tan kl, kB 1, and the ex- 
 pression for < is 
 
 sin fc 0-0 7 
 
 *" kcoskl 
 
 The motion is a minimum, when cos kl = 1, that is, when the 
 length of the tube is a multiple of JX. 
 
 When I is an odd multiple of ^X, the place occupied by the 
 piston would be a node, if the open end were really a loop, but in 
 this case the solution fails. The escape of energy from the tube 
 prevents the energy from accumulating beyond a certain point ; 
 but no account can be taken of this so long as the open end is 
 treated rigorously as a loop. We shall resume the question of 
 resonance after we have considered in greater detail the theory of 
 the open end, when we shall be able to deal with it more satis- 
 factorily. 
 
 In like manner if the point x = I be a node, instead of a loop, 
 the expression for <f> is 
 
 , cos k (I - x) , . 
 
 * = -TksinJM 
 
 and thus the motion is a minimum when I is an odd multiple of X, 
 in which case the origin is a loop. When I is an even multiple of 
 JX, the origin should be a node, which is forbidden by the condi- 
 tions of the question. In this case according to (8) the motion 
 becomes infinite, which means that in the absence of dissipative 
 forces the vibration would increase without limit. 
 
 260. The experimental investigation of aerial waves within 
 pipes has been effected with considerable success by Kundt 2 . To 
 generate waves is easy enough ; but it is not so easy to invent a 
 method by which they can be effectually examined. Kundt dis- 
 covered that the nodes of stationary waves can be made evident 
 by dust. A little fine sand or lycopodium seed, shaken over the 
 interior of a glass tube containing a vibrating column of air 
 
 1 These problems are considered by Poisson, Mem. de VInstitut, t. u. p. 305, 1819. 
 
 2 Pogg. Ann. t. cxxxv. p. 337, 1868. 
 
58 KU^DT'S EXPERIMENTS. [260. 
 
 disposes itself in recurring patterns, by means of which it is easy 
 to determine the positions of the nodes and to measure the 
 intervals between them. In Kundt's experiments the origin of 
 the sound was in the longitudinal vibration of a glass tube called 
 the sounding- tube, and the dust-figures were formed in a second 
 and larger tube, called the wave-tube, the latter being provided 
 with a moveable stopper for the purpose of adjusting its length. 
 The other end of the wave- tube was fitted with a cork through 
 which the sounding-tube passed half way. By suitable friction 
 the sounding-tube was caused to vibrate in its gravest mode, so 
 that the central point was nodal, and its interior extremity (closed 
 with a cork) excited aerial vibrations in the wave-tube. By means 
 of the stopper the length of the column of air could be adjusted so 
 as to make the vibrations as vigorous as possible, which happens 
 when the interval between the stopper and the end of the 
 sounding-tube is a multiple of half the wave-length of the 
 sound. 
 
 With this apparatus Kundt was able to compare the wave- 
 lengths of the same sound in various gases, from which the rela- 
 tive velocities of propagation are at once deducible, but the results 
 were not entirely satisfactory. It was found that the intervals 
 of recurrence of the dust-patterns were not strictly equal, and, 
 what was worse, that the pitch of the sound was not constant 
 from one experiment to another. These defects were traced to a 
 communication of motion to the wave-tube through the cork, by 
 which the dust-figures were disturbed, and the pitch made irregular 
 in consequence of unavoidable variations in the mounting of the 
 apparatus. To obviate them, Kundt replaced the cork, which 
 formed too stiff a connection between the tubes, by layers of sheet 
 indiarubber tied round with silk, obtaining in this way a flexible 
 and perfectly air-tight joint ; and in order to avoid any risk of the 
 comparison of wave-lengths being vitiated by an alteration of pitch, 
 the apparatus was modified so as to make it possible to excite 
 the two systems of dust-figures simultaneously and in response to 
 the same sound. A collateral advantage of the new method con- 
 sisted in the elimination of temperature-corrections. 
 
 In the improved " Double Apparatus " the sounding-tube was 
 caused to vibrate in its second mode by friction applied near 
 the middle ; and thus the nodes were formed at the points distant 
 from the ends by one-fourth of the length of the tube. At each 
 
260.] KUNDT'S EXPERIMENTS. 59 
 
 of these points connection was made with an independent wave- 
 tube, provided with an adjustable stopper, and with branch tubes 
 and stop-cocks suitable for admitting the various gases to be 
 experimented upon. It is evident that dust-figures formed in the 
 two tubes correspond rigorously to the same pitch, and that there- 
 fore a comparison of the intervals of recurrence leads to a correct 
 determination of the velocities of propagation, under the circum- 
 stances of the experiment, for the two gases with which the tubes 
 are filled. 
 
 The results at which Kundt arrived were as follows : 
 (a) The velocity of sound in a tube diminishes with the 
 diameter. Above a certain diameter, however, the change is not 
 perceptible. 
 
 (6) The diminution of velocity increases with the wave- 
 length of the tone employed. 
 
 (c) Powder, scattered in a tube, diminishes the velocity of 
 sound in narrow tubes, but in wide ones is without effect. 
 
 (d) In narrow tubes the effect of powder increases, when 
 it is very finely divided, and is strongly agitated in consequence. 
 
 (e) Roughening the interior of a narrow tube, or increasing 
 its surface, diminishes the velocity. 
 
 (/) In wide tubes these changes of velocity are of no im- 
 portance, so that the method may be used in spite of them for 
 exact determinations. 
 
 (g) The influence of the intensity of sound on the velocity 
 cannot be proved. 
 
 (h) With the exception of the first, the wave-lengths of a 
 tone as shewn by dust are not affected by the mode of excitation. 
 
 (i) In wide tubes the velocity is independent of pressure, 
 but in small tubes the velocity increases with the pressure. 
 
 (j) All the observed changes in the velocity were due to 
 friction and especially to exchange of heat between the air and 
 the sides of the tube. 
 
 (k) The velocity of sound at 100 agrees exactly with that 
 given by theory 1 . 
 
 1 From some expressions in the memoir already cited, from which the notice 
 in the text is principally derived, Kundt appears to have contemplated a continua- 
 tion of his investigations; but I am unable to find any later publication on the 
 subject. 
 
60 KUNDT'S EXPERIMENTS. [260. 
 
 We shall return to the question of the propagation of sound in 
 narrow tubes as affected by the causes mentioned above (j), and 
 shall then investigate the formulae given by Helmholtz and 
 Kirchhoff. 
 
 [The genesis of the peculiar transverse striation which forms 
 a leading feature of the dust-figures has already been considered 
 2536. According to the observations of Dvorak 1 the powerful 
 vibrations which occur in a Kundt's tube are accompanied by 
 certain mean motions of the gas. Thus near the walls there is a 
 flow from the loops to the nodes, and in the interior a return flow 
 from the nodes to the loops. This is a consequence of viscosity 
 acting with peculiar advantage upon the parts of the fluid con- 
 tiguous to the walls 2 . We may perhaps return to this subject in 
 a later chapter.] 
 
 261. In the experiments described in the preceding section the 
 aerial vibrations are forced, the pitch being determined by the 
 external source, and not (in any appreciable degree) by the length 
 of the column of air. Indeed, strictly speaking, all sustained 
 vibrations are forced, as it is not in the power of free vibrations 
 to maintain themselves, except in the ideal case when there is 
 absolutely no friction. Nevertheless there is an important prac- 
 tical distinction between the vibrations of a column of air as 
 excited by a longitudinally vibrating rod or by a tuning-fork, and 
 such vibrations as those of the organ-pipe or chemical harmonicon. 
 In the latter cases the pitch of the sound depends principally on 
 the length of the aerial column, the function of the wind or of the 
 flame 3 being merely to restore the energy lost by friction and by 
 communication to the external air. The air in an organ-pipe is to 
 be considered as a column swinging almost freely, the lower end, 
 across which the wind sweeps, being treated roughly as open, and 
 the upper end as closed, or open, as the case may be. Thus the 
 wave-length of the principal tone of a stopped pipe is four times 
 the length of the pipe ; and, except at the extremities, there is 
 neither node nor loop. The overtones of the pipe are the odd 
 
 1 Pogfj. Ann. t. CLVII. p. 61, 1876. 
 
 2 On the Circulation of Air observed in Kundt's Tubes, and on some allied 
 Acoustical Problems, Phil. Trans, vol. CLXXV. p. 1, 1884. 
 
 3 The subject of sensitive flames with and without pipes is treated in con- 
 siderable detail by Prof. Tyndall in his work on Sound ; but the mechanics of 
 this class of phenomena is still very imperfectly understood. We shall return to 
 it in a subsequent chapter. 
 
261.] EXPERIMENTS OF SAVART AND KONIG. 61 
 
 harmonics, twelfth, higher third, &c., corresponding to the various 
 subdivisions of the column of air. In the case of the twelfth, for 
 example, there is a node at the point of trisection nearest to the 
 open end, and a loop at the other point of trisection midway 
 between the first and the stopped end of the pipe. 
 
 In the case of the open organ -pipe both ends are loops, and 
 there must be at least one internal node. The wave-length of the 
 principal tone is twice the length of the pipe, which is divided 
 into two similar parts by a node in the middle. From this we see 
 the foundation of the ordinary rule that the pitch of an open pipe 
 is the same as that of a stopped pipe of half its length. For reasons 
 to be more fully explained in a subsequent chapter, connected 
 with our present imperfect treatment of the open end, the rule is 
 only approximately correct. The open pipe, differing in this re- 
 spect from the stopped pipe, is capable of sounding the whole series 
 of tones forming the harmonic scale founded upon its principal 
 tone. In the case of the octave there is a loop at the centre of the 
 pipe and nodes at the points midway between the centre and the 
 extremities. 
 
 Since the frequency of the vibration in a pipe is proportional 
 to the velocity of propagation of sound in the gas with which the 
 pipe is filled, the comparison of the pitches of the notes obtained 
 from the same pipe in different gases is an obvious method of 
 determining the velocity of propagation, in cases where the impos- 
 sibility of obtaining a sufficiently long column of the gas precludes 
 the use of the direct method. In this application Chladni with his 
 usual sagacity led the way. The subject was resumed at a later 
 date by Dulong 1 and by Wertheim 2 , who obtained fairly satisfac- 
 tory results. 
 
 262. The condition of the air in the interior of an organ -pipe 
 was investigated experimentally by Savart 3 , who lowered into the 
 pipe a small stretched membrane on which a little sand was 
 scattered. In the neighbourhood of a node the sand remained 
 sensibly undisturbed, but, as a loop was approached, it danced with 
 more and more vigour. But by far the most striking form of the 
 
 1 Recherches sur les chaleurs sp6cifiques des fluides 61astiques. Ann. de Chlm.> 
 t. XLI. p. 113, 1829. 
 
 2 Ann. de Chim., 3 ieme s^rie, t. xxm. p. 434, 1848. 
 
 3 Ann. de Chim., t. xxiv. p. 56, 1823. 
 
62 CURVED PIPE. [262. 
 
 experiment is that invented by Konig. In this method the vibra- 
 tion is indicated by a small gas flame, fed through a tube which 
 is in communication with a cavity called a manometric capsule. 
 This cavity is bounded on one side by a membrane on which 
 the vibrating air acts. As the membrane vibrates, rendering the 
 capacity of the capsule variable, the supply of gas becomes un- 
 steady and the flame intermittent. The period is of course too 
 small for the intermittence to manifest itself as such when the 
 flame is looked at steadily. By shaking the head, or with the aid 
 of a moveable mirror, the resolution into more or less detached 
 images may be effected ; but even without resolution the altered 
 character of the flame is evident from its general appearance. In 
 the application to organ-pipes, one or more capsules are mounted 
 on a pipe in such a manner that the membranes are in contact 
 with the vibrating column of air ; and the difference in the flame 
 is very marked, according as the associated capsule is situated at 
 a node or at a loop. 
 
 263. Hitherto we have supposed the pipe to be straight, but 
 it will readily be anticipated that, when the cross section is small 
 and does not vary in area, straightness is not a matter of impor- 
 tance. Conceive a curved axis of x running along the middle of 
 the pipe, and let the constant section perpendicular to this axis 
 be 8. When the greatest diameter of 8 is very small in comparison 
 with the wave-length of the sound, the velocity-potential <f> 
 becomes nearly invariable over the section; applying Green's 
 theorem to the space bounded by the interior of the pipe and by 
 two cross sections, we get 
 
 Now by the general equation of motion 
 
 and in the limit, when the distance between the sections is made 
 to vanish, 
 
 so that 
 
263.] 
 
 BRANCHED PIPES. 
 
 63 
 
 shewing that < depends upon x in the same way as if the pipe 
 were straight. By means of equation (1) the vibrations of air in 
 curved pipes of uniform section may be easily investigated, and the 
 results are the rigorous consequences of our fundamental equations 
 (which take no account of friction), when the section is supposed to 
 be infinitely smajl. In the case of thin tubes such as would be 
 used in experiment, they suffice at any rate to give a very good 
 representation of what actually happens. 
 
 264. We now pass on to the consideration of certain cases of 
 connected tubes. In the accompanying figure AD represents a 
 thin pipe, which divides at D into two branches DB, DC. At E 
 the branches reunite and form a single tube EF. The sections 
 of the single tubes and of the branches are assumed to be uniform 
 
 as well as very small. 
 
 Fig. 55. 
 
 In the first instance let us suppose that a positive wave of 
 arbitrary type is advancing in A. On its arrival at the fork D, it 
 will give rise to positive waves in B and C, and, unless a certain 
 condition be satisfied, to a negative reflected wave in A. Let the 
 potential of the positive waves be denoted by f A ,f B ,f ct f being in 
 each case a function of x - at] and let the reflected wave be 
 F (x + at). Then the conditions to be satisfied at D are first that 
 the pressures shall be the same for the three pipes, and secondly 
 that the whole velocity of the fluid in A shall be equal to the sum 
 of the whole velocities of the fluid in B and C. Thus, using 
 A, B, C to denote the areas of the sections, we have, 244, 
 
 fi-F -f. -f. 
 
 whence 
 
 ~ 
 
 C-A 
 
 .(i); 
 
 (2), 
 
 _2A_ 
 
 J* > B + C + A 
 
 (3) 1 . 
 
 1 These formulas, as applied to determine the reflected and refracted waves 
 at the junction of two tubes of sections B+C, and A respectively, are given by 
 
64 BRANCHED PIPES. 
 
 It appears that/ 5 and/ c are always the same. There is no reflec- 
 
 tion, if 
 
 G=A ............................ (4), 
 
 that is, if the combined sections of the branches be equal to the 
 section of the trunk ; and, when this condition is satisfied, 
 
 The wave then advances in B and C exactly as it would have 
 done in A, had there been no break. If the lengths of the 
 branches between D and E be equal, and the section of F be equal 
 to that of A , the waves on arrival at E combine into a wave pro- 
 pagated along F, and again there is no reflection. The division 
 of the tube has thus been absolutely without effect ; and since the 
 same would be true for a negative wave passing from F to A, 
 we may conclude generally that a tube may be divided into two, 
 or more, branches, all of the same length, without in any way 
 influencing the law of aerial vibration, provided that the whole 
 section remain constant. If the lengths of the branches from D 
 to E be unequal, the result is different. Besides the positive wave 
 in F) there will be in general negative reflected waves in B and G. 
 The most interesting case is when the wave is of harmonic type 
 and one of the branches is longer than the other by a multiple of 
 X. If the difference be an even multiple of J X, the result will be 
 the same as if the branches were of equal length, and no reflection 
 will ensue. But suppose that, while B and G are equal in section, 
 one of them is longer than the other by an odd multiple of \ X. 
 Since the waves arrive at E in opposite phases, it follows from 
 symmetry that the positive wave in F must vanish, and that the 
 pressure at E, which is necessarily the same for all the tubes, 
 must be constant. The waves in B and G are thus reflected as 
 from an open end. That the conditions of the question are thus 
 satisfied may also be seen by supposing a barrier taken across the 
 tube F in the neighbourhood of E in such a way that the tubes 
 B and G communicate without a change of section. The wave in 
 each tube will then pass on into the other without interruption, 
 and the pressure-variation at E, being the resultant of equal and 
 opposite components, will vanish. This being so, the barrier may 
 be removed without altering the conditions, and no wave will be 
 propagated along F, whatever its section may be. The arrange- 
 
 Poisson, M6m. de VInstitut, t. n. p. 305, 1819. The reader will not forget that both 
 diameters must be small in comparison with the wave-length. 
 
264.] BRANCHED PIPES. 65 
 
 ment now under consideration was invented by Herschel, and has 
 been employed by Quincke and others for experimental purposes, 
 an application that we shall afterwards have occasion to describe. 
 The phenomenon itself is often referred to as an example of inter- 
 ference, to which there can be no objection, but the same cannot 
 be said when the reader is led to suppose that the positive waves 
 neutralise each other in F, and that there the matter ends. It must 
 never be forgotten that there is no loss of energy in interference, 
 but only a different distribution ; when energy is diverted from 
 one place, it reappears in another. In the present case the positive 
 wave in A conveys energy with it. If there is no wave along F, 
 there are two possible alternatives. Either energy accumulates 
 in the branches, or else it passes back along A in the form of a 
 negative wave. In order to see what really happens, let us trace 
 the progress of the waves reflected back at E. 
 
 These waves are equal in magnitude and start from E in 
 opposite phases ; in the passage from E to D one has to travel 
 a greater distance than the other by an odd multiple of \ ; and 
 therefore on arrival at D they will be in complete accordance. 
 Under these circumstances they combine into a single wave, which 
 travels negatively along A, and there is no reflection. When the 
 negative wave reaches the end of the tube A, or is otherwise dis- 
 turbed in its course, the whole or a part may be reflected, and then 
 the process is repeated. But however often this may happen there 
 will be no wave along F, unless by accumulation, in consequence 
 of a coincidence of periods, the vibration in the branches becomes 
 so great that a small fraction of it can no longer be neglected. 
 
 Or we may reason thus. Suppose the tube F cut off by a 
 barrier as before. The motion in the Fig. 56. 
 
 ring being due to forces acting at D is 
 necessarily symmetrical with respect to 
 D, and D' the point which divides 
 DBCD into equal parts. Hence D' is 
 a node, and the vibration is stationary. 
 This being the case, at a point E distant 
 ^ X from D' on either side, there must be 
 a loop; and if the barrier be removed 
 there will still be no tendency to produce 
 vibration in F. If the perimeter of the 
 ring be a multiple of X, there may be 
 R. II. 
 
66 BRANCHED PIPES. [264. 
 
 vibration within it of the period in question, independently of 
 any lateral openings. 
 
 Any combination of connected tubes may be treated in a 
 similar manner. The general Fig. 57. 
 
 principle is that at any junction 
 a space can be taken large enough 
 to include all the region through 
 which the want of uniformity 
 affects the law of the waves, and 
 yet so small that its longest 
 dimension may be neglected in comparison with X. Under these 
 circumstances the fluid within the space in question may be 
 treated as if the wave-length were infinite, or the fluid itself 
 incompressible, in which case its velocity-potential would satisfy 
 V 3 < = 0, following the same laws as electricity. 
 
 265. When the section of a pipe is variable, the problem of the 
 vibrations of air within it cannot generally be solved. The case 
 of conical pipes will be treated on a future page. At present we 
 will investigate an approximate expression for the pitch of a nearly 
 cylindrical pipe, taking first the case where both ends are closed. 
 The method that will be employed is similar to that used for a string 
 whose density is not quite constant, 91, 140, depending on the 
 principle that the period of a free vibration fulfils the stationary 
 condition, and may therefore be calculated from the potential and 
 kinetic energies of any hypothetical motion not departing far from 
 the actual type. In accordance with this plan we shall assume that 
 the velocity normal to any section S is constant over the section, 
 as must be very nearly the case when the variation of $ is slow. 
 Let X represent the total transfer of fluid at time t across the 
 section at #, reckoned from the equilibrium condition ; then X 
 represents the total velocity of the current, and X -f- 8 represents 
 the actual velocity of the particles of fluid, so that the kinetic 
 energy of the motion within the tube is expressed by 
 
 im ........................ (1). 
 
 The potential energy 245 (12) is expressed in general by 
 
265.] VARIABLE SECTION. 67 
 
 or, since dV ' Sdx, by 
 
 F=ia 2 /3 ISs 2 dx (2). 
 
 Again, by the condition of continuity, 
 
 s=- dX 
 and thus 
 
 F=ia 2 p f-sf-j- \ dx... ,..(4). 
 
 / jS \ dec J 
 
 If we now assume for X an expression of the same form as 
 would obtain if 8 were constant, viz. 
 
 X = sin -j- cos nt (5), 
 
 we obtain from the values of T and Fin (1) and (4), 
 ' l TTX dx C l . % TTX dx 
 
 or, if we write '8 = $ + A$ and neglect the square of 
 
 AS<fo{ 
 
 -- ............ (7). 
 
 The result may be expressed conveniently in terms of AZ, the cor- 
 rection that must be made to I in order that the pitch may be 
 calculated from the ordinary formula, as if S were constant. For 
 the value of AZ we have 
 
 f l 
 | 
 
 J 
 
 j 
 cos. --- ~-dx ..................... (8). 
 
 I *9g 
 
 The effect of a variation of section is greatest near a node or near 
 a loop. An enlargement of section in the first case lowers the 
 pitch, and in the second case raises it. At the points midway 
 between the nodes and loops a slight variation of section is with- 
 out effect. The pitch is thus decidedly altered by an enlargement 
 or contraction near the middle of the tube, but the influence of a 
 slight conicality would be much less. 
 
 The expression for A/ given by (8) is applicable as it stands to 
 the gravest tone only; but we may apply it to the m th tone 
 of the harmonic scale, if we modify it by the substitution of 
 for cos(27n/7). 
 
 52 
 
68 VARIABLE SECTION. [265. 
 
 In the case of a tube open at both ends (5) is replaced by 
 
 X = cos -=- cos nt ........................ (9), 
 
 i 
 
 which leads to 
 
 ZTTX AS 
 - cos -- ~-d# 
 
 instead of (8). The pitch of the sound is now raised by an 
 enlargement at the ends, or by a contraction at the middle, of the 
 tube ; and, as before, it is unaffected by a slight general conicality 
 ( 281). 
 
 266. The case of progressive waves moving in a tube of vari- 
 able section is also interesting. In its general form the problem 
 would be one of great difficulty ; but where the change of section 
 is very gradual, so that no considerable alteration occurs within a 
 distance of a great many wave-lengths, the principle of energy 
 will guide us to an approximate solution. It is not difficult to see 
 that in the case supposed there will be no sensible reflection of the 
 wave at any part of its course, and that therefore the energy of the 
 motion must remain unchanged 1 . Now we know, 245, that for 
 a given area of wave-front, the energy of a train of simple waves 
 is as the square of the amplitude, from which it follows that as 
 the waves advance the amplitude of vibration varies inversely as 
 the square root of the section of the tube. In all other respects 
 the type of vibration remains absolutely unchanged. From these 
 results we may get a general idea of the action of an ear-trumpet. 
 It appears that according to the ordinary approximate equations, 
 there is no limit to the concentration of sound producible in a 
 tube of gradually diminishing section. 
 
 The same method is applicable, when the density of the 
 medium varies slowly from point to point. For example, the 
 amplitude of a sound-wave moving upwards in the atmosphere 
 may be determined by the condition that the energy remains 
 unchanged. From 245 it appears that the amplitude is in- 
 versely as the square root of the density 2 . 
 
 1 Phil. Mag. (5) i. p. 261, 1876. 
 
 2 A delicate question arises as to the ultimate fate of sonorous waves propagated 
 upwards. It should be remarked that in rare air the deadening influence of 
 viscosity is much increased. 
 
CHAPTER XIII. 
 
 SPECIAL PROBLEMS. REFLECTION AND REFRACTION OF 
 PLANE WAVES. 
 
 267. BEFORE undertaking the discussion of the general equa- 
 tions for aerial vibrations we may conveniently turn our attention 
 to a few special problems, relating principally to motion in two 
 dimensions, which are susceptible of rigorous and yet compara- 
 tively simple solution. In this way the reader, to whom the 
 subject is new, will acquire some familiarity with the ideas and 
 methods employed before attacking more formidable difficulties. 
 
 In the previous chapter ( 255) we investigated the vibrations in 
 one dimension, which may take place parallel to the axis of a tube, 
 of which both ends are closed. We will now inquire what vibrations 
 are possible within a closed rectangular box, dispensing with the 
 restriction that the motion is to be in one dimension only. For 
 each simple vibration of which the system is capable, < varies as 
 a circular function of the time, say coskat, where k is some 
 constant ; hence <jj> = & 2 a 2 <, and therefore by the general differen- 
 tial equation (9) 244 
 
 ........................... (1). 
 
 Equation (1) must be satisfied throughout the whole of the 
 included volume. The surface condition to be satisfied over the 
 six sides of the box is simply 
 
 where dn represents an element of the normal to the surface. It 
 is only for special values of k that it is possible to satisfy (1) and 
 (2) simultaneously. 
 
70 AERIAL VIBRATIONS [267. 
 
 Taking three edges which meet as axes of rectangular co-ordi- 
 nates, and supposing that the lengths of the edges are respectively 
 , P, V, we know ( 255) that 
 
 where p, q, r are integers, are particular solutions of the problem. 
 By any of these forms equation (2) is satisfied, and provided that 
 k be equal to p-rr/a, <?7r//3, or mr/y, as the case may be, (1) is also 
 satisfied. It is equally evident that the boundary equation (2) is 
 satisfied over all the surface by the form 
 
 / irx\ ( iry\ ( TTZ\ 
 <f) = coslp J cos I q-jt 1 cos f r J ............ (3), 
 
 a form which also satisfies (1), if k be taken such that 
 
 where as before p, q, r are integers 1 . 
 
 The general solution, obtained by compounding all particular 
 solutions included under (3), is 
 
 <f> 2 2 S (A cos kat + B sin kat) 
 
 ( TTX\ / 7ry\ ( irz\ 
 x cos (p J cos (q ^J cos (r J ............... (o), 
 
 in which A and B are arbitrary constants, and the summation is 
 extended to all integral values of p, q, r. 
 
 This solution is sufficiently general to cover the case of any 
 initial state of things within the box, not involving molecular 
 rotation. The initial distribution of velocities depends upon the 
 initial value of <, or f(u dx + v dy + w<>dz), and by Fourier's 
 theorem can be represented by (5), suitable values being ascribed 
 to the coefficients A. In like manner an arbitrary initial distribu- 
 tion of condensation (or rarefaction), depending on the initial 
 value of <f>, can be represented by ascribing suitable values to the 
 coefficients B. 
 
 The investigation might be presented somewhat differently 
 by commencing with assuming in accordance with Fourier's 
 
 1 Duhamel, Liouville Journ. Math., vol. xiv. p. 84, 1849. 
 
267.] IN A RECTANGULAR CHAMBER. 71 
 
 theorem that the general value of </> at time t can be expressed in 
 the form 
 
 vw n ( ^ x \ f >n y\ ( ^ 
 
 = 2 S 2, Ccos (p 1 cos ( q-g I cos (r 1 , 
 
 in which the coefficients G may depend upon t, but not upon 
 x t y, z. The expressions for T and V would then be formed, and 
 shewn to involve only the squares of the coefficients C, and from 
 these expressions would follow the normal equations of motion 
 connecting each normal co-ordinate G with the time. 
 
 The gravest mode of vibration is that in which the entire 
 motion is parallel to the longest dimension of the box, and there 
 is no internal node. Thus, if a be the greatest of the three sides 
 > A 7> we are to take p = 1, q = 0, r = 0. 
 
 In the case of a cubical box, a = ft = 7, and then instead of 
 (4) we have 
 
 or, if X be the wave-length of plane waves of the same period, 
 
 X = 2a-rV(F J + 2 2 + r 2 ) ..................... (7). 
 
 For the gravest mode p = 1, q = 0, r = 0, or p = 0, q = 1, r 0, &c., 
 and X = 2a. The next gravest is when p = 1, q = l, r = 0, &c., and 
 then X=V2a. When p=l, q = l, r=l, \ = 2a/V3. For the 
 fourth gravest mode p = 2, q = 0, r = 0, &c., and then X = 4a. 
 
 As in the case of the membrane ( 197), when two or more 
 primitive modes have the same period of vibration, other modes 
 of like period may be derived by composition. 
 
 The trebly infinite series of possible simple component vibra- 
 tions is not necessarily completely represented in particular cases 
 of compound vibrations. If, for example, we suppose the contents 
 of the box in its initial condition to be neither condensed nor 
 rarefied in any part, and to have a uniform velocity, whose 
 components parallel to the axes of co-ordinates are respectively 
 u , v , w , no simple vibrations are generated for which more 
 than one of the three numbers p, q, r is finite. In fact each 
 component initial velocity may be considered separately, and the 
 problem is similar to that solved in 258. 
 
 In future chapters we shall meet with other examples of the 
 vibrations of air within completely closed vessels. 
 
72 NOTES OF NARROW PASSAGES. [267. 
 
 Some of the natural notes of the air contained within a room 
 may generally be detected on singing the scale. Probably it is 
 somewhat in this way that blind people are able to estimate the 
 size of rooms 1 . 
 
 In long and narrow passages the vibrations parallel to the 
 length are too slow to affect the ear, but notes due to transverse 
 vibrations may often be heard. The relative proportions of the 
 various overtones depend upon the place at which the disturbance 
 is created 2 . 
 
 In some cases of this kind the pitch of the vibrations, whose 
 direction is principally transverse, is influenced by the occurrence 
 of longitudinal motion. Suppose, for example, in (3) and (4), that 
 q 1, r = 0, and that a is much greater than /3. For the principal 
 transverse vibration p = 0, and k = 7T//3. But besides this there 
 are other modes of vibration in which the motion is principally 
 transverse, obtained by ascribing to p small integral values. Thus, 
 when p = 1, 
 
 shewing that the pitch is nearly the same as before 3 . 
 
 268. If we suppose 7 to become infinitely great, the box of 
 the preceding section is transformed into an infinite rectangular 
 tube, whose sides are a and fi. Whatever may be the motion of 
 the air within this tube, its velocity-potential may be expressed 
 by Fourier's theorem in the series 
 
 where the coefficients A are independent of as and y. By the use 
 of this form we secure the fulfilment of the boundary condition 
 
 1 A remarkable instance is quoted in Young's Natural Philosophy, n. p. 272, 
 from Darwin's Zoonomia, n. 487. "The late blind Justice Fielding walked for the 
 first time into my room, when he once visited me, and after speaking a few words 
 said, ' This room is about 22 feet long, 18 wide, and 12 high ' ; all which he guessed 
 by the ear with great accuracy." 
 
 2 Oppel, Die harmonischen Obertone des durch parallels Wdnde erregten Re- 
 flexionstones. ForUchritte der Physik, xx. p. 130. 
 
 3 There is an underground passage in my house in which it is possible, by 
 singing the right note, to excite free vibrations of many seconds' duration, and it 
 often happens that the resonant note is affected with distinct beats. The breadth 
 of the passage is about 4 feet, and the height about 6 feet. 
 
268.] RECTANGULAR TUBE. 73 
 
 that there is to be no velocity across the sides of the tube; the 
 nature of -A as a function of z and t depends upon the other 
 conditions of the problem. 
 
 Let us consider the case in which the motion at every point is 
 harmonic, and due to a normal motion imposed upon a barrier 
 stretching across the tube at z 0. Assuming < to be proportional 
 to e ikat at all points, we have the usual differential equation 
 
 which by the conjugate property of the functions must be satisfied 
 separately by each term of (1). Thus to determine A pq as a 
 function of z, we get 
 
 (3). 
 
 The solution of this equation differs in form according to the sign 
 of the coefficient of A pq . When p and q are both zero, the coeffi- 
 cient is necessarily positive, but as p and q increase the coefficient 
 changes sign. If the coefficient be positive and be called ft 2 , 
 the general value of A pq may be written 
 
 A pq = Bpg e ^ kat+ ^ + G pq e *(to-M*> ............... (4), 
 
 where, as the factor e ikat is expressed, B pq , C pq are absolute 
 constants. However, the first term in (4) expresses a motion 
 propagated in the negative direction, which is excluded by the 
 conditions of the problem, and thus we are to take simply as the 
 term corresponding to p, q, 
 
 In this expression C pq may be complex ; passing to real quantities 
 and taking two new real arbitrary constants, we obtain 
 
 <f> = [D pq cos (kat fiz) + E pq sin (kat - /^)] cos cos ~- . . .(5). 
 
 a p 
 
 We have now to consider the form of the solution in cases 
 where the coefficient of A pq in (3) is negative. If we call it v 2 , 
 the solution corresponding to (4) is 
 
 (6), 
 
74 RECTANGULAR TUBE. [268. 
 
 of which the first term is to be rejected as becoming infinite with z. 
 We thus obtain corresponding to (5) 
 
 $ = e ~ vz [D pq cos kat + E pq sin kat] cos ^ cos ^ (7). 
 
 The solution obtained by combining all the particular solutions 
 given by (5) and (7) is the general solution of the problem, and 
 allows of a value of d<j>/dz over the section z 0, arbitrary at 
 every point in both amplitude and phase. 
 
 At a great distance from the source the terms given in (7) 
 become insensible, and the motion is represented by the terms of 
 (5) alone. The effect of the terms involving high values of p and q 
 is thus confined to the neighbourhood of the source, and at 
 moderate distances any sudden variations or discontinuities in the 
 motion at z = are gradually eased off and obliterated. 
 
 If we fix our attention on any particular simple mode of vibra- 
 tion (for which p and q do not both vanish), and conceive the 
 frequency of vibration to increase from zero upwards, we see that 
 the effect, at first confined to the neighbourhood of the source, 
 gradually extends further and further and, after a certain value 
 is passed, propagates itself to an infinite distance, the critical 
 frequency being that of the two dimensional free vibrations of the 
 corresponding mode. Below the critical point no work is required 
 to maintain the motion ; above it as much work must be done at 
 z = as is carried off to infinity in the same time. 
 
 268 a. If in the general formulae of 267 we suppose that 
 r = 0, we fall back upon the case of a motion purely two-dimen- 
 sional. The third dimension (7) of the chamber is then a matter 
 of indifference ; and the problem may be supposed to be that of 
 the vibrations of a rectangular plate of air bounded, for example, 
 by two parallel plates of glass, and confined at the rectangular 
 boundary. In this form it has been treated both theoretically 
 and experimentally by Kundt 1 . The velocity-potential is simply 
 
 irx\ f 7ry\ 
 
 jcos^-|J (1), 
 
 where p and q are integers ; and the frequency is determined by 
 
 & 2 ==7r 2 (p 2 /a 2 +2 2 /W (2). 
 
 1 Pogg. Ann. vol. XL. pp. 177, 337, 1873. 
 
268 a.] RECTANGULAR PLATE. 75 
 
 If the plate be open at the boundary, an approximate solution 
 may be obtained by supposing that <f> is there evanescent. In 
 this case the expression for < is derived from (1) by writing sines 
 instead of cosines, while the frequency equation retains the same 
 form (2). This has already been discussed under the head of 
 membranes in 197. If a=/3, so that the rectangle becomes a 
 square, the various normal modes of the same pitch may be 
 combined, as explained in 197. 
 
 In Kundt's experiments the vibrations were excited through a 
 perforation in one of the glass plates, to which was applied the 
 extremity of a suitably tuned rod vibrating longitudinally, and 
 the division into segments was indicated by the behaviour of cork 
 filings. As regards pitch there was a good agreement with 
 calculation in the case of plates closed at the boundary. When 
 the rectangular boundary was open, the observed frequencies were 
 too small, a discrepancy to be attributed to the merely approxi- 
 mate character of the assumption that the pressure is there 
 invariable (see 307). 
 
 The theory of the circular plate of air depends upon Bessel's 
 functions, and is considered in 339. 
 
 269. We will now examine the result of the composition of 
 two trains of plane waves of harmonic type, whose amplitudes and 
 wave-lengths are equal, but whose directions of propagation are 
 inclined to one another at an angle 2a. The problem is one of 
 two dimensions only, inasmuch as everything is the same in 
 planes perpendicular to the lines of intersection of the two sets of 
 wave-fronts. 
 
 At any moment of time the positions of the planes of maximum 
 condensation for each train of waves may be represented by pa- 
 rallel lines drawn at equal intervals X on the plane of the paper, 
 and these lines must be supposed to move with a velocity a in a 
 direction perpendicular to their length. If both sets of lines be 
 drawn, the paper will be divided into a system of equal parallelo- 
 grams, which advance in the direction of one set of diagonals. At 
 each corner of a parallelogram the condensation is doubled by the 
 superposition of the two trains of waves, and in the centre of each 
 parallelogram the rarefaction is a maximum for the same reason. 
 On each diagonal there is therefore a series of maxima and minima 
 condensations, advancing without change of relative position and 
 
76 TWO EQUAL TRAINS OF WAVES. [269. 
 
 with velocity a /cos a. Between each adjacent pair of lines of 
 maxima and minima there is a parallel line of zero condensation, 
 on which the two trains of waves neutralize one another. It is 
 especially remarkable that, if the wave-pattern were visible (like 
 the corresponding water wave-pattern to which the whole of the 
 preceding argument is applicable), it would appear to move for- 
 wards without change of type in a direction different from that of 
 either component train, and with a velocity different from that 
 with which both component trains move. 
 
 In order to express the result analytically, let us suppose 
 that the two directions of propagation are equally inclined at an 
 angle a to the axis of x. The condensations themselves may be 
 denoted by 
 
 2?r / \ 
 
 cos - (a t x cos a y sm a) 
 
 A/ 
 
 and cos (a t x cos a + y sin a) 
 
 A, 
 
 respectively, and thus the expression for the resultant is 
 
 2__ o_ 
 
 s = cos (a t x cos a y sin a) + cos (a t x cos a + y sin a) 
 
 A, A, 
 
 = 2 cos (a t x cos a) cos - (y sin a) ......... (1). 
 
 A/ A. 
 
 It appears from (1) that the distribution of s on the plane xy 
 advances parallel to the axis of x y unchanged in type, and with a 
 uniform velocity a/cos a. Considered as depending on y, s is a 
 maximum, when y sin a is equal to 0, X, 2X, 3X, &c., while for the 
 intermediate values, viz. \ X, f X, &c., s vanishes. 
 
 If a = J TT, so that the two trains of waves meet one another 
 directly, the velocity of propagation parallel to x becomes infinite, 
 and (1) assumes the form 
 
 (-- y] .................. (2); 
 
 \ A. / 
 
 which represents stationary waves. 
 
 The problem that we have just been considering is in reality 
 the same as that of the reflection of a train of plane waves by an 
 infinite plane wall. Since the expression on the right-hand side 
 of equation (1) is an even function of y, s is symmetrical with 
 respect to the axis of a), and consequently there is no motion 
 
 5= 2 cos - at cos 
 
269.] REFLECTION FROM FIXED WALL. 77 
 
 across that axis. Under these circumstances it is evident that the 
 motion could in no way be altered by the introduction along the 
 axis of x of an absolutely immovable wall. If a be the angle 
 between the surface and the direction of propagation of the inci- 
 dent waves, the velocity with which the places of maximum con- 
 densation (corresponding to the greatest elevation of water-waves) 
 move along the wall is a/ cos . It may be noticed that the aerial 
 pressures have no tendency to move the wall as a whole, except in 
 the case of absolutely perpendicular incidence, since they are at 
 any moment as much negative as positive. 
 
 269 a. When sound waves proceeding from a distant source 
 are reflected perpendicularly by a solid wall, the superposition of 
 the direct and reflected waves gives rise to a system of nodes and 
 loops, exactly as in the case of a tube considered in 255. The 
 nodal planes, viz. the surfaces of evanescent motion, occur at 
 distances from the wall which are even multiples of the quarter 
 wave length, and the loops bisect the intervals between the nodes. 
 In exploring experimentally it is usually best to seek the places 
 of minimum effect, but whether these will be nodes or loops 
 depends upon the apparatus employed, a consideration of which 
 the neglect has led to some confusion 1 . Thus a resonator will 
 cease to respond when its mouth coincides with a loop, so that 
 this method of experimenting gives the loops whether the 
 resonator be in connection with the ear or with a " manometric 
 capsule " ( 282). The same conclusion applies also to the use of 
 the unaided ear, except that in this case the head is an obstacle 
 large enough to disturb sensibly the original distribution of the 
 loop and nodes 2 . If on the other hand the indicating apparatus 
 be a small stretched membrane exposed upon both sides, or a 
 sensitive smoke jet or flame, the places of vanishing disturbance 
 are the nodes 3 . 
 
 The complete establishment of stationary vibrations with 
 nodes and loops occupies a certain time during which the sound is 
 to be maintained. When a harmonium reed is sounding steadily 
 in a room free from carpets and curtains, it is easy, listening with 
 a resonator, to find places where the principal tone is almost 
 entirely subdued. But at the first moment of putting down the 
 
 1 N. Savart, Ann. d. Chim. LXXI. p. 20, 1839 ; XL. p. 385, 1845. 
 
 2 Phil. Mag. vn. p. 150, 1879. 
 z Phil. Mag. loc. cit. p. 153. 
 
78 REFRACTION OF PLANE WAVES. [269 a. 
 
 key, or immediately after letting it go, the tone in question asserts 
 itself, often with surprising vigour. 
 
 The formation of stationary nodes and loops in front of a 
 reflecting wall may be turned to good account when it is desired 
 to determine the wave-lengths of aerial vibrations. The method 
 is especially valuable in the case of very acute sounds and of 
 vibrations of frequency so high as to be inaudible. With the aid 
 of a high pressure sensitive name vibrations produced by small 
 41 bird-calls" may be traced down to a complete wave-length of 
 6 mm., corresponding to a frequency of about 55,000 per second. 
 
 270. So long as the medium which is the vehicle of sound 
 continues of unbroken uniformity, plane waves may be propagated 
 in any direction with constant velocity and with type unchanged ; 
 but a disturbance ensues when the waves reach any part where the 
 mechanical properties of the medium undergo a change. The 
 general problem of the vibrations of a variable medium is probably 
 quite beyond the grasp of our present mathematics, but many of 
 the points of physical interest are raised in the case of plane 
 waves. Let us suppose that the medium is uniform above and 
 below a certain infinite plane (# = 0), but that in crossing that 
 plane there is an abrupt variation in the mechanical properties on 
 which the propagation of sound depends namely the compressi- 
 bility and the density. On the upper side of the plane (which for 
 distinctness of conception we may suppose horizontal) a train of 
 plane waves advances so as to meet it more or less obliquely ; the 
 problem is to determine the (refracted) wave which is propagated 
 onwards within the second medium, and also that thrown back 
 into the first medium, or reflected. We have in the first place 
 to form the equations of motion and to express the boundary 
 conditions. 
 
 In the upper medium, if p be the natural density and s the 
 condensation, 
 
 density = p (1 + s), 
 
 and pressure = P (1 + A s), 
 
 where A is a coefficient depending on the compressibility, and P 
 is the undisturbed pressure. In like manner in the lower medium 
 
 density = p 1 (1 + s^, 
 pressure = P (1 + A 1 s^, 
 
270.] REFRACTION OF PLANE WAVES. 79 
 
 the undisturbed pressure being the same on both sides of x ~ 0. 
 Taking the axis of z parallel to the line of intersection of the 
 plane of the waves with the surface of separation x 0, we have 
 for the upper medium ( 244), 
 
 _ 
 ~ V 
 
 and + F 2 5 = .......................... (2), 
 
 where V* = PA+p .......................... (3). 
 
 Similarly, in the lower medium, 
 
 " 
 
 and + 7^ = .......................... (5), 
 
 where V^ = PA 1 ^-p l .......................... (6). 
 
 These equations must be satisfied at all points of the fluid. Further 
 the boundary conditions require (i) that at all points of the 
 surface of separation the velocities perpendicular to the surface 
 shall be the same for the two fluids, or 
 
 d<f>ldx dfajdx, when x ................ (7) ; 
 
 (ii) that the pressures shall be the same, whence A^^As, or by 
 (2), (3), (5) and (6), 
 
 pd<f>/dt = p 1 d(j) 1 /dt, when x ............... (8). 
 
 In order to represent a train of waves of harmonic type, we 
 may assume </> and fa to be proportional to e i(ax+by+et) t where 
 ax + by = const, gives the direction of the plane of the waves. If 
 we assume for the incident wave, 
 
 the reflected and refracted waves may be represented respectively 
 
 by 
 
 The coefficient of t is necessarily the same in all three waves 
 on account of the periodicity, and the coefficient of y must be the 
 same, since the traces of all the waves on the plane of separation 
 
80 GREEN'S INVESTIGATION [270. 
 
 must move together. With regard to the coefficient of #, it ap- 
 pears by substitution in the differential equations that its sign is 
 changed in passing from the incident to the reflected wave ; in 
 fact 
 
 C 2=F 2 [(a) 2 + 6 2 ]=F 1 2 K+6 2 ] ............ (12). 
 
 Now b -7- V( ft2 + & 2 ) is t ne s i ne f tne angle included between the 
 axis of x and the normal to the plane of the waves in optical 
 language, the sine of the angle of incidence, and b -r V(#i 2 + b 2 ) is in 
 like manner the sine of the angle of refraction. If these angles 
 be called 0, O l} (12) asserts that sin# : sin^ is equal to the con- 
 stant ratio V : V lt the well-known law of sines. The laws of 
 refraction and reflection follow simply from the fact that the velo- 
 city of propagation normal to the wave-fronts is constant in each 
 medium, that is to say, independent of the direction of the wave- 
 front, taken in connection with the equal velocities of the traces of 
 all the waves on the plane of separation (F-e-sin0 = V 1 
 It remains to satisfy the boundary conditions (7) and (8). 
 
 These give 
 
 whence 
 
 This completes the symbolical solution. If a 2 (and #j) be real, we 
 see that if the incident wave be 
 
 </> = cos (ax + by + ct), 
 
 or in terms of V, X, and 0, 
 
 & 
 < = cos (a? cos 6 + ysmd + Vt) ............ (15), 
 
 A. 
 
 the reflected wave is 
 
 PI cot 6 l 
 
 PI 
 
 ~ 
 
 ~p cot 
 and the refracted wave is 
 
 Pi , 
 
 p COt 
 
 cos (x cos 0, + y sin ft + Vj) . . .(17). 
 
270.] OF REFLECTION AND REFRACTION. 81 
 
 The formula for the amplitude of the reflected wave, viz. 
 
 p l cot ft 
 
 4>_-p^0 
 
 +'~lh cotfl." 8)> 
 
 p COt 6 
 
 is here obtained on the supposition that the waves are of harmonic 
 type ; but since it does not involve X, and there is no change of 
 phase, it may be extended by Fourier's theorem to waves of any 
 type whatever. 
 
 If there be no reflected wave, cot ft : cot 6 p l : p, from which 
 and (1 + cot 2 0,) : (1 + cot 2 0) = F 2 : Vf t we deduce 
 
 which shews that, provided the refractive index V 1 : V be inter- 
 mediate in value between unity and p : p l} there is always an 
 angle of incidence at which the wave is completely intromitted ; 
 but otherwise there is no such angle. 
 
 Since (18) is not altered (except as to sign) by an interchange 
 of 0, ft ; p, p l ; &c., we infer that a wave incident in the second 
 medium at an angle ft is reflected in the same proportion as a 
 wave incident in the first medium at an angle 6. 
 
 As a numerical example let us suppose that the upper medium 
 is air at atmospheric pressure, and the lower medium water. 
 Substituting for cot ft its value in terms of 6 and the refractive 
 index, we get 
 
 or, since V l : V = 4*3 approximately, 
 
 cot ft /cot 0= '23 V(l - 17-5 tan 2 0), 
 
 which shews 'that the ratio of cotangents diminishes to zero, as 6 
 increases from zero to about 13, after which it becomes imaginary, 
 indicating total reflection, as we shall see presently. It must be 
 remembered that in applying optical terms to acoustics, it is the 
 water that must be conceived to be the ' rare ' medium. The ratio 
 of densities is about 770 : 1 ; so that 
 
 $' _ 1 - -0003 V(l - 17-5 tan 2 6) 
 </>' ~ 1 + '0003 V(l - 17'5 tan 2 0) 
 
 = 1 - '0006 V(l - 17*5 tan 2 0) vety nearly. 
 Even at perpendicular incidence the reflection is sensibly perfect. 
 
 R. II. ff*& 
 
 [ U-N 
 
82 FRESNEL'S EXPRESSIONS. [270. 
 
 If both media be gaseous, A 1 = A, if the temperature be con- 
 stant ; and even if the development of heat by compression be 
 taken into account, there will be no sensible difference between 
 A and A l in the case of the simple gases. Now, if A l = A, 
 p l : p = sin 2 6 : sin 2 ft, and the formula for the intensity of the 
 reflected wave becomes 
 
 ft" = sin 20 - sin 2ft = tan (0 - ft) 
 </>' ~ sin 26 + sin 2ft ~ tan (6 + ft) " 
 
 coinciding with that given by Fresnel for light polarized perpen- 
 dicularly to the plane of incidence. In accordance with Brewster's 
 law the reflection vanishes at the angle of incidence, whose 
 tangent is F/F X . 
 
 But, if on the other hand p l = p, the cause of disturbance 
 being the change of compressibility, we have 
 
 <j>" = tan ft - tan 6 = sin (ft - 0) 
 $ tan ft + tan sin (ft + 0) " 
 
 agreeing with Fresnel's formula for light polarized in the plane 
 of incidence. In this case the reflected wave does not vanish at 
 any angle of incidence. 
 In general, when 6 = 0, 
 
 I*'*'-?-*'?** (23); 
 
 so that there is no reflection, if p t : p = V : V-^. In the case of 
 gases F 2 : Fj 2 = p l : p, and then 
 
 *"_Vft-Vp_F-r. 
 f -Jfr + Jp F+Fr 
 
 Suppose, for example, that after perpendicular incidence re- 
 flection takes place at a surface separating air and hydrogen. We 
 have 
 
 p = -001 276, />! = -00008837; 
 
 whence Vp : Vfi = 3'800, giving 
 
 <" = - -5833 f. 
 
 The ratio of intensities, which is as the square of the amplitudes, 
 is '3402 : 1, so that about one-third part is reflected. 
 
 If the difference between the two media be very small, and we 
 write F!= F+SF, (24) becomes 
 
270.] REFLECTION DUE TO TEMPERATURE AND MOISTURE. 83 
 
 If the first medium be air at Cent., and the second medium be 
 air at t Cent., V+ SV = V*J(l + '00366 t) ; so that 
 
 </>"/<' = - -0009U 
 
 The ratio of the intensities of the reflected and incident sounds is 
 therefore '83 x 10~ 6 x t 2 : 1. 
 
 As another example of the same kind we may take the case in 
 which the first medium is dry air and the second is air of the 
 same temperature saturated with moisture. At 10 Cent, air 
 saturated with moisture is lighter than dry air by about one part 
 in 220, so that S V = ^ V nearly. Hence we conclude from (25) 
 that the reflected sound is only about one 774,000 th part of the 
 incident sound. 
 
 From these calculations we see that reflections from warm or 
 moist air must generally be very small, though of course the effect 
 may accumulate by repetition. It must also be remembered that 
 in practice the transition from one state of things to the other 
 would be gradual, and not abrupt, as the present theory supposes. 
 If the space occupied by the transition amount to a considerable 
 fraction of the wave-length, the reflection would be materially 
 lessened. On this account we might expect grave sounds to travel 
 through a heterogeneous medium less freely than acute sounds. 
 
 The reflection of sound from surfaces separating portions of 
 gas of different densities has engaged the attention of Tyndall, 
 who has devised several striking experiments in illustration of the 
 subject 1 . For example, sound from a high-pitched reed was con- 
 ducted through a tin tube towards a sensitive flame, which served 
 as an indicator. By the interposition of a coal-gas flame issuing 
 from an ordinary bat's-wing burner between the tube and the 
 sensitive flame, the greater part of the effect could be cut off. 
 Not only so, but by holding the flame at a suitable angle, the 
 sound could be reflected through another tube in sufficient quantity 
 to excite a second sensitive flame, which but for the interposition 
 of the reflecting flame would have remained undisturbed. 
 
 [The refraction of Sound has been demonstrated experimentally 
 by Sondhauss 2 with the aid of a collodion balloon charged with 
 carbonic acid.] 
 
 
 1 Sound, 3rd edition, p. 282, 1875. 
 
 2 Pogg. Ann. t. 85, p. 378, 1852. Phil Mag. vol. v. p. 73, 1853. 
 
 62 
 
84 TOTAL REFLECTION. [270. 
 
 The preceding expressions (16), (17), (18) hold good in every 
 case of reflection from a 'denser' medium; but if the velocity of 
 sound be greater in the lower medium, and the angle of incidence 
 exceed the critical angle, a 1 becomes imaginary, and the formulae 
 require modification. In the latter case it is impossible that a 
 refracted wave should exist, since, even if the angle of refraction 
 were 90, its trace on the plane of separation must necessarily 
 outrun the trace of the incident wave. 
 
 If i a/ be written in place of a 1} the symbolical equations are 
 
 Incident wave 
 
 $ 
 
 Reflected wave 
 
 Refracted wave 
 
 <k - f e i(-foi'*+&y+c) . 
 
 p~~ l a 
 from which by discarding the imaginary parts, we obtain 
 
 Incident wave 
 
 (f> = cos (ax + by + ct) (26), 
 
 Reflected wave 
 
 < = cos (- ax + ~by + ct + 2e) (27), 
 
 Refracted wave 
 
 cos (6y + c + e) (28), 
 
 . / 2 
 
 ^ + ^V 
 - a 2 / 
 
 p- 
 
 where tane = i ......................... (29). 
 
 api 
 
 These formulae indicate total reflection. The disturbance in the 
 second medium is not a wave at all in the ordinary sense, and at 
 a short distance from the surface of separation (x negative) be- 
 comes insensible. Calculating a/ from (12) and expressing it in 
 terms of 9 and X, we find 
 
 shewing that the disturbance does not penetrate into the second 
 medium more than a few wave-lengths. 
 
270.] LAW OF ENERGY VERIFIED. 85 
 
 The difference of phase between the reflected and the incident 
 waves is 2e, where 
 
 tane= . J tan 2 - ^ sec 2 (31). 
 
 If the media have the same compressibilities, p : p : = V? : V' 2 , and 
 
 ...(32). 
 
 Since there is no loss of energy in reflection and refraction, the 
 work transmitted in any time across any area of the front of the 
 incident wave must be equal to the work transmitted in the same 
 time across corresponding areas of the reflected and refracted 
 waves. These corresponding areas are plainly in the ratio 
 
 cos 6 : cos 6 : cos l ; 
 and thus by 245 (T being the same for all the waves), 
 
 or since V : V l = sin 6 : sin lf 
 
 p cot (#*-<!>" *) = pl cot 6i<tf ............... (33), 
 
 which is the energy condition, and agrees with the result of multi- 
 plying together the two boundary equations (13). 
 
 When the velocity of propagation is greater in the lower than 
 in the upper medium, and the angle of incidence exceeds the 
 critical angle, no energy is transmitted into the second medium ; 
 in other words the reflection is total. 
 
 The method of the present investigation is substantially the 
 same as that employed by Green in a paper on the Reflection and 
 Refraction of Sound 1 . The case of perpendicular incidence was 
 first investigated by Poisson 2 , who obtained formulae corresponding 
 to (23) and (24), which had however been already given by Young 
 for the reflection of Light. In a subsequent memoir 8 Poisson 
 considered the general case of oblique incidence, limiting himself, 
 however, to gaseous media for which Boyle's law holds good, and 
 by a very complicated analysis arrived at a result equivalent to 
 
 1 Cambridge Transactions, vol. vi. p. 403, 1838. 
 
 2 Mem. de VInstitut, t. u. p. 305. 1819. 
 
 3 "Memoire sur le mouvement de deux fluides elastiques superposes." Mem. 
 de rinstitut, t. x. p. 317. 1831. 
 
86 REFLECTION FROM A [270. 
 
 (21). He also verified that the energies of the reflected and re- 
 fracted waves make up that of the incident wave 1 . 
 
 271. If the second medium be indefinitely extended down- 
 wards with complete uniformity in its mechanical properties, the 
 transmitted wave is propagated onwards continually. But if at 
 x l there be a further change in the compressibility, or density, 
 or both, part of the wave will be thrown back, and on arrival at 
 the first surface (x 0) will be divided into two parts, one trans- 
 mitted into the first medium, and one reflected back, to be again 
 divided at x = l, and so on. By following the progress of these 
 waves the solution of the problem may be obtained, the resultant 
 reflected and transmitted waves being compounded of an infinite 
 convergent series of components, all parallel and harmonic. This 
 is the method usually adopted in Optics for the corresponding 
 problem, and is quite rigorous, though perhaps not always suf- 
 ficiently explained ; but it does not appear to have any advantage 
 over a more straightforward analysis. In the following investi- 
 gation we shall confine ourselves to the case where the third 
 medium is similar in its properties to the first medium. 
 
 In the first medium 
 
 In the second medium 
 
 ^ _ ^'^( 
 
 In the third medium 
 
 with the conditions 
 
 c 2 =F 2 (a 2 + 6 2 )=F 1 2 (a 1 2 + 6 2 ) .................. (1). 
 
 At the two surfaces of separation we have to secure the 
 equality of normal motions and pressures ; for x 0, 
 
 (2); 
 
 p ((p ~\~ <p ) == P\ (. *y ~t~ "V ^ J 
 for x = Z, 
 
 a i (V - y c. - / w.Y'i^ ( /g\ 
 
 pi (^jr'e~ iai1 + ^"e 11 ) = pibitT**)' 
 
 1 [It is interesting and encouraging to note Laplace's remark in a correspondence 
 with T. Young. The great analyst writes (1817) "Je persiste a croire que le 
 probleme de la propagation des ondes, lorsqu'elles traversent diffe'rens milieux, n'a 
 jamais 6t6 re"solu, et qu'il surpasse peut-etre les forces actuelles de 1'analyse " 
 (Young's Works, vol. i. p. 374).] 
 
271.] PLATE OF FINITE THICKNESS. 87 
 
 from which ty' and ty" are to be eliminated. We get 
 
 (<&' - 6") cos aj-i ^- (# 
 ' 
 
 (4); 
 ") cos aj-i (<' - <#>" 
 
 and from these, if for brevity ap 1 /a l p = a, 
 
 C = a-or 1 (5) 
 
 <' a + or 1 - 2i cot aj" 
 
 ...(6). 
 
 . 
 <' 2 cos a^ 4- i sin a^ (a + a x ) 
 
 In order to pass to real quantities, these expressions must be 
 put into the form Be ie . If a* be real, we find corresponding to 
 
 the incident wave 
 
 = cos (ax + by + ct), 
 the reflected wave 
 
 (or 1 a) sin (- ax + by+ct e) , 
 
 and the transmitted wave 
 
 _ 2 cos (ax + by + ct + al g) /g\ 
 
 ~~ V{4 cos 2 a^ + sin 2 ^ I (a + a" 1 ) 2 } 
 where 
 
 , (9)- 
 
 If a = p l cot 0/p cot ^ = 1, there is no reflected wave, and the 
 transmitted wave is represented by 
 
 </> = cos (ax -f by + ct + al a^), 
 
 shewing that, except for the alteration of phase, the whole of the 
 medium might as well have been uniform. 
 
 If I be small, we have approximately for the reflected wave 
 < = i^Z (a" 1 - a) sin (- ax + by + ct), 
 
 a formula applying when the plate is thin in comparison with 
 the wave-length. Since a x = ^TT/XJ) cos O l , it appears that for a 
 given angle of incidence the amplitude varies inversely as \ lt or 
 as X. 
 
 In any case the reflection vanishes, if cot 2 a^ =00 , that is, if 
 
 21 cos #! = m\!, 
 m being an integer. The wave is then wholly transmitted. 
 
88 REFLECTION FROM A PLATE [271. 
 
 At perpendicular incidence, the intensity of the reflection is 
 expressed by 
 
 Let us now suppose that the second medium is incompressible, so 
 that FJ = x ; our expression becomes 
 
 shewing how the amount of reflection depends upon the relative 
 masses of such quantities of the media as have volumes in the ratio 
 of I : X. It is obvious that the second medium behaves like a 
 rigid body and acts only in virtue of its inertia. If this be suf- 
 ficient, the reflection may become sensibly total. 
 
 We have now to consider the case in which a-^ is imaginary. 
 In the symbolical expressions (5) and (6) coso^ and isma^l are 
 real, while a, a + a" 1 , a a" 1 are pure imaginaries. Thus, if we 
 suppose that o 1 = za 1 / , a = ia', and introduce the notation of the 
 hyperbolic sine and cosine ( 170), we get 
 
 ' 2 cosh a^l i (a' a'" 1 ) sinh a^'l ' 
 
 _ _ 
 <' 2 cosh a/J - i (a' - a'- 1 ) sinh afl * 
 
 Hence, if the incident wave be 
 
 $ = cos (ax + by + ct), 
 the reflected wave is expressed by 
 
 , _ (' + a'" 1 ) sinh a^'l cos (- ax + by + ct + e) 
 
 V{4 coshVJ + (a' - a 7 - 1 ) 2 sinh 2 a//} 
 where cot e = ^ (a 7 - 1 - a') tanh a,' I ............... (13), 
 
 and the transmitted wave is expressed by 
 
 2 sin (ax + by + ct + al + e) .. 
 
 " V{4 cosh 2 a// + (a x - a'- 1 ) 2 sinhV/} " 
 
 It is easy to verify that the energies of the reflected and 
 transmitted waves account for the whole energy of the incident 
 wave. Since in the present case the corresponding areas of wave- 
 front are equal for all three waves, it is only necessary to add the 
 squares of the amplitudes given in equations (7), (8), or in equa- 
 tions (12), (14). 
 
272.] OF FINITE THICKNESS. 89 
 
 272. These calculations of reflection and refraction under 
 various circumstances might be carried further, but their interest 
 would be rather optical than acoustical. It is important to bear 
 in mind that no energy is destroyed by any number of reflections 
 and refractions, whether partial or total, what is lost in one direc- 
 tion always reappearing in another. 
 
 On account of the great difference of densities reflection is 
 usually nearly total at the boundary between air and any solid or 
 liquid matter. Sounds produced in air are not easily communi- 
 cated to water, and. wee versa sounds, whose origin is under water, 
 are heard with difficulty in air. A beam of wood, or a metallic 
 wire, acts like a speaking tube, conveying sounds to considerable 
 distances with very little loss. 
 
 272 a. In preceding sections the surface of separation, at 
 which reflection takes place, is supposed to be absolutely plane. 
 It is of interest, both from an acoustical and from an optical point 
 of view, to inquire what effect would be produced by roughnesses, 
 or corrugations, in the reflecting surface ; and the problem thus 
 presented may be solved without difficulty to a certain extent by 
 the method of 268, especially if we limit ourselves to the case of 
 perpendicular incidence. The equation of the reflecting surface 
 will be supposed to be z = f, where f is a periodic function of x 
 whose mean value is zero. As a particular case we may take 
 
 % = ccospx ........................... (1); 
 
 but in general we should have to supplement the first term of the 
 series expressed in (1) by cosines and sines of the multiples of px. 
 The velocity-potential of the incident wave (of amplitude unity) 
 may be written 
 
 For the regularly reflected wave we have </> = A Q e~ ikz } the time 
 factor being dropped for the sake of brevity ; but to this must be 
 added terms in cospx, cos2px, &c. Thus, as the complete value 
 of </> in the upper medium, 
 
 = e i*x + A,e- ikz + Atf-** cospx + A 2 e-^ z cos 2px+ ...... (3), 
 
 in which 
 
 ^ = A; 2 -^, /* 2 2 = & 2 -V, .................. (4). 
 
 The expression (3), in which for simplicity sines of multiples 
 of px have been omitted from the first, would be sufficiently 
 
90 REFLECTION FROM [272 a. 
 
 general even though cosines of multiples of px accompanied 
 c cos poo in (1). 
 
 As explained in 268, much turns upon whether the quanti- 
 ties /*!, /^j,... are real or imaginary. In the latter case the 
 corresponding terms are sensible only in the neighbourhood of 
 2 = 0. If all the values of ^ be imaginary, as happens when 
 p>k, the reflected wave soon reduces itself to its first term. 
 
 For any real value of p, say /^ r , the corresponding part of the 
 velocity-potential is 
 
 representing plane waves inclined to z at angles whose sines are 
 rp/k. These are known in Optics as the spectra of the rth 
 order. When the wave-length of the corrugation is less than that 
 of the vibration, there are no lateral spectra. 
 
 In the lower medium we have 
 
 <t> 1 = B e ik ^ + l e i ^' z cospx + B 2 e i ^' z cos 2px + ...... (5), 
 
 where ^ = k? -p 2 , /v 2 = k? - 4p 2 , ............... (6). 
 
 In each exponential the coefficient of z is to be taken positive; 
 if it be imaginary, because the wave is propagated in the negative 
 direction; if it be real, because the disturbance must decrease, 
 and not increase, in penetrating the second medium. 
 
 The conditions to be satisfied at the boundary are ( 270) 
 that 
 
 (7), 
 
 and that d<f>/dn = dfa/dn, where dn is perpendicular to the surface 
 z f. Hence 
 
 .(8). 
 
 Thus far there is no limitation upon either the amplitude (c) 
 or the wave-length (2?r/p) of the corrugation. We will now 
 suppose that the wave-length is very large, so that p 2 - may be 
 neglected throughout. Under these conditions, (8) reduces to 
 
 ........................ (9). 
 
 In the differentiation of (3) and (5) with respect to z, the 
 various terms are multiplied by the coefficients yuj, ^ 2 ,.../ii', fa ,-'> 
 
272 a.] A CORRUGATED SURFACE. 91 
 
 but when p 2 is neglected these quantities may be identified with 
 k t &j respectively. Thus at the boundary 
 
 (JLZ 
 
 and 
 
 az p l 
 
 by (7). Accordingly, 
 
 -d. e Atf- * cospx j, 
 
 By this equation J. , A 1} &c. are determined when f is known. 
 
 If we put = 0, we fall back on previous results (23) 270 for 
 a truly plane surface. Thus A lt A Zt ... vanish, while 
 
 expressing the amplitude of the wave regularly reflected. 
 
 We will now apply (10) to the case of a simple corrugation, as 
 expressed in (1), and for brevity we will denote the right hand 
 member of (11) by R. The determination of A , A l ,... requires 
 the expression of e 2 *^ in Fourier's series. We have (compare 
 343) 
 
 (2&c) 2 Jjj (2&c) cos Zp% + 2 J 4 (2&c) cos &px + ... 
 cospx 2J" 3 (2&c) cos 3px + %J 5 (2&c) cos 5px . . .} 
 
 (12), 
 
 where J , /i,... are the Bessel's functions of the various orders. 
 Thus 
 
 A /R= J (2kc), Ai/R= 2i'J 1 (2&c), 
 
 ^/ = -2V,(2Aw), I f . 
 = 2*V,(2*c), ^ (13) ' 
 
 the coefficients of even order being real, and those of odd order 
 pure imaginaries. The complete solution of the problem of 
 reflection, under the restriction that p is small, is then obtained 
 by substitution in (3); and it may be remarked that it is the same 
 as would be furnished by the usual optical methods, which take 
 account only of phase retardations. Thus, as regards the wave 
 
92 CASE WHERE THE SECOND [272 a. 
 
 reflected parallel to z t the retardation at any point of the surface 
 due to the corrugation is 2 or 2ccos_pa?. The influence of the 
 corrugations is therefore to change the amplitude of the reflected 
 vibration in the ratio 
 
 / cos (2kc cospx) dx : Jdx, or J (2&c). 
 
 In like manner the amplitude of each of the lateral spectra of 
 the first order is J^&c), and so on. The sum of the intensities 
 of all the reflected waves is 
 
 2 2 +...}=.ft 2 ............... (14) 
 
 by a known theorem ; so that, in the case supposed (of p infinitely 
 small), the fraction of the whole energy thrown back is the same 
 as if the surface were smooth. 
 
 It should be remarked that in this theory there is no limitation 
 upon the value of 2&c. If 2kc be small, only the earlier terms of 
 the series are sensible, the Bessel's function J n (2kc) being of order 
 (2kc) n . When on the other hand 2&c is large, the early terms are 
 small, while the series is less convergent. The values of J and 
 J-i are tabulated in 200. For certain values of 2kc individual 
 reflected waves vanish. In the case of the regularly reflected wave, 
 or spectrum of zero order, this first occurs when 2kc = 2 '404, 206, 
 or c = '2X. 
 
 The full solution of the problem of the present section would 
 require the determination of the reflection when k is given for all 
 values of c and for all values of p. We have considered the case 
 of p infinitely small, and we shall presently deal with the case 
 where p > k. For intermediate values of p the problem is more 
 difficult, and in considering them we shall limit ourselves to the 
 simpler boundary conditions which obtain when no energy pene- 
 trates the second medium. The simplest case of all arises when 
 p l = 0, so that the boundary equation (7) reduces to 
 
 4> = ............................. (15), 
 
 the condition for an " open end," 256. We may also refer to 
 the case of a rigid wall, or "closed" end, where the surface condi- 
 tion is 
 
 ........................... (16). 
 
 By (3) and (15) the condition to be satisfied at the surface is 
 A ie i(*-M* cos px + A 2 e i(k -^* cos 2px + . . . = 0. . .(16). 
 
27 2 a.] MEDIUM IS IMPENETRABLE. 93 
 
 In our problem z is given by (1) as a function of # ; and the 
 equations of condition are to be found by equating to zero the 
 coefficients of the various terms involving cospx, cos2px, &c., 
 when the left hand member of (16) is expanded in Fourier's series. 
 The development of the various exponentials is effected as in (12); 
 and the resulting equations are 
 
 2 iJ, (2k) + A, { / (k - ri - J* (k - MI)} 
 
 ...=0 ......... (18), 
 
 f ...Q ......... (19), 
 
 and so on, where for the sake of brevity c has been made equal to 
 unity. So far as (k yu,) may be treated as real, as happens for a 
 large number of terms when p is small relatively to k, the various 
 Bessel's functions are all real, and thus the A's of even order are 
 real and the A's of odd order are pure imaginaries. Accordingly 
 the phase of the perpendicularly reflected wave is the same as if 
 c = 0; but it must be remembered that this conclusion is in reality 
 only approximate, because, however small p may be, the JJLS end 
 by becoming imaginary. 
 
 From the above equations it is easy to obtain the value of A 
 as far as the term in p 4 . From (19) 
 
 from (18) 
 
 1^ = 2^(2*0 + (A? -/*,)/, 
 
 and finally from (17) 
 
 -A = J (2k) + (k-f J , l )J 1 (2k) 
 
 ....... (20). 
 
 From (4) 
 
 k --& + & + "'> 
 
 so that, as expanded in powers of p with reintroduction of c, 
 
 1 Brit. Ass. Eep. 1893, p. 691. 
 
94 FIXED WALL. [272 a. 
 
 This gives the amplitude of the perpendicularly reflected wave, 
 with omission of p 6 and higher powers of p. 
 
 The case of reflection from a fixed wall is a little more compli- 
 cated. By (8) the boundary condition is 
 
 d<f>/dz + pc sin px . d^fdx = 0, 
 which gives 
 
 x- A 2 e i(k ~^ z cos 2x - ... 
 
 sin 2 +... = 
 
 i/c 
 
 ......... (22) 
 
 as the equation to be satisfied when z c cospx. The first approxi- 
 mation to A! gives 
 
 A^MJ^Zkc) ........................ (23); 
 
 whence to a second approximation 
 A. = J. ( 
 
 .. 1 .................. (24). 
 
 The first approximation to the various coefficients may be found 
 by putting R = + l in (13). 
 
 When p > k, there are no diffracted spectra, and the whole 
 energy of the wave incident upon an impenetrable medium must 
 be represented in the wave directly reflected. The modulus of A 
 is therefore unity. When p<k, the energy is divided between 
 the various spectra, including that of zero order. There is thus a 
 relation between the squares of the moduli of A , A lf A 2) ..., the 
 series being continued as long as //, is real. 
 
 A more analytical investigation may be based upon v. Helm- 
 holtz's theorem ( 293), according to which 
 
 where 8 is any closed surface, and T/T and ^ satisfy the equation 
 
 V 2 + & 2 = 0. 
 In order to apply this we take for ^r and % the real and 
 
272 a.] ENERGY EQUATION. 95 
 
 imaginary parts respectively of <j> as given by (3). Thus repre- 
 senting each complex coefficient A n in the form C n + iD n , we get 
 
 o/r = cos kz + (7 cos kz + D sin kz 
 
 -f (Ci cos ^2 + Asm fr^cospx + ..................... (25), 
 
 % = sin kz C Q sin &s + D cos kz 
 
 + (- Cisin /A! 2 4- A cos /z^cos ># + ..................... (26). 
 
 In (25), (26), when the series are carried sufficiently far, the 
 terms change their form on account of //, becoming imaginary ; 
 but for the present purpose these terms will not be required, as 
 they disappear when z is very great. The surface of integration 
 8 is made up of the reflecting surface and of a plane parallel to it 
 at a great distance. Although this surface is not strictly closed, 
 it may be treated as such, since the part still remaining open 
 laterally at infinity does not contribute sensibly to the result. 
 Now the part of the integral corresponding to the reflecting 
 surface vanishes, either because 
 
 t-x-o, 
 
 or else because dty/dn = d%/dn = ; 
 
 and we conclude that when z is great 
 
 The application of (27) to the values of ^r and % in (25), (26) 
 gives 
 
 Cf + A 2 + &(Cf + A 2 ) + g(CV + A 2 ) + .- = 1 ...... (28), 
 
 the series in (28) being continued so far as to include every real 
 value of fji. 
 
 In (28) J (C n 2 + D n 2 ) represents the intensity of each spectrum 
 of the nth order. 
 
 The coefficient fj, n /k is equal to cos O n , where 6 n is the 
 obliquity of the diffracted rays. The meaning of this factor 
 will be evident when it is remarked that to each unit of area 
 of the waves incident and directly reflected, there corresponds an 
 area cos O n of the waves which constitute the spectrum of the nth 
 order. 
 
 If all the values of p are imaginary, as happens when p > k, 
 (28) reduces to 
 
 C * + D * = I ............................ (29), 
 
 or the intensity of the wave directly reflected is unity. It is of 
 
96 OBLIQUE INCIDENCE. [272 a. 
 
 importance to notice the full significance of this result. However 
 deep the corrugations may be, if only they are periodic in a period 
 less than the wave-length of the vibration, the regular reflection is 
 total. An extremely rough wall will thus reflect sound waves of 
 moderate pitch as well as if it were theoretically smooth. 
 
 The above investigation is limited to the case where the second 
 medium is impenetrable, so that the whole energy of the incident 
 wave is thrown back in the regularly reflected wave and in the 
 diffracted spectra. It is an interesting question whether the 
 conclusion that corrugations of period less than X have no effect 
 can be extended so as to apply when there is a wave regularly 
 transmitted. It is evident that the principle of energy does not 
 suffice to decide the question, but it is probable that the answer 
 should be in the negative. If we suppose the corrugations of 
 given period to become very deep and involved, it would seem 
 that the condition of things would at last approach that of a very 
 gradual transition between the media, in which case ( 148 b) the 
 reflection tends to vanish. 
 
 Our limits will not allow us to treat at length the problem of 
 oblique incidence upon a corrugated surface; but one or two 
 remarks may be made. 
 
 If p* may be neglected, the solution corresponding to (13) is 
 
 A Q = RJ 9 (2kccos0) (30), 
 
 6 being the angle of incidence and reflection, and R the value of 
 J*o, 270, corresponding to c = 0. The factor expressing the 
 effect of the corrugations is thus a function of c cos ; so that a 
 deep corrugation when 6 is large may have the same effect as a 
 shallow one when 6 is small. 
 
 Whatever be the angle of incidence, there are no reflected 
 spectra (except of zero order) when the wave-length of the 
 corrugation is less than the half of that of the vibrations. Hence, 
 if the second medium be impenetrable, the regular reflection 
 under the above condition is total. 
 
 The reader who wishes to pursue the study of the theory of 
 gratings is referred to treatises on optics, and to papers by the 
 Author 1 , and by Prof. Rowland 2 . 
 
 1 The Manufacture and Theory of Diffraction Gratings, Phil. Mag. vol. XLVII. 
 pp. 81, 193, 1874 ; On Copying Diffraction Gratings, and on some Phenomena con- 
 nected therewith, Phil. Mag. vol. xi. p. 196, 1881 ; Enc. Brit. Wave Theory of Light. 
 
 2 Gratings in Theory and Practice, Phil. Mag. vol. xxxv. p. 397, 1893. 
 
CHAPTER XIV. 
 
 GENERAL EQUATIONS. 
 
 273. IN connection with the general problem of aerial 
 vibrations in three dimensions one of the first questions, which 
 naturally offers itself, is the determination of the motion in an 
 unlimited atmosphere consequent upon arbitrary initial dis- 
 turbances. It will be assumed that the disturbance is small, so 
 that the ordinary approximate equations are applicable, and further 
 that the initial velocities are such as can be derived from a velocity- 
 potential, or ( 240) that there is no circulation. If the latter con- 
 dition be violated, the problem is one of vortex motion, on which 
 we do not enter. We shall also suppose in the first place that no 
 external forces act upon the fluid, so that the motion to be 
 investigated is due solely to a disturbance actually existing at 
 a time (t = 0), previous to which we do not push our inquiries. 
 The method that we shall employ is not very different from that 
 of Poisson 1 , by whom the problem was first successfully attacked. 
 
 If u , v , W be the initial velocities at the point x, y, 2, and s 
 the initial condensation, we have ( 244), 
 
 = { 
 
 V dy + w dz) (1), 
 
 <o = -a 2 s (2), 
 
 by which the initial values of the velocity-potential </> and of its 
 differential coefficient with respect to time c/> are determined. 
 The problem before us is to determine (/> at time t from the above 
 
 1 Sur Pint6gration de quelques equations lineaires aux differences partielles, 
 et particulierement de 1'equation generate du mouvement des fluides elastiques. 
 Mem. de Vlnstitut, t. in. p. 121. 1820. 
 
 R. II. 7 
 
98 ARBITRARY INITIAL DISTURBANCE. [273. 
 
 initial values, and the general equation applicable at all times and 
 places, 
 
 When < is known, its derivatives give the component velocities at 
 any point. 
 
 The symbolical solution of (3) may be written 
 
 ............... (4), 
 
 where 6 and % are two arbitrary functions of x, y, z and i = \/( !) 
 To connect and ^ with the initial values of <f) and <f>, which we 
 shall denote by / and F respectively, it is only necessary to observe 
 that when t = 0, (4) gives 
 
 so that our result may be expressed 
 
 sin(mVi) 
 
 in which equation the question of the interpretation of odd powers 
 of V need not be considered, as both the symbolic functions are 
 wholly even. 
 
 In the case where was a function of x only, we saw ( 245) 
 that its value for any point x at time t depended on the initial 
 values of <f> and </> at the points whose co-ordinates were x at 
 and x -\- at, and was wholly independent of the initial circumstances 
 at all other points. In the present case the simplest supposition 
 open to us is that the value of <j> at a point depends on the 
 initial values of < and <j> at points situated on the surface of the 
 sphere, whose centre is and radius at ; and, as there can be no 
 reason for giving one direction a preference over another, we are 
 thus led to investigate the expression for the mean value of a 
 function over a spherical surface in terms of the successive differ- 
 ential coefficients of the function at the centre. 
 
 By the symbolical form of Maclaurin's theorem the value of 
 F(x, y, z) at any point P on the surface of the sphere of radius r 
 may be written 
 
 _d_ d_ + d^ 
 
 F(x, y, z) = e dx '** ***.*(*, y , *o), 
 the centre of the sphere being the origin of co-ordinates. In 
 
273.] ARBITRARY INITIAL DISTURBANCE. 99 
 
 the integration over the surface of the sphere d/dx Q> d/dy , 
 behave as constants ; we may denote them temporarily by I, m, n, 
 so that V 2 = I* + ??t 2 + n 2 . 
 
 Thus, r being the radius of the sphere, and dS an element of 
 its surface, since, by the symmetry of the sphere, we may replace 
 
 any function of , - - - T by the same function of z without 
 V( 2 + ra 2 + 7i 2 ) 
 
 altering the result of the integration, 
 
 r r r 1" lx + my+nz 
 
 I Li*+,!/+nz dS = I |( e v) V(P+'+') dS 
 
 dS = 
 
 The mean value of F over the surface of the sphere of radius r is 
 thus expressed by the result of the operation on F of the symbol 
 sin (iVr)/iVr, or, if ffdcr denote integration with respect to angular 
 space, 
 
 By comparison with (5) we now see that so far as </> depends 
 on the initial values of $, it is expressed by 
 
 or in words, < at any point at time t is the mean of the initial 
 values of < over the surface of the sphere described round the 
 point in question with radius at, the whole multiplied by t. 
 
 By Stokes' rule ( 95), or by simple inspection of (5), we see 
 that the part of </> depending on the initial values of < may be 
 derived from that just written by differentiating with respect to t 
 and changing the arbitrary function. The complete value of cf> at 
 time t is therefore 
 
 which is Poisson's result *. 
 
 On account of the importance of the present problem, it may 
 
 1 Another investigation will be found in Kirchhoff's Vorlesungen uber Mathe- 
 matische Physik, p. 317. 1876. [See also Note to 273 at the end of this volume.] 
 
 72 
 
100 VERIFICATION OF SOLUTION. [273. 
 
 be well to verify the solution a posteriori. We have first to prove 
 that it satisfies the general differential equation (3). Taking for 
 the present the first term only, and bearing in mind the general 
 symbolic equation 
 
 -t-* 
 
 9* t&-& 
 
 we find from (8) 
 
 1 d ..d 
 
 dS being the surface element of the sphere r = at. 
 But by Green's theorem 
 
 and thus 
 
 Now II V 2 Fda- is the same as V 2 nFdar, and thus (3) is in fact 
 satisfied. 
 
 Since the second part of < is obtained from the first by differen- 
 tiation, it also must satisfy the fundamental equation. 
 
 With respect to the initial conditions we see that when t is 
 made equal to zero in (8), 
 
 (<*)* (< = 0)=/(0); 
 
 of which the first term becomes in the limit F(Q). When ^ = 0, 
 
 = 2ajjf'(at)do- (t = 0) = 0, 
 
 since the oppositely situated elements cancel in the limit, when 
 the radius of the spherical surface is indefinitely diminished. The 
 expression in (8) therefore satisfies the prescribed initial con- 
 ditions as well as the general differential equation. 
 
274.] LIMITED INITIAL DISTURBANCE; /101 
 
 274. If the initial disturbance be confined to a space; IV the* 
 integrals in (8) 273 are zero, unless some part of the surface ol 
 the sphere r at be included within T. Let be a point external 
 to T, T! and r 2 the radii of the least and greatest spheres described 
 about which cut it. Then so long as at<r l ,(f) remains equal 
 to zero. When at lies between r x and r 2 , </> may be finite, but for 
 values greater than r a (/> is again zero. The disturbance is thus at 
 any moment confined to those parts of space for which a t is inter- 
 mediate between r : and r. 2 . The limit of the wave is the envelope 
 of spheres with radius at, whose centres are situated on the surface 
 of T. " When t is small, this system of spheres will have an 
 exterior envelope of two sheets, the outer of these sheets being 
 exterior, and the inner interior to the shell formed by the as- 
 semblage of the spheres. The outer sheet forms the outer limit 
 to the portion of the medium in which the dilatation is different 
 from zero. As t increases, the inner sheet contracts, and at last its 
 opposite sides cross, and it changes its character from being ex- 
 terior, with reference to the spheres, to interior. It then expands, 
 and forms the inner boundary of the shell in which the wave of 
 condensation is comprised 1 ." The successive positions of the 
 boundaries of the wave are thus a series of parallel surfaces, and 
 each boundary is propagated normally with a velocity equal to a. 
 
 If at the time t = there be no motion, so that the initial 
 disturbance consists merely in a variation of density, the subse- 
 quent condition of things is expressed by the first term of (8) 273. 
 Let us suppose that the original disturbance, still limited to a 
 finite region T, consists of condensation only, without rarefaction. 
 It might be thought that the same peculiarity would attach to the 
 resulting wave throughout the whole of its subsequent course; but, 
 as Prof. Stokes has remarked, such a conclusion would be erroneous. 
 For values of the time less than rja the potential at is zero ; 
 it then becomes negative (s being positive), and continues nega- 
 tive until it vanishes again when t = r 2 /a, after which it always 
 remains equal to zero. While </> is diminishing, the medium at 
 is in a state of condensation, but as </> increases again to zero, the 
 state of the medium at is one of rarefaction. The wave propa- 
 gated outwards consists therefore of two parts at least, of which 
 the first is condensed and the last rarefied. Whatever may be the 
 character of the original disturbance within T, the final value of < 
 
 1 Stokes, "Dynamical Theory of Diffraction," Cam/;. Trans, ix. p. 15, 1849. 
 
102 CASE OF PLANE WAVES. [274. 
 
 ; as ^any external point is the same as the initial value, and there- 
 fore, since &*s = <j>, the mean condensation during the passage of 
 the wave, depending on the integral fsdt, is zero. Under the 
 head of spherical waves we shall have occasion to return to this 
 subject ( 279). 
 
 The general solution embodied in (8) 273 must of course 
 embrace the particular case of plane waves, but a few words on 
 this application may not be superfluous, for it might appear at 
 first sight that the effect at a given point of a disturbance initially 
 confined to a slice of the medium enclosed between two parallel 
 planes would not pass off in any finite time, as we know it ought 
 to do. Let us suppose for simplicity that < is zero throughout, 
 and that within the slice in question the initial value <f> is 
 constant. From the theory of plane waves we know that at any 
 arbitrary point the disturbance will finally cease after the lapse of 
 a time t, such that at is equal to the distance (d) of the point 
 under consideration from the further boundary of the initially 
 disturbed region ; while on the other hand, since the sphere of 
 radius at continues to cut the region, it would appear from the 
 general formula that the disturbance continues. It is true indeed 
 that </> remains finite, but this is not inconsistent with rest. It 
 will in fact appear on examination that the mean value of (f> 
 multiplied by the radius of the sphere is the same whatever may 
 be the position and size of the sphere, provided only that it 
 cut completely through the region of original disturbance. If 
 at>d, <f> is thus constant with respect both to space and time, 
 and accordingly the medium is at rest. 
 
 [The same principles may find an application to the phenomena 
 of thunder. Along the path of the lightning we may perhaps 
 suppose that the generation of heat is uniform, equivalent to a 
 uniform initial distribution of condensation. It appears that the 
 value of </> at the point of observation can change rapidly only 
 when the sphere r = at meets the path of the discharge at its 
 extremities or very obliquely.] 
 
 275. In two dimensions, when </> is independent of z t it might 
 be supposed that the corresponding formula would be obtained by 
 simply substituting for the sphere of radius at the circle of equal 
 radius. This, however, is not the case. It may be proved that 
 
275.] TWO DIMENSIONS. 103 
 
 the mean value of a function F (x, y) over the circumference of a 
 circle of radius r is J (irV) F , where i = V( 1), 
 
 and J is Bessel's function of zero order ; so that 
 
 differing from what is required to satisfy the fundamental equation. 
 The correct result applicable to two dimensions may be obtained 
 from the general formula. The element of spherical surface dS 
 may be replaced by rdrdO/costy, where r t are plane polar 
 co-ordinates, and ^ is the angle between the tangent plane and 
 that in which the motion takes place. Thus 
 
 COS 
 
 , 
 
 at 
 F(at) is replaced by F(r, 0), and so 
 
 F(r,0)rdrd0 
 
 
 where the integration extends over the area of the circle rat. 
 The other term might be obtained by Stokes' rule. 
 
 This solution is applicable to the motion of a layer of gas 
 between two parallel planes, or to that of an unlimited stretched 
 membrane, which depends upon the same fundamental equation. 
 
 276. From the solution in terms of initial conditions we may, 
 as usual ( 66), deduce the effect of a continually renewed dis- 
 turbance. Let us suppose that throughout the space T (which 
 will ultimately be made to vanish), a uniform disturbance <f>, 
 equal to <I> (t') dt', is communicated at time t'. The resulting value 
 of c/> at time t is 
 
 where S denotes the part of the surface of the sphere r = a(t t') 
 intercepted within T, a quantity which vanishes, unless a(t t') be 
 comprised between the narrow limits r x and r a . Ultimately t t' 
 may be replaced by r/a, and <E> (t') by < (t - r/a) ; and the result 
 of the integration with respect to dt' is found by writing T (the 
 volume) for fa S dt'. Hence 
 
104 SOURCES OF SOUND. [276. 
 
 shewing that the disturbance originating at any point spreads itself 
 symmetrically in all directions with velocity a, and with amplitude 
 varying inversely as the distance. Since any number of particular 
 solutions may be superposed, the general solution of the equation 
 
 < = a 2 V 2 (f) + <I> (2) 
 
 may be written 
 
 r denoting the distance of the element dV situated at x, y, z from 
 (at which <j> is estimated), and <I> (t r/a) the value of <1> for the 
 point x, y, z at the time t r/a. Complementary terms, satisfying 
 through all space the equation $ a 2 V 2 </>, may of course occur 
 independently. 
 
 In our previous notation ( 244) 
 
 3> = i[(Xdm + Ydy + Zdz) ; 
 
 and it is assumed that Xdx + Ydy + Zdz is a complete differential. 
 Forces, under whose action the medium could not adjust itself to 
 equilibrium, are excluded ; as for instance, a force uniform in mag- 
 nitude and direction within a space T, and vanishing outside that 
 space. The nature of the disturbance denoted by <E> is perhaps best 
 seen by considering the extreme case when <l> vanishes except 
 through a small volume, which is supposed to diminish without 
 limit, while the magnitude of <1> increases in such a manner that 
 the whole effect remains finite. If then we integrate equation (2) 
 through a small space including the point at which <X> is ulti- 
 mately concentrated, we find in the limit 
 
 shewing that the effect of <I> may be represented by a proportional 
 introduction or abstraction of fluid at the place in question. The 
 simplest source of sound is thus analogous to a focus in the theory 
 of conduction of heat, or to an electrode in the theory of electricity. 
 
 277. The preceding expressions are general in respect of the 
 relation to time of the functions concerned ; but in almost all the 
 applications that we shall have to make, it will be convenient to 
 analyse the motion by Fourier's theorem and treat separately the 
 
277.] HARMONIC TYPE. 105 
 
 simple harmonic motions of various periods, afterwards, if necessary, 
 compounding the results. The values of </> and <3>, if simple har- 
 monic at every point of space, may be expressed in the form 
 R cos (nt + e), R and e being independent of time, but variable 
 from point to point. But as in such cases it often conduces to 
 simplicity to add the term iR sin (nt 4- e), making altogether 
 Re i(nt+e] , or Re ie .e int , we will assume simply that all the functions 
 which enter into a problem are proportional to e int , the coeffi- 
 cients being in general complex. After our operations are com- 
 pleted, the real and imaginary parts of the expressions can be 
 separated, either of them by itself constituting a solution of the 
 question. 
 
 Since (/> is proportional to e int , <J> = ri*<f>\ and the differential 
 equation becomes 
 
 2 = ........................ (1), 
 
 where, for the sake of brevity, k is written in place of n/a. If X 
 denote the wave-length of the vibration of the period in question, 
 
 (2). 
 
 To adapt (3) of the preceding section to the present case, it is 
 only necessary to remark that the substitution of t r/a for t is 
 effected by introducing the factor e~ inrfa , or e~ ikr : thus 
 
 and the solution of (1) is 
 
 to which may be added any solution of V 2 + k 2 <f> = 0. 
 
 If the disturbing forces be all in the same phase, and the 
 region through which they act be very small in comparison with 
 the wave-length, e~ ikr may be removed from under the integral 
 sign, and at a sufficient distance we may take 
 
 or in real quantities, on restoring the time factor and replacing 
 dV by * 1; 
 
106 VERIFICATION OF SOLUTION. [277. 
 
 In order to verify that (3) satisfies the differential equation (1), 
 we may proceed as in the theory of the common potential. Con- 
 sidering one element of the integral at a time, we have first to 
 shew that 
 
 oikr 
 
 </>= .............................. (5) 
 
 satisfies V 2 <+& 2 < = 0, at points for which r is finite. The 
 simplest course is to express V 2 in polar co-ordinates referred to 
 the element itself as pole, when it appears that 
 
 ( 
 
 / 
 
 \ 
 
 2 d \ e~ ikr 1 3? e~ ikr e~ ikr 
 
 _ _ _ _ ___ _ __ M _ _-..,_. L A/2 _ 
 
 dr 2 r drj r ~ r dr* ' r r 
 
 We infer that (3) satisfies V 2 </> + & 2 = 0, at all points for 
 which <I> vanishes. In the case of a point at which <I> does not 
 vanish, we may put out of account all the elements situated at a 
 finite distance (as contributing only terms satisfying V 2 < + k z (f> = 0), 
 and for the element at an infinitesimal distance replace e~ ikr by 
 unity. Thus on the whole 
 
 exactly as in Poisson's theorem for the common potential 1 . 
 
 278. The effect of a force <J> X distributed over a surface S may 
 be obtained as a limiting case from (3) 277. O dV is replaced by 
 <> bdS, b denoting the thickness of the layer ; and in the limit we 
 may write <l> 6 = Q. Thus 
 
 The value of <f> is the same on the two sides of S, but there is 
 discontinuity in its derivatives. If dn be drawn outwards from S 
 normally, (4) 276 gives 
 
 ' T,*,. -(2) 2 - 
 
 If the surface S be plane, the integral in (1) is evidently 
 symmetrical with respect to it, and therefore 
 
 1 See Thomson and Tait's Natural Philosophy, 491. 
 
 2 Helmholtz. Crelle, t. 57, p. 21, 1860. 
 
278.] SURFACE DISTRIBUTIONS. 107 
 
 Hence, if d(f>/dn be the given normal velocity of the fluid in 
 contact with the plane, the value of <j> is determined by 
 
 dn r 
 
 which is a result of considerable importance. To exhibit it in 
 terms of real quantities, we may take 
 
 Pe^^ (4), 
 
 P and 6 being real functions of the position of dS. The symbolical 
 solution then becomes 
 
 (5). 
 
 from which, if the imaginary part be rejected, we obtain 
 
 *dS.. ..(6), 
 
 corresponding to 
 
 = Pcos(nt + e) ..................... (7). 
 
 The same method is applicable to the general case when the 
 motion is not restricted to be simple harmonic. We have 
 
 where by V(t r/a) is denoted the normal velocity at the plane 
 for the element dS at the time t r/a, that is to say, at a time 
 r/a antecedent to that at which (f> is estimated. 
 
 In order to complete the solution of the problem for the 
 unlimited mass of fluid lying on one side of an infinite plane, we 
 have to add the most general value of $, consistent with V = 0. 
 This part of the question is identical with the general problem of 
 reflection from an infinite rigid plane 1 . 
 
 It is evident that the effect of the constraint will be represented 
 by the introduction on the other side of the plane of fictitious 
 initial displacements and forces, forming in conjunction with those 
 actually existing on the first side a system perfectly symmetrical 
 with respect to the plane. Whatever the initial values of (f> and 
 <j> may be belonging to any point on the first side, the same must 
 be ascribed to its image, and in like manner whatever function of 
 
 1 Poisson, Journal de Vecole poly technique, t. vn. 1808. 
 
108 INFINITE PLANE WALL. [278. 
 
 the time <I> may be at the first point, it must be conceived to be the 
 same function of the time at the other. Under these circumstances 
 it is clear that for all future time < will be symmetrical with 
 respect to the plane, and therefore the normal velocity zero. So 
 far then as the motion on the first side is concerned, there will be 
 no change if the plane be removed, and the fluid continued 
 indefinitely in all directions, provided the circumstances on the 
 second side are the exact reflection of those on the first. This 
 being understood, the general solution of the problem for a 
 fluid bounded by an infinite plane is contained in the formulae 
 (8) 273, (3) 277, and (8) of the present section. They give the 
 result of arbitrary initial conditions (< and < ), arbitrary applied 
 forces (<3>), and arbitrary motion of the plane ( V). 
 
 Measured by the resulting potential, a source of given magni- 
 tude, i.e. a source at which a given introduction and withdrawal 
 of fluid takes place, is thus twice as effective when close to a rigid 
 plane, as if it were situated in the open ; and the result is ulti- 
 mately the same, whether the source be concentrated in a point 
 close to the plane, or be due to a corresponding normal motion 
 of the surface of the plane itself. 
 
 The operation of the plane is to double the effective pressures 
 which oppose the expansion and contraction at the source, and 
 therefore to double the total energy emitted ; and since this energy 
 is diffused through only the half of angular space, the intensity of 
 the sound is quadrupled, which corresponds to a doubled amplitude, 
 or potential ( 245). 
 
 We will now suppose that instead of d<f>/dn = 0, the prescribed 
 condition at the infinite plane is that < = 0. In this case the 
 fictitious distribution of < , <j> , <>, on the second side of the plane 
 must be the opposite of that on the first side, so that the sum of the 
 values at two corresponding points is always zero. This secures 
 that on the plane of symmetry itself <f) shall vanish throughout. 
 
 Let us next suppose that there are two parallel surfaces S lt 
 $ 2 , separated by the infinitely small interval dn, and that the 
 value of <!>! on the second surface is equal and opposite to the value 
 of <J>j on the first. In crossing S l} there is by (2) a finite change 
 in the value of d<f>/dn to the amount of 3^/a 2 , but in crossing S 2 the 
 same finite change occurs in the reverse direction. When dh is 
 reduced without limit, and ^dn replaced by 4> n , d<j>/dn will be 
 
278.] DOUBLE SHEETS. 109 
 
 the same on the two sides of the double sheet, but there will be 
 discontinuity in the value of < to the amount of 4> u /a 2 . At the 
 same time (1) becomes 
 
 > a dS (9). 
 
 . 
 
 4>7ra 2 
 
 If the surface S be plane, the values of </> on the two sides of it 
 are numerically equal, and therefore close to the surface itself 
 
 Hence (9) may be written 
 
 where < under the integral sign represents the surface-potential, 
 positive on the one side and negative on the other, due to the 
 action of the forces at 8. The direction of dn must be under- 
 stood to be towards the side at which < is to be estimated. 
 
 279. The problem of spherical waves diverging from a point 
 has already been forced upon us and in some degree considered, 
 but on account of its importance it demands a more detailed 
 treatment. If the centre of symmetry be taken as pole the velo- 
 city-potential is a function of r only, and ( 241) V 2 reduces to 
 
 d* 2 d 1 d 2 
 
 5-H --- =-, or to - --,- r. The equation of free motion (3) S 273 
 
 dr 2 r dr r dr* 
 
 thus becomes 
 
 whence, as in 245, 
 
 r<f>=f(at-r) + F(at + r) .................. (2). 
 
 The values of the velocity and condensation are to be found by 
 differentiation in accordance with the formulae 
 
 _, ._-!. ..................... (3). 
 
 dr a 2 dt 
 
 As in the case of one dimension, the first term represents a wave 
 advancing in the direction of r increasing, that is to say, a diver- 
 gent wave, and the second term represents a wave converging upon 
 the pole. The latter does not in itself possess much interest. If 
 we confine our attention to the divergent wave, we have 
 
110 SPHERICAL WAVES. [279. 
 
 When r is very great the term divided by r 2 may be neglected, 
 and then approximately 
 
 u = as ................................. (5), 
 
 the same relation as obtains in the case of a plane wave, as might 
 have been expected. 
 
 If the type be harmonic, 
 
 r( f ) = Ae ik ^ t - r ^ ............................ (6), 
 
 or, if only the real part be retained, 
 
 (7). 
 
 If a divergent disturbance be confined to a spherical shell, 
 within and without which there is neither condensation nor 
 velocity, the character of the wave is limited by a remarkable re- 
 lation, first pointed out by Stokes 1 . From equations (4) we have 
 
 shewing that the value of f(at - r) is the same, viz. zero, both 
 inside and outside the shell to which the wave is limited. Hence 
 by (4), if a and @ be radii less and greater than the extreme 
 radii of the shell, 
 
 ........................... (8), 
 
 which is the expression of the relation referred to. As in 274, 
 we see that a condensed or a rarefied wave cannot exist alone. 
 When the radius becomes great in comparison with the thickness, 
 the variation of r in the integral may be neglected, and (8) then 
 expresses that the mean condensation is zero. 
 
 [Availing himself of Foucault's method for rendering visible 
 minute optical differences, Topler 2 succeeded in observing spherical 
 sonorous waves originating in small electric sparks, and their 
 reflection from a plane wall. Subsequently photographic records 
 of similar phenomena have been obtained by Mach 8 .] 
 
 In applying the general solution (2) to deduce the motion 
 resulting from arbitrary initial circumstances, we must remember 
 that in its present form it is too general for the purpose, since it 
 covers the case in which the pole is itself a source, or place where 
 
 1 Phil. Mag. xxxiv. p. 52. 1849. 
 
 2 Pogg. Ann. vol. cxxxi. pp. 33, 180. 1867. 
 
 3 Sitzber. der Wiener Akad., 1889. 
 
279.] CONTINUITY THROUGH POLE. Ill 
 
 fluid is introduced or withdrawn in violation of the equation of 
 continuity. The total current across the surface of a sphere of 
 radius r is 4t7rr 2 u, or by (2) and (3) 
 
 - 4-rr [f(at -r) + F(at + r)} + 4-rrr {F' (at + r) -f (at - r)}, 
 
 so that, if the pole be not a source, f(at r) + F(at + r), or r<, 
 must vanish with r. Thus 
 
 f(at) + F(at) = ........................ (9), 
 
 an equation which must hold good for all positive values of the 
 argument 1 . 
 
 By the known initial circumstances the values of u and s are 
 determined for the time t = 0, and for all (positive) values of r. 
 If these initial values be represented by u and s , we obtain from 
 (2) and (3) 
 
 by which the function f is determined for all negative arguments, 
 and the function F for all positive arguments. The form of / for 
 positive arguments follows by means of (9), and then the whole 
 subsequent motion is determined by (2). The form of F for 
 negative arguments is not required. 
 
 The initial disturbance divides itself into two parts, travelling 
 in opposite directions, in each of which r< is propagated with 
 constant velocity a, and the inwards travelling wave is continually 
 reflected at the pole. Since the condition to be there satisfied is 
 r< = 0, the case is somewhat similar to that of a parallel tube 
 terminated by an open end, and we may thus perhaps better 
 understand why the condensed wave, arising from the liberation 
 of a mass of condensed air round the pole, is followed immediately 
 by a wave of rarefaction. 
 
 [The composite character of the wave resulting from an initial 
 condensation may be invoked to explain a phenomenon which has 
 often occasioned surprise. When windows are broken by a violent 
 explosion in their neighbourhood, they are frequently observed to 
 
 1 The solution for spherical vibrations may be obtained without the use of (1) 
 by superposition of trains of plane waves, related similarly to the pole, and tra- 
 velling outwards in all directions symmetrically. 
 
112 SIMPLE SOURCE. [279. 
 
 have fallen outwards as if from exposure to a wave of rarefaction. 
 This effect may be attributed to the second part of the compound 
 wave; but it may be asked why should the second part preponderate 
 over the first ? If the window were freely suspended, the 
 momentum acquired from the waves of condensation and rare- 
 faction would be equal. But under the actual conditions it may 
 well happen that the force of the condensed wave is spent in 
 overcoming the resistance of the supports, and then the rarefied 
 wave is left free to produce its full effect] 
 
 280. Returning now to the case of a train of harmonic waves 
 travelling outwards continually from the pole as source, let us 
 investigate the connection between the velocity-potential and the 
 quantity of fluid which must be supposed to be introduced and 
 withdrawn alternately. If the velocity-potential be 
 
 A 
 
 r) .................... (1), 
 
 we have, as in the preceding section, for the total current crossing 
 a sphere of radius r, 
 
 4?rr 2 -^- = A {cos k (at r) kr sin k (at r)}=A cos kat, 
 
 where r is small enough. If the maximum rate of introduction of 
 fluid be denoted by A, the corresponding potential is given by (1). 
 
 It will be observed that when the source, as measured by A, is 
 finite, the potential and the pressure-variation (proportional to <j>) 
 are infinite at the pole. But this does not, as might for a moment 
 be supposed, imply an infinite emission of energy. If the pressure 
 be divided into two parts, one of which has the same phase as 
 the velocity, and the other the same phase as the acceleration, it 
 will be found that the former part, on which the work depends, 
 is finite. The infinite part of the pressure does no work on the 
 whole, but merely keeps up the vibration of the air immediately 
 round the source, whose effective inertia is indefinitely great. 
 
 We will now investigate the energy emitted from a simple 
 source of given magnitude, supposing for the sake of greater 
 generality that the source is situated at the vertex of a rigid cone 
 of solid angle co. If the rate of introduction of fluid at the source 
 be A cos kat, we have 
 
 cor 2 d(j)/dr A cos kat 
 
280.] ENERGY EMITTED FROM GIVEN SOURC 
 ultimately, corresponding to 
 
 </> = cos k(at r) (2); 
 
 whence cf> = - sin k (at r) (3), 
 
 and cor 2 -- = A jcos k (at r) krsmk(at r)} (4). 
 
 Thus, as in 245, if dW be the work transmitted in time dt, 
 we get, since Bp = pj>, 
 
 dW pkaA* . . . . 
 
 ~-j = sin k (at r) cos k (at - r) 
 
 + p sin 2 k (at r). 
 
 O) 
 
 Of the right-hand member the first term is entirely periodic, and 
 in the second the mean value of sm 2 k(at r) is J. Thus in the 
 long run 
 
 W=^t (5)'. 
 
 Za) 
 
 It will be remarked that when the source is given, the ampli- 
 tude varies inversely as <w, and therefore the intensity inversely 
 as &) 2 . For an acute cone the intensity is greater, not only on 
 account of the diminution in the solid angle through which the 
 sound is distributed, but also because the total energy emitted 
 from the source is itself increased. 
 
 When the source is in the open, we have only to put o> = 4-Tr, 
 and when it is close to a rigid plane, o> = 2?r. 
 
 The results of this article find an interesting application in the 
 theory of the speaking trumpet, or (by the law of reciprocity 
 109, 294) hearing trumpet. If the diameter of the large open 
 end be small in comparison with the wave-length, the waves on 
 arrival suffer copious reflection, and the ultimate result, which 
 must depend largely on the precise relative lengths of the tube 
 and of the wave, requires to be determined by a different process. 
 But by sufficiently prolonging the cone, this reflection may be 
 diminished, and it will tend to cease when the diameter of the 
 open end includes a large number of wave-lengths. Apart from 
 friction it would therefore be possible by diminishing &> to obtain 
 from a given source any desired amount of energy, and at the 
 
 1 Cambridge Mathematical Tripos Examination, 1876. 
 K. II. 8 
 
114 SPEAKING TRUMPET. [280. 
 
 same time by lengthening the cone to secure the unimpeded 
 transference of this energy from the tube to the surrounding air. 
 
 From the theory of diffraction it appears that the sound will 
 not fall off to any great extent in a lateral direction, unless the 
 diameter at the large end exceed half a wave-length. The 
 ordinary explanation of the effect of a common trumpet, depending 
 on a supposed concentration of rays in the axial direction, is thus 
 untenable. 
 
 281. By means of Euler's equation, 
 
 dr* 
 
 we may easily establish a theory for conical pipes with open ends, 
 analogous to that of Bernoulli for parallel tubes, subject to the 
 same limitation as to the smallness of the diameter of the tubes in 
 comparison with the wave-length of the sound 1 . Assuming that 
 the vibration is stationary, so that r$ is everywhere proportional 
 to cos to, we get from (1) 
 
 (2), 
 
 of which the general solution is 
 
 r(f) = A cos kr + B sin kr ................... (3). 
 
 The condition to be satisfied at an open end, viz., that there is 
 to be no condensation or rarefaction, gives rcj) = 0, so that, if the 
 extreme radii of the tube be r-^ and r 2 , we have 
 
 A cos ATJ + B sin kr t = 0, A cos Ar 2 + B sin kr 2 = 0, 
 
 whence by elimination of A : B, sin k (r 2 TJ) = 0, or r 2 1\ = J m\, 
 where ra is an integer. In fact since the form of the general 
 solution (3) and the condition for an open end are the same as for 
 a parallel tube, the result that the length of the tube is a multiple 
 of the half wave-length is necessarily also the same. 
 
 A cone, which is complete as far as the vertex, may be treated 
 as if the vertex were an open end, since, as we saw in 279, the 
 condition rc/> = is there satisfied. 
 
 The resemblance to the case of parallel tubes does not extend 
 to the position of the nodes. In. the case of the gravest vibration 
 
 1 D. Bernoulli, Mem. d. VAcad. d. Sci. 1762; Duhamel, Liouville Journ. 
 Math. vol. xiv. p. 98, 1849. 
 
281.] THEORY OF CONICAL TUBES. 115 
 
 of a parallel tube open at both ends, the node occupies a central 
 position, and the two halves vibrate synchronously as tubes open 
 at one end and stopped at the other. But if a conical tube were 
 divided by a partition at its centre, the two parts would have 
 different periods, as is evident, because the one part differs from a 
 parallel tube by being contracted at its open end where the effect 
 of a contraction is to depress the pitch, while the other part is 
 contracted at its stopped end, where the effect is to raise the pitch. 
 In order that the two periods may be the same, the partition must 
 approach nearer to the narrower end of the tube. Its actual 
 position may be determined analytically from (3) by equating to 
 zero the value of dcf)/dr. 
 
 When both ends of a conical pipe are closed, the corresponding 
 notes are determined by eliminating A : B between the equations, 
 
 A (cos kr v + kr-L sin kr^) + B (sin kr-^ kr cos kr^ = 0, 
 A (cos kr a + kr 2 sin Ar 2 ) + B (sin kr z kr 2 cos kr 2 ) = 0, 
 of which the result may be put into the form 
 
 kr z tan"" 1 kr% = k^ tan" 1 kr ............... (4). 
 
 If TI = () } we have simply 
 
 tan kr 2 = kr 2 ......................... (5) 1 ; 
 
 if T! and r 2 be very great, tan" 1 ^ and tan" 1 ^ are both odd 
 multiples of ^ir, so that r* 2 r is a multiple of JX, as the theory 
 of parallel tubes requires. 
 
 [If r 2 TI = I, r 2 + r 1 = r, (4) may be written 
 
 When r is great in comparison with I, the approximate solution 
 of (6) gives 
 
 m being an integer. The influence of conicality upon the pitch is 
 thus of the second order. 
 
 Experiments upon conical pipes have been made by Boutet 2 
 and by Blaikley 3 .] 
 
 1 For the roots of this equation see 207. 
 
 2 Ann. d. Chim. vol. xxi. p. 150, 1870. 
 
 3 Phil. Mag. vi. p. 119, 1878. 
 
 82 
 
116 TWO SOURCES OF LIKE PITCH. [282. 
 
 282. If there be two distinct sources of sound of the same 
 pitch, situated at O l and 2 , the velocity-potential <f> at a point 
 P whose distances from 1} 2 are ^ and r 2 , may be expressed 
 
 A cosk(at-r 1 ) ^cos k(at-r 2 -a) 
 
 n r z 
 
 where A and B are coefficients representing the magnitudes of 
 the sources (which without loss of generality may be supposed to 
 have the same sign), and a represents the retardation (considered 
 as a distance) of the second source relatively to the first. The two 
 trains of spherical waves are in agreement at any point P, if 
 r 2 + a TJ = mX, where m is an integer, that is, if P lie on any 
 one of a system of hyperboloids of revolution having foci at 
 Oi and 2 . At points lying on the intermediate hyperboloids, 
 represented by r 2 4- a r x = + \ (2m + 1) X, the two sets of waves 
 are opposed in phase, and neutralize one another as far as their 
 actual magnitudes permit. The neutralization is complete, if 
 r x : r 2 = A : B, and then the density at P continues permanently 
 unchanged. The intersections of this sphere with the system of 
 hyperboloids will thus mark out in most cases -several circles of 
 absolute silence. If the distance O^ between the sources be great 
 in comparison with the length of a wave, and the sources themselves 
 be not very unequal in power, it will be possible to depart from 
 the sphere r x : r z = A : B for a distance of several wave-lengths, 
 without appreciably disturbing the equality of intensities, and 
 thus to obtain over finite surfaces several alternations of sound 
 and of almost complete silence. 
 
 There is some difficulty in actually realising a satisfactory 
 interference of two independent sounds. Unless the unison be 
 extraordinarily perfect, the silences are only momentary and are 
 consequently difficult to appreciate. It is therefore best to employ 
 sources which are mechanically connected in such a way that the 
 relative phases of the sounds issuing from them cannot vary. The 
 simplest plan is to repeat the first sound by reflection from a flat 
 wall ( 269, 278), but the experiment then loses something in 
 directness owing to the fictitious character of the second source. 
 Perhaps the most satisfactory form of the experiment is that 
 described in the Philosophical Magazine for June 1877 by myself. 
 "An intermittent electric current, obtained from a fork interrupter 
 making 128 vibrations per second, excited by means of electro- 
 magnets two other forks, whose frequency was 256, ( 63, 64). 
 
282.] POINTS OF SILENCE. 117 
 
 These latter forks were placed at a distance of about ten yards 
 apart, and were provided with suitably tuned resonators, by which 
 their sounds were reinforced. The pitch of the forks was 
 necessarily identical, since the vibrations were forced by electro- 
 magnetic forces of absolutely the same period. With one ear 
 closed it was found possible to define the places of silence with 
 considerable accuracy, a motion of about an inch being sufficient 
 to produce a marked revival of sound. At a point of silence, from 
 which the line joining the forks subtended an angle of about 60, 
 the apparent striking up of one fork, when the other was stopped, 
 had a very peculiar effect." 
 
 Another method is to duplicate a sound coming along a tube 
 by means of branch tubes, whose open ends act as sources. But 
 the experiment in this form is not a very easy one. 
 
 It often happens that considerations of symmetry are sufficient 
 to indicate the existence of places of silence. For example, it is 
 evident that there can be no variation of density in the continua- 
 tion of the plane of a vibrating plate, nor in the equatorial plane 
 of a symmetrical solid of revolution vibrating in the direction of 
 its axis. More generally, any plane is a plane of silence, with 
 respect to which the sources are symmetrical in such a manner 
 that at any point and at its image in the plane there are sources 
 of equal intensities and of opposite phases, or, as it is often more 
 conveniently expressed, of the same phase and of opposite ampli- 
 tudes. 
 
 If any number of sources in the same phase, whose amplitudes 
 are on the whole as much negative as positive, be placed on the 
 circumference of a circle, they will give rise to no disturbance of 
 pressure at points on the straight line which passes through the 
 centre of the circle and is directed at right angles to its plane. 
 This is the case of the symmetrical bell ( 232), which emits no 
 sound in the direction of its axis 1 . 
 
 The accurate experimental investigation of aerial vibrations is 
 beset with considerable difficulties, which have been only partially 
 surmounted hitherto. In order to avoid unwished for reflections 
 it is generally necessary to work in the open air, where delicate 
 apparatus, such as a sensitive flame, is difficult of management. 
 Another impediment arises from the presence of the experimenter 
 himself, whose person is large enough to disturb materially the 
 
 1 Phil. Mag. (5), in. p. 460. 1877. 
 
118 EXPERIMENTAL METHODS. [282. 
 
 state of things which he wishes to examine. Among indicators of 
 sound may be mentioned membranes stretched over cups, the agita- 
 tion being made apparent by sand, or by small pendulums resting 
 lightly against them. If a membrane be simply stretched across a 
 hoop, both its faces are acted upon by nearly the same forces, and 
 consequently the motion is much diminished, unless the membrane 
 be large enough to cast a sensible shadow, in which its hinder face 
 may be protected. Probably the best method of examining the 
 intensity of sound at any point in the air is to divert a portion of 
 it by means of a tube ending in a small cone or resonator, the 
 sound so diverted being led to the ear, or to a manometric 
 capsule. In this way it is not difficult to determine places of 
 silence with considerable precision. 
 
 By means of the same kind of apparatus it is possible to 
 examine even the phase of the vibration at any point in air, and to 
 trace out the surfaces on which the phase does not vary 1 . If the 
 interior of a resonator be connected by flexible tubing with a 
 manometric capsule, which influences a small gas flame, the motion 
 of the flame is related in an invariable manner (depending on the 
 apparatus itself) to the variation of pressure at the mouth of the 
 resonator; and in particular the interval between the lowest drop 
 of the flame and the lowest pressure at the resonator is independent 
 of the absolute time at which these effects occur. In Mayer's 
 experiment two flames were employed, placed close together in one 
 vertical line, and were examined with a revolving mirror. So long 
 as the associated resonators were undisturbed, the serrations of the 
 two flames occupied a fixed relative position, and this relative 
 position was also maintained when one resonator was moved about 
 so as to trace out a surface of invariable phase. For further 
 details the reader must be referred to the original paper. 
 
 283. When waves of sound impinge upon an obstacle, a 
 portion of the motion is thrown back as an echo, and under cover 
 of the obstacle there is formed a sort of sound shadow. In order, 
 however, to produce shadows in anything like optical perfection, 
 the dimensions of the intervening body must be considerable. 
 The standard of comparison proper to the subject is the wave- 
 length of the vibration ; it requires almost as extreme conditions 
 to produce rays in the case of sound, as it requires in optics to 
 avoid producing them. Still, sound shadows thrown by hills, or 
 1 Mayer, Phil. Mag. (4), XLIV. p. 321. 1872. 
 
283.] SOUND SHADOWS. 119 
 
 buildings, are often tolerably complete, and must be within the 
 experience of all. 
 
 For closer examination let us take first the case of plane waves 
 of harmonic type impinging upon an immovable plane screen, of 
 infinitesimal thickness, in which there is an aperture of any form, 
 the plane of the screen (x = 0) being parallel to the fronts of the 
 waves. The velocity-potential of the undisturbed train of waves 
 may be taken, 
 
 $ cos (nt kx) ........................ (1). 
 
 If the value of d<j>/dx over the aperture be known, formulae (6) 
 and (7) 278 allow us to calculate the value of $ at any point on 
 the further side. In the ordinary theory of diffraction, as given 
 in works on optics, it is assumed that the disturbance in the plane 
 of the aperture is the same as if the screen were away. This 
 hypothesis, though it can never be rigorously exact, will suffice 
 when the aperture is very large in comparison with the wave- 
 length, as is usually the case in optics. 
 
 For the undisturbed wave we have 
 
 ^(0 = 0) = k sin nt ....................... (2), 
 
 dx 
 
 and therefore on the further side, we get 
 
 k 
 
 the integration extending over the area of the aperture. Since 
 k = 27r/\ we see by comparison with (1) that in supposing a 
 primary wave broken up, with the view of applying Huygens' 
 principle, dS must be divided by Xr, and the phase must be 
 accelerated by a quarter of a period. 
 
 When r is large in comparison with the dimensions of the 
 aperture, the composition of the integral is best studied by the aid 
 of Fresnel's 1 zones. With the point 0, for which <t> is to be 
 estimated, as centre describe a series of spheres of radii increasing 
 by the constant difference X, the first sphere of the series being 
 of such radius (c) as to touch the plane of the screen. On this 
 plane are thus marked out a series of circles, whose radii p are 
 
 1 [These zones are usually spoken of as Huygens' zones by optical writers (e.g. 
 Billet, Traite d'Optique physique, vol. i. p. 102, Paris, 1858) ; but, as has been 
 pointed out by Schuster (Phil. Mag. vol. xxxi. p. 85, 1891), it is more correct to 
 name them after Fresnel.] 
 
120 FRESNEL'S ZONES. [283. 
 
 given by p 2 -f c 2 = (c + J wX) 2 , or p 2 = nc\, very nearly ; so that the 
 rings into which the plane is divided, being of approximately 
 equal area, make contributions to < which are approximately 
 equal in numerical magnitude and alternately opposite in sign. 
 If lie decidedly within the projection of the area, the first term 
 of the series representing the integral is finite, and the terms 
 which follow are alternately opposite in sign and of numerical 
 magnitude at first nearly constant, but afterwards diminishing 
 gradually to zero, as the parts of the rings intercepted within the 
 aperture become less and less. The case of an aperture, whose 
 boundary is equidistant from 0, is excepted. 
 
 In a series of this description any term after the first is 
 neutralized almost exactly [that is, so far as first differences are 
 concerned] by half the sum of those which immediately precede 
 and follow it, so that the sum of the whole series is represented 
 approximately by half the first term, which stands over uncom- 
 pensated. We see that, provided a sufficient number of zones be 
 included within the aperture, the value of (f> at the point is 
 independent of the nature of the aperture, and is therefore the 
 same as if there had been no screen at all. Or we may calculate 
 directly the effect of the circle with which the system of zones 
 begins; a course which will have the advantage of bringing out 
 more clearly the significance of the change of phase which we 
 found it necessary to introduce when the primary wave was broken 
 up. Thus, let us conceive the circle in question divided into 
 infinitesimal rings of equal area. The parts of </> due to each of 
 these rings are equal in amplitude and of phase ranging uniformly 
 over half a complete period. The phase of the resultant is there- 
 fore midway between those of the extreme elements, that is to 
 say, a quarter of a period behind that due to the element at 
 the centre of the circle. The amplitude of the resultant will be 
 less than if all its components had been in the same phase, in 
 the ratio J7 sin x dx : TT, or 2 : TT ; and therefore since the area 
 of the circle is TrXc, half the effect of the first zone is 
 
 - 2 sin (nt kc JTT) , x 
 
 <f) = \ . - . ^ - . ir\c = cos (nt kc), 
 
 7T AC 
 
 the same as if the primary wave were to pass on undisturbed. 
 
 When the point is well away from the projection of the 
 aperture, the result is quite different. The series representing the 
 integral then converges at both ends, and by the same reasoning 
 
283.] FRESNEL'S ZONES. 121 
 
 as before its sum is seen to be approximately zero. We conclude 
 that if the projection of on the plane # = fall within the 
 aperture, and be nearer to by a great many wave-lengths than 
 the nearest point of the boundary of the aperture, then the 
 disturbance at is nearly the same as if there were no obstacle at 
 all ; but, if the projection of fall outside the aperture and be 
 nearer to by a great many wave-lengths than the nearest point of 
 the boundary, then the disturbance at practically vanishes. 
 This is the theory of sound rays in its simplest form. 
 
 The argument is not very different if the screen be oblique to 
 the plane of the waves. As before, the motion on the further side 
 of the screen may be regarded as due to the normal motion of the 
 particles in the plane of the aperture, but this normal motion now 
 varies in phase from point to point. If the primary waves proceed 
 from a source at Q, Fresnel's zones for a point P are the series of 
 ellipses represented by r\ + r z = P Q + J n X, where r t and r z are 
 the distances of any point on the screen from Q and P respectively, 
 and n is an integer. On account of the assumed smallness of X in 
 comparison with r-^ and r 2 , the zones are at first of equal area and 
 make equal and opposite contributions to the value of </> ; and 
 thus by the same reasoning as before we may conclude that at any 
 point decidedly outside the geometrical projection of the aperture 
 the disturbance vanishes, while at any point decidedly within the 
 geometrical projection the disturbance is the same as if the 
 primary wave had passed the screen unimpeded. It may be 
 remarked that the increase of area of the Fresnel's zones due to 
 obliquity is compensated in the calculation of the integral by the 
 correspondingly diminished value of the normal velocity of the 
 fluid. The enfeeblement of the primary wave between the screen 
 and the point P due to divergency is represented by a diminution 
 in the area of the Fresnel's zones below that corresponding to 
 plane incident waves in the ratio r-L + r^ : r : . 
 
 There is a simple relation between the transmission of sound 
 through an aperture in a screen and its reflection from a plane 
 reflector of the same form as the aperture, of which advantage may 
 sometimes be taken in experiment. Let us imagine a source 
 similar to Q and in the same phase to be placed at Q', the image of 
 Q in the plane of the screen, and let us suppose that the screen is 
 removed and replaced by a plate whose form and position is exactly 
 that of the aperture ; then we know that the effect at P of the two 
 
122 CONDITIONS OF COMPLETE REFLECTION. [283. 
 
 sources is uninfluenced by the presence of the plate, so that the 
 vibration from Q' reflected from the plate and the vibration from 
 Q transmitted round the plate together make up the same vibra- 
 tion as would be received from Q if there were no obstacle at all. 
 Now according to the assumption which we made at the begin- 
 ning of this section, the unimpeded vibration from Q may be 
 regarded as composed of the vibration that finds its way round the 
 plate and of that which would pass an aperture of the same form 
 in an infinite screen, and thus the vibration from Q as transmitted 
 through the aperture is equal to the vibration from Q' as reflected 
 from the plate. 
 
 In order to obtain a nearly complete reflection it is not neces- 
 sary that the reflecting plate include more than a small number of 
 Fresnel's zones. In the case of direct reflection the radius p of 
 the first zone is determined by the equation 
 
 P *(l/c 1 + l/c 2 ) = \ (4), 
 
 where ^ and c 2 are the distances from the reflector of the source 
 and of the point of observation. When the distances concerned 
 are great, the zones become so large that ordinary walls are 
 insufficient to give a complete reflection, but at more moderate 
 distances echos are often nearly perfect. The area necessary for 
 complete reflection depends also upon the wave-length ; and thus 
 it happens that a board or plate, which would be quite inadequate 
 to reflect a grave musical note, may reflect very fairly a hiss or 
 the sound of a high whistle. In experiments on reflection by 
 screens of moderate size, the principal difficulty is to get rid 
 sufficiently of the direct sound. The simplest plan is to reflect 
 the sound from an electric bell, or other fairly steady source, round 
 the corner of a large building 1 . 
 
 284. In the preceding section we have applied Huygens' 
 principle to the case where the primary wave is supposed to be 
 broken up at the surface of an imaginary plane. If we really 
 know what the normal motion at the plane is, we can calculate 
 the disturbance at any point ,on the further side by a rigorous 
 process. For surfaces other than the plane the problem has not 
 been solved generally ; nevertheless, it is not difficult to see that 
 when the radii of curvature of the surface are very great in com- 
 parison with the wave-length, the effect of a normal motion of an 
 
 1 Phil. Mag. (5), m. p. 458. 1877. 
 
 
284.] DIVERGING WAVES. 123 
 
 element of the surface must be very nearly the same as if the 
 surface were plane. On this understanding we may employ the 
 same integral as before to calculate the aggregate result. As a 
 matter of convenience it is usually best to suppose the wave to be 
 broken up at what is called in optics a wave-surface, that is, a 
 surface at every point of which the phase of the disturbance is the 
 same. 
 
 Let us consider the application of Huygens' principle to cal- 
 culate the progress of a given divergent wave. With any point 
 P, at which the disturbance is required, as centre, describe a series 
 of spheres of radii continually increasing by the constant difference 
 ^ X, the first of the series being of such radius (c) as to touch the 
 given wave-surface at C. If R be the radius of curvature of the 
 surface in any plane through P and (7, the corresponding radius p 
 of the outer boundary of the n th zone is given by the equation 
 
 R + c = V{ 2 - p 2 } + V{(c + J ^X) 2 - p*}, 
 from which we get approximately 
 
 (1). 
 
 If the surface be one of revolution round PC, the area of the first 
 n zones is TT/T, and since p 2 is proportional to n, it follows that the 
 zones are of equal area. If the surface be not of revolution, the 
 area of the first n zones is represented ^fp 2 dO, where 6 is the 
 azimuth of the plane in which p is measured, but it still remains 
 true that the zones are of equal area. Since by hypothesis the 
 normal motion does not vary rapidly over the wave-surface, the 
 disturbances at P due to the various zones are nearly equal in 
 magnitude and alternately opposite in sign, and we conclude that, 
 as in the case of plane waves, the aggregate effect is the half of 
 that due to the first zone. The phase at P is accordingly retarded 
 behind that prevailing over the given wave-surface by an amount 
 corresponding to the distance c. 
 
 The intensity of the disturbance at P depends upon the area of 
 the first Fresnel's zone, and upon the distance c. In the case of 
 
 symmetry, we have 
 
 ?rp 2 
 
 which shews that the disturbance is less than if R were infinite in 
 the ratio R -f c : R. This diminution is the effect of divergency, 
 
 i 
 
124 VARIATION OF INTENSITY. [284. 
 
 and is the same as would be obtained on the supposition that the 
 motion is limited by a conical tube whose vertex is at the centre of 
 curvature ( 266). When the surface is not of revolution, the 
 value of ^/Q n p^dO -=- c may be expressed in terms of the principal 
 radii of curvature RI and R 2 , with which R is connected by the 
 relation 
 
 l/R = cos^O/R, + sm*0/R 2 . 
 
 We obtain on effecting the integration 
 
 ^rpde~ 
 
 2cJ r 
 
 c) (R, + c) 
 so that the amplitude is diminished by divergency in the ratio 
 
 a, a result which might be anticipated by 
 supposing the motion limited to a tube formed by normals drawn 
 through a small contour traced on the wave-surface. 
 
 Although we have spoken hitherto of diverging waves only, 
 the preceding expressions may also be applied to waves converging 
 in one or in both of the principal planes, if we attach suitable 
 signs to R! and R^. In such a case the area of the first Fresnel's 
 zone is greater than if the wave were plane, and the intensity 
 of the vibration is correspondingly increased. If the point P 
 coincide with one of the principal centres of curvature, the 
 expression (2) becomes infinite. The investigation, on which (2) 
 was founded, is then insufficient ; all that we are entitled to affirm 
 is that the disturbance is much greater at P than at other points 
 on the same normal, that the disproportion increases with the 
 frequency, and that it would become infinite for notes of infinitely 
 high pitch, whose wave-length would be negligible in comparison 
 with the distances concerned. 
 
 285. Huygens' principle may also be applied to investigate 
 the reflection of sound from curved surfaces. If the material 
 surface of the reflector yielded so completely to the aerial 
 pressures that the normal motion at every point were the same as 
 it would have been in the absence of the reflector, then the sound 
 waves would pass on undisturbed. The reflection which actually 
 ensues when the surface is unyielding may therefore be regarded 
 as due to a normal motion of each element of the reflector, equal 
 and opposite to that of the primary waves at the same point, and 
 may be investigated by the formula proper to plane surfaces in 
 the manner of the preceding section, and subject to a similar 
 
285.] BEFLECTION FROM CURVED SURFACES. 125 
 
 limitation as to the relative magnitudes of the wave-length and of 
 the other distances concerned. 
 
 The most interesting case of reflection occurs when the 
 surface is so shaped as to cause a concentration of rays upon a 
 particular point (P). If the sound issue originally from a simple 
 source at Q, and the surface be an ellipsoid of revolution having 
 its foci at P and Q, the concentration is complete, the vibration 
 reflected from every element of the surface being in the same 
 phase on arrival at Q. If Q be infinitely distant, so that the 
 incident waves are plane, the surface becomes a paraboloid having 
 its focus at P, and its axis parallel to the incident rays. We must 
 not suppose, however, that a symmetrical wave diverging from 
 Q is converted by reflection at the ellipsoidal surface into a 
 spherical wave converging symmetrically upon P ; in fact, it is 
 easy to see that the intensity of the convergent wave must be 
 different in different directions. Nevertheless, when the wave- 
 length is very small in comparison with the radius, the different 
 parts of the convergent wave become approximately independent 
 of one another, and their progress is not materially affected by 
 the failure of perfect symmetry. 
 
 The increase of loudness due to curvature depends upon the 
 area of reflecting surface, from which disturbances of uniform 
 phase arrive, as compared with the area of the first Fresnel's 
 zone of a plane reflector in the same position. If the distances of 
 the reflector from the source and from the point of observation be 
 considerable, and the wave-length be not very small, the first 
 Fresnel's zone is already rather large, and therefore in the case 
 of a reflector of moderate dimensions but little is gained by 
 making it concave. On the other hand, in laboratory experiments, 
 when the distances are moderate and the sounds employed are of 
 high pitch, e.g. the ticking of a watch or the cracking of electric 
 sparks, concave reflectors are very efficient and give a distinct 
 concentration of sound on particular spots. 
 
 286. We have seen that if a ray proceeding from Q passes 
 after reflection at a plane or curved surface through P, the point 
 R at which it meets the surface is determined by the condition 
 that QR + RP is a minimum (or in some cases a maximum). 
 The point R is then the centre of the system of Fresnel's zones ; 
 the amplitude of the vibration at P depends upon the area of the 
 
126 FERMAT'S PRINCIPLE. [286. 
 
 first zone, and its phase depends upon the distance QR + RP. If 
 there be no point on the surface of the reflector, for which 
 QR + RP is a maximum or a minimum, the system of Fresnel's 
 zones has no centre, and there is no ray proceeding from Q which 
 arrives at P after reflection from the surface. In like manner if 
 sound be reflected more than once, the course of a ray is deter- 
 mined by the condition that its whole length between any two 
 points is a maximum or a minimum. 
 
 The same principle may be applied to investigate the refraction 
 of sound in a medium, whose mechanical properties vary gradually 
 from point to point. The variation is supposed to be so slow 
 that no sensible reflection occurs, and this is not inconsistent 
 with decided refraction of the rays in travelling distances which 
 include a very great number of wave-lengths. It is evident 
 that what we are now concerned with is not merely the length 
 of the ray, but also the velocity with which the wave travels 
 along it, inasmuch as this velocity is no longer constant. The 
 condition to be satisfied is that the time occupied by a wave 
 in travelling along a ray between any two points shall be a 
 maximum or a minimum ; so that, if V be the velocity of propa- 
 gation at any point, and ds an element of the length of the ray, 
 the condition may be expressed, S j V~ l ds = 0. This is Fermat's 
 principle of least time. 
 
 The further developement of this part of the subject would 
 lead us too far into the domain of geometrical optics. The funda- 
 mental assumption of the smallness of the wave-length, on which 
 the doctrine of rays is built, having a far wider application to the 
 phenomena of light than to those of sound, the task of developing 
 its consequences may properly be left to the cultivators of the 
 sister science. In the following sections the methods of optics 
 are applied to one or two isolated questions, whose acoustical 
 interest is sufficient to demand their consideration in the present 
 work. 
 
 287. One of the most striking of the phenomena connected 
 with the propagation of sound within closed buildings is that 
 presented by "whispering galleries," of which a good and easily 
 accessible example is to be found in the circular gallery at the 
 base of the dome of St Paul's cathedral. As to the precise mode 
 of action acoustical authorities are not entirely agreed. In the 
 
287.] WHISPERING GALLERIES. 127 
 
 opinion of the Astronomer Royal 1 the effect is to be ascribed to 
 reflection from the surface of the dome overhead, and is to be 
 observed at the point of the gallery diametrically opposite to the 
 source of sound. Every ray proceeding from a radiant point and 
 reflected from the surface of a spherical reflector, will after 
 reflection intersect that diameter of the sphere which contains the 
 radiant point. This diameter is in fact a degraded form of one of 
 the two caustic surfaces touched by systems of rays in general, 
 being the loci of the centres of principal curvature of the surface 
 to which the rays are normal. The concentration of rays on one 
 diameter thus effected, does not require the proximity of the 
 radiant point to the reflecting surface. 
 
 Judging from some observations that I have made in St Paul's 
 whispering gallery, I am disposed to think that the principal 
 phenomenon is to be explained somewhat differently. The ab- 
 normal loudness with which a whisper is heard is not confined 
 to the position diametrically opposite to that occupied by the 
 whisperer, and therefore, it would appear, does not depend 
 materially upon the symmetry of the dome. The whisper seems 
 to creep round the gallery horizontally, not necessarily along the 
 shorter arc, but rather along that arc towards which the whisperer 
 faces. This is a consequence of the very unequal audibility of a 
 whisper in front of and behind the speaker, a phenomenon which 
 may easily be observed in the open air 2 . 
 
 Let us consider the course of the rays diverging from a radiant 
 point P, situated near the surface of a reflecting sphere, and let us 
 denote the centre of the sphere by 0, and the diameter passing 
 through P by A A', so that A is the point on the surface nearest 
 to P. If we fix our attention on a ray which issues from P at an 
 angle + with the tangent plane at A, we see that after any 
 number of reflections it continues to touch a concentric sphere of 
 radius OP cos 6, so that the whole conical pencil of rays which 
 originally make angles with the tangent plane at A numerically 
 less than 6, is ever afterwards included between the reflecting 
 surface and that of the concentric sphere of radius OP cos 6. The 
 usual divergence in three dimensions entailing a diminishing 
 intensity varying as r~ 2 is replaced by a divergence in two dimen- 
 sions, like that of waves issuing from a source situated between 
 
 1 Airy On Sound, 2nd edition, 1871, p. 145. 
 
 2 Phil. Mag. (5), in. p. 458, 1877. 
 
128 WHISPERING GALLERIES. [287. 
 
 two parallel reflecting planes, with an intensity varying as r~ l . 
 The less rapid enfeeblement of sound by distance than that usually 
 experienced is the leading feature in the phenomena of whispering 
 galleries. 
 
 The thickness of the sheet included between the two spheres 
 becomes less and less as A approaches P, and in the limiting case 
 of a radiant point situated on the surface of the reflector is 
 expressed by OA (1 cos 0), or, if 6 be small. ^OA.0* approxi- 
 mately. The solid angle of the pencil, which determines the 
 whole amount of radiation in the sheet, is 4-7T0 ; so that as is 
 diminished without limit the intensity becomes infinite, as com- 
 pared with the intensity at a finite distance from a similar source 
 in the open. 
 
 It is evident that this clinging, so to speak, of sound to the 
 surface of a concave wall does not depend upon the exactness of 
 the spherical form. But in the case of a true sphere, or rather of 
 any surface symmetrical with respect to A A, there is in addition 
 the other kind of concentration spoken of at the commencement of 
 the present section which is peculiar to the point A diametrically 
 opposite to the source. It is probable that in the case of a nearly 
 spherical dome like that of St Paul's a part of the observed effect 
 depends upon the symmetry, though perhaps the greater part is 
 referable simply to the general concavity of the walls. 
 
 The propagation of earthquake disturbances is probably affected 
 by the curvature of the surface of the globe acting like a whisper- 
 ing gallery, and perhaps even sonorous vibrations generated at the 
 surface of the land or water do not entirely escape the same kind 
 of influence. 
 
 In connection with the acoustics of public buildings there are 
 many points which still remain obscure. It is important to bear 
 in mind that the loss of sound in a single reflection at a smooth 
 wall is very small, whether the wall be plane or curved. In order 
 to prevent reverberation it may often be necessary to introduce 
 carpets or hangings to absorb the sound. In some cases the 
 presence of an audience is found sufficient to produce the desired 
 effect. In the absence of all deadening material the prolongation 
 of sound may be very considerable, of which perhaps the most 
 striking example is that afforded by the Baptistery at Pisa, where 
 the notes of the common chord sung consecutively may be heard 
 
287.] RESONANCE IN BUILDINGS. 129 
 
 ringing on together for many seconds 1 . According to Henry 2 it is 
 important to prevent the repeated reflection of sound backwards 
 and forwards along the length of a hall intended for public speak- 
 ing, which may be accomplished by suitably placed oblique 
 surfaces. In this way the number of reflections in a given time is 
 increased, and the undue prolongation of sound is checked. 
 
 288. Almost the only instance of acoustical refraction, which 
 has a practical interest, is the deviation of sonorous rays from a 
 rectilinear course due to heterogeneity of the atmosphere. The 
 variation of pressure at different levels does not of itself give rise 
 to refraction, since the velocity of sound is independent of density ; ' 
 but, as was first pointed out by Prof. Osborne Reynolds 3 , the case 
 is different with the variations of temperature which are usually 
 to be met with. The temperature of the atmosphere is determined 
 principally by the condensation or rarefaction, which any portion 
 of air must undergo in its passage from one level to another, and 
 its normal state is one of " convective equilibrium 4 ,'' rather than of 
 uniformity. According to this view the relation between pressure 
 and density is that expressed in (9) 246, and the velocity of 
 sound is given by 
 
 p.i'.yft^ (1). 
 
 dp p \pj 
 
 To connect the pressure and density with the elevation (z), we 
 have the hydrostatical equation 
 
 dp = -gpdz (2), 
 
 from which and (1) we find 
 
 F=F.-(7-l)0* (3), 
 
 if F be the velocity at the surface. The corresponding relation 
 between temperature and elevation obtained by means of equation 
 (10) 246 is 
 
 J-l-Zp^. (4), 
 
 where is the temperature at the surface. 
 
 1 [Some observations of my own, made in 1883, gave the duration as 12 seconds. 
 If a note changes pitch, both sounds are heard together and may give rise to a 
 combination-tone, 68. See Haberditzl, Ueber die von Dvorak beobachteten Vari- 
 ationston. Wien, AJcad. Sitzber., 77, p. 204, 1878.] 
 
 2 Amer. Assoc. Proc. 1856, p. 119. 
 
 3 Proceedings of the Royal Society, Vol. xxn. p. 531. 1874. 
 
 4 Thomson, On the convective equilibrium of temperature in the atmosphere. 
 Manchester Memoirs, 1861 62. 
 
 
 
130 ATMOSPHERIC REFRACTION. [288. 
 
 According to (4) the fall of temperature would be about 
 1 Cent, in 330 feet [100 m.], which does not differ much from the 
 results of Glaisher's balloon observations. When the sky is clear, 
 the fall of temperature during the day is more rapid than when 
 the sky is cloudy, but towards sunset the temperature becomes 
 approximately constant 1 . Probably on clear nights it is often 
 warmer above than below. 
 
 The explanation of acoustical refraction as dependent upon a 
 variation of temperature with height is almost exactly the same as 
 that of the optical phenomenon of mirage. The curvature (p~ l ) of 
 a ray, whose course is approximately horizontal, is easily estimated 
 by the method given by Prof. James Thomson 2 . Normal planes 
 drawn at two consecutive points along the ray meet at the centre of 
 curvature and are tangential to the wave-surface in its two con- 
 secutive positions. The portions of rays at elevations z and z + Sz 
 respectively intercepted between the normal planes are to one 
 another in the ratio p : p Sz, and also, since they are described 
 in the same time, in the ratio V : V+SV. Hence in the limit 
 
 p z 
 
 In the normal state of the atmosphere a ray, which starts 
 horizontally, turns gradually upwards, and at a sufficient distance 
 passes over the head of an observer whose station is at the same 
 level as the source. If the source be elevated, the sound is heard 
 at the surface of the earth by means of a ray which starts with 
 a downward inclination ; but, if both the observer and the 
 source be on the surface, there is no direct ray, and the sound is 
 heard, if at all, by means of diffraction. The observer may then 
 be said to be situated in a sound shadow, although there may be 
 no obstacle in the direct line between himself and the source. 
 According to (3) 
 
 sothat p= - y =-^ . ' ................ (6); 
 
 or the radius of curvature of a horizontal ray is about ten times 
 the height through which a body must fall under the action of 
 
 1 Nature, Sept. 20, 1877. 
 
 2 See Everett, On the Optics of Mirage. Phil Mag. (4) XLV. pp. 161, 248. 
 
288.] CONVECTIVE EQUILIBBIUM. 131 
 
 gravity in order to acquire a velocity equal to the velocity of 
 sound. If the elevations of the observer and of the source be ^ 
 and # 2 , the greatest distance at which the sound can be heard 
 otherwise than by diffraction is 
 
 It is not to be supposed that the condition of the atmosphere 
 is always such that the relation between velocity and elevation is 
 that expressed in (3). When the sun is shining, the variation of 
 temperature upwards is more rapid ; on the other hand, as Prof. 
 Reynolds has remarked, when rain is falling, a much slower varia- 
 tion is to be expected. In the arctic regions, where the nights 
 are long and still, radiation may have more influence than convec- 
 tion in determining the equilibrium of temperature, and if so the 
 propagation of sound in a horizontal direction would be favoured 
 by the approximately isothermal condition of the atmosphere. 
 
 The general differential equation for the path of a ray, when 
 the surfaces of equal velocity are parallel planes, is readily obtained 
 from the law of sines. If 6 be the angle of incidence, F/sin 6 is 
 not altered by a refracting surface, and therefore in the case 
 supposed remains constant along the whole course of a ray. If x 
 be the horizontal co-ordinate, and the constant value of F/sin 6 
 be called c, we get dxjdz = F/V (c 2 - F 2 ), 
 
 Vdx 
 
 If the law of velocity be that expressed in (3), 
 
 CtZ ~ 
 
 v*dv 
 
 and thus 
 
 2 r v*d 
 
 = -. - ~ 
 
 (v-i)gJ Vc 2 - 
 
 . , 
 
 Vc 2 -F 2 
 
 or, on effecting the integration, 
 
 (y-l)gac = constant + V J(& - F 2 ) - c 2 sin" 1 (F/c) ...... (9), 
 
 in which F may be expressed in terms of z by (3). 
 
 A simpler result will be obtained by taking an approximate 
 form of (3), which will be accurate enough to represent the cases 
 of practical interest. Neglecting the square and higher powers of 
 z, we may take 
 
 92 
 
132 PATH OF A RAY. [288. 
 
 Writing for brevity /3 in place of \g (7 1)/F 3 , we have 
 
 #df 
 
 By substitution in (8) 
 
 the origin of x being taken so as to correspond with F=c, that is 
 at the place where the ray is horizontal. Expressing V in terms 
 of x, we find 
 
 whence /3z = - Ftf 1 + ^- (e c ^ + e?-^*) ............. (12). 
 
 ZiC 
 
 The path of each ray is therefore a catenary whose vertex is 
 
 2F 3 
 downwards ; the linear parameter is . -- Q -- , and varies from 
 
 0(7-l)c' 
 ray to ray. 
 
 289. Another cause of atmospheric refraction is to be found 
 in the action of wind. It has long been known that sounds are 
 generally better heard to leeward than to windward of the source ; 
 but the fact remained unexplained until Stokes 1 pointed out that 
 the increasing velocity of the wind overhead must interfere with 
 the rectilinear propagation of sound rays. From Fermat's law of 
 least time it follows that the course of a ray in a moving, but 
 otherwise homogeneous, medium, is the same as it would be in a 
 medium, of which all the parts are at rest, if the velocity of 
 propagation be increased at every point by the component of 
 the wind-velocity in the direction of the ray. If the wind be 
 horizontal, and do not vary in the same horizontal plane, the 
 course of a ray, whose direction is everywhere but slightly inclined 
 to that of the wind, may be calculated on the same principles as 
 were applied in the preceding section to the case of a variable 
 temperature, the normal velocity of propagation at any point being 
 increased, or diminished, by the local wind-velocity, according as 
 the motion of the sound is to leeward or to windward. Thus, 
 when the wind increases overhead, which may be looked upon as 
 the normal state of things, a horizontal ray travelling to windward 
 is gradually bent upwards, and at a moderate distance passes over 
 the head of an observer; rays travelling with the wind, on the 
 
 1 Brit. Assoc. Rep. 1857, p. 22. 
 
289.] REFRACTION BY WIND. 133 
 
 other hand, are bent downwards, so that an observer to leeward of 
 the source hears by a direct ray which starts with a slight upward 
 inclination, and has the advantage of being out of the way of 
 obstructions for the greater part of its course. 
 
 The law of refraction at a horizontal surface, in crossing which 
 the velocity of the wind changes discontinuously, is easily investi- 
 gated. It will be sufficient to consider the case in which the 
 direction of the wind and the ray are in the same vertical plane. 
 If 6 be the angle of incidence, which is also the angle between the 
 plane of the wave and the surface of separation, U be the velocity 
 of the air in that direction which makes the smaller angle with 
 the ray, and V be the common velocity of propagation, the velocity 
 of the trace of the plane of the wave on the surface of separa- 
 tion is 
 
 which quantity is unchanged by the refraction. If therefore U' be 
 the velocity of the wind on the second side, and & be the angle of 
 refraction, 
 
 V V 
 
 JL-+U =S -J: + U' ..................... (2), 
 
 sin sm 6 
 
 which differs from the ordinary optical law. If the wind-velocity 
 vary continuously, the course of a ray may be calculated from the 
 condition that the expression (1) remains constant. 
 
 If we suppose that U = 0, the greatest admissible value of 
 U' is 
 
 U' = FJcosec 6-1} ..................... (3). 
 
 At a stratum where U' has this value, the direction of the ray 
 which started at an angle 6 has become parallel to the refracting 
 surfaces, and a stratum where U' has a greater value cannot be 
 penetrated at all. Thus a ray travelling upwards in still air at an 
 inclination (^TT 6) to the horizon is reflected by a wind overhead 
 of velocity exceeding that given in (3), and this independently of 
 the velocities of intermediate strata. To take a numerical example, 
 all rays whose upward inclination is less than 11, are totally 
 reflected by a wind of the same azimuth moving at the moderate 
 speed of 15 miles per hour. The effects of such a wind on the 
 propagation of sound cannot fail to be very important. Over the 
 surface of still water sound moving to leeward, being confined 
 
134 TOTAL REFLECTION BY WIND. [289. 
 
 between parallel reflecting planes, diverges in two dimensions 
 only, and may therefore be heard at distances far greater than 
 would otherwise be possible. Another possible effect of the reflector 
 overhead is to render sounds audible which in still air would 
 be intercepted by hills or other obstacles intervening. For the 
 production of these phenomena it is not necessary that there be 
 absence of wind at the source of sound, but, as appears at once 
 from the form of (2), merely that the difference of velocities U' U 
 attain a sufficient value. 
 
 The differential equation to the path of a ray, when the wind- 
 velocity U is continuously variable, is 
 
 , Vdz 
 
 In comparing (5) with (8) of the preceding section, which 
 is the corresponding equation for ordinary refraction, we must 
 remember that V is now constant. If, for the sake of obtaining a 
 definite result, we suppose that the law of variation of wind at 
 different levels is that expressed by 
 
 U=a + /3z ........................... (6), 
 
 we have 
 
 = V j d ^ f _ ^ .................. (7), 
 
 which is of the same form as (11) of the preceding section. The 
 course of a ray is accordingly a catenary in the present case also, 
 but there is a most important distinction between the two problems. 
 When the refraction is of the ordinary kind, depending upon a 
 variable velocity of propagation, the direction of a ray may be 
 reversed. In the case of atmospheric refraction, due to a diminu- 
 tion of temperature upwards, the course of a ray is a catenary, 
 whose vertex is downwards, in whichever direction the ray may be 
 propagated. When the refraction is due to wind, whose velocity 
 increases upwards, according to the law expressed in (6) with 
 positive, the path of a ray, whose direction is upwind, is also along 
 a catenary with vertex downwards, but a ray whose direction is 
 downwind cannot travel along this path. In the latter case the 
 vertex of the catenary along which the ray travels is directed 
 upwards. 
 
290.] REYNOLDS' OBSERVATIONS. 135 
 
 290. In the paper by Reynolds already referred to, an account 
 is given of some interesting experiments especially directed to test 
 the theory of refraction by wind. It was found that "In the 
 direction of the wind, when it was strong, the sound (of an electric 
 bell) could be heard as well with the head on the ground as when 
 raised, even when in a hollow with the bell hidden from view by 
 the slope of the ground ; and no advantage whatever was gained 
 either by ascending to an elevation or raising the bell. Thus, with 
 the wind over the grass the sound could be heard 140 yards, and 
 over snow 360 yards, either with the head lifted or on the ground ; 
 whereas at right angles to the wind on all occasions the range was 
 extended by raising either the observer or the bell." 
 
 " Elevation was found to affect the range of sound against the 
 wind in a much more marked manner than at right angles." 
 
 " Over the grass no sound could be heard with the head on the 
 ground at 20 yards from the bell, and at 30 yards it was lost with 
 the head 3 feet from the ground, and its full intensity was lost 
 when standing erect at 30 yards. At 70 yards, when standing 
 erect, the sound was lost at long intervals, and was only faintly 
 heard even then ; but it became continuous again when the ear 
 was raised 9 feet from the ground, and it reached its full intensity 
 at an elevation of 12 feet." 
 
 * 
 
 Prof. Reynolds thus sums up the results of his experiments : 
 
 1. "When there is no wind, sound proceeding over a rough 
 surface is more intense above than below." 
 
 2. " As long as the velocity of the wind is greater above than 
 below, sound is lifted up to windward and is not destroyed." 
 
 3. "Under the same circumstances it is brought down to 
 leeward, and hence its range extended at the surface of the 
 ground." 
 
 Atmospheric refraction has an important bearing on the 
 audibility of fog-signals, a subject which within the last few years 
 has occupied the attention of two eminent physicists, Prof. Henry 
 in America and Prof. Tyndall in this country. Henry 1 attributes 
 almost all the vagaries of distant sounds to refraction, and has 
 shewn how it is possible by various suppositions as to the motion 
 of the air overhead to explain certain abnormal phenomena which 
 have come under the notice of himself and other observers, while 
 
 1 Eeport of the Lighthouse Board of the United States for the year 1874. 
 
136 TYNDALL'S OBSERVATIONS [290. 
 
 Tyndall 1 , whose investigations have been equally extensive, 
 considers the very limited distances to which sounds are sometimes 
 audible to be due to an actual stopping of the sound by a flocculent 
 condition of the atmosphere arising from unequal heating or 
 moisture. That the latter cause is capable of operating in this 
 direction to a certain extent cannot be doubted. Tyndall has 
 proved by laboratory experiments that the sound of an electric bell 
 may be sensibly intercepted by alternate layers of gases of different 
 densities ; and, although it must be admitted that the alternations 
 of density were both more considerable and more abrupt than 
 can well be supposed to occur in the open air, except perhaps in 
 the immediate neighbourhood of the solid ground, some of the 
 observations on fog-signals themselves seem to point directly to 
 the explanation in question. 
 
 Thus it was found .that the blast of a siren placed on the 
 summit of a cliff overlooking the sea was followed by an echo 
 of gradually diminishing intensity, whose duration sometimes 
 amounted to as much as 15 seconds. This phenomenon was 
 observed "when the sea was of glassy smoothness," and cannot 
 apparently be attributed to any other cause than that assigned to 
 it by Tyndall. It is therefore probable that refraction and 
 acoustical opacity are both concerned in the capricious behaviour 
 of fog-signals. A priori we should certainly be disposed to attach 
 the greater importance to refraction, and Reynolds has shewn that 
 some of Tyndall's own observations admit of explanation upon this 
 principle. A failure in reciprocity can only be explained in 
 accordance with theory by the action of wind ( 111). 
 
 According to the hypothesis of acoustic clouds, a difference 
 might be expected in the behaviour of sounds of long and of short 
 duration, which it may be worth while to point out here, as it does 
 not appear to have been noticed by any previous writer. Since 
 energy is not lost in reflection and refraction, the intensity of 
 radiation at a given distance from a continuous source of sound (or 
 light) is not altered by an enveloping cloud of spherical form and of 
 uniform density, the loss due to the intervening parts of the cloud 
 being compensated by reflection from those which lie beyond the 
 source. When, however, the sound is of short duration, the 
 intensity at a distance may be very much diminished by the cloud 
 on account of the different distances of its reflecting parts and the 
 
 1 Phil. Trans. 1874. Sound, 3rd edition, Ch. vn. 
 
290.] ON FOG-SIGNALS. 137 
 
 consequent drawing out of the sound, although the whole intensity, 
 as measured by the time-integral, may be the same as if there had 
 been no cloud at all. This is perhaps the explanation of Tyndall's 
 observation, that different kinds of signals do not always preserve 
 the same order of effectiveness. In some states of the weather a 
 " howitzer firing a 3-lb. charge commanded a larger range than the 
 whistles, trumpets, or syren," while on other days " the inferiority 
 of the gun to the syren was demonstrated in the clearest manner." 
 It should be noticed, however, that in the same series of experi- 
 ments it was found that the liability of the sound of a gun " to be 
 quenched or deflected by an opposing wind, so as to be practically 
 useless at a very short distance to windward, is very remarkable." 
 The refraction proper must be the same for all kinds of sounds, 
 but for the reason explained above, the diffraction round the edge 
 of an obstacle may be less effective for the report of a gun than for 
 the sustained note of a siren. 
 
 Another point examined by Tyndall was the influence of fog on 
 the propagation of sound. In spite of isolated assertions to the 
 contrary 1 , it was generally believed on the authority of Derham 
 that the influence of fog was prejudicial. Tyndall's observations 
 prove satisfactorily that this opinion is erroneous, and that the 
 passage of sound is favoured by the homogeneous condition of the 
 atmosphere which is the usual concomitant of foggy weather. 
 When the air is saturated with moisture, the fall of temperature 
 with elevation according to the law of convective equilibrium is 
 much less rapid than in the case of dry air, on account of the 
 condensation of vapour which then accompanies expansion. From 
 a calculation by Thomson 2 it appears that in warm fog the effect 
 of evaporation and condensation would be to diminish the fall of 
 temperature by one-half. The acoustical refraction due to tem- 
 perature would thus be lessened, and in other respects no doubt 
 the condition of the air would be favourable to the propagation of 
 sound, provided no obstruction were offered by the suspended 
 particles themselves. In a future chapter we shall investigate the 
 disturbance of plane sonorous waves by a small obstacle, and we 
 shall find that the effect depends upon the ratio of the diameter of 
 the obstacle to the wave-length of the sound. 
 
 The reader who is desirous of pursuing this subject may 
 
 1 See for example Desor, Fortschritte der Fhysik, xi. p. 217. 1855. 
 
 2 Manchester Memoirs, 186162. 
 
138 LAW OF DIVERGENCE OF SOUND. [290. 
 
 consult a paper by Reynolds " On the Refraction of Sound by the 
 Atmosphere 1 ," as well as the authorities already referred to. It 
 may be mentioned that Reynolds agrees with Henry in consider- 
 ing refraction to be the really important cause of disturbance, but 
 further observations are much needed. See also 294. 
 
 291. On the assumption that the disturbance at an aperture 
 in a screen is the same as it would have been at the same place in 
 the absence of the screen, we may solve various problems respecting 
 the diffraction of sound by the same methods as are employed for 
 the corresponding problems in physical optics. For example, the 
 disturbance at a distance on the further side of an infinite plane 
 wall, pierced with a circular aperture on which plane waves of 
 sound impinge directly, may be calculated as in the analogous 
 problem of the diffraction pattern formed at the focus of a circular 
 object-glass. Thus in the case of a symmetrical speaking trumpet 
 the sound is a maximum along the axis of the instrument, where 
 all the elementary disturbances issuing from the various points 
 of the plane of the mouth are in one phase. In oblique direc- 
 tions the intensity is less; but it does not fall materially short 
 of the maximum value until the obliquity is such that the 
 difference of distances of the nearest and furthest points of the 
 mouth amounts to about half a wave-length. At a somewhat 
 greater obliquity the mouth may be divided into two parts, of 
 which the nearer gives an aggregate effect equal in magnitude, 
 but opposite in phase, to that of the further ; so that the intensity 
 in this direction vanishes. In directions still more oblique the 
 sound revives, increases to an intensity equal to "017 of that 
 along the axis 2 , again diminishes to zero, and so on, the alternations 
 corresponding to the bright and dark rings which surround the 
 central patch of light in the image of a star. If R denote the 
 radius of the mouth, the angle, at which the first silence occurs, is 
 sin" 1 ('610X/JR). When the diameter of the mouth does not exceed 
 |- A, the elementary disturbances combine without any considerable 
 antagonism of phase, and the intensity is nearly uniform in all 
 directions. It appears that concentration of sound along the axis 
 requires that the ratio R : \ should be large, a condition not 
 usually satisfied in the ordinary use of speaking trumpets, whose 
 efficiency depends rather upon an increase in the original volume 
 
 1 Phil. Trans. Vol. 166, p. 315. 1876. 
 
 2 Verdet, Lemons cCoptique physique, t. i. p. 306. 
 
291.] SPEAKING TRUMPET. 139 
 
 of sound ( 280). When, however, the vibrations are of very short 
 wave-length, a trumpet of moderate size is capable of effecting a 
 considerable concentration along the axis, as I have myself verified 
 in the case of a hiss. 
 
 292. Although such calculations as those referred to in the 
 preceding section are useful as giving us a general idea of the 
 phenomena of diffraction, it must not be forgotten that the 
 auxiliary assumption on which they are founded is by no means 
 strictly and generally true. Thus in the case of a wave directly 
 incident upon a screen the normal velocity in the plane of the 
 aperture is not constant, as has been supposed, but increases from 
 the centre towards the edge, becoming infinite at the edge itself. 
 In order to investigate the conditions by which the actual velocity 
 is determined, let us for the moment suppose that the aperture is 
 filled up. The incident wave </> = cos (nt lex) is then perfectly 
 reflected, and the velocity-potential on the negative side of the 
 screen (x = 0) is 
 
 cf> = cos (nt kx) + cos (nt + Tex) ............... (1), 
 
 giving, when x = 0, </> = 2 cos nt This corresponds to the vanish- 
 ing of the normal velocity over the area of the aperture ; the 
 completion of the problem requires us to determine a variable 
 normal velocity over the aperture such that the potential due to it 
 ( 278) shall increase by the constant quantity 2 cos nt in crossing 
 from the negative to the positive side; or, since the crossing 
 involves simply a change of sign, to determine a value of the 
 normal velocity over the area of the aperture which shall give on 
 the positive side (j> = cos nt over the same area. The result of 
 superposing the two motions thus defined satisfies all the condi- 
 tions of the problem, giving the same velocity and pressure on the 
 two sides of the aperture, and a vanishing normal velocity over the 
 remainder of the screen. 
 
 If P cos (nt + e) denote the value of d(f>/dx at the various points 
 of the area ($) of the aperture, the condition for determining 
 P and e is by (6) 278, 
 
 (2), 
 
 where r denotes the distance between the element dS and any 
 fixed point in the aperture. When P and e are known, the 
 
140 DIFFRACTION THROUGH SMALL APERTURE. [292. 
 
 complete value of </> for any point on the positive side of the screen 
 is given by 
 
 and for any point on the negative side by 
 
 1 [C n cos(nt kr + e) .. 1 
 
 < = + HP- - - dS + 2 cos nt cos ksc ...... (4). 
 
 The expression of P and e for a finite aperture, even if of circular 
 form, is probably beyond the power of known methods ; but in the 
 case where the dimensions are very small in comparison with the 
 wave-length the solution of the problem may be effected for the 
 circle and the ellipse. If r be the distance between two points, 
 both of which are situated in the aperture, kr may be neglected, 
 and we then obtain from (2) 
 
 -. 
 
 shewing that P/2?r is the density of the matter which must be 
 distributed over 8 in order to produce there the constant potential 
 unity. At a distance from the opening on the positive side we 
 may consider r as constant, and take 
 
 
 where M ^ 1 1 PdS, denoting the total quantity of matter 
 
 which must be supposed to be distributed. It will be shewn 
 on a future page ( 306) that for an ellipse of semimajor axis a, 
 and eccentricity e, 
 
 (7), 
 
 where F is the symbol of the complete elliptic function of the first 
 kind. In the case of a circle, F(e) = JTT, and 
 
 <> 
 
 This result is quite different from that which we should obtain on 
 the hypothesis that the normal velocity in the aperture has the 
 value proper to the primary wave. In that case by (3) 283 
 
 ?ra 2 sin (nt kr) 
 
292.] ELLIPTIC APERTURE. 141 
 
 If there be several small apertures, whose distances apart are 
 much greater than their dimensions, the same method gives 
 
 = ^ cos (nt - fa-Q + ^ cos (nt - kr,) + _ _ _ (IQ) 
 T-L r% 
 
 The diffraction of sound is a subject which has attracted but 
 little attention either from mathematicians or experimentalists. 
 Although the general character of the phenomena is well under- 
 stood, and therefore no very startling discoveries are to be 
 expected, the exact theoretical solution of a few of the simpler 
 problems, which the subject presents, would be interesting; and, 
 even with the present imperfect methods, something probably 
 might be done in the way of experimental examination. 
 
 292 a. By means of a bird-call giving waves of about 1 cm. 
 wave-length and a high pressure sensitive flame it is possible to 
 imitate many interesting optical experiments. With this apparatus 
 the shadow of an obstacle so small as the hand may be made 
 apparent at a distance of several feet. 
 
 An experiment shewing the antagonism between the parts of a 
 wave corresponding to the first and second Fresnel's zones ( 283) 
 
 SOURCE BURNER 
 
 o ~ 
 
 is very effective. A large glass screen (Fig. 57 a) is perforated 
 with a circular hole 20 cm. in diameter, and is so situated between 
 the source of sound and the burner that the aperture corresponds 
 to the first two zones. By means of a zinc plate, held close to the 
 
142 EXPERIMENTS ON DIFFRACTION. [292 a. 
 
 glass, the aperture may be reduced to 14 cm., and then admits 
 only the first zone. If the adjustments are well made, the flame, 
 unaffected by the waves which penetrate the larger aperture, 
 flares violently when the aperture is further restricted by the 
 zinc plate. Or, as an alternative, the perforated plate may be 
 replaced by a disc of 14 cm. diameter, which allows the second 
 zone to be operative while the first is blocked off. 
 
 If a, b denote the distances of the screen from the source and 
 from the point of observation, the external radius p of the nth 
 zone is given by 
 
 or approximately 
 
 '-"^ ........................... < 
 
 When a = 6, 
 
 /o 2 = ^Xa ........................... (2). 
 
 With the apertures specified above, p 2 = 49 for n = 1 ; p 2 = 100 
 for n = 2 ; so that 
 
 Xa = 100, 
 
 the measurements being in centimetres. This gives the suitable 
 distances when X is known. In an actual experiment X = 1*2, 
 a = 83. 
 
 The process of augmenting the total effect by blocking out the 
 alternate zones may be carried much further. Thus when a 
 suitable circular grating, cut out of a sheet of zinc, is interposed 
 between the source of sound and the flame, the effect is many 
 times greater than when the screen is removed altogether 1 . As 
 in Soret's corresponding optical experiment, the grating plays the 
 part of a condensing lens. 
 
 The focal length of the lens is determined by (1), which may 
 be written in the form 
 
 1 1 1_X 
 f-a + bf ........................ ( ' 
 
 so that 
 
 In an actual grating constructed upon this plan eight zones the 
 first, third, fifth &c. are occupied by metal. The radius of the 
 first zone, or central circle, is 7'6 cm., so that p 2 /n = 58. Thus, if 
 "X = 1'2 cm.,/= 48 cm. If a and b are equal, each must be 96 cm. 
 
 1 "Diffraction of Sound," Proc. Roy. Inst. Jan. 20, 1888. 
 
292 a.] SHADOW or A CIRCULAR DISC. 143 
 
 The condition of things at the centre of the shadow of a 
 circular disc is still more easily investigated. If we construct in 
 imagination a system of zones beginning with the circular edge of 
 the disc, we see, as in 283, that the total effect at a point upon 
 the axis, being represented by the half of that of the first zone, is 
 the same as if no obstacle at all were interposed. This analogue 
 of a famous optical phenomenon is readily exhibited 1 . In one 
 experiment a glass disc 38 cm. in diameter was employed, and its 
 distances from the source and from the flame were respectively 
 70 cm. and 25 cm. A bird-call giving a pure tone (A, = 1*5 cm.) is 
 suitable, but may be replaced by a toy reed or other source giving 
 short, though not necessarily simple, waves. In private work the 
 ear furnished with a rubber tube may be used instead of a sensitive 
 flame. 
 
 The region of no sensible shadow, though not confined to a 
 mathematical point upon the axis, is of small dimensions, and a 
 very moderate movement of the disc in its own plane suffices to 
 reduce the flame to quiet. Immediately surrounding the central 
 spot there is a ring of almost complete silence, and beyond that 
 again a moderate revival of effect. The calculation of the in- 
 tensity of sound at points off the axis of symmetry is too com- 
 plicated to be entered upon here. The results obtained by 
 Lommel 2 may be readily adapted to the acoustical problem. With 
 the data specified above the diameter of the silent ring immediately 
 surrounding the central region of activity is about 1'7 cm. 
 
 293. The value of a function < which satisfies V 2 (/> = through- 
 out the interior of a simply-connected closed space S can be 
 expressed as the potential of matter distributed over the surface 
 of S. In a certain sense this is also true of the class of functions 
 with which we are now occupied, which satisfy V 2 < + & 2 c/> = 0. 
 The following is Helmholtz's proof 3 . By Green's theorem, if < 
 and ty denote any two functions of x, y, z, 
 
 1 "Acoustical Observations," Phil. Mag. Vol. ix. p. 281, 1880; Proc. Roy. lust. 
 loc. cit. 
 
 2 Abh. der bayer. Akad. der Wiss. ii. CL, xv. Bd., ii. Abth. See also Encyclo- 
 paedia Britannica, Article " Wave Theory." 
 
 3 Theorie der Luftschwingungen in Rohren mit offenen Enden. Crelle, Bd. LVII. 
 p. 1. 1860. 
 
144 EXTENSION OF GREEN'S THEOREM. [293. 
 
 To each side add - 1 1 1 fc^dV; then if 
 
 = 0, a 2 (V 2 -^ + & 2 f ) + W = 0, 
 
 If <3> and ^ vanish within 8, we have simply 
 
 //S 
 
 Suppose, however, that 
 
 (3), 
 
 where r represents the distance of any point from a fixed origin 
 within 8. At all points, except 0, <I> vanishes ; and the last term 
 in (1) becomes 
 
 -* a? v* dV 
 
 referring to the point 0. Thus 
 
 in which, if ^ vanish, we have an expression for the value of M* at 
 any interior point in terms of the surface values of -^ and of 
 dtjr/dn. In the case of the common potential, on which we fall 
 back by putting & = 0, t/r would be determined by the surface 
 values of d-^/dn only. But with k finite, this law ceases to be 
 universally true. For a given space 8 there is, as in the case 
 investigated in 267, a series of determinate values of k, corre- 
 sponding to the periods of the possible modes of simple harmonic 
 vibration which may take place within a closed rigid envelope 
 having the form of 8. With any of these values of k, it is obvious 
 that o/r cannot be determined by its normal variation over 8, and 
 the fact that it satisfies throughout S the equation V 2 -^ + k 2 ^ = 0. 
 But if the supposed value of k do not coincide with one of the 
 series, then the problem is determinate ; for the difference of any 
 two possible solutions, if finite, would satisfy the condition of 
 giving no normal velocity over 8, a condition which by hypothesis 
 cannot be satisfied with the assumed value of k. 
 
293.J HELMHOLTZ'S THEOREM. 145 
 
 If the dimensions of the space S be very small in comparison 
 with X (= 2-7T/&), e~ ikr may be replaced by unity ; and we learn 
 that ty differs but little from a function which satisfies throughout 
 8 the equation V 2 < = 0. 
 
 294. On his extension of Green's theorem (1) Helmholtz 
 founds his proof of the important theorem contained in the following 
 statement : If in a space filled with air which is partly bounded by 
 finitely extended fixed bodies and is partly unbounded, sound waves 
 be excited at any point A, the resulting velocity -potential at a second 
 point B is the same both in magnitude and phase, as it would have 
 been at A, had B been the source of the sound. 
 
 If the equation 
 
 in which <f> and -x/r are arbitrary functions, and 
 
 - a 2 
 
 be applied to a space completely enclosed by a rigid boundary and 
 containing any number of detached rigid fixed bodies, and if <f>, ty 
 be velocity-potentials due to sources within 8, we get 
 
 in 
 
 (2). 
 
 Thus, if <f> be due to a source concentrated in one point A, <3> = 
 except at that point, and 
 
 where I ll&dV represents the intensity of the source. Similarly, 
 if T|T be due to a source situated at B, 
 
 Accordingly, if the sources be finite and equal, so that 
 
 .................. (3), 
 
 it follows that 
 
 *, = *, ........................... (4), 
 
 which is the symbolical statement of Helmholtz's theorem. 
 
 R. II. 10 
 
146 HELMHOLTZ'S THEOREM. [294. 
 
 If the space S extend to infinity, the surface integral still 
 vanishes, and the result is the same ; but it is not necessary to go 
 into detail here, as this theorem is included in the vastly more 
 general principle of reciprocity established in Chapter V. The 
 investigation there given shews that the principle remains true in 
 the presence of dissipative forces, provided that these arise from 
 resistances varying as the first power of the velocity, that the 
 fluid need not be homogeneous, nor the neighbouring bodies rigid 
 or fixed. In the application to infinite space, all obscurity is 
 avoided by supposing the vibrations to be slowly dissipated after 
 having escaped to a distance from A and B, the sources under 
 contemplation. 
 
 The reader must carefully remember that in this theorem 
 equal sources of sound are those produced by the periodic intro- 
 duction and abstraction of equal quantities of fluid, or something 
 whose effect is the same, and that equal sources do not necessarily 
 evolve equal amounts of energy in equal times. For instance, a 
 source close to the surface of a large obstacle emits twice as much 
 energy as an equal source situated in the open. 
 
 As an example of the use of this theorem we may take the 
 case of a hearing, or speaking, trumpet consisting of a conical tube, 
 whose efficiency is thus seen to be the same, whether a sound pro- 
 duced at a point outside is observed at the vertex of the cone, or 
 a source of equal strength situated at the vertex is observed at the 
 external point. 
 
 It is important also to bear in mind that Helmholtz's form of 
 the reciprocity theorem is applicable only to simple sources of sound, 
 which in the absence of obstacles would generate symmetrical 
 waves. As we shall see more clearly in a subsequent chapter, it is 
 possible to have sources of sound, which, though concentrated in 
 an infinitely small region, do not satisfy this condition. It will be 
 sufficient here to consider the case of double sources, for which the 
 modified reciprocal theorem has an interest of its own. 
 
 Let us suppose that A is a simple source, giving at a point B 
 the potential ^, and that A' is an equal and opposite source 
 situated at a neighbouring point, whose potential at B is i/r + A-^r. 
 If both sources be in operation simultaneously, the potential at B 
 is Ai/r. Now let us suppose that there is a simple source at B, 
 
294.] APPLICATION TO DOUBLE SOURCES. 147 
 
 whose intensity and phase are the same as those of the sources at 
 A and A' ; the resulting potential at A is ty, and at A' T/T + A-^r. 
 If the distance A A' be denoted by h, and be supposed to diminish 
 without limit, the velocity of the fluid at A in the direction AA' 
 is the limit of At/r/A, Hence, if we define a unit double source 
 as the limit of two equal and opposite simple sources whose dis- 
 tance is diminished, and whose intensity is increased without 
 limit in such a manner that the product of the intensity and 
 the distance is the same as for two unit simple sources placed at 
 the unit distance apart, we may say that the velocity of the fluid 
 at A in direction A A due to a unit simple source at B is numeri- 
 cally equal to the potential at B due to a unit double source at A, 
 whose axis is in the direction A A. This theorem, be it observed, 
 is true in spite of any obstacles or reflectors that may exist in the 
 neighbourhood of the sources. 
 
 Again, if A A and BE' represent two unit double sources of the 
 same phase, the velocity at B in direction BB' due to the source 
 AA is the same as the velocity at A in direction A A' due to the 
 source BB'. These and other results of a like character may also 
 be obtained on an immediate application of the general principle of 
 108. These examples will be sufficient to shew that in applying 
 the principle of reciprocity it is necessary to attend to the character 
 of the sources. A double source, situated in an open space, is in- 
 audible from any point in its equatorial plane, but it does not 
 follow that a simple source in the equatorial plane is inaudible 
 from the position of the double source. On this principle, I believe, 
 may be explained a curious experiment by Tyndall 1 , in which 
 there was an apparent failure of reciprocity 2 . The source of sound 
 employed was a reed of very high pitch, mounted in a tube, along 
 whose axis the intensity was considerably greater than in oblique 
 directions. 
 
 295. The kinetic energy T of the motion within a closed 
 surface S is expressed by 
 
 1 Proceedings of the Royal Institution, Jan. 1875. Also Tyndall, On Sound, 3rd 
 edition, p. 405. 
 
 2 See a note "On the- Application of the Principle of Eeciprocity to Acoustics." 
 Royal Society Proceedings, Vol. xxv. p. 118, 1876, or Phil. Mag. (5), in. p. 300. 
 
 102 
 
148 VARIATION OF TOTAL ENERGY. [295. 
 
 dT 
 
 so that = 
 
 (2), 
 
 by Green's theorem. For the potential energy F a we have by 
 (12) 245 
 
 by the general equation of motion (9) 244. Thus, if E denote 
 the whole energy within the space 8, 
 
 ........... (5), 
 
 of which the first term represents the work transmitted across the 
 boundary S, and the second represents the work done by internal 
 sources of sound. 
 
 If the boundary S be a fixed rigid envelope, and there be no 
 internal sources, E retains its initial value throughout the motion. 
 This principle has been applied by Kirchhoff 1 to prove the deter- 
 minateness of the motion resulting from given arbitrary initial 
 conditions. Since every element of E is positive, there can be no 
 motion within S, if E be zero. Now, if there were two motions 
 possible corresponding to the same initial conditions, their differ- 
 ence would be a motion for which the initial value of E was zero ; 
 but by what has just been said such a motion cannot exist. 
 
 1 Vorlesungen iiber Math. Physik, p. 311. 
 
CHAPTER XV. 
 
 FURTHER APPLICATION OF THE GENERAL EQUATIONS. 
 
 296. WHEN a train of plane waves, otherwise unimpeded, 
 impinges upon a space occupied by matter, whose mechanical pro- 
 perties differ from those of the surrounding medium, secondary 
 waves are thrown off, which may be regarded as a disturbance due 
 to the change in the nature of the medium a point of view more 
 especially appropriate, when the region of disturbance, as well 
 as the alteration of mechanical properties, is small. If the 
 medium and the obstacle be fluid, the mechanical properties 
 spoken of are two the compressibility and the density: no 
 account is here taken of friction or viscosity. In the chapter on 
 spherical harmonic analysis we shall consider the problem here 
 proposed on the supposition that the obstacle is spherical, without 
 any restriction as to the smallness of the change of mechanical 
 properties ; in the present investigation the form of the obstacle 
 is arbitrary, but we assume that the squares and higher powers 
 of the changes of mechanical properties may be omitted. 
 
 If > V> ? denote the displacements parallel to the axes of 
 co-ordinates of the particle, whose equilibrium position is denned 
 by a?, y, z, and if a be the normal density, and m the constant 
 of compressibility so that &p = ms, the equations of motion are 
 
 d*% d(ms)_ 
 
 ** ' 
 
 and two similar equations in 77 and On the assumption 
 that the whole motion is proportional to e ikat , where as usual 
 k = STT/X, and ( 244) a 2 = m/cr, (1) may be written 
 
 =0 ..................... (2). 
 
150 SECONDARY WAVES [296. 
 
 The relation between the condensation s, and the displace- 
 ments f, rj, f, obtained by integrating (3) 238 with respect 
 to the time, is 
 
 _,.jjf + *? + j*r ..................... (3) . 
 
 dx dy dz 
 
 For the system of primary waves advancing in the direction 
 of x, 77 and f vanish ; if f , s be the values of f and s, and 
 m , CT O be the mechanical constants for the undisturbed medium, 
 we have as in (2) 
 
 ^-Wf. = .................. (4); 
 
 but f > S do not satisfy (2) at the region of disturbance on account 
 of the variation in m and cr, which occurs there. Let us assume 
 that the complete values are f + f , 77, f, s + s 1 ) and substitute 
 in (2). Then taking account of (4), we get 
 
 d (ms) jo <. / .ds dm . 
 
 * (m - m ) - + S - - (a- - er ) k*a*& = 0, 
 
 or, as it may also be written, 
 
 -j- (ms) - cr& 2 a 2 f + -g (Am.So) - A<7.& 2 a 2 f = (5), 
 
 if Am, ACT stand respectively for m m , cr CT O . The equations 
 in 77 and f are in like manner 
 
 I (*.) -*.. I (Am.*) = | 
 
 , , f 
 
 -g (ms) - vtftfZ+j- (Am.s ) = 
 
 It is to be observed that Am, ACT vanish, except through a 
 small space, which is regarded as the region of disturbance ; 
 f, 77, f, s, being the result of the disturbance are to be treated 
 as small quantities of the order Am, ACT ; so that in our ap- 
 proximate analysis the variations of m and cr in the first two 
 terms of (5) and (6) are to be neglected, being there multiplied 
 by small quantities. We thus obtain from (5) and (6) by differ- 
 entiation and addition, with use of (3), as the differential equation 
 in s, 
 
 V 2 (ms) + & 2 ms = & 2 a 2 -y- (Ao-. f ) - V 2 (Am.s ) (7). 
 
 1 [This notation was adopted for brevity. It might be clearer to take 
 = s + As, &c. ; so that , s, &c. should retain their former meanings.] 
 
296.] DUE TO VARIATION OF MEDIUM. 151 
 
 As in 277, the solution of (7) is 
 
 47rm s = \\\ 6 ^- | V2 (Am. s ) - & 2 a 2 ^ (A<r. f ) i dV ...... (8), 
 
 in which the integration extends over a volume completely in- 
 cluding the region of disturbance. The integrals in (8) may be 
 transformed with the aid of Green's theorem. Calling the two 
 parts respectively P and Q, we have 
 
 P = ~ V* (Am.s ) dF = Am.s V 2 dV 
 
 - ikr d . - ikr 
 
 where S denotes the surface of the space through which the triple 
 integration extends. Now on S, Am and -7- (Am.s ) vanish, 
 so that both the surface integrals disappear. Moreover 
 
 \ r J ~ r dr z r ' 
 
 and thus 
 
 rrr e -ur 
 P k 2 1 1 1 - - Am.s cF (9). 
 
 If the region of disturbance be small in comparison with X, 
 
 we may write 
 
 ikr err 
 
 (10). 
 
 In like manner for the second integral in (8), we find 
 
 dV = &*fo ~<rdr . . .(11), 
 
 where //, denotes the cosine of the angle between x and r. The 
 linear dimension of the region of disturbance is neglected in 
 comparison with X, and X is neglected in comparison with r. 
 
 If T be the volume of the space through which Am, A<r are 
 sensible, we may write 
 
 = T. Am, &ff dV = T. Ao-, 
 
152 LAW OF DEPENDENCE ON WAVE-LENGTH. [296. 
 
 if on the right-hand sides Am, A<J refer to the mean values of 
 the variations in question. Thus from (8) 
 
 s = , 4 Am.s ika z ACT. 
 
 To express f in terms of s > we have from (3), %o = fs dx} and 
 thus, if the condensation for the primary waves be s = e ik (at+x) , 
 ikg = s , and (12) may be put into the form 
 
 irTer** (Am , A<r ) 
 
 r + P \ v 1 ' 3 )' 
 
 in which s denotes the condensation of the primary waves at 
 the place of disturbance at time t, and s denotes the condensa- 
 tion of the secondary waves at the same time at a distance r from 
 the disturbance. Since the difference of phase represented by the 
 factor e~ ikr corresponds simply to the distance r, we may consider 
 that a simple reversal of phase occurs at the place of disturbance. 
 The amplitude of the secondary waves is inversely proportional 
 to the distance r, and to the square of the wave-length X. Of 
 the two terms expressed in (13) the first is symmetrical in all 
 directions round the place of disturbance, while the second varies 
 as the cosine of the angle between the primary and the secondary 
 rays. Thus a place at which m varies behaves as a simple source, 
 and a place at which a varies behaves as a double source ( 294). 
 
 That the secondary disturbance must vary as X~ 2 may be 
 proved immediately by the method of dimensions. Am and Ao- 
 being given, the amplitude is necessarily proportional to T, and in 
 accordance with the principle of energy must also vary inversely 
 as r. Now the only quantities (dependent upon space, time, and 
 mass) of which the ratio of amplitudes can be a function, are 
 T, r, X, a (the velocity of sound), and cr, of which the last cannot 
 occur in the expression of a simple ratio, as it is the only one of 
 the five which involves a reference to mass. Of the remaining 
 four quantities. T, r, X, and ,a, the last is the only one which 
 involves a reference to time, and is therefore excluded. We are 
 left with T, r, and X, of which the only combination varying 
 as Tr~ l , and independent of the unit of length, is Tr~ l X" 2 . 1 
 
 An interesting application of the results of this section may 
 be made to explain what have been called harmonic echoes'*. 
 
 1 " On the Light from the Sky," Phil. Mag. Feb. 1871, and " On the scattering 
 of Light by small Particles," Phil Mag. June, 1871. 
 
 2 Nature, 1873, vm. 319. 
 
296.] SOUNDS ALTERED IN CHARACTER. 153 
 
 If the primary sound be a compound musical note, the various 
 component tones are scattered in unlike proportions. The octave, 
 for example, is sixteen times stronger relatively to the funda- 
 mental tone in the secondary than it was in the primary sound. 
 There is thus no difficulty in understanding how it may happen 
 that echoes returned from such reflecting bodies as groups of trees 
 may be raised an octave. The phenomenon has also a comple- 
 mentary side. If a number of small bodies lie in the path of 
 waves of sound, the vibrations which issue from them in all direc- 
 tions are at the expense of the energy of the main stream, and 
 where the sound is compound, the exaltation of the higher har- 
 monics in the scattered waves involves a proportional deficiency 
 of them in the direct wave after passing the obstacles. This is 
 perhaps the explanation of certain echoes which are said to return 
 a sound graver than the original ; for it is known that the pitch of 
 a pure tone is apt to be estimated too low. But the evidence 
 is conflicting, and the whole subject requires further careful expe- 
 rimental investigation ; it may be commended to the attention of 
 those who may have the necessary opportunities. While an altera- 
 tion in the character of a sound is easily intelligible, and must 
 indeed generally happen to a limited extent, a change in the 
 pitch of a simple tone would be a violation of the law of forced 
 vibrations, and hardly to be reconciled with theoretical ideas. 
 
 In obtaining (13) we have neglected the effect of the variable 
 nature of the medium on the disturbance. When the disturb- 
 ance on this supposition is thoroughly known, we might approxi- 
 mate again in the same manner. The additional terms so obtained 
 would be necessarily of the second order in Am, A<r, so that our 
 expressions are in all cases correct as far as the first powers of 
 those quantities. 
 
 Even when the region of disturbance is not small in com- 
 parison with X, the same method is applicable, provided the 
 squares of Am, Ao- be really negligible. The total effect of any 
 obstacle may then be calculated by integration from those of its 
 parts. In this way we may trace the transition from a small 
 region of disturbance whose surface does not come into considera- 
 tion, to a thin plate of a few or of a great many square wave- 
 lengths in area, which will ultimately reflect according to the 
 regular optical law. But if the obstacle be at all elongated in the 
 direction of the primary rays, this method of calculation soon 
 
154 SECONDARY SOURCES. [296. 
 
 ceases to be practically available, because, even although the 
 change of mechanical properties be very small, the interaction 
 of the various parts of the obstacle cannot be left out of account. 
 This caution is more especially needed in dealing with the case of 
 light, where the wave-length is so exceedingly small in comparison 
 with the dimensions of ordinary obstacles. 
 
 297. In some degree similar to the effect produced by a 
 change in the mechanical properties of a small region of the fluid, 
 is that which ensues when the square of the motion rises any- 
 where to such importance that it can be no longer neglected. 
 V 2 < + & 2 then acquires a finite value dependent upon the square 
 of the motion. Such places therefore act like sources of sound; 
 the periods of the sources including the submultiples of the ori- 
 ginal period. Thus any part of space, at which the intensity 
 accumulates to a -sufficient extent, becomes itself a secondary 
 source, emitting the harmonic tones of the primary sound. If 
 there be two primary sounds of sufficient intensity, the secondary 
 vibrations have frequencies which are the sums and differences of 
 the frequencies of the primaries ( 68) \ 
 
 298. The pitch of a sound is liable to modification when the 
 source and the recipient are in relative motion. It is clear, for 
 instance, that an observer approaching a fixed source will meet 
 the waves with a frequency exceeding that proper to the sound, by 
 the number of wave-lengths passed over in a second of time. Thus 
 if v be the velocity of the observer and a that of sound, the 
 frequency is altered in the ratio a v : a, according as the motion 
 is towards or from the source. Since the alteration of pitch is 
 constant, a musical performance would still be heard in tune, 
 although in the second case, when a and v are nearly equal, the 
 fall in pitch would be so great as to destroy all musical character. 
 If we could suppose v to be greater than a, a sound produced after 
 the motion had begun would never reach the observer, but sounds 
 previously excited would be gradually overtaken and heard in the 
 reverse of the natural order. If v = 2a, the observer would hear 
 a musical piece in correct time and tune, but backwards. 
 
 Corresponding results ensue when the source is in motion and 
 the observer at rest ; the alteration depending only on the relative 
 motion in the line of hearing. If the source and the observer move 
 with the same velocity there is no alteration of frequency, whether 
 
 1 Helmholtz iiber Combinationstone. Pogg. Ann. Bd. xcix. s. 497. 1856. 
 
298.] DOPPLER'S PRINCIPLE. 155 
 
 the medium be in motion, or not. With a relative motion of 
 40 miles [64 kilometres] per hour the alteration of pitch is very 
 conspicuous, amounting to about a semitone. The whistle of a loco- 
 motive is heard too high as it approaches, and too low as it recedes 
 from an observer at a station, changing rather suddenly at the 
 moment of passage. 
 
 The principle of the alteration of pitch by relative motion was 
 first enunciated by Doppler 1 , and is often called Doppler's prin- 
 ciple. Strangely enough its legitimacy was disputed by Petzval 2 , 
 whose objection was the result of a confusion between two 
 perfectly distinct cases, that in which there is a relative motion 
 of the source and recipient, and that in which the medium is in 
 motion while the source and the recipient are at rest. In the 
 latter case the circumstances are mechanically the same as if the 
 medium were at rest and the source and the recipient had a 
 common motion, and therefore by Doppler's principle no change 
 of pitch is to be expected. 
 
 Doppler's principle has been experimentally verified by Buijs 
 Ballot 3 and Scott Russell, who examined the alterations of pitch 
 of musical instruments carried on locomotives. A laboratory in- 
 strument for proving the change of pitch due to motion has been 
 invented by Mach 4 . It consists of a tube six feet [183 cm.] in 
 length, capable of turning about an axis at its centre. At one end is 
 placed a small whistle or reed, which is blown by wind forced 
 along the axis of the tube. An observer situated in the plane of 
 rotation hears a note of fluctuating pitch, but if he places himself 
 in the prolongation of the axis of rotation, the sound becomes , 
 steady. Perhaps the simplest experiment is that described by 
 Konig 5 . Two c" tuning-forks mounted on resonance cases are 
 prepared to give with each other four beats per second. If the 
 graver of the forks be made to approach the ear while the other 
 remains at rest, one beat is lost for each two feet [61 cm.] of 
 approach ; if, however, it be the more acute of the two forks which 
 approaches the ear, one beat is gained in the same distance. 
 
 1 Theorie des farbigen Lichtes der Doppelsterne. Prag, 1842. See Pisko, Die 
 neueren Apparate der Akustik. Wien, 1865. 
 
 2 Wien. Ber. viu. 134. 1852. Fortschritte der Physik, vin. 167. 
 
 3 Pogg. Ann. LXVI. p. 321. 
 
 4 Pogg. Ann. cxii. p. 66, 1861, and cxvi. p. 333, 1862. 
 
 5 Konig's Catalogue des Appareils d'Acomtique. Paris, 1865. 
 
156 DOPPLER'S PRINCIPLE. [298. 
 
 A modification of this experiment due to Mayer 1 may also be noticed. 
 In this case one fork excites the vibrations of a second in unison 
 with itself, the excitation being made apparent by a small pendulum, 
 whose bob rests against the extremity of one of the prongs. If the 
 exciting fork be at rest, the effect is apparent up to a distance 
 of 60 feet [1830 cm.], but it ceases when the exciting fork is 
 moved rapidly to or fro in the direction of the line joining the two 
 forks. 
 
 There is some difficulty in treating mathematically the problem 
 of a moving source, arising from the fact that any practical source 
 acts also as an obstacle. Thus in the case of a bell carried 
 through the air, we should require to solve a problem difficult 
 enough without including the vibrations at all. But the solution 
 of such a problem, even if it could be obtained, would throw no 
 particular light on Doppler's law, and we may therefore advan- 
 tageously simplify the question by idealizing the bell into a simple 
 source of sound. 
 
 In 147 we considered the problem of a moving source of 
 disturbance in the case of a stretched string. The theory for 
 aerial waves in one dimension is precisely similar, but for the 
 general case of three dimensions some extension is necessary, in 
 order to take account of the possibility of a motion across the 
 direction of the sound rays. From 273, 276 it appears that the 
 effect at any point of a source of sound is the same, whether the 
 source be at rest, or whether it move in any manner on the surface 
 of a sphere described about as centre. If the source move in 
 such a manner as to change its distance (r) from 0, its effect is 
 altered in two ways. Not only is the phase of the disturbance on 
 arrival at affected by the variation of distance, but the amplitude 
 also undergoes a change. The latter complication however may 
 be put out of account, if we limit ourselves to the case in which 
 the source is sufficiently distant. On this understanding we may 
 assert that the effect at of a disturbance generated at time t and 
 at distance r is the same as that of a similar disturbance generated 
 at the time t + St and at the distance r a St. In the case of a 
 periodic disturbance a velocity of approach (v) is equivalent to an 
 increase of frequency in the ratio a : a + v. 
 
 299. We will now investigate the forced vibrations of the 
 air contained within a rectangular chamber, due to internal sources 
 
 1 Phil. Mag. (4), XLIII. p. 278, 1872. 
 
299.] RECTANGULAR CHAMBER. 157 
 
 of sound. By 267 it appears that the result at time t of an 
 initial condensation confined to the neighbourhood of the point 
 
 4> = %2,'2<kaB pqr coskatcos Ip J cos Iq -^j cos Ir J ...(1), 
 where 
 
 = cos ( p - ) cos ( q ^ 1 cos ( T - 
 
 from which the effect of an impressed force may be deduced, 
 
 as in 276. The disturbance ffjfadxdyd* communicated at 
 
 time t' being denoted by ffi<b(t')dt! dxdydz, or ^(lT)dif t the 
 resultant disturbance at time t is 
 
 iry\ ( I JTZ\ 
 | J cos ir J x 
 
 cos (p "^ } cos ( q - 1 ) cos ( r ^- ) I O, (f) cos ka(t - 1') dt f . . .(3). 
 
 \ * /* / \ * l*\ I \ r\/ / / 
 
 p / 
 
 The symmetry of this expression with respect to x, y, z and 
 f, ?;, f is an example of the principle of reciprocity ( 107). 
 
 In the case of a harmonic force, for which ^ (tf) = A cos matf, 
 we have to consider the value of 
 
 -t 
 
 cos mat' coska(t t') dt' (4). 
 
 00 
 
 Strictly speaking, this integral has no definite value ; but, if 
 we wish for the expression of the forced vibrations only, we must 
 omit the integrated function at the lower limit, as may be seen 
 by supposing the introduction of very small dissipative forces. 
 We thus obtain 
 
 [ 
 J 
 
 ma sin mat 
 
 As might have been predicted, the expressions become infinite 
 in case of a coincidence between the period of the source and one 
 of the natural periods of the chamber. Any particular normal 
 vibration will not be excited, if the source be situated on one 
 of its loops. 
 
 The effect of a multiplicity of sources may readily be inferred 
 by summation or integration. 
 
158 UNLIMITED TUBE. [300. 
 
 300. When sound is excited within a cylindrical pipe, the 
 simplest kind of excitation that we can suppose is by the forced 
 vibration of a piston. In this case the waves are plane from 
 the beginning. But it is important also to inquire what happens 
 when the source, instead of being uniformly diffused over the 
 section, is concentrated in one point of it. If we assume (what, 
 however, is not unreservedly true) that at a sufficient distance 
 from the source the waves become plane, the law of reciprocity 
 is sufficient to guide us to the desired information. 
 
 Let A be a simple source in an unlimited tube, B, B f two 
 points of the same normal section in the region of plane waves. 
 Ex hypothesi, the potentials at B and B' due to the source A 
 are the same, and accordingly by the law of reciprocity equal 
 sources at B and B' would give the same potential at A. From 
 this it follows that the effect of any source is the same at a 
 distance, as if the source were uniformly diffused over the section 
 which passes through it. For example, if B and B' were equal 
 sources in opposite phases, the disturbance at A would be nil. 
 
 The energy emitted by a simple source situated within a 
 tube may now be calculated. If the section of the tube be <r, 
 and the source such that in the open the potential due to it 
 
 would be 
 
 A cos k (at r) 
 
 the velocity-potential at a distance within the tube will be 
 the same as if the cause of the disturbance were the motion 
 of a piston at the origin, giving the same total displacement, 
 and the energy emitted will also be the same. Now from (1) 
 
 2?rr 2 -^ ^A cos kat ultimately, 
 
 and therefore if ty be the velocity-potential of the plane waves 
 in the tube (supposed parallel to z), we may take 
 
 z) ............... (2), 
 
 corresponding to which 
 
 .................. (3). 
 
300.] ENERGY EMITTED. 159 
 
 Hence, as in 245, the energy (W) emitted on each side of 
 the source is given by 
 
 dW f : d^\ pa A 2 
 
 = <r ( p TIT -^- = ' - cos feat ; 
 
 at \ dzJz = Q 4cr 
 
 so that in the long run 
 
 ^ pa A 2 
 
 W== 8^T t W- 
 
 If the tube be stopped by an immovable piston placed close to 
 the source, the whole energy is emitted in one direction ; but 
 this is not all. In consequence of the doubled pressure, twice 
 as much energy as before is developed, and thus in this case 
 
 ! ..(5). 
 
 The narrower the tube, the greater is the energy issuing from 
 a given source. It is interesting to compare the efficiency of 
 a source at the stopped end of a cylindrical tube with that of 
 an equal source situated at the vertex of a cone. From 280 
 we have in the latter case, 
 
 so that W : W = a> : fro- ..................... (7). 
 
 The energies emitted in the two cases are the same when co = & 2 cr, 
 that is, when the section of the cylinder is equal to the area 
 cut off by the cone from a sphere of radius lc~ l . 
 
 301. We have now to examine how far it is true that vibra- 
 tions within a cylindrical tube become approximately plane at a 
 sufficient distance from their source. Taking the axis of z parallel 
 to the generating lines of the cylinder, let us investigate the 
 motion, whose potential varies as e ikat t on the positive side of a 
 source, situated at z = 0. If <j> be the potential and V 2 stand for 
 4- d 2 ldy 2 the equation of the motion is 
 
 ...................... (1). 
 
 If </> be independent of z, it represents vibrations wholly 
 transverse to the axis of the cylinder. If the potential be then 
 proportional to e ipat , it must satisfy 
 
 (2), 
 
160 VIBRATIONS IN UNLIMITED TUBES. [301. 
 
 as well as the condition that over the boundary of the section 
 
 In order that these equations may be compatible, p is restricted 
 to certain definite values corresponding to the periods of the 
 natural vibrations. A zero value of p gives < = constant, which 
 solution, though it is of no significance in the two dimension pro- 
 blem, we shall presently have to consider. For each admissible 
 value of p, there is a definite normal function u of x and y ( 92), 
 such that a solution is 
 
 (4). 
 
 Two functions u, u', corresponding to different values of p, are 
 conjugate, viz. make 
 
 ffuu'dxdy = Q ........................... (5), 
 
 and any function of x and y may be expanded within the contour 
 in the series 
 
 (/> = A u + A^ + A 2 u 2 + ........................ (6), 
 
 in which U Q , corresponding to p = 0, is constant. 
 
 In the actual problem </> may still be expanded in the same 
 series, provided that A , A lt -&c. be regarded as functions of z. 
 By substitution in (1) we get, having regard to (2), 
 
 in which, by virtue of the conjugate property of the normal func- 
 tions, each coefficient of u must vanish separately. Thus 
 
 = (8). 
 
 The solution of the first of these equations is 
 
 A 
 giving 
 
 (9). 
 
 The solution of the general equation in A assumes a different 
 form, according as & 2 p 2 is positive or negative. If the forced 
 
301.] DISCRIMINATION OF CASES. 
 
 vibration be graver in pitch than the gravest of the purely trans- 
 verse natural vibrations, every finite value of p* is greater than } 
 or & p 2 is always negative. Putting 
 
 k 2 -p* = -fji* (10), 
 
 we have A = 
 
 whence (f> = (ae* z + /3 e-* z ) ue ikat .................. (11). 
 
 Now under the circumstances supposed, it is evident that the 
 motion does not become infinite with z, so that all the coefficients 
 a. vanish. For a somewhat different reason the same is true of , 
 as there can be no wave in the negative direction. We may 
 therefore take 
 
 ikat + &u 2 tr** e ikat + ......... (12), 
 
 an expression which reduces to its first term when z is sufficiently 
 great. We conclude that in all cases the waves ultimately become 
 plane, if the forced vibration be graver than the gravest of the 
 natural transverse vibrations. 
 
 In the case of a circular cylinder, of radius r, the gravest trans- 
 verse vibration has a wave-length equal to 27rr-r- 1*841 = 3'413r 
 ( 339). If then the wave-length of the forced vibration exceed 
 3'413r, the waves ultimately become plane. It may happen 
 however that the waves ultimately become plane, although the 
 wave-length fall short of the above limit. For example, if the 
 source of vibration be symmetrical with respect to the axis of the 
 tube, e.g. a simple source situated on the axis itself, the gravest 
 transverse vibration with which we should have to deal would 
 be more than an octave higher than in the general case, and 
 the wave-length of the forced vibration might have less than half 
 the above value. 
 
 From (12), whens = 0, 
 
 whence da- = - ik& a- e ikat .................. (13), 
 
 inasmuch as jju^da, ffu. 2 dcr, &c., all vanish. 
 
 It appears accordingly that the plane waves at a distance are 
 the same as would be produced by a rigid piston at the origin, 
 R. II. 11 
 
162 REACTION OF AIR [301. 
 
 giving the same mean normal velocity as actually exists. Any 
 normal motion of which the negative and positive parts are equal, 
 produces ultimately no effect. 
 
 When there is no restriction on the character of the source, and 
 when some of the transverse natural vibrations are graver than 
 the actual one, some of the values of & 2 p 2 are positive, and then 
 terms enter of the form 
 
 or in real quantities 
 
 <f> = /3ucos{kat- V(& 2 -p 2 )*} .................. (14), 
 
 indicating that the peculiarities of the source are propagated to 
 an infinite distance. 
 
 The problem here considered may be regarded as a generaliza- 
 tion of that of 268. For the case of a circular cylinder it may 
 be worked out completely with the aid of Bessel's functions, but 
 this must be left to the reader. 
 
 302. In 278 we have fully determined the motion of the 
 air due to the normal periodic motion of a bounding plane plate of 
 infinite extent. If d<f>/dn be the given normal velocity at the 
 element dS, 
 
 gives the velocity-potential at any point P distant r from dS. The 
 remainder of this chapter is devoted to the examination of the 
 particular case of this problem which arises when the normal 
 velocity has a given constant value over a circular area of radius 
 jR, while over the remainder of the plane it is zero. In particular 
 we shall investigate what forces due to the reaction of the air will 
 act on a rigid circular plate, vibrating with a simple harmonic 
 motion in an equal circular aperture cut out of a rigid plane plate 
 extending to infinity. 
 
 For the whole variation of pressure acting on the plate we 
 have ( 244) 
 
302.] ON A VIBRATING CIRCULAR PLATE. 163 
 
 where a- is the natural density, and < varies as e ikat . Thus by (1) 
 
 v ,~ 1/0(1(7 d<p ^^ Q J&JW /o\ 
 
 )pdfo = - , - 2,2, dbdio (4). 
 
 In the double sum 
 
 p ikr 
 
 22 -dSdS' (3), 
 
 which we have now to evaluate, each pair of elements is to be 
 taken once only, and the product is to be summed after multipli- 
 cation by the factor r~ l e~ ikr , depending on their mutual distance. 
 The best method is that suggested by Prof. Maxwell for the 
 common potential 1 . The quantity (3) is regarded as the work 
 that would be consumed in the complete dissociation of the 
 matter composing the disc, that is to say, in the removal of every 
 element from the influence of every other, on the supposition that 
 the potential of two elements is proportional to r~ l e~ ikr . The 
 amount of work required, which depends only on the initial 
 and final states, may be calculated by supposing the operation 
 performed in any way that may be most convenient. For this 
 purpose we suppose that the disc is divided into elementary rings, 
 and that each ring is carried away to infinity before any of the 
 interior rings are disturbed. 
 
 The first step is the calculation of the potential (V) at the 
 edge of a disc of radius c. Taking polar co-ordinates (p, 6) with 
 any point of the circumference for pole, we have 
 
 This quantity must be multiplied by Zircdc, and afterwards 
 integrated with respect to c between the limits and R. But 
 it will be convenient first to effect a transformation. We have 
 
 9 r jw 9 /"i* 
 
 e-*a**o<i0 = 
 
 TrJo irJ o 
 
 = -/ cos(2kcsm&)d0- sin (2kc sin 6) dO 
 
 = J.(z}-iK(z} ............................................. (4), 
 
 where z is written for 2kc. J (z) is the Bessel's function of zero 
 
 1 Theory of Eesonance. Phil. Trans. 1870. 
 
 112 
 
164 REACTION OF AIR [302. 
 
 order ( 200), and K (z) is a function defined by the equation 
 
 2 fl- . 
 
 = - ["sin (z sin 0)dO 
 
 7TJ 
 
 77- . I 2 . 3 2 . 5 2 I 2 . 3 2 . 5 2 . 7 2 
 
 Deferring for the moment the further consideration of the 
 function K, we have 
 
 V = ^[K(z)-i{\-J,(z)}} (6), 
 
 and thus 
 
 Now by (6) 200 and (8) 204 
 
 *>*./!(*)..: -..(7); 
 
 
 
 and thus, if K l be defined by 
 
 KI(Z)= I* zdzK(z} (8), 
 
 Jo 
 we may write 
 
 From this the total pressure is derived by introduction of the 
 
 ikao- d6 
 
 factor -~ , so that 
 
 TT an 
 
 The reaction of the air on the disc may thus be divided into 
 two parts, of which the first is proportional to the velocity of the 
 disc, and the second to the acceleration. If denote the dis- 
 
 placement of the disc, so that f = , , we have f = ika f = ika -~* ; 
 and therefore in the equation of motion of the disc, the reaction of 
 the air is represented by a frictional force ao- . TrR 2 .Ml- JLD) 
 retarding the motion, and by an accession to the inertia equal to 
 
302.] ON A VIBRATING CIRCULAR, PLATE. 165 
 
 When kR is small, we have from the ascending series for Ji 
 (5) 200, 
 
 ,__ 
 
 y^ft ~ 1.2 1.2 2 .3 + 1.2 2 .3 2 .4 1.2 2 .32.4 2 .5 + '" ^ '' 
 
 so that the frictional term is approximately 
 
 a<r.7rR 2 .k*R 2 .% ........................ (12). 
 
 From the nature of the case the coefficient of f must be 
 positive, otherwise the reaction of the air would tend to augment, 
 instead of to diminish, the motion. That J T (z) is in fact always less 
 than J z may be verified as follows. If 6 lie between and TT, and 
 z be positive, sin (z sin 6) z sin is negative, and therefore also 
 
 1 [* 
 
 (sin (z sin 6) z sin 6} sin 6 dO 
 
 TTJ 
 
 is negative. But this integral is J l (z) \z> which is accordingly 
 negative for all positive values of z. 
 
 When kR is great, J^ZkR) tends to vanish, and then the 
 frictional term becomes simply acr . irR* . . This result might 
 have been expected ; for when kR is very large, the wave motion 
 in the neighbourhood of the disc becomes approximately plane. 
 We have then by (6) and (8) 245, dp = ap %, in which p is the 
 density (<j); so that the retarding force is 7rR 2 8p = acr.7rR 2 ^. 
 
 We have now to consider the term representing an alteration 
 of inertia, and among other things to prove that this alteration is 
 an increase, or that K (z) is positive. By direct integration of the 
 ascending series (5) for K (which is always convergent), 
 
 . 3 12 m 32 < 5 p 1 32 . 52 m 7 
 
 When therefore kR is small, we may take as the expression for 
 the increase of inertia 
 
 SR 
 
 This part of the reaction of the air is therefore represented by 
 supposing the vibrating plate to carry with it a mass of air equal 
 to that contained in a cylinder whose base is the plate, and whose 
 height is equal to 8jR/3?r ; so that, when the plate is sufficiently 
 small, the mass to be added is independent of the period of 
 vibration. 
 
166 REACTION OF AIR [302. 
 
 From the series (5) for K(z), it may be proved immediately 
 that 
 
 1 d ( d\ ,,. x 2 ,, , x 
 
 - -r-\Z-T} K(z) = K(z) (lo), 
 
 zdz dz TTZ 
 
 or +i(*)-- (16). 
 
 \dz* zdz J TTZ 
 
 From the first form (15) it follows that 
 
 By means of this expression for K l (z) we may readily prove that 
 the function is always positive. For 
 
 dK(z) d 2 [* 
 
 7 = J---I sm 
 dz dz 
 
 so that 
 
 (z sin 0)d0=- f COB (z sin 0) sin dO. . .(18); 
 
 TTJO 
 
 %z ( r- w 
 
 ^ (z) = \ 1 cos (^ sin 0) sm 
 TT ( Jo 
 
 = I sirf(zsm0)sin0d0 (19), 
 
 an integral of which every element is positive. When z is very 
 large, cos (z sin 0) fluctuates with great rapidity, and thus K l (z) 
 tends to the form 
 
 ;(*)-:*.. ...(20). 
 
 7T 
 
 When z is great, the ascending series for K and K l} though always 
 ultimately convergent, become useless for practical calculation, and 
 it is necessary to resort to other processes. It will be observed 
 that the differential equation (16) satisfied by K is the same as 
 that belonging to the Bessel's function J" , with the exception of 
 the term on the right-hand side, viz. 2/7rz. The function K is 
 therefore included in the form obtained by adding to the general 
 solution of Bessel's equation containing two arbitrary constants any 
 particular solution of (16). Such a particular solution is 
 
 %Tr.K(z) = Z- l Z-*+1*.&.Z-* l*.&.&.Z-'' + I 2 . 3 2 . 5 2 . 7 2 . 2~ 9 - . . .(21), 
 
 as may be readily verified on substitution. The series on the 
 right of (21), notwithstanding its ultimate divergency, may be 
 used successfully for computation when z is great. It is in fact 
 
302.] ON A VIBRATING CIRCULAR PLATE. 167 
 
 the analytical equivalent of j^ e~? (z z + /3' 2 )~$dl3, and we might take 
 
 2 r 
 K (z) = I if-^ ,Q2\ + Complementary Function, 
 
 determining the two arbitrary constants by an examination of the 
 forms assumed when z is very great. But it is perhaps simpler to 
 follow the method used by Lipschitz * for Bessel's functions. 
 
 By (4) we have 
 
 9 
 
 (22> 
 
 r &~ zw dw 
 
 Consider the integral I -. ' , where w is a complex variable of 
 
 the form u + iv. Representing, as usual, simultaneous pairs of 
 values of u and v by the co-ordinates of a point, we see that the 
 value of the integral will be zero, if the integration with respect 
 to w range round the rectangle, whose angular points are respec- 
 tively 0, h, h + i, i, where h is any real positive quantity. Thus 
 
 /* er^du^ V e -z(h+iv) (i v ) ro e -z(u+i)d u ro e -izvd (i v ) 
 
 Jo Vl + u 9 + Jo Vl + (h + ivy + J A Vl +"(t"+?) + Ji Vl-v 2 " 
 from which, if we suppose that h = x , 
 
 .Te- zu du .r e~ z ^du 
 1 -7 -------- hi ............ (23). 
 
 Replacing uz by /3, we may write (23) in the form 
 e~ izv dv _ . r e-?d{3 er^v-to p e-^^d/3 
 
 f 8 ) " 'Jo ^ v / i+ 2 - ^2^ Jo v -i2f'r' 
 
 o 
 
 The first term on the right in (24) is entirely imaginary ; it 
 therefore follows by (22) that \ f jrJ Q (z) is the real part of the 
 second term. By expanding the binomial under the integral sign, 
 and afterwards integrating by the formula 
 
 we obtain as the expansion for JQ (z) in negative powers of z, 
 
 "r A / " ~ 
 
 +....} on (*-) (25). 
 
 \i.S2 1.2. a. 
 
 1 Crelle, Bd. LVI. 1859. Loinmel, Studien iiber die Bessel'schen Functionen, p. 59. 
 
168 REACTION OF AIR [302. 
 
 By stopping the expansion after any desired number of terms, 
 and forming the expression for the remainder, it may be proved 
 that the error committed by neglecting the remainder cannot 
 exceed the last term retained ( 200). 
 
 In like manner the imaginary part of the right-hand member 
 of (24) is the equivalent of \inrK (z), so that 
 
 -v. 32 .5 2 . 2- 
 
 + A/ 
 
 The value of K(z) may now be determined by means of (17). 
 We find 
 
 dK 9 
 
 -T- = - - {z~* - 3 . Z~* + I 2 . 3 2 . 5 . Z-* I 2 . 3 2 . 5 2 . 7 . Z~ 8 + ...... } 
 
 3. 5. 7. 9. 11. 1.3. 5. 7_ / 2 >7\ 
 
 5 " ' 
 
 The final expression for K l (z) may be put into the form 
 
 2 
 
 (! 2 -4)(3 2 -4)(5 2 -4)(7 2 -4)_ 
 1.2.3.4. (82) 4 
 
 It appears then that K^ does not vanish when z is great, but 
 approximates to 2z/7r. But although the accession to the inertia, 
 
 1 As was to be expected, the series within brackets are the same as those that 
 occur in the expression of the function J l (z). 
 
302.] ON A VIBRATING CIRCULAR PLATE. 169 
 
 which is proportional to K l} becomes infinite with R, it vanishes 
 ultimately when compared with the area of the disc, and with the 
 other term which represents the dissipation. And this agrees 
 with what we should anticipate from the theory of plane waves. 
 
 If, independently of the reaction of the air, the mass of the 
 plate be M, and the force of restitution be yuf, the equation of 
 motion of the plate when acted on by an impressed force F, pro- 
 portional to e ikat , will be 
 
 or by (13), if, as will be usual in practical applications, kR be 
 small, 
 
 j .jr. ........... (30). 
 
 Two particular cases of this problem deserve notice. First let 
 M and //, vanish, so that the plate, itself devoid of mass, is subject 
 to no other forces than F and those arising from aerial pressures. 
 Since % = ika%, the frictional term is relatively negligible, and we 
 get when kR is very small, 
 
 = -iF ..................... (31). 
 
 O7T 
 
 Next let M and jj, be such that the natural period of the plate, 
 when subject to the reaction of the air, is the same as that imposed 
 upon it. Under these circumstances 
 
 and therefore 
 
 ^2 7?2 
 
 acT7rR*.^. = F .................. (32). 
 
 2i 
 
 Comparing with (31), we see that the amplitude of vibration is 
 greater in the case when the inertia of the air is balanced, in the 
 ratio of 16 : SirkR, shewing a large increase when kR is small. In 
 the first case the phase of the motion is such that comparatively 
 very little work is done by the force F\ while in the second, the 
 inertia of the air is compensated by the spring, and then F, being 
 of the same phase as the velocity, does the maximum amount of 
 work. 
 
CHAPTER XVI. 
 
 THEORY OF RESONATORS. 
 
 303. IN the pipe closed at one end and open at the other we had 
 an example of a mass of air endowed with the property of vibrating 
 in certain definite periods peculiar to itself in more or less com- 
 plete independence of the external atmosphere. If the air beyond 
 the open end were entirely without mass, the motion within the 
 pipe would have no tendency to escape, and the contained column 
 of air would behave like any other complex system not subject to 
 dissipation. In actual experiment the inertia of the external air 
 cannot, of course, be got rid of, but when the diameter of the pipe 
 is small, the effect produced in the course of a few periods may be 
 insignificant, and then vibrations once excited in the pipe have a 
 certain degree of persistence. The narrower the channel of com- 
 munication between the interior of a vessel and the external 
 medium, the greater does the independence become. Such 
 cavities constitute resonators; in the presence of an external 
 source of sound, the contained air vibrates in unison, and with an 
 amplitude dependent upon the relative magnitudes of the natural 
 and forced periods, rising to great intensity in the case of approxi- 
 mate isochronism. When the original cause of sound ceases, the 
 resonator yields back the vibrations stored up as it were within it, 
 thus becoming itself for a short time a secondary source. The 
 theory of resonators constitutes an important branch of our 
 subject. 
 
 As an introduction to it we may take the simple case of a 
 stopped cylinder, in which a piston moves without friction. On 
 the further side of the piston the air is supposed to be devoid of 
 inertia, so that the pressure is absolutely constant. If now the 
 piston be set into vibration of very long period, it is clear that 
 the contained air will be at any time very nearly in the equi- 
 librium condition (of uniform density) corresponding to the 
 
303.] POTENTIAL ENERGY OF COMPRESSION. 171 
 
 momentary position of the piston. If the mass of the piston be 
 very considerable in comparison with that of the included air, the 
 natural vibrations resulting from a displacement will occur nearly 
 as if the air had no inertia ; and in deriving the period from the 
 kinetic and potential energies, the former may be calculated with- 
 out allowance for the inertia of the air, and the latter as if the 
 rarefaction and condensation were uniform. Under the circum- 
 stances contemplated the air acts merely as a spring in virtue of 
 its resistance to compression or dilatation ; the form of the contain- 
 ing vessel is therefore immaterial, and the period of vibration 
 remains the same, provided the capacity be not varied. 
 
 When a gas is compressed or rarefied, the mechanical value of 
 the resulting displacement is found by multiplying each infinitesi- 
 mal increment of volume by the corresponding pressure and 
 integrating over the range required. In the present case it is of 
 course only the difference of pressure on the two sides of the 
 piston which is really operative, and this for a small change is 
 proportional to the alteration of volume. The whole mechanical 
 value of the small change is the same as if the expansion were 
 opposed throughout by the mean, that is half the final,, pressure ; 
 thus corresponding to a change of volume from S to 8 + SS, 
 since p a?p, 
 
 If A denote the area of the piston, M its mass, and x its linear 
 displacement, SS = Ax, and the equation of motion is 
 
 (2), 
 
 indicating vibrations, whose periodic time is 
 
 Let us now imagine a vessel containing air, whose interior 
 communicates with the external atmosphere by a narrow aperture 
 or neck. It is not difficult to see that this system is capable of 
 vibrations similar to those just considered, the air in the neigh- 
 bourhood of the aperture supplying the place of the piston. By 
 sufficiently increasing S, tjie period of the vibration may be made 
 as long as we please, and we obtain finally a state of things in 
 
 1 Compare (12) 245. 
 
172 KINETIC ENERGY OF MOTION [303. 
 
 which the kinetic energy of the motion may be neglected except 
 in the neighbourhood of the aperture, and the potential energy 
 may be calculated as if the density in the interior of the vessel 
 were uniform. In flowing through the aperture under the operation 
 of a difference of pressure on the two sides, or in virtue of its own 
 inertia after such pressure has ceased, the air moves approximately 
 as an incompressible fluid would do under like circumstances, 
 provided that the space through which the kinetic energy is 
 sensible be very small in comparison with the length of the wave. 
 The suppositions on which we are about to proceed are not of 
 course strictly correct as applied to actual resonators such as are 
 used in experiment, but they are near enough to the mark to afford 
 an instructive view of the subject and in many cases a foundation 
 for a sufficiently accurate calculation of the pitch. They become 
 rigorous only in the limit when the wave-length is indefinitely 
 great in comparison with the dimensions of the vessel. 
 
 [On the above principles we may at once calculate the pitch of 
 a resonator of volume S, whose cavity communicates with the 
 external air by a long cylindrical neck of length L and area A. 
 The mass of the aerial piston is pAL\ so that (3) gives as the 
 period of vibration 
 
 or, if X be the length of plane waves of the same pitch, 
 
 (5). 
 
 If the cross-section of the neck be a circle of radius R, A = irR", 
 and we obtain the formula (8) of 307.] 
 
 304. The kinetic energy of the motion of an incompressible 
 fluid through a given channel may be expressed in terms of the 
 density p, and the rate of transfer, or current, X , for under the cir- 
 cumstances contemplated the character of the motion is always 
 the same. Since T necessarily varies as p and as X 2 , we may put 
 
 T=\P~ (D- 
 
 where the constant c, which depends only on the nature of the 
 channel, is a linear quantity, as may be inferred from the fact that 
 
304.] 
 
 THROUGH NARROW PASSAGES. 
 
 173 
 
 the dimensions of X are 3 in space and 1 in time. In fact, if 
 be the velocity-potential, 
 
 by Green's theorem, where the integration is to be extended over 
 a surface including the whole region through which the motion is 
 sensible. At a sufficient distance on either side of the aperture, <j> 
 becomes constant, and if the constant values be denoted by fa and 
 fa, and the integration be now limited to that half of S towards 
 which the fluid flows, we have 
 
 Now, since within 8 <f> is determined linearly by its surface 
 values, 1 1 -~- dS, or X, is proportional to (fa fa). If we put 
 X = c (fa fa), we get as before T = 
 
 Fig. 58. 
 
 The nature of the constant c will be better understood by con- 
 sidering the electrical problem, whose 
 conditions are mathematically identical 
 with those of that under discussion. 
 Let us suppose that the fluid is re- 
 placed by uniformly conducting ma- 
 terial, and that the boundary of the 
 channel or aperture is replaced by in- 
 sulators. We know that if by battery 
 power or otherwise, a difference of 
 electric potential be maintained on the 
 two sides, a steady current through the 
 aperture of proportional magnitude 
 will be generated. The ratio of the 
 
 total current to the electromotive force is called the conductivity 
 of the channel, and thus we see that our constant c represents 
 simply this conductivity, on the supposition that the specific 
 conducting power of the hypothetical substance is unity. The 
 same thing may be otherwise expressed by saying that c is the 
 side of the cube, whose resistance between opposite faces is the 
 same as that of the channel. In the sequel we shall often avail 
 ourselves of the electrical analogy. 
 
174 NATURAL PITCH OF RESONATORS. [304. 
 
 When c is known, the proper tone of the resonator can be 
 easily deduced. Since 
 
 V = $pa?^, T = p^- .................. (2), 
 
 the equation of motion is 
 
 X = ........................... (3), 
 
 indicating simple oscillations performed in a time 
 
 (4). 
 
 If N be the frequency, or number of complete vibrations 
 executed in the unit time, 
 
 The wave-length X, which is the quantity most closely con- 
 nected with the dimensions of the cavity, is given by 
 
 F 
 
 and varies directly as the linear dimension. The wave-length, it 
 will be observed, is a function of the size and shape of the 
 resonator only, while the frequency depends also upon the nature 
 of the gas ; and it is important to remark that it is on the nature 
 of the gas in and near the channel that the pitch depends and not 
 on that occupying the interior of the vessel, for the inertia of the 
 air in the latter situation does not come into play, while the com- 
 pressibility of all gases is very approximately the same. Thus in 
 the case of a pipe, the substitution of hydrogen for air in the 
 neighbourhood of a node would make but little difference, but its 
 effect in the neighbourhood of a loop would be considerable. 
 
 Hitherto we have spoken of the channel of communication as 
 single, but if there be more than one channel, the problem is not 
 essentially altered. The same formula for the frequency is still 
 applicable, if as before we understand by c the whole conduc- 
 tivity between the interior and exterior of the vessel. When the 
 channels are situated sufficiently far apart to act independently 
 one of another, the resultant conductivity is the simple sum of 
 those belonging to the separate channels ; otherwise the resultant 
 is less than that calculated by mere addition. 
 
304.] SUPERIOR AND INFERIOR LIMITS. 175 
 
 If there be two precisely similar channels, which do not 
 interfere, and whose conductivity taken separately is c, we have 
 
 shewing that the note is higher than if there were only one 
 channel in the ratio *J2 : 1, or by rather less than a fifth a law 
 observed by Sondhauss and proved theoretically by Helmholtz in 
 the case, where the channels of communication consist of simple 
 holes in the infinitely thin sides of the reservoir. 
 
 305. The investigation of the conductivity for various kinds 
 of channels is an important part of the theory of resonators ; but 
 in all except a very few cases the accurate solution of the problem 
 is beyond the power of existing mathematics. Some general 
 principles throwing light on the question may however be laid 
 down, and in many cases of interest an approximate solution, 
 sufficient for practical purposes, may be obtained. 
 
 We know ( 79, 242) that the energy of a fluid flowing 
 through a channel cannot be greater than that of any fictitious 
 motion giving the same total current. Hence, if the channel be, 
 narrowed in any way, or any obstruction be introduced, the con- 
 ductivity is thereby diminished, because the alteration is of the 
 nature of an additional constraint. Before the change the fluid 
 was free to adopt the distribution of flow finally assumed. In 
 cases where a rigorous solution cannot be obtained we may use the 
 minimum property to estimate an inferior limit to the conductivity; 
 the energy calculated from a hypothetical law of flow can never be 
 less than the truth, and must exceed it unless the hypothetical 
 and the actual motion coincide. 
 
 Another general principle, which is of frequent use, may be 
 more conveniently stated in electrical language. The quantity 
 with which we are concerned is the conductivity of a certain con- 
 ductor composed of matter of unit specific conductivity. The 
 principle is that if the conductivity of any part of the conductor 
 be increased that of the whole is increased, and if the conductivity 
 of any part be diminished that of the whole is diminished, 
 exception being made of certain very particular cases, where no 
 alteration ensues. In its passage through a conductor electricity 
 distributes itself, so that the energy dissipated is for a given total 
 
 
176 SIMPLE APERTURES. [305. 
 
 current the least possible ( 82). If now the specific resistance of 
 any part be diminished, the total dissipation would be less than 
 before, even if the distribution of currents remained unchanged. A 
 
 o 
 
 fortiori will this be the case, when the currents redistribute them- 
 selves so as to make the dissipation a minimum. If an infinitely 
 thin lamina of matter stretching across the channel be made 
 perfectly conducting, the resistance of the whole will be diminished, 
 unless the lamina coincide with one of the undisturbed equipoten- 
 tial surfaces. In the excepted case no effect will be produced. 
 
 306. Among different kinds of channels an important place 
 must be assigned to those consisting of simple apertures in un- 
 limited plane walls of infinitesimal thickness. In practical appli- 
 cations it is sufficient that a wall be very thin in proportion to the 
 dimensions of the aperture, and approximately plane within a 
 distance from the aperture large in proportion to the same 
 quantity. 
 
 On account of the symmetry on the two sides of the wall, the 
 motion of the fluid in the plane of the aperture must be normal, 
 and therefore the velocity-potential must be constant ; over the 
 remainder of the plane the motion must be exclusively tangential, 
 so that to determine <p on one side of the plane we have the 
 conditions (i) </> = constant over the aperture, (ii) d$/dn = over 
 the rest of the plane of the wall, (iii) < = constant at infinity. 
 
 Since we are concerned only with the differences of </> we may 
 suppose that at infinity <f> vanishes. It will be seen that conditions 
 (ii) and (iii) are satisfied by supposing $ to be the potential of 
 attracting matter distributed over the aperture ; the remainder of 
 the problem consists in determining the distribution of matter so 
 that its potential may be constant over the same area. The 
 problem is mathematically the same as that of determining the 
 distribution of electricity on a charged conducting plate situated 
 in an open space, whose form is that of the aperture under con- 
 sideration, and the conductivity of the aperture may be expressed 
 in terms of the capacity of the plate of the statical problem. If 
 <f> denote the constant potential in the aperture, the electrical 
 resistance (for one side only) will be 
 
 , . ff<ty 
 
 h+jjfa 
 
 -j-d<*> 
 an 
 
 the integration extending over the area of the opening. 
 
306.] ELLIPTIC APERTURE. 177 
 
 Now 1 1 T-^ dcr = %TT x (whole quantity of matter distributed), 
 
 and thus, if M be the capacity, or charge corresponding to unit- 
 potential, the total resistance is ( r jrM)~ l . Accordingly for the con- 
 ductivity, which is the reciprocal of the resistance, 
 
 (1). 
 
 So far as I am aware, the ellipse is the only form of aperture 
 for which c or M can be determined theoretically x , in which case 
 the result is included in the known solution of the problem of 
 determining the distribution of charge on an ellipsoidal conductor. 
 From the fact that a shell bounded by two concentric, similar and 
 similarly situated ellipsoids exerts no force on an internal particle, 
 it is easy to see that the superficial density at any point of an ellip- 
 soid necessary to give a constant potential is proportional to the 
 perpendicular (p) let fall from the centre upon the tangent plane 
 at the point in question. Thus if p be the density, p = tcp\ the 
 whole quantity of matter Q is given by 
 
 In the usual notation 
 
 or, since z*c z = 1- 
 
 If we now suppose that c is infinitely small, we obtain the par- 
 ticular case of an elliptic plate, and if we no longer distinguish 
 between the two surfaces, we get 
 
 Q l\ * y* 
 
 = 2^6^V ~a*~V 
 
 1 The case of a resonator with an elliptic aperture was considered by Helmholtz 
 (Crelle, Bd. 57, 1860), whose result is equivalent to (8). 
 
 2 2c being for the moment the third principal axis of the ellipsoid. ^r^ f 
 
 R. II. 
 
 UN IV 
 
178 ELLIPTIC APERTURE. [306. 
 
 We have next to find the value of the constant potential (P). 
 By considering the value of P at the centre of the plate, we see 
 that 
 
 Integrating first with respect to r, we have 
 
 f pdr = Q + 4a V(l - e~ cos 2 (9), 
 
 Jo 
 
 e being the eccentricity; and thus 
 
 dff _Q F() 
 
 ***)" a 
 
 where P is the symbol of the complete elliptic function of the first 
 order. Putting P = 1, we find 
 
 a 
 
 as the final expression for the capacity of an ellipse, whose semi- 
 major axis is a and eccentricity is e. In the particular case of the 
 circle, e = 0, F(e) = JTT, and thus for a circle of radius R, 
 
 c = 2R ........................... ...(6). 
 
 If the capacity of the resonator be S, we find from (6) 304 
 
 The area of the ellipse (<r) is given by 
 
 <7 = 7ra 2 V(l-e 2 ), 
 and hence in terms of cr 
 
 c 
 
 When e is small, we obtain by expanding in powers of e pre- 
 vious to integration, 
 
 whence 
 
 _ _ _ 
 
 +> " 
 
 TT 64 64 
 
306.] COMPARISON WITH CIRCULAR APERTURE. 179 
 
 Neglecting e 8 and higher powers, we have therefore 
 
 From this result we see that, if its eccentricity be small, the 
 conductivity of an elliptic aperture is very nearly the same as 
 that of a circular aperture of equal area. Among various forms 
 of aperture of given area there must be one which has a minimum 
 conductivity, and, though a formal proof might be difficult, it is 
 easy to recognise that this can be no other than the circle. An 
 inferior limit to the value of c is thus always afforded by the con- 
 ductivity of the circle of equal area, that is 2v'(o-/7r), and when 
 the true form is nearly circular, this limit may be taken as a close 
 approximation to the real value. 
 
 The value of X is then given by 
 
 (11). 
 
 In order to shew how slightly a moderate eccentricity affects 
 the value of c, I have calculated the following short table with the 
 aid of Legendre's values of F(e). Putting e = sin ty, we have 
 cos -\Jr as the ratio of axes, and for the conductivity 
 
 c = 
 
 *Y. 
 
 7T 
 
 J ' 2V(cos \lr) . F (sin 
 
 *. 
 
 e = sin \j/. 
 
 b : a cos if/. 
 
 7 r-2^(e)(l-e 2 ) i . 
 
 
 
 00000 
 
 1-00000 
 
 1-0000 
 
 20 
 
 34204 
 
 93969 
 
 1-0002 
 
 30 
 
 50000 
 
 86603 
 
 1-0013 
 
 40 
 
 64279 
 
 76604 
 
 1-0044 
 
 50 
 
 76604 
 
 64279 
 
 1-0122 
 
 60 
 
 86603 
 
 50000 
 
 1-0301 
 
 70 
 
 93969 
 
 34202 
 
 1 0724 
 
 80 
 
 98481 
 
 17365 
 
 1-1954 
 
 90 
 
 1-00000 
 
 00000 
 
 oo 
 
 The value of the last factor given in the fourth column is the 
 ratio of the conductivity of the ellipse to that of a circle of equal 
 area. It appears that even when the ellipse is so eccentric that 
 
 122 
 
180 CALCULATION BASED ON AREA. [306. 
 
 the ratio of the axes is 2:1, the conductivity is increased by 
 only about 3 per cent., which would correspond to an alteration 
 of little more than a comma ( 18) in the pitch of a resonator. 
 There seems no reason to suppose that this approximate inde- 
 pendence of shape is a property peculiar to the ellipse, and we 
 may conclude with some confidence that in the case of any mode- 
 rately elongated oval aperture, the conductivity may be calculated 
 from the area alone with a considerable degree of accuracy. 
 
 If -the area be given, there is no superior limit to c. For sup- 
 pose the area a to be distributed over n equal circles sufficiently 
 far apart to act independently. The area of each circle is a-/n, 
 and its conductivity is 2 (mr)~~a^. The whole conductivity is n 
 times as great, and therefore increases indefinitely with n. As a 
 general rule, the more the opening is elongated or broken up, the 
 greater will be the conductivity for a given area. 
 
 To find a superior limit to the conductivity of a given aperture 
 we may avail ourselves of the principle that any addition to the 
 aperture must be attended by an increase in the value of c. Thus 
 in the case of a square, we may be sure that c is less than for the 
 circumscribed circle, and we have already seen that it is greater 
 than for the circle of equal area. If b be the side of the square, 
 
 ^ < c < V2 b. 
 
 VTT 
 
 The tones of a resonator with a square aperture calculated from 
 these two limits would differ by about a whole tone ; the graver of 
 them would doubtless be much the nearer to the truth. This 
 example shews that even when analysis fails to give a solution in 
 the mathematical sense, we need not be altogether in the dark as 
 to the magnitudes of the quantities with which we are dealing. 
 
 In the case of similar orifices, or systems of orifices, c varies as 
 the linear dimension. 
 
 307. Most resonators used in practice have necks of greater or 
 less length, and even when there is nothing that would be called a 
 neck, the thickness of the side of the reservoir cannot always be 
 neglected. We shall therefore examine the conductivity of a 
 channel formed by a cylindrical boring through an obstructing 
 plate bounded by parallel planes, and, though we fail to solve the 
 problem rigorously, we shall obtain information sufficient for most 
 
307.] 
 
 CONDUCTIVITY OF NECKS. 
 
 181 
 
 Fig. 59. 
 
 practical purposes. The thickness of the plate we shall call L, and 
 the radius of the cylindrical channel R. 
 
 Whatever the resistance of the channel may be, 
 it will be lessened by the introduction of infinitely 
 thin discs of perfect conductivity at A and B, fig. 59. 
 The effect of the discs is to produce constant potential 
 over their areas, and the problem thus modified is 
 susceptible of rigorous solution. Outside A and B 
 the motion is the same as that previously investi- 
 gated, when the obstructing plate is infinitely thin ; 
 between A and B the flow is uniform. The resist- 
 ance is therefore on the whole 
 
 whence 
 
 (i). 
 
 If a denote the correction, which must be added to L on 
 account of an open end, 
 
 a^l-rrR ........................ .....(2). 
 
 This correction is in general under the mark, but, when L is 
 very small in comparison with R, the assumed motion coincides 
 more and more nearly with the actual motion, and thus the value 
 of a in (2) tends to become correct. 
 
 A superior limit to the resistance may be calculated from a 
 hypothetical motion of the fluid. For this purpose we will suppose 
 infinitely thin pistons introduced at A and B, the effect of which 
 will be to make the normal velocity constant at those places. 
 Within the tube the flow will be uniform as before, but for the 
 external space we have a new problem to consider : To determine 
 the motion of a fluid bounded by an infinite plane, the normal 
 velocity over a circular area of the plane having a given constant 
 value, and over the remainder of the plane being zero. 
 
 The potential may still be regarded as due to matter distributed 
 over the disc, but it is no longer constant over the area; the density 
 of the matter, however, being proportional to d<j>/dn is constant. 
 
 The kinetic energy of the motion 
 
 the integration going over the area of the circle. 
 
182 CONDUCTIVITY OF NECKS. [307. 
 
 The total current through the plane 
 
 f> d<p 
 
 - d(T = 7T1 Z -r- . 
 
 n dn 
 
 2 kinetic energy {[Ado- 
 Hence ,., &J = JJ < ,, . 
 
 (current) 2 2 ^ 4 d<f> 
 
 dn 
 
 If the density of the matter be taken as unity, d(f>/dn = 2?r, and 
 the required ratio is expressed by P/Tr 3 ^ 4 , where P denotes the 
 potential on itself of a circular layer of matter of unit density and 
 of radius R. 
 
 The simplest method of calculating P depends upon the con- 
 sideration that it represents the work required to break up the 
 disc into infinitesimal elements and to remove them from each 
 other's influence *. If we take polar co-ordinates (p, 6), the pole 
 being at the edge of the disc whose radius is a, we have for the 
 potential at the pole, V = ffd6dp, the limits of p being and 
 Za cos 0, and those of being J TT and -f- J TT. 
 
 Thus F=4a (3). 
 
 Now let us cut off a strip of breadth da from the edge of the disc. 
 The work required to remove this to an infinite distance is 
 Zjrada . 4a. If we gradually pare the disc down to nothing and 
 carry all the parings to infinity 2 , we find for the total work by 
 integrating with respect to a from to R, 
 
 P = 
 
 3 
 
 The limit to the resistance (for one side) is thus 8/37r 2 ^; we 
 conclude that the resistance of the whole channel is less than 
 
 L 16 
 
 irr> (4). 
 
 Collecting our results, we see that 
 
 1 A part of 302 is repeated here for the sake of those who may wish to avoid 
 the difficulties of the more complete investigation. 
 
 2 This method of calculating P was suggested to the author by Professor 
 Clerk Maxwell. 
 
307.] CORRECTION TO LENGTH. 183 
 
 or in decimals, 
 
 a>-785E| 
 
 a < -849^1" 
 
 It must be observed that a here denotes the correction for one 
 end. The whole resistance corresponds to a length L + 2a of 
 tube having the section 
 
 When L is very great in relation to R, we may take simply 
 
 <=*? w- 
 
 In this case we have from (6) 304 
 
 ft 
 
 (8). 
 
 _f.t 
 
 The correction for an open end (a) is a function of L, coinciding 
 with the lower limit, viz. %7rR, when L vanishes. As L increases, 
 a increases with it ; but does not, even when L is infinite, attain 
 the superior limit SR/STT. For consider the motion going on in 
 any middle piece of the tube. The kinetic energy is greater than 
 corresponds merely to the length of the piece. If therefore the 
 piece be removed, and the free ends brought together, the motion 
 otherwise continuing as before, the kinetic energy will be dimin- 
 ished more than corresponds to the length of the piece subtracted. 
 A fortiori will this be true of the real motion which would exist in 
 the shortened tube. That, when L = oo , a does not become SR/Sw 
 is evident, because the normal velocity at the end, far from being 
 constant, as was assumed in the calculation of this result, must 
 increase from the centre outwards and become infinite at the edge. 
 
 A further approximation to the value of a may be obtained by 
 assuming a variable velocity at the plane of the mouth. The 
 calculation will be found in Appendix A. It appears that in the 
 case of an infinitely long tube a cannot be so great as '82422 R. 
 The real value of a is probably not far from "82 R. 
 
 308. Besides the cylinder there are very few forms of 
 channel whose conductivity can be determined mathematically. 
 When however the form is approximately cylindrical we may 
 obtain limits, which are useful as allowing us to estimate the 
 
184 TUBES OF REVOLUTION. [308. 
 
 effect of such departures from mathematical accuracy as must 
 occur in practice. 
 
 An inferior limit to the resistance of any elongated and approxi- 
 mately straight conductor may be obtained immediately by the 
 imaginary introduction of an infinite number of plane perfectly 
 conducting layers perpendicular to the axis. If a* denote the area 
 of the section at any point x, the resistance between two layers 
 distant dx will be o-~ l dx, and therefore the whole actual resistance 
 is certainly greater than 
 
 f ^dx (1), 
 
 unless indeed the conductor be truly cylindrical. 
 
 In order to find a superior limit we may calculate the kinetic 
 energy of the current on the hypothesis that the velocity parallel 
 to the axis is uniform over each section. The hypothetical motion 
 is that which would follow from the introduction of an infinite 
 number of rigid pistons moving freely, and the calculated result is 
 necessarily in excess of the truth, unless the section be absolutely 
 constant. We shall suppose for the sake of simplicity that the 
 channel is symmetrical about an axis, in which case of course the 
 motion of the fluid is symmetrical also. 
 
 If U denote the total current, we have ex hypothesi for the 
 axial velocity at any point x 
 
 u = ^U.. ..(2), 
 
 from which the radial velocity v is determined by the equation of 
 continuity (6 238), 
 
 d(ru) d(rv) _ 
 dx dr 
 
 Thus TO const. \ Ur* -= , 
 
 ax 
 
 or, since there is no source of fluid on the axis, 
 
308.] SUPERIOR LIMIT. 185 
 
 The kinetic energy may now be calculated by simple integra- 
 tion : 
 
 fu 2 crdx = U* (a- l dx, 
 
 if y be the radius of the channel at the point x, so that <r = Try*. 
 
 Thug 2 kinetic energy = |-: L f /dyyl^ ......... 
 
 (current) 2 / Try* ( J \dx) } 
 
 This is the quantity which gives a superior limit to the resist- 
 ance. The first term, which corresponds to the component velocity 
 u, is the same as that previously obtained for the lower limit, as 
 might have been foreseen. The difference between the two, which 
 gives the utmost error involved in taking either of them as the 
 true value, is 
 
 In a nearly cylindrical channel dyjdx is a small quantity and 
 so the result found in this manner is closely approximate. It is 
 not necessary that the section should be nearly constant, but only 
 that it should vary slowly. The success of the approximation in 
 this and similar cases depends upon the fact that the quantity to 
 be estimated is at a minimum. Any reasonable approximation to 
 the real motion will give a result very near the truth according to 
 the principles of the differential calculus. 
 
 By means of the properties of the potential and stream 
 functions ( 238) the present problem admits of actual approxi- 
 mate solution. If < and ty denote the values of these functions 
 at any point x, r\ u, v denote the axial and transverse velocities, 
 
 dx r dr ' dr ~ r dx" 
 
 whence by elimination 
 
186 APPROXIMATE CALCULATION. [308. 
 
 If F denote the value of </> as a function of x when r = 0, the 
 general values of </> and ^ may be expressed in terms of F by 
 means of (7) and (8) in the series 
 
 92 A 2 fis 
 
 0), 
 + .. 
 
 ~ 2 2 2 .42 2 .4 2 .6 "2? . 4 2 . 6 2 . 8 
 
 where accents denote differentiation with respect to x. At the 
 boundary of the channel where r = y, ty is constant, say fa. Then 
 
 2*6 
 
 is the equation connecting y and F. In the present problem y is 
 given, and we have to express F by means of it. By successive 
 approximation we obtain from (10) 
 
 2fa f f#/2\ 1 
 ~ 
 
 8 (rfa; 2 \ip & da? dx* \ 
 
 (ID 
 
 The total stream is given by the integral 
 
 o r dr 
 
 and therefore the resistance between any two equipotential surfaces 
 is represented by 
 
 i_f 
 
 rrfa) 
 
 9 , ,F'dx. 
 2-rrfa 
 
 The expression for the resistance admits of considerable simpli- 
 fication by integration by parts in the case when the channel is 
 truly cylindrical in the neighbourhood of the limits of integration. 
 In this way we find for the final result, 
 
 '" (12)', 
 
 y', y" denoting the differential coefficients of y with respect to x. 
 
 It thus appears that the superior limit of the preceding 
 investigation is in fact the correct result to the second order of 
 
 1 Proceedings of the London Mathematical Society, Vol. vn. p. 70, 1876. 
 
308.] COMPARISON WITH EXPERIMENT. 187 
 
 approximation. If we regard y as a function of o>#, where &> is a 
 small quantity, (12) is correct as far as terms containing o> 4 . 
 
 309. Our knowledge of the laws on which the pitch of 
 resonators depends, is due to the labours of several experimenters 
 and mathematicians. 
 
 The observation that for a given mouthpiece the pitch of a 
 resonator depends mainly upon the volume S is due to Liscovius, 
 who found that the pitch of a flask partly filled with water was 
 not altered when the flask was inclined. This result was con- 
 firmed by Sondhauss 1 . The latter observer found further, that in 
 the case of resonators without necks, the influence of the aperture 
 depended mainly upon its area, although when the shape was very 
 elongated, a certain rise of pitch ensued. He gave the formula 
 
 ........................... (1), 
 
 the unit of length being the millimetre. 
 
 The theory of this kind of resonator we owe to Helmholtz 2 , 
 whose formula is 
 
 
 applicable to circular apertures. 
 
 For flasks with long necks, Sondhauss 3 found 
 
 corresponding to the theoretical 
 
 JIT-46705 ........................... (3), 
 
 In practice it does not often happen either that the neck 
 is so long that the correction for the open ends can be neglected, 
 as (4) supposes, or, on the other hand, so short that it can 
 itself be neglected, as supposed in (2). Wertheim 4 was the first 
 
 1 Ueber den Brummkreisel und das Schwingungsgesetz der cubischen Pfeifen. 
 Pogg. Ann. LXXXI. pp. 235, 347. 1850. 
 
 2 Crelle, Bd. LVII. 172. 1860. 
 
 3 Ueber die Schallschwingungen der Luft in erhitzten Glasrohren und in gedeck- 
 ten Pfeifen von ungleicher Weite. Pogg. Ann. LXXIX. p. 1. 1850. 
 
 4 Me'moire sur les vibrations sonores de 1'air. Ann. d. Chim. (3) xxxi. p. 385. 
 1851. 
 
188 HELMHOLTZ'S INVESTIGATION. [309. 
 
 to shew that the effect of an open end could be represented by 
 an addition (a) to the length, independent, or nearly so, of L 
 and X. 
 
 The approximate theoretical determination of a is due to 
 Helmholtz, who gave %7?R as the correction for an open end 
 fitted with an infinite flange. His method consisted in inventing 
 forms of tube for which the problem was soluble, and selecting 
 that one which agreed most nearly with a cylinder. The cor- 
 rection ^TrR is rigorously applicable to a tube whose radius at the 
 open end and at a great distance from it is R, but which in the 
 neighbourhood of the open end bulges slightly. 
 
 From the fact that the true cylinder may be derived by in- 
 troducing an obstruction, we may infer that the result thus obtained 
 is too small. 
 
 It is curious that the process followed in this work, which was 
 first given in the memoir on resonance, leads to exactly the same 
 result, though it would be difficult to conceive two methods more 
 unlike each other. 
 
 The correction to the length will depend to some extent upon 
 whether the flow of air from the open end is obstructed, or not. 
 When the neck projects into open space, there will be less ob- 
 struction than when a backward flow is prevented by a flange as 
 supposed in our approximate calculations. However, the un- 
 certainty introduced in this way is not very important, and we 
 may generally take a=^7rR as a sufficient approximation. In 
 practice, when the necks are short, the hypothesis of the flange 
 agrees pretty well with fact, and when the necks are long, the 
 correction is itself of subordinate importance. 
 
 The general formula will then run 
 
 , r a I a 
 
 y -teVS\L + '-" (0)> 
 
 where a- is the area of the section of the neck, or in numbers 
 
 AT- 
 * 
 
 6-2832" S*J(L +.4868 Y*)" 
 
 A formula not differing much from this was given, as the em- 
 bodiment of the results of his measurements, by Sondhauss 1 who 
 
 1 Pogg. Ann. CXL. pp. 53, 219. 1870. 
 
309.] MULTIPLE RESONANCE. 189 
 
 at the same time expressed a conviction that it was no mere 
 empirical formula of interpolation, but the expression of a natural 
 law. The theory of resonators with necks was given about the 
 same time 1 in a memoir 'on Resonance' published in the Philo- 
 sophical Transactions for 1871, from which most of the last few 
 pages is derived. 
 
 310. The simple method of calculating the pitch of resonators 
 with which we have been occupied is applicable to the gravest 
 mode of vibration only, the character of which is quite distinct. 
 The overtones of resonators with contracted necks are relatively 
 very high, and the corresponding modes of vibration are by no 
 means independent of the inertia of the air in the interior of the 
 reservoir. The character of these modes will be more evident, 
 when we come to consider the vibrations of air within a com- 
 pletely closed vessel, such as a sphere, but it will rarely happen 
 that the pitch can be calculated theoretically. 
 
 There are, however, cases of multiple resonance to which our 
 theory is applicable. These occur when two or more vessels com- 
 municate by channels with each other and with the external air ; 
 and are readily treated by Lagrange's method, provided of course 
 that the wave-length of the vibration is sufficiently large in com- 
 parison with the dimensions of the vessels. 
 
 Suppose that there are two reservoirs, 8, 8', communicating 
 with each other and with the external air by narrow passages or 
 
 Fig. 60. 
 
 necks. If we were to consider 88' as a, single reservoir and apply 
 our previous formula, we should be led to an erroneous result ; for 
 that formula is founded on the assumption that within the reservoir 
 the inertia of the air may be left out of account, whereas it is 
 evident that the energy of the motion through the connecting 
 passage may be as great as through the two others. However, an 
 
 1 Proceedings of the Royal Society, Nov. 24, 1870. 
 
190 DOUBLE RESONATOR. [310. 
 
 investigation on the same general plan as before meets the case 
 perfectly. Denoting by X lt X. 2 , X 3 the total transfers of fluid 
 through the three passages, we have as in (2) 304 for the kinetic 
 energy the expression 
 
 and for the potential energy, 
 
 An application of Lagrange's method gives as the differential 
 equations of motion, 
 
 f + a^-** = 
 
 By addition and integration, 
 
 f-' + J + f' = ....................... (4). 
 
 l/l 1/2 ^3 
 
 Hence on elimination of X z , 
 
 , + <*) JT,+^ 
 
 Assuming X^ = AeP*, X 3 = Be pt , we obtain on substitution 
 and determination of A : B, 
 
 ^o. ..(6), 
 
 as the equation to determine the natural tones. If N be the 
 frequency of vibration, N' 2 = - > 2 /4?r 2 , the two values of p* being 
 of course real and negative. The formula simplifies considerably 
 if c s = Cj, S' = S ; but it will be more instructive to work out this 
 case from the beginning. Let d = c 3 = mc 2 = me. 
 
310.] DOUBLE RESONATOR. 191 
 
 The differential equations take the form 
 
 8 
 
 y , y 
 
 while from (4) X<> = * 
 
 m 
 
 Hence 
 
 (8). 
 
 The whole motion may be divided into two parts. For the first of 
 these 
 
 X l + X 3 = ........................... (9), 
 
 which requires that X z = 0. The motion is therefore the same as 
 might take place were the communication between S and 8' cut 
 off, and has its frequency given by 
 
 
 The density of the air is the same in both reservoirs. 
 For the other component part, X t X 3 = 0, so that 
 
 m 
 
 The vibrations are thus opposed in phase. The ratio of frequencies 
 is given by N'* : N 2 = m + 2 : m, shewing that the second mode 
 has the shorter period. In this mode of vibration the connecting 
 passage acts in some measure as a second opening to both vessels, 
 and thus raises the pitch. If the passage be contracted, the interval 
 of pitch between the two notes is small. 
 
 A particular case of the general formula worthy of notice is 
 obtained by putting C 3 = 0, which amounts to suppressing one of 
 the communications with the external air. We thus obtain 
 
192 PARTICULAB CASE. [310. 
 
 or, if 8 = S', G! = mc 2 = me, 
 
 whence N 2 = ^{m + 2 V(m 2 + 4)} ............. (14). 
 
 If we further suppose m = 1, or c 2 = C 1} 
 
 If J\r be the frequency for a simple resonator (8, c), 
 
 and thus Nf : N' 2 = > = 2 618, 
 
 -,* 2-618. 
 
 It appears that the interval from N^ to N' is the same as from 
 N' to JV a , namely, V(2'618) = T618, or rather more than a fifth. 
 It will be found that whatever the value of m may be, the interval 
 between the two tones cannot be less than 2'414, which is about 
 an octave and a minor third. The corresponding value of m is 2. 
 
 A similar method is applicable to any combination, however 
 complicated, of reservoirs and connecting passages under the 
 single restriction as to the comparative magnitudes of the reser- 
 voirs and wave-lengths; but the example just given is sufficient 
 to illustrate the theory of multiple resonance. A few measure- 
 ments of the pitch of double resonators are detailed in my memoir 
 on resonance, already referred to. 
 
 311. The equations which we have employed hitherto take 
 no account of the escape of energy from a resonator. If there 
 were really no transfer of energy between a resonator and the 
 external atmosphere, the motion would be isolated and of little 
 practical interest : nevertheless the characteristic of a resonator 
 consists in its vibrations being in great measure independent. 
 Vibrations, once excited, will continue for a considerable number of 
 periods without much loss of energy, and their frequency will be 
 almost entirely independent of the rate of dissipation. The rate 
 of dissipation is, however, an important feature in the character 
 
311.] 
 
 COMMUNICATION OF ENERGY. 
 
 193 
 
 of a resonator, on which its behaviour under certain circumstances 
 materially depends. It will be understood that the dissipation 
 here spoken of means only the escape of energy from the vessel 
 and its neighbourhood, and its diffusion in the surrounding 
 medium, and not the transformation of ordinary energy into heat. 
 Of such transformation our equations take no account, unless 
 special terms be introduced for the purpose of representing the 
 effects of viscosity, and of the conduction and radiation of heat. 
 
 [The influence of the conduction of heat has been considered 
 by Kolacek 1 .] 
 
 Fig. 61. 
 
 \.A. 
 
 In a previous chapter ( 278) we saw how to express the motion 
 on the right of the infinite flange (Fig. 61), in terms of the normal 
 velocity of the fluid over the disc A. We found, 278 (3), 
 
 27TJJ dn r 
 
 int 
 
 where $ is proportional to e 
 
 If r be the distance between any two points of the disc, kr is a 
 small quantity, and e~ ikr = 1 ikr approximately. 
 
 Thus 
 
 ''d(j)do 
 dn r 
 
 dn 
 
 .(i). 
 
 The first term depends upon the distribution of the current. If 
 we suppose that dcj>/dn is constant, we obtain ultimately a term 
 representing an increase of inertia, or a correction to the length, 
 equal to SK/Str. This we have already considered, under the 
 supposition of a piston at A. The second term, on which the 
 dissipation depends, is independent of the distribution of current, 
 
 1 Wied. Ann. t. 12, p. 353, 1881. 
 
 R. II. 
 
 13 
 
194 RATE OF DISSIPATION. [311. 
 
 being a function of the total current (X) only. Confining our 
 attention to this term, we have 
 
 Assuming now that <f> oc e int , we have for the part of the varia- 
 tion of pressure at A , on which dissipation depends, 
 
 pnkX pn*X 
 
 ^' (3). 
 
 The corresponding work done during a transfer of fluid BX is 
 ^ SX ; and since, as in 304, the expressions for the potential 
 and kinetic energies are 
 
 2 X 2 
 
 -, y=i/>~ ................ (4), 
 
 the equation of motion ( 80) is 
 
 in place of (3) 304. In the valuation of c an allowance must be 
 included for the inertia of the fluid on the right-hand side of A, 
 corresponding to the term omitted in the expression for $p. 
 
 Equation (5) is of the standard form for the free vibrations 
 of dissipative systems of one degree of freedom ( 45). The 
 amplitude varies as e~ n ' 2ct/ ' iira } being diminished in the ratio e : 1 
 after a time equal to 4?ra/n 2 c. If the pitch (determined by n) be 
 given, the vibrations have the greatest persistence when c is 
 smallest, that is, when the neck is most contracted. 
 
 If S be given, we have on substituting for c its value in terms 
 of 8 and n, 
 
 shewing that under these circumstances the duration of the motion 
 increases rapidly as n diminishes. 
 
 In the case of similar resonators c oc n~ l , and then 
 
 4-Tra 1 
 ~^c n j 
 
 1 Equation (5) is only approximate, inasmuch as the dissipative force is calcu- 
 lated on the supposition that the vibration is permanent ; but this will lead to no 
 material error when the dissipation is small. 
 
311.] NUMERICAL EXAMPLE. 195 
 
 which shews that in this case the same proportional loss of 
 amplitude always occurs after the lapse of the same number of 
 periods. This result may be obtained by the method of di- 
 .mensions, as a consequence of the principle of dynamical 
 similarity. 
 
 As an example of (5), I may refer to the case of a globe with 
 a neck, intended for burning phosphorus in oxygen gas, whose 
 capacity is '251 cubic feet [7100 c.c.]. It was found by experiment 
 that the note of maximum resonance made 120 vibrations per 
 second, so that n = 120 x 2?r. Taking the velocity of sound (a) at 
 1120 feet [34200 cent.] per second, we find from these data 
 
 = ^ of a second nearly. 
 
 Judging from the sound produced when the globe is struck, 
 I think that this estimate must be too low; but it should be 
 observed that the absence of the infinite flange assumed in the 
 theory must influence very materially the rate of dissipation. 
 
 We will now examine the forced vibrations due to a source 
 of sound external to the resonator. If the pressure 8p at the 
 mouth of the resonator due to the source, i.e. calculated on the 
 supposition that the mouth is closed, be Fe ikat , the equation of 
 motion corresponding to (5), but applicable to the forced vibration 
 only, is 
 
 If X = X &&*+), where X. is real, 
 
 = 'S"l +1 1 
 
 ,fj L> , 
 
 The maximum variation of pressure (G) inside the resonator 
 is connected with X by the equation 
 
 (8), 
 
 since X + S is the maximum condensation. Thus 
 
 which agrees with the equation obtained by Helmholtz for the 
 case where the communication with the external air is by a 
 simple aperture ( 306). The present problem is nearly, but not 
 
 132 
 
196 FORCED VIBRATIONS. [311. 
 
 quite, a case of that treated in 46, the difference depending 
 upon the fact that the coefficient of dissipation in (7) is itself 
 a function of the period, and not an absolutely constant quantity. 
 If the period, determined by k, and $ be given, (9) shews that 
 the internal variation of pressure (G) is a maximum when c = & 2 $, 
 that is, when the natural note of the resonator (calculated without 
 allowance for dissipation) is the same as that of the generating 
 sound. The maximum vibration, when the coincidence of periods 
 is perfect, varies inversely as $; but, if S be small, a very slight 
 inequality in the periods is sufficient to cause a marked falling 
 off in the intensity of the resonance ( 49). In the practical 
 use of resonators it is not advantageous to carry the reduction 
 of S and c very far, probably because the arrangements necessary 
 for connecting the interior with the ear or other sensitive ap- 
 paratus involve a departure from the suppositions on which the 
 calculations are founded, which becomes more and more important 
 as the dimensions are reduced. When the sensitive apparatus 
 is not in connection with the interior, as in the experiment of 
 reinforcing the sound of a tuning-fork by means of a resonator, 
 other elements enter into the question, and a distinct investigation 
 is necessary ( 319). 
 
 In virtue of the principle of reciprocity the investigation of the 
 preceding paragraph may be applied to calculate the effect of a 
 source of sound situated in the interior of a resonator. 
 
 312. We now pass on to the further discussion of the problem 
 of the open pipe. We shall suppose that the open end of the 
 pipe is provided with an infinite flange, and that its diameter 
 is small in comparison with the wave-length of the vibration 
 under consideration. 
 
 As an introduction to the question, we will further suppose 
 that the mouth of the pipe is fitted with a freely moving piston 
 without thickness and mass. The preceding problems, from 
 which the present differs in reality but little, have already given 
 us reason to think that the presence of the piston will cause 
 no important modification. Within the tube we suppose ( 255) 
 that the velocity-potential is 
 
 <j> = (Acoskx + Bsiuka;)e int (1), 
 
 where, as usual, k = 2-Tr/X = n/a. At the mouth, where x 0, 
 
 2).-***" (2) - 
 
312.] OPEN PIPE. 197 
 
 On the right of. the piston the relation between < and f ~- ) 
 is by 302 
 
 l 
 
 FJ 1 ' 
 
 R being the radius of the pipe. From this the solution of the 
 problem may be obtained without any restriction as to the 
 smallness of kR: since, however, it is only when kR is small 
 that the presence of the piston would not materially modify 
 the question, we may as well have the benefit of the simplification 
 at once by taking as in (1) 311 
 
 SR 3 ... 
 
 Now, since the piston occupies no space, the values o 
 must be the same on both sides of it ; and since there is no mass, 
 the like must be true of the values of ff<j> d<r. Thus 
 
 8R 3 
 
 SkR 
 
 Substituting in (1), we find on rejecting the imaginary part, 
 and putting for brevity B = I, 
 
 (/> = \ sin Tex -~ cos kx \ cos nt ^k z R* cos kx sin nt ...... (6). 
 
 In this expression the term containing sinnt depends upon the 
 dissipation, and is the same as if there were no piston, while that 
 involving 8kR/37r represents the effect of the inertia of the external 
 air in the neighbourhood of the mouth. In order to compare with 
 previous results, let a be such that 
 
 SkR 
 sin KX - cos kx = sin K (x a) ; 
 
 O7T 
 
 then, the squares of small quantities being neglected, 
 
 and 
 
 (j> = sin k (x a) cos nt ^k 2 R 2 cos kx sin nt ............ (8). 
 
 These formulae shew that, if the dissipation be left out of account, 
 the velocity-potential is the same as if the tube were lengthened 
 
198 THEORY OF OPEN ENDS. [312. 
 
 by 8/3-7T of the radius, and the open end then behaved as a loop. 
 The amount of the correction agrees with what previous investi- 
 gations would have led us to expect as the result of the intro- 
 duction of the piston. We have seen reason to know that the 
 true value of a lies between \TrR and 8^/3?r, and that the presence 
 of the piston does not affect the term representing the dissipation. 
 But, before discussing our results, it will be advantageous to in- 
 vestigate them afresh by a rather different method, which besides 
 being of somewhat greater generality, will help to throw light on 
 the mechanics of the question. 
 
 313. For this purpose it will be convenient to shift the origin 
 in the negative direction to such a distance from the mouth that 
 the waves are there approximately plane, a displacement which 
 according to our suppositions need not amount to more than a 
 small fraction of the wave-length. The difficulty of the question 
 consists in finding the connection between the waves in the pipe, 
 which at a sufficient distance from the mouth are plane, and the 
 diverging waves outside, which at a moderate distance may be 
 treated as spherical. If the transition take place within a space 
 small compared with the wave-length, which it must evidently do, 
 if the diameter be small enough, the problem admits of solution, 
 whatever may be the form of the pipe in the neighbourhood of 
 the mouth. 
 
 Fig. 62. 
 
 At a point P, whose distance from A is moderate, the velocity- 
 potential is (279) 
 
 ty = 
 
 whence 
 
 Let us consider the behaviour of the mass of air included be- 
 tween the plane section at and a hemispherical surface whose 
 
313.] THEORY OF OPEN ENDS. 199 
 
 centre is A, and radius r, r being large in comparison with the 
 diameter of the pipe, but small in comparison with the wave- 
 length. Within this space the air must move approximately as an 
 incompressible fluid would do. Now the current across the hemi- 
 spherical surface 
 
 int ......... (3), 
 
 U> 
 
 if the square of kr be neglected. 
 
 If, as before, we take for the velocity-potential within the pipe 
 = ( A cos kx + B sin kx) e int ..................... (4), 
 
 we have for the current across the section at 0, 
 
 O" I TT 
 
 and thus 
 
 (6). 
 
 This is the first condition ; the second is to be found from the 
 consideration that the total current (whose two values have just 
 been equated) is proportional to the difference of potential at the 
 terminals. Thus, if c denote the conductivity of the passage be- 
 tween the terminal surfaces, 
 
 - 
 
 c r 
 
 On substituting for A' its value from (6), we have 
 
 P -ikr\ 
 
 c 2-Trr 
 
 In this expression the second term is negligible in comparison with 
 the first, for c is at most a quantity of the same order as the radius 
 of the tube, and when the mouth is much contracted it is smaller 
 still. Thus we may take 
 
 (-!+) 
 
 
 
 Substituting this in (4), we have for the imaginary expression of 
 the velocity-potential within the tube, if B be put equal to unity, 
 
 </> = jsin kx + crk ( --- h ~ J cos kx > e i 
 
200 CORRECTION TO LENGTH. [313. 
 
 or, if only the real part be retained, 
 
 < = -jsin kx cos kx\ cos nt -=- cos kx sin nt (9). 
 
 ( c } ZTT 
 
 Following Helm hoi tz, we may simplify our results by introducing 
 a quantity a defined by the equation 
 
 ka 
 
 tan ka. = (10). 
 
 c 
 
 Thus 
 
 6 = *-s cos nt x cos kx sin nt .. . . .(11), 
 
 cos ka. 2-7T 
 
 and the corresponding potential outside the mouth is 
 
 ^ 
 
 >s (lit kr) (1.2). 
 
 If R be the radius of the tube, we may replace cr by 
 
 When the tube is a simple cylinder, and the origin lies at a 
 distance AZ from the mouth, we know that crc~ l = AZ + pR, where 
 yw, is a number rather greater than JTT. In such a case (the origin 
 being taken sufficiently near the mouth) ka. is a small quantity, 
 and therefore from (10) 
 
 * = - = &l+pR (13). 
 
 c 
 
 At the same time cos ka. may be identified with unity. 
 The principal term in </>, involving cosnt, may then be calcu- 
 lated, as if the tube were prolonged, and there were a loop at a 
 point situated at a distance ^R beyond the actual position of the 
 mouth, in accordance with what we found before. These results, 
 approximate for ordinary tubes, become rigorous when the diameter 
 is reduced without limit, friction being neglected. 
 
 If there be no flange at A, the value of c is slightly modified 
 by the removal of what acts as an obstruction, but the principal 
 effect is on the term representing the dissipation. If we suppose 
 as an approximation that the waves diverging from A are 
 spherical, we must take for the current 4<7rr* d-fr/dr instead of 
 2?rr a dtyjdr. The ultimate effect of the alteration will be to halve 
 the expression for the velocity-potential outside the mouth, as well 
 as the corresponding second term in < (involving sin nt). The 
 amount of dissipation is thus seen to depend materially on the 
 degree in which the waves are free to diverge, and our analytical 
 expressions must not be regarded as more than rough estimates. 
 
313.] OPEN AND CLOSED PIPES. 201 
 
 The correct theory of the open organ-pipe, including equations 
 (11) and (12), was discovered by Helmholtz 1 , whose method, 
 however, differs considerably from that here adopted. The 
 earliest solutions of the problem by Lagrange, D. Bernoulli, and 
 Euler, were founded on the assumption that at an open end 
 the pressure could not vary from that of the surrounding atmo- 
 sphere, a principle which may perhaps even now be considered 
 applicable to an end whose openness is ideally perfect. The fact 
 that in all ordinary cases energy escapes is a proof that there is 
 not anywhere in the pipe an absolute loop, and it might have been 
 expected that the inertia of the air just outside the mouth would 
 have the effect of an increase in the length. The positions of the 
 nodes in a sounding pipe were investigated experimentally by 
 Savart 2 and Hopkins 3 , with the result that the interval between 
 the mouth and the nearest node is always less than the half of that 
 separating consecutive nodes. 
 
 [The correction necessary for an open end is the origin of a 
 departure from the simple law of octaves, which according to 
 elementary theory would connect the notes of closed and open pipes 
 of the same length. Thus in the application to an organ-pipe let 
 a.R denote the correction for the upper end when open, and I the 
 length of the pipe including the correction for the mouth at the 
 lower end. The whole effective length of the open pipe is then 
 I + OiR, while the effective length of the pipe if closed at the upper 
 end is I simply. The open pipe is practically the longer, and the 
 interval between the notes is less than the octave of the simple 
 theory 4 . 
 
 It may be worthy of remark that the correction, assumed to be 
 independent of wave-length, does not disturb the harmonic rela- 
 tions between the partial tones, whether a pipe be open or closed.] 
 
 314. Experimental determinations of the correction for an 
 open end have generally been made without the use of a flange, 
 and it therefore becomes important to form at any rate a rough 
 estimate of its effect. No theoretical solution of the problem of 
 an unflanged open end has hitherto been given, but it is easy to 
 
 1 Crelle, Bd. 57, p. 1. 1860. 
 
 2 Eecherches sur les vibrations de 1'air. Ann. d. Chim. t. xxiv. 1823. 
 
 3 Aerial vibrations in cylindrical tubes. Cambridge Transactions, Vol. v. p. 231. 
 1833. 
 
 4 Bosanquet, Phil. Mag. vi. p. 63, 1878. 
 
202 INFLUENCE OF FLANGE. [314. 
 
 see ( 79, 307) that the removal of the flange will reduce the 
 correction materially below the value '82 R (Appendix A). In the 
 absence of theory I have attempted to determine the influence 
 of a flange experimentally 1 . Two organ -pipes nearly enough in 
 unison with one another to give countable beats were blown from 
 an organ bellows ; the effect of the flange was deduced from the 
 difference in the frequencies of the beats according as one of the 
 pipes was flanged or not. The correction due to the flange was 
 about '2B. A (probably more trustworthy) repetition of this 
 experiment by Mr Bosanquet gave '25J2. If we subtract '22J? 
 from '82.R, we obtain '6R, which may be regarded as about the 
 probable value of the correction for an unflanged open end, on the 
 supposition that the wave-length is great in comparison with the 
 diameter of the pipe. 
 
 Attempts to determine the correction entirely from experiment 
 have not led hitherto to very precise results. Measurements by 
 Wertheim 2 on doubly open pipes gave as a mean (for each end) 
 "663 R, while for pipes open at one end only the mean result was 
 '746 R. In two careful experiments by Bosanquet 3 on doubly 
 open pipes the correction for one end was "635 R, when \ = 12R, 
 and "543 R, when X = 30 R. Bosanquet lays it down as a general 
 rule that the correction (expressed as a fraction of R) increases 
 with the ratio of diameter to wave-length ; part of this increase 
 may however be due to the mutual reaction of the ends, which 
 causes the plane of symmetry to behave like a rigid wall. When 
 the pipe is only moderately long in proportion to its diameter, a 
 state of things is approached which may be more nearly repre- 
 sented by the presence than by the absence of a flange. The 
 comparison of theory and observation on this subject is a matter 
 of some difficulty, because when the correction is small, its value, 
 as calculated from observation, is affected by uncertainties as to 
 absolute pitch and the velocity of sound, while for the case, when 
 the correction is relatively larger, which experiment is more com- 
 petent to deal with, there is at present no theory. Probably a more 
 accurate value of the correction could be obtained from a resonator 
 of the kind considered in 306, where the communication with 
 
 1 Phil. Mag. (5) in. 456. 1877. [The earliest experiments of the kind are 
 those of Gripon (Ann. d. Chim. in. p. 384, 1874) who shewed that the effect of a 
 large flange is proportional to the diameter of the pipe.] 
 
 2 Ann. d. Chim. (3) t. xxxi. p. 394, 1851. 
 
 3 Phil. Mag. (5) iv. p. 219. 1877. 
 
314.] EXPERIMENTAL METHODS. 203 
 
 the outside air is by a simple aperture ; the " length " is in that 
 case zero, and the " correction " is everything. Some measurements 
 of this kind, in which, however, no great accuracy was attempted, 
 will be found in my memoir on resonance 1 . 
 
 [Careful experimental determinations of the correction for an 
 unflanged open end have been made by Blaikley 2 , who employed 
 a vertical tube of thin brass 2'08 inches (5'3 cm.) in diameter. 
 The lower part of the tube was immersed in water, the surface of 
 which defined the " closed end," and the experiment consisted in 
 varying the degree of immersion until the resonance to a fork of 
 known pitch was a maximum. If the two shortest distances of 
 the water surface from the open end thus found be ^ and 1%, 
 (L li) represents the half wave-length, and the " correction for 
 the open end" is ^(1 2 h) /i. The following are the results 
 obtained by Blaikley, expressed as a fraction of the radius. They 
 relate to the same tube resounding to forks of various pitch. 
 
 c 253-68 -565 
 
 e 317-46 -595 
 
 g' 380-81 -564 
 
 W 444-72 -587 
 
 c" 507-45 -568 
 
 The mean correction is thus '576 R.] 
 
 Various methods have been used to determine the pitch of 
 resonators experimentally. Most frequently, perhaps, the resonators 
 have been made to speak after the manner of organ-pipes by a 
 stream of air blown obliquely across their mouths. Although good 
 results have been obtained in this way, our ignorance as to the 
 mode of action of the wind renders the method unsatisfactory. In 
 Bosanquet's experiments the pipes were not actually made to 
 speak, but short discontinuous jets of air were blown across the 
 open end, the pitch being estimated from the free vibrations as 
 the sound died away. A method, similar in principle, that I have 
 sometimes employed with advantage consists in exciting free vibra- 
 tions by means of a blow. In order to obtain as well defined a note 
 as possible, it is of importance to accommodate the hardness of the 
 substance with which the resonator comes into contact to the pitch, 
 
 1 Phil. Trans. 1871. See also Sondhauss, Pogg. Ann. t. 140, 53, 219 (1870), and 
 some remarks thereupon by myself (Phil. Mag., Sept. 1870). 
 
 2 Phil. Mag. vol. 7, p. 339, 1879. 
 
204 DISCUSSION OF MOTION [314. 
 
 a low pitch requiring a soft blow. Thus the pitch of a test-tube 
 may be determined in a moment by striking it against the bent 
 knee. 
 
 In using this method we ought not entirely to overlook the 
 fact that the natural pitch of a vibrating body is altered by a 
 term depending upon the square of the dissipation. With the 
 notation of 45, the frequency is diminished from n to 
 n(l ^k 2 n~~ 2 ), or if x be the number of vibrations executed while 
 the amplitude falls in the ratio e : 1, from n to 
 
 The correction, however, would rarely be worth taking into 
 account. 
 
 The measurements given in my memoir on resonance were 
 conducted upon a different principle by estimating the note of 
 maximum resonance. The ear was placed in communication with 
 the interior of the cavity, while the chromatic scale was sounded. 
 In this way it was found possible with a little practice to estimate 
 the pitch of a good resonator to about a quarter of a semitone. In 
 the case of small flasks with long necks, to which the above method 
 would not be applicable, it was found sufficient merely to hold the 
 flask near the vibrating wires of a pianoforte. The resonant note 
 announced itself by a quivering of the body of the flask, easily per- 
 ceptible by the fingers. In using this method it is important that 
 the mind should be free from bias in subdividing the interval 
 between two consecutive semitones. When the theoretical result 
 is known, it is almost impossible to arrive at an independent 
 opinion by experiment. 
 
 315. We will now, following Helmholtz, examine more closely 
 the nature of the motion within the pipe, represented by the 
 formula (11) 313. We have 
 
 0) .................... ....(1), 
 
 ro sin 2 k (x a) k*a- 
 where Z 2 = -+ . cos*kx ............... (2), 
 
 cos 2 ka. 4-7T- 
 
 Jc-cr cos ka. cos kx 
 - f . - ------- .................. (3). 
 
 2?r sin k(x a) 
 
315.] ORIGINATING WITHIN AN OPEN PIPE. 205 
 
 In the expression for Z 2 the second term is very small, and 
 therefore the maximum values of <f> occur very nearly when 
 
 k (x a) = ( m + $) TT, 
 or x = \m\ J A. a ..................... (4), 
 
 where m is a positive integer. 
 
 The distance between consecutive maxima is thus ^X, and the 
 value of the maximum is sec 2 ka. The minimum values of L 2 occur 
 approximately when k (x a) = mir, 
 or x = ^m\ a .......................... (5), 
 
 and their magnitude is given by 
 
 Z, 2 = -r - cos 2 kx = - cos 2 ka ............... (6). 
 
 4?r 2 4-Tr 2 
 
 In like manner, 
 
 X) ........................ (7), 
 
 where J^ +sin'fa ...... (8), 
 
 cos 2 ka 2 
 
 & 2 <r cos ka sin kas 
 tany = 7r -^, -- v ..................... (9). 
 
 * 2?r cos A; (# - a) 
 
 The maximum values of J 2 occur when 
 
 -x = %m\-oi ........................ (10), 
 
 and the minimum values, when 
 
 x = %m\ l\ a ....................... (11). 
 
 The approximate magnitude of the maximum is k* sec 2 ka., and 
 that of the minimum & 6 <r 2 cos 2 ka. -f- 4?r 2 . It appears that the 
 maxima of velocity occur in the same parts of the tube as the 
 minima of condensation (and rarefaction), and the minima of 
 velocity in the same places as the maxima of condensation. The 
 series of loops and nodes are arranged as if the first loop were at a 
 distance a beyond the mouth. 
 
 With regard to the phases, we see that both 6 and % are in 
 general small; and therefore with the exception of the places 
 where Z 2 and J 2 are near their minima the whole motion is 
 synchronous, as if there were no dissipation. 
 
 Hitherto we have considered the problem of the passage of 
 plane waves along the pipe and their gradual diffusion from the 
 mouth, without regard to the origin of the plane waves them- 
 
206 MOTION DUE TO [315. 
 
 selves. All that we have assumed is that the origin of the motion 
 is somewhere within the pipe. We will now suppose that the 
 motion is due to the known vibration of a piston, situated 
 at x l, the origin of co-ordinates being at the mouth. Thus, 
 when x l, 
 
 = cos .......................... (12), 
 
 and this must be made to correspond with the expression for the 
 plane waves, generalized by the introduction of arbitrary amplitude 
 and phase. 
 
 We may take 
 
 e - x ) ............... (13), 
 
 where J and ^ have the values given in (8), (9), while B and e are 
 arbitrary. Comparing (12) and (13) we conclude that 
 
 k^crcosko. sinkl 
 
 __ 
 
 cos*&a 4-7T 2 
 
 by which B and e are determined. 
 
 In accordance with (12) 313, the corresponding divergent 
 wave is represented by 
 
 
 
 cos (nt e kr) ............... (16). 
 
 If G be given, B is greatest, when cos k (I + a) = 0, that is 
 when the piston is situated at an approximate node. In that case 
 
 ...... ..(17), 
 
 shewing that the magnitude of the resulting vibration is very 
 great, though not infinite, since cos ka. cannot vanish. When 
 the mouth is much contracted, cos ka. may become small, but 
 in this case it is necessary that the adjustment of periods be 
 very exact in order that the first term of (15) may be negligible in 
 comparison with the second. In ordinary pipes cos ka. is nearly 
 equal to unity. 
 
 The minimum of vibration occurs when I is such that 
 
315.] EXTERNAL SOURCES. 207 
 
 cos k (I + a) = 1, that is, when the piston is situated at a loop. In 
 that case 
 
 5 = ^^ ......................... (18). 
 
 The vibration outside the tube is then, according to the value of 
 a, equal to or smaller than the vibration which there would be 
 if there were no tube and the vibrating plate were made part of 
 the yz plane. 
 
 316. Our equations may also be applied to the investigation 
 of the motion excited in a tube by external sources of sound. 
 Let us suppose in the first place that the mouth of the tube is 
 closed by a fixed plate forming part of the yz plane, and that the 
 potential due to the external sources (approximately constant 
 over the plate) is under these circumstances 
 
 ^r = H cos nt ........................... (1), 
 
 where ty is composed of the potential due to each source and its 
 image in the yz plane, as explained in 278. Inside the tube let 
 the potential be 
 
 <f> = H cos kx cos nt ..................... (2), 
 
 so that (f> and its differential coefficient are continuous across the 
 barrier. The physical meaning of this is simple. We imagine 
 within the tube such a motion as is determined by the conditions 
 that the velocity at the mouth is zero, and that the condensation 
 at the mouth is the same as that due to the sources of sound when 
 the mouth is closed. It is obvious that under these circumstances 
 the closing plate may be removed without any alteration in the 
 motion. Now, however, there is in general a finite velocity at 
 x = l ) and therefore we cannot suppose the pipe to be there 
 stopped. But when there happens to be a node at x = I, that is 
 to say when I is such that [sin kl\ = 0, all the conditions are 
 satisfied, and the actual motion within the pipe is that expressed 
 by (2) 1 . This motion is evidently the same as might obtain if the 
 pipe were closed at both ends; and in external space the potential 
 is the same as if the mouth of the pipe were closed with the rigid 
 plate. 
 
 In the general case in order to reduce the air at x = I to rest, 
 we must superpose on the motion represented by (2) another of 
 
 1 [An error, pointed out by Dr Burton, is here corrected.] 
 
208 ENLARGEMENT AT A CLOSED END. [316. 
 
 the kind investigated in 313, so determined as to give at x = I 
 a velocity equal and opposite to that of the first. Thus, if the 
 second motion be given by 
 
 d(f>/dx = BJ cos (nt e %), 
 we have e + % = 0, and 
 
 ! sin , kl = 
 
 When sin kl = 0, we have, as above explained, B = 0. The maxi- 
 mum value of B occurs when cos k (I + a) = 0, and then 
 
 
 It appears, as might have been expected, that the resonance is 
 greatest when the reduced length is an odd multiple of JX. 
 
 317. From the principle that in the neighbourhood of a node 
 the inertia of the air does not come much into play, we see that 
 in such places the form of a tube is of little consequence, and that 
 only the capacity need be attended to. This consideration allows 
 us to calculate the pitch of a pipe which is cylindrical through 
 most of its length (I), but near the closed end expands into a 
 bulb of small capacity ($). The reduced length is then evi- 
 dently 
 
 Z + a+Scr- 1 .............................. (1), 
 
 where a is the correction for the open end, and <r is the area of 
 the transverse section of the cylindrical part. This formula is 
 often useful, and may be applied also when the deviation from the 
 cylindrical form does not take the shape of an enlargement. 
 
 When the enlargement represented by S is too large to allow 
 of the above treatment, we may proceed as follows. The dissipa- 
 tion being neglected, the velocity-potential in the tube may be 
 taken to be 
 
 (f> = sin k (x a.) cos nt, 
 
 the origin being at the mouth, while a = \-rrR approximately. At 
 x = I, we have 
 
 </> = n sin k (I + a) sin nt, 
 
 and -J- = k cos k (I + a) cos nt. 
 
 i Helmholtz, CreUe, Bd. 57, 1860. 
 
317.] ABSORPTION OF SOUND BY RESONATORS. \ 209 
 
 Now the condensation is given by s = a~ 2 <, and the conditioi 
 to be satisfied at x I is 
 
 ds d<f> 
 
 if it be assumed that the condensation within S is sensibly 
 uniform. Thus 
 
 8 n 2 or 2 sin k (I + a) = ak cos k (I + a), 
 or, since n ak, 
 
 ........................... (3) 
 
 is the equation determining the pitch. Numerical examples of 
 the application of (3) are given in my memoir on resonance 
 (Phil. Trans. 1871, p. 117). 
 
 Similar reasoning proves that in any case of stationary vibra- 
 tions, for which the wave-length is several times as great as the 
 diameter of the bulb, the end of the tube adjoining the bulb 
 behaves approximately as an open end if kS be much greater 
 than cr, and as a stopped end if kS be much less than <r. 
 
 318. The action of a resonator when under the influence of a 
 source of sound in unison with itself is a point of considerable 
 delicacy and importance, and one on which there has been a 
 good deal of confusion among acoustical writers, the author not 
 excepted. 
 
 There are cases where a resonator absorbs sound, as it were 
 attracting the vibrations to itself and so diverting them from 
 regions where otherwise they would be felt. For example, 
 suppose that there is a simple source of sound B situated in a 
 narrow tube at a distance J\ (or any odd multiple thereof) from a 
 closed end, and not too near the mouth : then at any distant 
 external point A, its effect is nil. This is an immediate conse- 
 quence of the principle of reciprocity, because if A were the 
 source, there could be no variation of potential at B. The 
 restriction, precluding too great a proximity to the mouth, may 
 be dispensed with, if we suppose the source B to be diffused 
 uniformly over the cross section, instead of concentrated in one 
 point. Then, whatever may be the size and shape of the section, 
 there is absolutely no disturbance on the further side. This 
 is clear from the theory of vibrations in one dimension; the 
 R. II. 14 
 
210 
 
 QUINCKE S TUBES. 
 
 [318. 
 
 reciprocal form of the proposition that whatever sources of 
 disturbance may exist beyond the section, fftyda- = may be 
 proved from Helmholtz's formula (2) 293, by taking for </> the 
 velocity potential of the purely axial vibration of the same period. 
 
 It is scarcely necessary to say that, whenever no energy 
 is. emitted, the source does no work ; and this requires, not 
 that there shall be no variation of pressure at the source, for that 
 in the case of a simple source is impossible, but that the variable 
 part of the pressure shall have exactly the phase of the accelera- 
 tion, and no component with the phase of the velocity. 
 
 Other examples of the absorption of sound by resonators are 
 afforded by certain modifications of Herschel's interference tube 
 used by Quincke 1 to stop tones of definite pitch from reaching 
 the ear. 
 
 In the combinations of pipes represented in Fig. 63, the sound 
 enters freely at A ; at B it finds itself at the mouth of a resonator 
 of pitch identical with its own. Under these circumstances 
 it is absorbed, and there is no vibration propagated along BD. 
 It is clear that the cylindrical tube BC may be replaced by any 
 other resonator of the same pitch (7), without prejudice to the 
 action of the apparatus. The ordinary explanation by interference 
 (so called) of direct and reflected waves is then less applicable. 
 
 Fig. 63. 
 A 
 
 C\ I 
 
 These cases where the source is at the mouth of a resonator 
 must not be confused with others where the source is in the 
 interior. If B be a source at the bottom of a stopped tube whose 
 
 1 Pogg. Ann. cxxvm. 177. 1866. 
 
318.] RESONATOR CLOSE TO SOURCE. 211 
 
 reduced length is JX, the intensity at an external point A may 
 be vastly greater than if there had been no tube. In fact the 
 potential at A due to the source at B is the same as it would be 
 at B were the source at A. 
 
 319. For a closer examination of the mechanics of resonance, 
 we shall obtain the problem in a form disembarrassed of unne- 
 cessary difficulties by supposing the resonator to consist of a 
 small circular plate, backed by a spring, and imbedded in an 
 indefinite rigid plane. It was proved in a previous chapter, (30) 
 302, that if M be the mass of the plate, f its displacement, 
 /-tf the force of restitution, R the radius, and cr the density of the 
 air, the equation of vibration is 
 
 f ............ a), 
 
 where F and f are proportional to e ikat . 
 
 If the natural period of vibration (the reaction of external air 
 included) coincide with that imposed, the equation reduces to 
 
 %ao-7T&R*% = F ........................ (2). 
 
 Let us now suppose that F is due to an external source of 
 sound, giving when the plate is at rest a potential ^r , which will 
 be nearly constant over the area of the plate. Thus 
 
 F=-Sp.7rR* = ika<T.7rR 2 .^ .................. (3); 
 
 so that 7rR^ = X = 2i7rk- l = i\ () .................. (4), 
 
 and the potential </> due to the motion of the plate at a distance 
 r will be 
 
 Xe~ ikr i^e~ ikr . e~ ikr 
 * = ~2^ --/T~ = +Tkr ............... (5) ' 
 
 independent, it should be observed, of the area of the plate. 
 
 Leaving for the present the case of perfect isochronism, let us 
 suppose that 
 
 .................. (6), 
 
 so that 2-7T/&' is the wave-length of the natural note of the 
 resonator. If M' be written for M + fcrR 3 , the equation corre- 
 sponding to (5) takes the form 
 
 Q ikr 
 
 < t > = +'-ikr 
 
 [ 
 l - 2iM 
 
 142 
 
212 REINFORCEMENT OF SOUND [319. 
 
 from which we may infer as before that if k' = k the efficiency of 
 the resonator as a source is independent of R. When the adjust- 
 ment is imperfect, the law of falling off depends upon M'R~*. 
 Thus if M ' be great and R small, although the maximum efficiency 
 of the resonator is no less, a greater accuracy of adjustment is 
 required in order to approach the maximum ( 49). In the case 
 of resonators with simple apertures M' = ^ &R 3 , so that M'R~ 4 
 varies as R~\ Accordingly resonators with small apertures re- 
 quire the greatest precision of tuning, but the difference is not 
 important. From a comparison of the present investigation with 
 that of 311 it appears that the conditions of efficiency are 
 different according as internal or external effects are considered. 
 
 We will now return to the case of isochronism and suppose 
 further that the external source of sound to which the resonator 
 A responds, is the motion of a similar plate B, whose distance 
 c from A is a quantity large in comparison with the dimensions 
 of the plates. The intensity of B may be supposed to be such 
 that its potential is 
 
 ikr 
 
 Accordingly >/r = c~ l e~ ikc , and therefore by (5) 
 
 shewing that at equal distances from their sources 
 
 < : ^ = e~ ikc : ike ..................... (10). 
 
 The relation of phases may be represented by regarding the 
 induced vibration <f> as proceeding from B by way of A, and as 
 being subject to an additional retardation of JX, so that the whole 
 retardation between B and A is c + X. In respect of amplitude 
 <f> is greater than ->|r in the ratio of 1 : kc. 
 
 Thus when kc is small, the induced vibration is much the 
 greater, and the total sound is much louder than if A were not 
 permitted to operate. In this case the phase is retarded by a 
 quarter of a period. 
 
 It is important to have a clear idea of the cause of this 
 augmentation of sound. In a previous chapter ( 280) we saw 
 that, when A is fixed, B gives out much less sound than might 
 at first have been expected from the pressure developed. The 
 explanation was that the phase of the pressure was unfavourable ; 
 
319.] BY RESONATORS. 213 
 
 the larger part of it is concerned only in overcoming the inertia 
 of the surrounding air, and is ineffective towards the performance 
 of work. Now the pressure which sets A in motion is the whole 
 pressure, and not merely the insignificant part that would of itself 
 do work. The motion of A is determined by the condition that 
 that component of the whole pressure upon it, which has the phase 
 of the velocity, shall vanish. But of the pressure that is due to 
 the motion of A, the larger part has the phase of the acceleration ; 
 and therefore the prescribed condition requires an equality 
 between the small component of the pressure due to A's motion, 
 and a pressure comparable with the large component of the 
 pressure due to B's motion. The result is that A becomes a 
 much more powerful source than B. Of course no work is done 
 by the piston A ; its effect is to augment the work done at B, 
 by modifying the otherwise unfavourable relation between the 
 phases of the pressure and of the velocity. 
 
 The infinite plane in the preceding discussion is only required 
 in order that we may find room behind it for our machinery of 
 springs. If we are content with still more highly idealized 
 sources and resonators, we may dispense with it. To each piston 
 must be added a duplicate, vibrating in a similar manner, but in 
 the opposite direction, the effect of which will be to make the 
 normal velocity of the fluid vanish over the plane AB. Under 
 these circumstances the plane is without influence and may be 
 removed. If the size of the plates be reduced without limit they 
 become ultimately equivalent to simple sources of fluid ; and we 
 conclude that a simple source B will become more efficient than 
 before in the ratio of 1 : kc, when at a small distance c from 
 it there is allowed to operate a simple resonator (as we may call 
 it) of like pitch, that is, a source in which the inertia of the 
 immediately surrounding fluid is compensated by some adequate 
 machinery, and which is set in motion by external causes only. 
 
 In the present state of our knowledge of the mechanics of 
 vibrating fluids, while the difficulties of deduction are for the 
 most part still to be overcome, any simplification of conditions 
 which allows progress to be made, without wholly destroying the 
 practical character of the question, may be a step of great 
 importance. Such, for example, was the introduction by Helm- 
 holtz of the idea of a source concentrated in one point, represented 
 analytically by the violation at that point of the equation of 
 
214 RESONATOR AND DOUBLE SOURCE. [319. 
 
 continuity. Perhaps in like manner the idea of a simple reso- 
 nator may be useful, although the thing would be still more 
 impossible to construct than a simple source. 
 
 320. We have seen that there is a great augmentation of 
 sound, when a suitably tuned resonator is close to a simple 
 source. Much more is this the case, when the source of sound is 
 compound. The potential due to a double source is ( 294, 324) 
 
 If the resonator be at a small distance c, 
 
 -ikc 
 
 and therefore the potential due to the resonator at a distance r' is 
 
 e -ikc e -ikr' e -ikc e -ikr' 
 
 **&#' ............... (2) - 
 
 If fj, vanish, the resonator is without effect ; but when fjL = 1, 
 that is, when the resonator lies on the axis of the double source, 
 we have 
 
 ike aikr' 
 
 At a distance from the double source its potential is 
 
 Thus we may consider that the potential due to the resonator 
 is greater than that due to the double source in the ratio & 2 c 2 : 1, 
 the angular variation being disregarded. 
 
 A vibrating rigid sphere gives the same kind of motion to the 
 surrounding air as a double source situated at its centre ; but the 
 substitution suggested by this fact is only permissible when the 
 radius of the sphere is small in comparison with c : otherwise 
 the presence of the sphere modifies the action of the resonator. 
 Nevertheless the preceding investigation shews how powerful 
 in general the action of a resonator is when placed in a suitable 
 position close to a compound source of sound, whose character 
 is such that it would of itself produce but little effect at a 
 distance. 
 
320.] TWO OR MORE RESONATORS. 215 
 
 One of the best examples of this use of a resonator is afforded 
 by a vibrating bar of glass, or metal, held at the nodes. A strip 
 of plate glass about a foot [30 cm.] long and an inch [2*5 cm.] 
 broad, of medium thickness (say ^ inch ['32 cm.]), supported at 
 about 3 inches [7 '6 cm.] from the ends by means of string twisted 
 round it, answers the purpose very well. When struck by a 
 hammer it gives but little sound except overtones ; and even these 
 may almost be got rid of by choosing a hammer of suitable softness. 
 This deficiency of sound is a consequence of the small dimensions 
 of the bar in comparison with the wave-length, which allows of the 
 easy transference of air from one side to the other. If now the 
 mouth of a resonator of the right pitch 1 be held over one of the 
 free ends, a sound of considerable force and purity may be 
 obtained by a well-managed blow. In this way an improved 
 harmonicon may be constructed, with tones much lower than 
 would be practicable without resonators. In the ordinary instru- 
 ment the wave-lengths are sufficiently short to permit the bar to 
 communicate vibrations to the air independently. 
 
 The reinforcement of the sound of a bell in a well-known 
 experiment due to Savart 2 is an example of the same mode of 
 action ; but perhaps the most striking instance is in the ar- 
 rangement adopted by Helmholtz in his experiments requiring 
 pure tones, which are obtained by holding tuning-forks over the 
 mouths of resonators. 
 
 321. When two simple resonators A l} A%, separately in tune 
 with the source, are close together, the effect is less than if there 
 were only one. If the potentials due respectively to A lt A z be 
 
 $i> <l>'2> we may take 
 
 e -ar l e -ikr, 
 
 (f) l = A 1 -, <p2 = ^- 2 --- 
 
 '1 ~2 
 
 Let It represent the distance -4^4 2 , and i/r 1? ^ 2 , the potentials 
 that would exist at A lt A 2) if there were no resonators; then the 
 conditions to determine A l , A 2 are by (5) 319 
 
 1 To get the best effect, the mouth of the resonator ought to be pretty close to 
 the bar ; and then the pitch is decidedly lower than it would be in the open. The 
 final adjustment may be made by varying the amount of obstruction. This use of 
 resonators is of great antiquity. 
 
 2 Ann. d. Ghim. t. xxiv. 1823. 
 
216 FORMATION OF JETS [321. 
 
 By hypothesis ^ and ^ 2 are nearly equal, and therefore 
 
 Since ikR is small, the effect is much less than if there were 
 only one resonator. It must be observed however that the 
 diminished effectiveness is due to the resonators putting one 
 another out of tune, and if this tendency be compensated by an 
 alteration in the spring, any number of resonators near together 
 have just the effect of one. This point is illustrated by 302, 
 where it will be seen (32) that though the resonance does not 
 depend upon the size of the plate, still the inertia of the air, which 
 has to be compensated by a spring, does depend upon it. 
 
 322. It will be proper to say a few words in this place on 
 an objection, which has been brought forward by Bosanquet 1 as 
 possibly invalidating the usual calculations of the pitch of re- 
 sonators and of the correction to the length of organ-pipes. When 
 fluid flows in a steady stream through a hole in a thin plate, the 
 motion on the low pressure side is by no means of the character 
 investigated in 306. Instead of diverging after passing the hole 
 so as to follow the surface of the plate, the fluid shapes itself into 
 an approximately cylindrical jet, whose form for the case of two 
 dimensions can be calculated 2 from formulae given by Kirchhoff. 
 On the high pressure side the motion does not deviate so widely 
 from that determined by the electrical law. In like manner fluid 
 passing outwards from a pipe continues to move in a cylindrical 
 stream. If the external pressure be the greater, the character of 
 the motion is different. In this case the stream lines converge 
 from all directions to the mouth of the pipe, afterwards gathering 
 themselves into a parallel bundle, whose section is considerably 
 less than that of the pipe. It is clear that, if the formation of jets 
 took place to any considerable extent during the passage of air 
 through the mouths of resonators, our calculations of pitch would 
 have to be seriously modified. 
 
 The precise conditions under which jets are formed is a subject 
 of great delicacy. It may even be doubted whether they would occur 
 at all in frictionless fluid moving with velocities so small that the 
 corresponding pressures, which are proportional to the squares of 
 
 1 Phil. Mag. Vol. iv. p. 125, 1877. 
 
 2 Phil. Mag. Vol. n. p. 441, 1876. 
 
322.] DURING SONOROUS MOTION. 217 
 
 the velocities, are inconsiderable. But with air, as we actually 
 have it, moving under the action of the pressures to be found in 
 resonators, it must be admitted that jets may sometimes occur. 
 While experimenting about two years ago with one of Konig's 
 brass resonators of pitch c', I noticed that when the corresponding 
 fork, strongly excited, was held to the mouth, a wind of consider- 
 able force issued from the nipple at the opposite side. This effect 
 may rise to such intensity as to blow out a candle upon whose 
 wick the stream is directed. It does not depend upon any peculiar 
 motion of the air near the ends of the fork, as is proved by 
 mounting the fork upon its resonance-box and presenting the open 
 end of the box, instead of the fork itself, to the mouth of the 
 resonator, when the effect is obtained with but slightly diminished 
 intensity. A similar result was obtained with a fork and re- 
 sonator, of pitch an octave lower (c). Closer examination revealed 
 the fact that at the sides of the nipple the outward flowing 
 stream was replaced by one in the opposite direction, so that a 
 tongue of flame from a suitably placed candle appeared to enter 
 the nipple at the same time that another candle situated 
 immediately in front was blown away. The two effects are of 
 course in reality alternating, and only appear to be simultaneous 
 in consequence of the inability of the eye to follow such rapid 
 changes. The formation of jets must make a serious draft on the 
 energy of the motion, and this is no doubt the reason why it is 
 necessary to close the nipple in order to obtain a powerful sound 
 from a resonator of this form, when a suitably tuned fork is 
 presented to it. 
 
 At the same time it does not appear probable that jet forma- 
 tion occurs to any appreciable extent at the mouths of resonators 
 as ordinarily used. The near agreement between the observed and 
 the calculated pitch is almost a sufficient proof of this. Another 
 argument tending to the same conclusion may be drawn from the 
 persistence of the free vibrations of resonators (311), whose dura- 
 tion seems to exclude any important cause of dissipation beyond 
 the communication of motion to the surrounding air. 
 
 In the case of organ-pipes, where the vibrations are very 
 powerful, these arguments are less cogent, but I see no reason for 
 thinking that the motion at the upper open end differs greatly 
 from that supposed in Helmholtz's calculation. No conclusion to 
 the contrary can, I think, safely be drawn from the phenomena of 
 
218 MAINTENANCE OF SOUND [322. 
 
 steady motion. In the opposite extreme case of impulsive motion 
 jets certainly cannot be formed, as follows from Thomson's prin- 
 ciple of least energy ( 79), and it is doubtful to which extreme 
 the case of periodic motion may with greatest plausibility be 
 assimilated. Observation by the method of intermittent illumi- 
 nation ( 42) might lead to further information upon this subject. 
 
 322 a. As has already been mentioned, the free vibrations of the 
 body of air contained in a resonator may be excited by a suitable 
 blow delivered to the latter. The gas does not at first partake of the 
 sudden movement imposed upon the walls, and the relative motion 
 thus initiated is the origin of free vibrations of the kind considered 
 in preceding sections. When corks are drawn from partially 
 empty bottles, or when the lids are suddenly removed from 
 tubular pasteboard pencil-cases, free vibrations of the resonating 
 air columns are initiated in like manner. 
 
 If the vibrations are to be maintained with a view to the 
 emission of a continued sound, the vibrating body must be in 
 communication with a source of energy ( 68 a), and the reaction 
 between the two must be rightly accommodated with respect to 
 phase. The question whether the source of energy or the resona- 
 tor is to be regarded as the origin of the sound is of no particular 
 significance and will be variously answered according to the point 
 of view of the moment. In the organ the pipe, rather than the 
 compressed air within the bellows or even the escaping wind, is 
 regarded as the parent of the sound, but when a similar pipe 
 is maintained in action by a flame the credit of the joint perfor- 
 mance is usually given to the latter. 
 
 Up to this point the explanation of maintained vibrations is 
 simple enough ; but the complete theory in any particular case 
 demands such an investigation of the reaction as will determine 
 the phase relation. On this depends the whole question whether 
 the reaction is favourable or unfavourable to the continuance of 
 the vibrations, and the determination is often a matter of diffi- 
 culty. 
 
 Before proceeding to discuss the action of the blast it will be 
 desirable to say something further upon the organ-pipe considered 
 simply as a resonator. We have seen (314) how to take account 
 of an upper open end, but according to the rule of Cavaille-Coll 
 the whole addition which must be made to the measured length 
 
322 a.] IN FLUTE ORGAN-PIPES. 219 
 
 of an open pipe in order to bring about agreement with 
 the simple formula (8) 255 amounts to as much as 3J R, 
 very much greater than the correction (l'2jR) necessary for a 
 simple tube of circular section open at both ends. This dis- 
 crepancy is sometimes attributed to the blast. But it must be 
 remembered that the lower end is very much less open than the 
 upper end, and that if a sensible correction on account of deficient 
 openness is required for the latter, a much more important correc- 
 tion will probably be necessary for the former. Observations by 
 the author 1 have shewn that this is the case. A pipe fitted with 
 a sliding prolongation was tuned to maximum resonance with a 
 given (256) fork as in Blaikley's experiment ( 314). It was then 
 blown from a well-regulated bellows with measured pressures of 
 wind, and the pitch of the sounds so obtained was referred to that 
 of the fork by the method of beats ( 30). The results shewed 
 that at practical pressures the pitch of the pipe as sounded by 
 wind was higher than its natural note of maximum resonance ; so 
 that the considerable correction to the length found by Cavaille- 
 Coll is not attributable to the blast, but to the contracted 
 character of the lower end treated as open in the elementary 
 theory. In order to estimate the natural note an even larger 
 " correction to the length " would be required. 
 
 The rise of pitch due to the wind increases with pressure. 
 Thus in the case referred to above the pipe under a pressure of 
 T06 inches (2*7 cm.) of water gave a note about 2 vibrations per 
 second sharper than that of the fork, but when the wind pressure 
 was raised to 4'2 inches (10'7 cm.) the excess was as much as 11 
 vibrations per second. When the pressure was raised much 
 further, the pipe was " over blown " and gave the octave of its 
 proper pitch. This, of course, corresponds to another mode of 
 vibration of the aerial column. 
 
 It remains to consider the maintaining action of the blast. 
 The vibrations of a column of air may be encouraged either by 
 the introduction of fluid at a place where the density varies and 
 at a moment of condensation (and by the similar abstraction of 
 fluid at a moment of rarefaction), or by a suitable acceleration of 
 the parts of the column situated near a loop. Since the blast of 
 an organ acts at an open end of the pipe, it is clear that here we 
 
 1 Phil. Mag. m. p. 462, 1877 ; xm. p. 340, 1882. 
 
220 FLUTE ORGAN-PIPES. [322 a. 
 
 have to do with the latter alternative. The sheet of wind directed 
 across the lip of the pipe is easily deflected. When during the 
 vibration the external air tends to enter the pipe, it carries the jet 
 with it more or less completely. Half a period later when the 
 natural flow is outwards, the jet is deflected in the corresponding 
 direction. In either case the jet encourages the prevailing motion, 
 and thus renders possible the maintenance of the vibration. 
 
 For ready speech it is necessary that the sheet of wind be 
 accurately adjusted. But Schneebeli 1 has shewn that when the 
 vibration is once started there is more latitude. In an experi- 
 mental arrangement the jet was so adjusted as to pass entirely 
 outside the pipe. Under these circumstances there was failure 
 to speak until by a temporary strong blast directed upon it from 
 outside the jet was bent inwards to the proper position. The 
 pipe then spoke arid continued in action until by a pressure in the 
 reverse direction the jet was bent back. The motion of the jet 
 may be made apparent with the aid of smoke or by means of 
 a piece of tissue paper held so as to vibrate with it. Both 
 Schneebeli and H. Smith 2 insist upon a comparison between the 
 jet and the tongue of a reed organ- pipe, but the modes of action 
 appear to be essentially different. 
 
 The above view of the matter, which is that adopted by 
 v. Helmholtz in the fourth edition of his great work, appears to be 
 satisfactory as a general explanation of the maintenance of a 
 continued vibration, but it cannot be regarded as complete. In 
 matters of this kind practice is usually in advance of theory; 
 and many generations of practical men have brought the organ- 
 pipe to a high degree of excellence. 
 
 Another view that has been favourably entertained by many 
 good authorities regards the pipe as merely reinforcing by its 
 resonance a sound primarily due to the friction of the jet playing 
 against the lip, and there seems to be no doubt that sounds may 
 thus originate 3 . Perhaps after all there is less difference than 
 might at first appear between the two views, and the latter may 
 be especially appropriate when the initiation of the sound rather 
 than its maintenance is under consideration. A detailed discussion 
 
 1 Pogg. Ann. Bd. 153, p. 301, 1874. 
 
 2 Nature, 1873, 1874, 1875. 
 
 3 See for example Melde's Akustik, p. 252; Sondhauss Pogg. Ann. xci. p. 126, 
 1854. 
 
322 a.] OVERTONES. 221 
 
 of the question will be found in an essay by Van Schaik 1 . For a 
 fuller explanation we must probably await a better knowledge of 
 the mechanics of jets. 
 
 322 6. The character of the sound emitted from a pipe 
 depends upon the presence or absence of the various overtones, a 
 matter which requires further consideration. When a system 
 vibrates freely, the overtones may be harmonic or inharmonic 
 according to the nature of the system, and the composition of the 
 sound depends upon the initial circumstances. But in the case 
 of a maintained vibration like that now before us the motion 
 is strictly periodic, and the overtones must be harmonic if present 
 at all. The frequency of the whole vibration will correspond 
 approximately with that natural to the pipe in its gravest mode 2 , 
 but the agreement between the pitch of an audible overtone and 
 that of any free vibration may be much less close. The strength of 
 any overtone thus depends upon two things : first upon the extent 
 to which the maintaining forces possess a component of the right 
 kind, and secondly upon the degree of approximation between the 
 overtone and some natural tone of the vibrating body. In organ- 
 pipes the sharpness of the upper lip and the comparative thinness 
 of the sheet of wind are favourable to the production of overtones ; 
 so that in narrow open pipes v. Helmholtz was able to hear plainly 
 the first six partial tones. In wider open pipes, on the other hand, 
 the agreement between the overtones and the natural tones is less 
 close. In consequence, pipes of this class, especially if of wood, 
 give a softer quality of sound, in which besides the fundamental 
 only the octave and twelfth are to be detected 3 . 
 
 When a bottle ( 26), or a spherical resonator, is blown by 
 wind after the manner of an organ-pipe, there are no natural 
 tones in the neighbourhood of the harmonics, and the resulting 
 sound is almost free from overtones. 
 
 322 c. When two organ-pipes of the same pitch stand side 
 by side, complications ensue which not unfrequently give trouble 
 in practice. In extreme cases the pipes may almost reduce one 
 another to silence. Even when the mutual influence is more 
 moderate, it may still go so far as to cause the pipes to speak 
 
 1 Ueber die Tonerregung in LabiaJpfeifen. Eotterdam, 1891. 
 
 2 We are not now speaking of " over blowing." 
 
 3 Tonempfindungen. Fourth edition, p. 155, 1877. 
 
222 MUTUAL INFLUENCE [322 C. 
 
 in absolute unison, in spite of inevitable small natural differences. 
 The simplest case that can be considered is that of a pipe, along 
 the median plane of which a thin resisting wall is supposed to be 
 introduced. If this wall occupy the whole plane, the original 
 pipe is divided into two, independent of, and perfectly similar 
 to one another. And the pitch of these segments is the same 
 as that of the original pipe, fluid friction being neglected, since 
 during the vibrations of the latter there is no motion across 
 the median plane of symmetry. But the case is altered if the 
 wall be limited to the part of the plane included within the 
 pipe, for then the two vibrating columns are free to react upon 
 one another. The system as a whole has two degrees of freedom 
 we are not now regarding overtones and free vibrations are per- 
 formed in two distinct periods. The first of these is character- 
 ised by synchronism of phase between the vibrations of the com- 
 ponent columns, and the pitch is accordingly the same as before 
 the separation into two parts. But in the second mode the 
 phases of vibration of the component columns are opposed, so 
 that the air which escapes from one open end is absorbed by 
 the contiguous open end of the other part. In consequence the 
 "correction for the open ends" is much diminished in amount, 
 and the pitch in this mode is correspondingly raised. So long 
 as the motion is free, temporary vibrations in both modes may 
 co-exist, and would give rise to beats; but it does not follow 
 that both can be maintained by the blast. This would indeed 
 seem improbable beforehand, and experiment shews that after 
 the first moment the vibrations are confined to the second 
 mode. The contiguous open ends act as opposed sources, and 
 but little sound escapes, although within the pipes, and indeed 
 outside in the immediate neighbourhood of their mouths, the 
 vigour of the vibrations is unimpaired. Effects of the same 
 kind are produced when two distinct but similar pipes are 
 mounted side by side, and under the influence of the blast the 
 compound system may vibrate in one mode only, in spite of 
 small differences of pitch between the notes of the pipes when 
 sounding separately 1 . 
 
 322 d. Direct observation of the state of things within a 
 vibrating air column is of course a matter of great difficulty, but 
 
 1 Proceedings of the Musical Association, Dec. 1878. 
 

 322 d] OF TWO ORGAN-PIPES. 223 
 
 interesting results have been obtained by Topler and Boltzmann 1 , 
 calling to their aid the method of optical interference to meet the 
 difficulty arising out of the invisibility of air and the method of 
 stroboscopic vision to meet that arising out of the rapidity of the 
 changes. The upper end of an organ-pipe, closed by a thin plate 
 of metal, was provided with sides of worked glass projecting above 
 beyond the metal plate, and by suitable optical arrangements inter- 
 ference was produced between light which passed above and 
 below. The space above being occupied by air at normal density 
 and that below by air in a state of increased or diminished density 
 according to the phase at the moment, the interference bands 
 undergo displacements synchronous with the aerial vibration. 
 Observed directly these displacements would escape the eye ; but 
 by the aid of a fork electrically maintained and provided with 
 suitable slits ( 42) the light may be rendered intermittent in a 
 period nearly coincident with that of the vibration, and then the 
 sequence of changes becomes apparent. From the observed move- 
 ment of the bands it is possible to infer not merely the total 
 change of density from maximum to minimum, but the law of 
 the variation of density as a function of time. 
 
 When a pipe of large section was but moderately blown,. the 
 change of density at the node amounted to '009 of an atmosphere, 
 and the law was very nearly simple harmonic. Under a greater 
 pressure of wind the simple harmonic law was widely departed 
 from, the bands shifting themselves almost suddenly from one 
 extreme position to the other. In this case the amplitude of the 
 first overtone (the twelfth) was about one quarter of that of the 
 fundamental tone. The whole variation of density was "019 
 atmosphere. 
 
 322 e. In some experimental investigations a form of pipe 
 more completely symmetrical with respect to the axis has been 
 employed 2 . The lip is constituted by the entire circular edge of 
 the pipe as defined by a plane perpendicular to the axis, and upon 
 this an annular sheet of wind is brought to bear. A similar 
 arrangement is adopted in the ordinary steam whistle. 
 
 Another way of applying wind to evoke the speech of small 
 pipes has been experimented upon by Sondhauss 3 , and the rationale 
 
 1 Fogg. Ann. CXLI. p. 321, 1870. 
 
 2 Gripon, Ann. d. Chemie, in. p. 384, 1874. 
 
 3 Pogg. Ann. xci. p. 126, 1854. 
 
224 WHISTLING. [322 e. 
 
 is even less understood. A tube entirely open at one end is partially 
 closed at the other by a plate of wood or metal 2 or 3 mm. thick 
 and pierced by a cylindrical aperture with sharp edges (Fig. 63 a). 
 
 Fig. 63 a. 
 
 To set the pipe into action it is only necessary to insert the open 
 end into a reservoir of wind. For rough purposes when it is not 
 required to register the pressure nor to preserve a constant tempe- 
 rature, the mouth suffices, and the sound may be evoked either by 
 pressure or by suction. The cylindrical aperture may be replaced 
 by one of conical form, but in that case the wind must flow from 
 the narrower towards the wider end. The sounds tabulated by 
 Sondhauss vary from of to / 6 , corresponding in all cases to proper 
 tones of the tube. 
 
 The whistling sounds of the unaided mouth are evidently of 
 this class, the adjustment of pitch (from about c" to c 5 ) being 
 effected mainly by varying the internal capacity ( 304). The 
 formation of sound in whistling is sometimes said to be connected 
 with a vibration of the lips, but this appears to be a mistake. I 
 have found it possible to whistle through a suitable conical aperture 
 in a piece of box-wood held tightly between the lips. 
 
 The occurrence of vibration may be taken as evidence that 
 the steady flow of air through the passages in question is unstable. 
 Contrary to what occurs in the organ-pipe and in sensitive flames, 
 the deformations of the jet would seem here to be of the symme- 
 trical sort. There is perhaps a tendency alternately to follow and 
 to depart from the course marked out by the walls. 
 
 322 f. An important part of our present subject relates to 
 the maintenance of vibrations by means of heat, and it will be 
 possible to give at least a general account of the manner in which 
 the effect takes place. In almost all cases where heat is com- 
 municated to a body expansion ensues, and this expansion may be 
 made to do mechanical work. If the phases of the forces thus 
 operative be favourable, a vibration may be maintained. 
 
 An instructive example is afforded by Trevelyan's rocker, con- 
 
322 yi] TKEVELYAN'S ROCKER. 225 
 
 sisting of a mass of iron or copper, so shaped that during vibration 
 the weight is alternately carried on one or other of two adjacent 
 and parallel ridges. When the instrument is heated and placed 
 upon a block of cold lead, the vibrations persist so long as the heat 
 remains sufficient. " Sir John Leslie first suggested that the 
 cause of these vibrations is to be found in the expansion of the 
 cold block by the heat which flows into it from the hot metal 
 at the points of contact. Faraday \ Seebeck 2 , and Tyndall 3 have 
 adopted this explanation ; and they have shewn that most of the 
 facts that they and others have ascertained respecting these 
 vibrations are easily explained upon this view of their cause, 
 supposing only that the expansion is sufficiently great to produce 
 any sensible effect. Forbes 4 , on the other hand, after an extensive 
 series of experiments, was led to reject Sir John Leslie's ex- 
 planation, one of his principal reasons for doing so being the 
 impossibility, as it appeared to him, that the expansion occasioned 
 by so slow a process as the conduction of heat could produce any 
 sensible mechanical effect." 
 
 Davis, from whom 5 the above sentences are quoted, has 
 examined the question mathematically, and has shewn that the 
 explanation is adequate. It is evidently important that the 
 lower body should possess a high rate of expansibility with 
 temperature. In this respect lead stands high among the metals, 
 and rock salt, which Tyndall found to answer well, is even more 
 expansible. 
 
 The objection taken by Forbes may be met by the reply that 
 the conduction of heat is not a slow process when small distances 
 and masses are in question ; and the special repulsion invoked by 
 him as the basis of an alternative explanation would be of 
 unsuitable character in respect of phase. It is essential that the 
 phase of the force should be in arrear of the phase of the negative 
 displacement. 
 
 In an experiment due to Page 6 the vibrations are made 
 independent of an initial difference of temperature, the local 
 heating at the points of contact being obtained with the aid of an 
 
 1 Proc. of Roy. Inst. vol. n. p. 119, 1831. 
 
 2 Fogg. Ann. vol. LI. p. 1, 1840. 3 Phil. Mag. vol. vin. p. 1, 1854. 
 
 4 Phil. Mag. vol. iv. pp. 15, 182, 1834. 
 
 5 Phil. Mag. vol. XLV. p. 296, 1873. 
 
 6 Silliman's Journal, vol. ix. p. 105, 1850. 
 
 /^15 
 
226 MAINTENANCE BY HEAT [322 f. 
 
 electric current caused to pass from one body to the other. In 
 this arrangement there is no contraction in the upper body to be 
 deducted from the expansion in the lower. On a similar principle 
 Gore 1 has contrived a continuous motion of a copper ball which 
 travels upon circular rails themselves connected with a powerful 
 battery. 
 
 322^7. But the most interesting examples of vibrations 
 maintained by heat are those which occur when the resonating 
 body is gaseous. " If heat be periodically communicated to, and 
 abstracted from, a mass of air vibrating (for example) in a 
 cylinder bounded by a piston, the effect produced will depend 
 upon the phase of the vibration at which the transfer of heat 
 takes place. If heat be given to the air at the moment of greatest 
 condensation, or be taken from it at the moment of greatest 
 rarefaction, the vibration is encouraged. On the other hand, 
 if heat be given at the moment of greatest rarefaction, or 
 abstracted at the moment of greatest condensation, the vibration 
 is discouraged. The latter effect takes place of itself ( 247) 
 when the rapidity of alternation is neither very great nor very 
 small in consequence of radiation ; for when air is condensed 
 it becomes hotter, and communicates heat to surrounding bodies. 
 The two extreme cases are exceptional, though for different 
 reasons. In the first, which corresponds to the suppositions of 
 Laplace's theory of the propagation of sound, there is not 
 sufficient time for a sensible transfer to be effected. In the 
 second, the temperature remains nearly constant, and the loss of 
 heat occurs during the process of condensation, and not when the 
 condensation is effected. This case corresponds to Newton's 
 theory of the velocity of sound. When the transfer of heat takes 
 place at the moment of greatest condensation or of greatest 
 rarefaction, the pitch is not affected. 
 
 If the air be at its normal density at the moment when the 
 transfer of heat takes place, the vibration is neither encouraged 
 nor discouraged, but the pitch is altered. Thus the pitch is raised 
 if heat be communicated to the air a quarter period before the 
 phase of greatest condensation ; and the pitch is lowered if the 
 heat be communicated a quarter period after the phase of greatest 
 condensation. 
 
 1 Phil Mag. vol. xv. p. 519, 1858 ; vol. xvm. p. 94, 1859. 
 
322 g.~] OF AERIAL VIBRATIONS. 227 
 
 In general both kinds of effects are produced by a periodic 
 transfer of heat. The pitch is altered, and the vibrations are 
 either encouraged or discouraged. But there is no effect of the 
 second kind if the air concerned be at a loop, i.e. a place where 
 the density does not vary, nor if the communication of heat be the 
 same at any stage of rarefaction as at the corresponding stage of 
 condensation V 
 
 Thus in any problem which may present itself of the main- 
 tenance of a vibration by heat, the principal question to be 
 considered is the phase of the communication of heat relatively to 
 that of the vibration. 
 
 322 h. The sounds emitted by a jet of hydrogen burning in a 
 pipe open at both ends, were noticed soon after the discovery of 
 the gas, and have been the subject of several elaborate inquiries. 
 The fact that the notes are substantially the same as those which 
 may be elicited in other ways, e.g. by blowing, was announced by 
 Chladni. Faraday 2 proved that other gases were competent to 
 take the place of hydrogen, though not without disadvantage. 
 But it is to Sondhauss 3 that we owe the most detailed examina- 
 tion of the circumstances under which the sound is produced. 
 His experiments prove the importance of the part taken by the 
 column of gas in the tube which supplies the jet. For example, 
 sound cannot be got with a supply tube which is plugged with 
 cotton in the neighbourhood of the jet, although no difference can 
 be detected by the eye between the flame thus obtained and 
 others which are competent to excite sound. When the supply 
 tube is unobstructed, the sounds obtainable by varying the 
 resonator are limited as to pitch, often dividing themselves into 
 distinct groups. In the intervals between the groups no coaxing 
 will induce a maintained sound ; and it may be added that, for a 
 part of the interval at any rate, the influence of the flame is 
 inimical, so that a vibration started by a blow is damped more 
 rapidly than if the jet were not ignited. 
 
 Forms of resonator other than the open pipe may be employed, 
 and sometimes with advantage. Very low notes can be got from 
 spherical resonators, such as the large globes employed for demon- 
 
 1 Proc. Roy. Inst. vol. vui. p. 536, 1878 ; Naturk, vol. xvm. p. 319, 1878. 
 
 2 Quart. Journ. Sci. vol. v. p. 274, 1818. 
 
 3 Pogg. Ann. vol. cix. pp. 1, 426, 1860. 
 
 152 
 
228 SINGING FLAMES. [322 Jl. 
 
 strating the combustion of phosphorus in oxygen gas. A globe 
 of this kind gave in its natural condition a deep and pure tone of 
 64 vibrations per second. When it was fitted with a longer and 
 narrower neck formed from a pasteboard tube, the calculated 
 frequency fell to 25, and the vibrations, though vigorous enough to 
 extinguish the flame, were hardly audible. When it is desired to 
 excite very deep sounds, the supply tube should be made of 
 considerable length, and the orifice must not be much con- 
 tracted. 
 
 Singing flames may sometimes replace electrically maintained 
 tuning-forks for the production of pure tones, when absolute 
 constancy of pitch is not insisted upon. Inr^order to avoid 
 progressive deterioration of the air, it is advisable to use a 
 resonator open above as well as below. A bulbous chimney, 
 such as are often used with paraffin lamps, meets this require- 
 ment, and at the same time emits a pure tone. Or an otherwise 
 cylindrical pipe may be blocked in the middle by a loosely fitting 
 plug 1 . 
 
 As Wheatstone shewed, the intermittence of a singing flame is 
 easily made manifest by an oscillating, or a revolving, mirror. A 
 more minute examination is best effected by the stroboscopic 
 method, 42. Drawings of the transformations thus observed 
 have been given by Topler 2 , from which it appears that at one 
 phase the flame may withdraw itself entirely within the supply 
 tube. 
 
 Vibrations capable of being maintained are not always self- 
 starting. The initial impulse may be given by a blow ad- 
 ministered to the resonator, or by a gentle blast directed across 
 the mouth. In the striking experiments of Schaff'gotsch and 
 Tyndall 3 a flame, previously silent, responds to a sound in unison 
 with its own. In some cases the vibrations thus initiated rise to 
 such intensity as to extinguish the flame. 
 
 The experiments of Sondhauss shew that a relationship of 
 proportionality subsists between the lengths of the supply tubes 
 and of the sounding columns. When the nature of the gas is 
 varied, the same supply tube being retained, the mean lengths of 
 
 1 Phil. Mag. vol. vii. p. 149, 1879. 
 
 2 Pogg. Ann. vol. cxxvm. p. 126, 1866. 
 8 Sound, 3rd edition, p. 224, 1875. 
 
322 A.] LENGTH OF SUPPLY-PIPE. 229 
 
 the speaking columns are approximately as the square roots of the 
 density of the gas. A connection is thus established between the 
 natural note of a supply tube and the notes which can be sounded 
 with its aid. 
 
 Partly in consequence of the peculiar and ill understood be- 
 haviour of flames, and partly for other reasons, the thorough 
 explanation of the phenomena now under consideration is a matter 
 of some difficulty ; but there can be no doubt that they fall under 
 the head of vibrations maintained by heat, the heat being com- 
 municated periodically to the mass of air confined in the sounding 
 tube at a place where, in the course of a vibration, the pressure 
 varies. Although some authors have shewn a tendency to lay 
 stress upon the effects of the draught of air through the pipe, the 
 sounds, as we have seen, can be readily produced, not only when 
 there is no through draught, but even when the flame is so 
 situated that there is no sensible periodic motion of the air in its 
 neighbourhood. 
 
 In consequence of the variable pressure within the resonator, 
 the issue of gas, and therefore the development of heat, varies 
 during the vibration. The question is under what circumstances 
 the variation is of the kind necessary for the maintenance of the 
 vibration. If we were to suppose, as we might at first be inclined 
 to do, that the issue of gas is greatest when the pressure in the 
 resonator is least, and that the phase of greatest development of 
 heat coincides with that of the greatest issue of gas, we should 
 have the condition of things the most unfavourable of all to the 
 persistence of the vibration. It is not difficult, however, to see 
 that both suppositions are incorrect. In the supply tube (sup- 
 posed to be unplugged, and of not too small bore) stationary, or 
 approximately stationary, vibrations are excited, whose phase is 
 either the same or the opposite of that of the vibration in the 
 resonator. If the length of the supply tube from the burner to 
 the open end in the gas-generating flask be less than a quarter of 
 the wave-length in hydrogen of the actual vibration, the greatest 
 issue of gas precedes by a quarter period the phase of greatest 
 condensation ; so that, if the development of heat is retarded 
 somewhat in comparison with the issue of gas, a state of things 
 exists favourable to the maintenance of the sound. Some such 
 retardation is inevitable, because a jet of inflammable gas can 
 burn only at the outside ; but in many cases a still more potent 
 
230 SINGING FLAMES. [322 h. 
 
 cause may be found in the fact that during the retreat of the gas 
 in the supply tube small quantities of air may enter from the 
 interior of the resonator, whose expulsion must be effected before 
 the inflammable gas can again begin to escape. 
 
 If the length of the supply tube amounts to exactly one 
 quarter of the wave-length, the stationary vibration within it will 
 be of such a character that a node is formed at the burner, the 
 variable part of the pressure just inside the burner being the same 
 as in the interior of the resonator. Under these circumstances 
 there is nothing to make the flow of gas, or the development of 
 heat, variable, and therefore the vibration cannot be maintained. 
 This particular case is free from some of the difficulties which 
 attach themselves to the general problem, and the conclusion is in 
 accordance with Sondhauss' observations. 
 
 When the supply tube is somewhat longer than a quarter of 
 the wave, the motion of the gas is materially different from that 
 first described. Instead of preceding, the greatest outward flow 
 of gas follows at a quarter period interval the phase of greatest 
 condensation, and therefore if the development of heat be some- 
 what retarded, the whole effect is unfavourable. This state of 
 things continues to prevail, as the supply tube is lengthened, until 
 the length of half a wave is reached, after which the motion again 
 changes sign, so as to restore the possibility of maintenance. 
 Although the size of the flame and its position in the tube (or 
 neck of resonator) are not without influence, this sketch of the 
 theory is sufficient to explain the fact, formulated by Sondhauss, 
 that the principal element in the question is the length of the 
 supply tube. 
 
 322 i. Another phenomenon of the class now under considera- 
 tion occasionally obtrudes itself upon the notice of glass-blowers. 
 When a bulb about 2 cm. in diameter is blown at the end of a 
 somewhat narrow tube, 12 or 15 cm. long, a sound is sometimes 
 heard proceeding from the heated glass. For experimental pur- 
 poses it is well to use hard glass, which can afterwards be reheated 
 at pleasure without losing its shape. As was found by De la Rive, 
 the production of sound is facilitated by the presence of vapour, 
 especially of alcohol and ether. 
 
 It was proved by Sondhauss 1 that a vibration of the glass 
 1 Pogg. Ann. vol. LXXIX. p. 1, 1850. 
 
322 i.~\ SONDHAUSS' INVESTIGATIONS. 231 
 
 itself is no essential part of the phenomenon, and the same 
 indefatigable observer was very successful in discovering the con- 
 nection ( 303, 309) between the pitch of the note and the 
 dimensions of the apparatus. But no adequate explanation of the 
 production of sound was given. 
 
 For the sake of simplicity, a simple tube, hot at the closed end 
 and getting gradually cooler towards the open end, may be con- 
 sidered. At a quarter of a period before the phase of greatest 
 condensation (which occurs almost simultaneously at all parts of 
 the column) the air is moving inwards, i.e. towards the closed end, 
 and therefore is passing from colder to hotter parts of the tube ; 
 but the heat received at this moment (of normal density) has no 
 effect either in encouraging or discouraging the vibration. The 
 same would be true of the entire operation of the heat, if the 
 adjustment of temperature were instantaneous, so that there was 
 never any sensible difference between the temperatures of the air 
 and of the neighbouring parts of the tube. But in fact the 
 adjustment of temperature takes time, and thus the temperature 
 of the air deviates from that of the neighbouring parts of the 
 tube, inclining towards the temperature of that part of the tube 
 from which the air has just come. From this it follows that at 
 the phase of greatest condensation heat is received by 'the air, and 
 at the phase of greatest rarefaction heat is given up from it, and 
 thus there is a tendency to maintain the vibrations. It must not 
 be forgotten, however, that apart from transfer of heat altogether, 
 the condensed air is hotter than the rarefied air, and that in order 
 that the whole effect of heat may be on the side of encourage- 
 ment, it is necessary that previous to condensation the air should 
 pass not merely towards a hotter part of the tube, but towards a 
 part of the tube which is hotter than the air will be when it 
 arrives there. On this account a great range of temperature is 
 necessary for the maintenance of vibration, and even with a great 
 range the influence of the transfer of heat is necessarily unfavour- 
 able at the closed end, where the motion is very small. This is 
 probably the reason of the advantage of a bulb. It is obvious that 
 if the open end of the tube were heated, the effect of the transfer 
 of heat would be even more unfavourable than in the case of a 
 temperature uniform throughout. 
 
 322 j. The last example of the production of sound by heat 
 which we shall here consider is a very striking phenomenon 
 
232 SOUNDS DISCOVERED [322 j. 
 
 discovered by Rijke 1 . When a piece of fine metallic gauze, 
 stretching across the lower part of a tube open at both ends and 
 held vertically, is heated by a gas flame placed under it, a sound 
 of considerable power and lasting for several seconds is observed 
 almost immediately after the removal of the flame. Differing in 
 this respect from the case of sonorous flames ( 322), the genera- 
 tion of sound was found by Rijke to be closely connected with the 
 formation of a through draught impinging upon the heated gauze. 
 In this form of the experiment the heat is soon abstracted, and 
 then the sound ceases; but by keeping the gauze hot by the 
 current from a powerful galvanic battery Rijke was able to obtain 
 the prolongation of the sound for an indefinite period. 
 
 These notes may be obtained upon a large scale and form a 
 very effective lecture experiment. For this purpose a cast iron 
 pipe 5 feet (152cm.) long and 4f inches (12cm.) in diameter may 
 be employed. The gauze (iron wire) is of about 32 meshes to the 
 linear inch (2*54 cm.), and may advantageously be used in two 
 thicknesses. It may be moulded with a hammer on a circular 
 wooden block of somewhat smaller diameter than that of the pipe, 
 and will then retain its position in the pipe by friction. When it 
 is desired to produce the sound, the gauze caps are pushed up 
 the pipe to a distance of about a foot (30'5 cm.), and a gas flame 
 from a large rose burner is adjusted underneath, at such a level as 
 to heat the gauze to bright redness. For this purpose the ver- 
 tical tube of the lamp should be prolonged, if necessary, by an 
 additional length of brass tubing. When a good red heat is 
 attained, the flame is suddenly removed, either by withdrawing 
 the lamp or by stopping the supply of gas. In about a second 
 the sound begins, and presently rises to such intensity as to shake 
 the room, after which it gradually dies away. The whole duration 
 of the sound may be about 10 seconds 2 . 
 
 In discussing the question of maintenance in accordance with 
 the views already explained, we have to examine the character of 
 the variable communication of heat from the gauze to the air. 
 So far as the communication is affected directly by variations of 
 pressure or density, the influence is unfavourable, inasmuch as 
 the air will receive less heat from the gauze when its own tem- 
 perature is raised by condensation. The maintenance depends 
 
 1 Pogg. Ann. vol. cvn. p. 339, 1859; Phil. Mag. vol. xvn. p. 419, 1859. 
 
 2 Phil. Mag. vol. vii. p. 155, 1879. 
 
322 jj] BY BIJKE AND BOSSCHA. 233 
 
 upon the variable transfer of heat due to the varying motions of 
 the air through the gauze, this motion being compounded of a 
 uniform motion upwards with a motion, alternately upwards and 
 downwards, due to the vibration. In the lower half of the tube 
 these motions conspire a quarter period before the phase of greatest 
 condensation, and oppose one another a quarter period after that 
 phase. The rate of transfer of heat will depend mainly upon the 
 temperature of the air in contact with the gauze, being greatest 
 when that temperature is lowest. Perhaps the easiest way to 
 trace the mode of action is to begin with the case of a simple 
 vibration without a steady current. Under these circumstances 
 the whole of the air which comes in contact with the metal, in 
 the course of a complete period, becomes heated ; and after this 
 state of things is established, there is comparatively little further 
 transfer of heat. The effect of superposing a small steady up- 
 wards current is now easily recognized. At the limit of the 
 inwards motion, i.e. at the phase of greatest condensation, a small 
 quantity of air comes into contact with the metal, which has not 
 done so before, and is accordingly cool; and the heat communicated 
 to this quantity of air acts in the most favourable manner for the 
 maintenance of the vibration. 
 
 A quite different result ensues if the gauze be placed in the 
 upper half of the tube. In this case the fresh air will come into 
 the field at the moment of greatest rarefaction, when the commu- 
 nication of heat has an unfavourable instead of a favourable 
 effect. The principal note of the tube therefore cannot be 
 sounded. 
 
 A complementary phenomenon discovered by Bosscha 1 and 
 Biess 2 may be explained upon the same principles. If a current 
 of hot air impinge upon cold gauze, sound is produced; but in 
 order to obtain the principal note of the tube the gauze must be 
 in the upper, and not as before in the lower, half of the tube. In 
 an experiment due to Biess the sound is maintained indefinitely. 
 The upper part of a brass tube is kept cool by water contained in 
 a surrounding vessel, through the bottom of which the tube passes. 
 In this way the gauze remains comparatively cool, although 
 exposed to the heat of a gas flame situated an inch or two below 
 it. The experiment sometimes succeeds better when the draught 
 
 1 Pogg. Ann. vol. cvn. p. 342, 1859. 
 
 2 Pogg. Ann. vol. cvm. p. 653, 1859 ; cix. p. 145, 1860. 
 
234 MAINTENANCE OF SOUND [322 j. 
 
 is checked by a plate of wood placed somewhat closely over the 
 top of the tube. 
 
 Both in Rijke's and Riess' experiments the variable transfer of 
 heat depends upon the motion of vibration, while the effect of the 
 transfer depends upon the variation of pressure. The gauze must 
 therefore be placed where both effects are sensible, i.e. neither 
 near a node nor near a loop. About a quarter of the length of 
 the tube, from the lower or upper end, as the case may be, appears 
 to be the most favourable position 1 . 
 
 322k. It remains to consider briefly another class of main- 
 tained aerial vibrations where the maintenance is provided for by 
 the actual mechanical introduction of fluid, taking effect at a node 
 and near the phase of maximum condensation. Well-known 
 examples are afforded by such reed instruments as the clarinette, 
 and by the various wind instruments actuated directly by the lips. 
 The notes obtained are determined in the main by the aerial 
 columns, modified, it may be, to some extent by the maintaining 
 appliances. The reeds of the harmonium and organ come under a 
 different head. The pitch is there determined almost entirely by 
 the tongues themselves vibrating under their own elasticity* 
 resonating air columns being either altogether absent or playing 
 but a subordinate part. 
 
 In the instruments now under discussion the air column and 
 the wind-pipe are connected by a narrow aperture, which is opened 
 and closed periodically by a vibrating tongue. Tongues are 
 distinguished by v. Helmholtz as in-beating and out-beating. 
 In the first case the passage is opened when the tongue moves 
 inwards, i.e. against the wind, as happens in the clarinette. Lip 
 instruments, such as the trombone, belong to the second class, the 
 passage being open when the lips are moved outwards or with the 
 wind. 
 
 Let us consider the case of a cylindrical pipe, open at the 
 further end, in which the air vibrates at such a pitch as to make 
 the quarter wave-length equal to the length of the pipe. The end 
 of the column where the tongue is situated thus coincides with an 
 approximate node, and the aerial vibration can be maintained if 
 the passage is open at the moment of greatest condensation, so 
 
 1 Proc. Roy. Inst. vol. vm. p. 536, 1879 ; Nature, vol. xvni. p. 319, 1878. 
 
322 k.] IN REED INSTRUMENTS. 235 
 
 that air from the wind-pipe is then forcibly injected. The tongue 
 is maintained in motion by the variable pressure within the pipe, 
 and the phase of its motion will depend upon whether it is in- 
 beating or out-beating. In the latter case its phase is nearly the 
 opposite to that of the forces operative upon it, being open when 
 the pressure tending to close it is greatest. This is the state of 
 things in lip instruments, the lips being heavy in relation to the 
 rapidity of the vibrations actually performed, 46. When the 
 tongue is light and stiff, it must be made in-beating, as in the 
 clarinette, and its phase is then in approximate agreement with 
 the phase of the forces. A slight departure in the proper direction 
 from precise opposition or precise agreement of phase, as the case 
 may be, will allow of the communication of sufficient energy to 
 maintain the motion in spite of dissipative influences. A more 
 complete analytical statement of the circumstances has been 
 given by v. Helmholtz 1 , to whom the whole theory is due. 
 
 The character of the sounds from the various wind instru- 
 ments used in music differs greatly. Strongly contrasted qualities 
 are obtained from the trombone and the euphonion, the former 
 brilliant and piercing, and the latter mellow. Blaikley 2 has 
 analysed the sounds from a number of instruments, and has called 
 attention to various circumstances, such as the size of the bell- 
 mouth, and the shape of the cup applied to the lips, upon which 
 the differences probably depend. The pressures used in practice, 
 rising to 40 inches (102 cm.) of water in the case of the euphonion, 
 have been measured by Stone 3 . 
 
 1 Tonempfindungen, 4th edition, appendix vn. 
 
 2 Phil. Mag. vol. vi. p. 119, 1878. 3 Phil. Mag. vol. XLVIII. p. 113, 1874. 
 
CHAPTER XVII. 
 
 APPLICATIONS OF LAPLACE'S FUNCTIONS. 
 
 323. THE general equation of a velocity potential, when 
 referred to polar co-ordinates, takes the form ( 241) 
 
 e 
 
 dr smvd0\ dd J sm 2 dta 2 
 
 If k vanish, we have the equation of the ordinary potential, 
 which, as we know, is satisfied, if i]r = r n S n , where S n denotes the 
 spherical surface harmonic 1 of order n. On substitution it appears 
 that the equation satisfied by 8 n is 
 
 sm 6 dd \ dd J sin 2 6 
 
 Now, whatever the form of -fy may be, it can be expanded in 
 a series of spherical harmonics 
 
 ^ = ^0 + ^1 + ^2+ ...... +^n+ ............... (3), 
 
 where yjr n will satisfy an equation such as (2). 
 
 Comparing (1) and (2) we see that to determine ^r n as a 
 function of r, we have 
 
 + 2r * - ( + 1) ^ + r>+ n = ; 
 or, as it may also be written, 
 
 1 On the theory of these functions the latest English works are Todhunter's 
 The Functions of Laplace, Lame, and Bessel, and Ferrers' Spherical Harmonics. 
 
323.] SOLUTION IN LAPLACE'S FUNCTIONS. 237 
 
 In order to solve this equation, we may observe that when r 
 is very great, the middle term is relatively negligible, and that 
 then the solution is 
 
 rtyn^A^+Be-^ ..................... (5). 
 
 The same form may be assumed to hold good for the complete 
 equation (4), if we look upon A and B no longer as constants, but 
 as functions of r, whose nature is to be determined. Substituting 
 in (4), we find for B, 
 
 d*B dB nn + l 
 
 Let us assume 
 
 + B, (ikr)~* +...+B, (ikr)~ s + . . . (7), 
 
 and substitute in (6). Equating to zero the coefficient of (ikr)~ s ~ 2 , 
 we obtain 
 
 K(n + l)-a(g + l) ( n - s) (n + 8 + 1) ( . 
 
 2 + l 2 *-;-! 
 
 2( 
 
 Thus 
 
 2 ~ * ~ 2 2 2.4 
 
 so that 
 
 z> p , , 
 * 
 
 p 
 
 2 . 4 . 6 . 
 
 1.2.3...2ro _ 1 
 
 4 "' 
 
 Denoting with Prof. Stokes 1 the series within brackets by 
 
 f n (ikr), we have 
 
 B = B f n (ikr) ........................... (10). 
 
 In like manner by changing the sign of i, we get 
 
 A=A f n (-ikr) ........................ (11). 
 
 The symbols A and B Q , though independent of r, are functions 
 of the angular co-ordinates : in the most general case, they are 
 any two spherical surface harmonics of order n. Equation (5) may 
 therefore be written 
 
 S n 'e+^f n (- ikr) ......... (12). 
 
 1 -On the Communication of Vibrations from a Vibrating Body to a surrounding 
 Gas. Phil. Trans. 1868. 
 
238 EXPRESSION FOR RADIAL VELOCITY. [323. 
 
 By differentiation of (12) 
 
 Hr = ~ 5 e ~ ikr Fn (ikr) ~ e+ihr Fn ( ~ ikr) ~ - (13) ' 
 
 where 
 
 F n (ikr) = (1 + ikr)f n (tier) - ikrf n ' (ikr) ...... (14). 
 
 The forms of the functions F, as far as n = 7, are exhibited in 
 the accompanying table : 
 
 525? 
 
 F l (y)=y + 2 + 22T 1 
 
 y+ 4 + $y~ l + 9*r 2 
 y + 7 + 2?7/- 1 + 60?r 2 
 
 + 34020y~ 4 + 72765y- 5 + 
 F 7 (t/) = y + 29 + 434 y~ l + 4284 jr? + 29925 y~ s + 148995 y~ 4 + 509355 y~ 5 + 1081080 y- 6 
 + 1081080 y~ 7 
 
 In order to find the leading terms in F n (ikr) when ikr is small, 
 we have on reversing the series in (9) 
 
 f n (ikr) = 1.3.5 ... (2n- l)(ikr)- \l+ikr + ^\ (ikr)*+ ...I 
 
 I ZTi 1 J 
 
 .............................. (15), 
 
 whence by (14) we find 
 
 F n (ikr) = 1 . 3 . 5 . . . (2n - 1) (n + 1) (ikr)~ n 
 
 324. An important case of our general formulae occurs when 
 yfr represents a disturbance which is propagated wholly outwards. 
 At a great distance from the origin, f n (ikr) =f n (ikr) = l } and 
 thus, if we restore the time factor (e ikat ), we have 
 
 (1), 
 
 of which the second part represents a disturbance travelling 
 inwards. Under the circumstances contemplated we are there- 
 fore to take S n f = 0, and thus 
 
 ..................... (2), 
 
 which represents in the most general manner the n th harmonic 
 component of a disturbance of the given period diffusing itself 
 outwards into infinite space. 
 
324.] DIVERGENT WAVES. 239 
 
 The origin of the disturbance may be in a prescribed normal 
 motion of the surface of a sphere of radius c. Let us suppose 
 that at any point on the sphere the outward velocity is repre- 
 sented by Ue ikat , U being in general a function of the position of 
 the point considered. 
 
 If U be expanded in the spherical harmonic series 
 
 U=U + U,+ U,+ ... + U n + ............... (3), 
 
 we must have by (13) 323 
 
 F. y**A<*0 ..................... (4). 
 
 The complete value of ty is thus 
 
 where the summation is to be extended to all (integral) values of 
 n. The real part of this equation will give the velocity potential 
 due to the normal velocity U cos kat 1 at the surface of the 
 sphere r = c. 
 
 Prof. Stokes has applied this solution to the explanation of a 
 remarkable experiment by Leslie, according to which it appeared 
 that the sound of a bell vibrating in a partially exhausted receiver 
 is diminished by the introduction of hydrogen. This paradoxical 
 phenomenon has its origin in the augmented wave-length due to 
 the addition of hydrogen, in consequence of which the bell loses 
 its hold (so to speak) on the surrounding gas. The general expla- 
 nation cannot be better given than in the words of Prof. Stokes : 
 
 " Suppose a person to move his hand to and fro through a small 
 space. The motion which is occasioned in the air is almost exactly 
 the same as it would have been if the air had been an incompres- 
 sible fluid. There is a mere local reciprocating motion, in which 
 the air immediately in front is pushed forward, and that imme- 
 diately behind impelled after the moving body, while in the 
 anterior space generally the air recedes from the encroachment of 
 the moving body, and in the posterior space generally flows in 
 from all sides to supply the vacuum which tends to be created ; so 
 that in lateral directions the flow of the fluid is backwards, a 
 
 1 The assumption of a real value for U is equivalent to limiting the normal 
 velocity to be in the same phase all over the sphere r = c. To include the most 
 general aerial motion U would have to be treated as complex. 
 
240 FORMATION OF SONOROUS WAVES. [324. 
 
 portion of the excess of fluid in front going to supply the de- 
 ficiency behind. Now conceive the periodic time of the motion 
 to be continually diminished. Gradually the alternation of move- 
 ment becomes too rapid to permit of the full establishment of the 
 merely local reciprocating flow ; the air is sensibly compressed and 
 rarefied, and a sensible sound wave (or wave of the same nature, 
 in case the periodic time be beyond the limits suitable to hearing) 
 is propagated to a distance. The same takes place in any gas ; 
 and the more rapid be the propagation of condensations and rare- 
 factions in the gas, the more nearly will it approach, in relation to 
 the motions we have under consideration, to the condition of an 
 incompressible fluid ; the more nearly will the conditions of the 
 displacement of the gas at the surface of the solid be satisfied by a 
 merely local reciprocating flow." 
 
 In discussing the solution (5), Prof. Stokes goes on to say, 
 
 "At a great distance from the sphere the function f n (ikr) 1 be- 
 comes ultimately equal to 1, and we have 
 
 ^ = _ * 6 tt(-r+c, 2^-, . . (6). 
 
 r F n (ike) 
 
 " It appears (from the value of dty/dr) that the component of 
 the velocity along the radius vector is of the order r~ l , and that in 
 any direction perpendicular to the radius vector of the order r~ 2 , 
 so that the lateral motion may be disregarded except in the 
 neighbourhood of the sphere. 
 
 " In order to examine the influence of the lateral motion in the 
 neighbourhood of the sphere, let us compare the actual disturb- 
 ance at a great distance with what it would have been if all lateral 
 motion had been prevented, suppose by infinitely thin conical 
 partitions dividing the fluid into elementary canals, each bounded 
 by a conical surface having its vertex at the centre. 
 
 " On this supposition the motion in any canal would evidently 
 be the same as it would be in all directions if the sphere vibrated 
 by contraction and expansion of the surface, the same all round, 
 and such that the normal velocity of the surface was the same as 
 it is at the particular point at which the canal in question abuts 
 on the surface. Now if U were constant the expansion of U would 
 
 1 I have made some slight changes in Prof. Stokes' notation. 
 
324.1 EFFECT OF LATERAL MOTION. \ 241 
 
 J X^c 
 
 be reduced to its first term U , and seeing that f (ikr) = l, we 
 should have from (5), 
 
 r 2 U 
 
 ^ -- _ e ik (at-r+c) U <> 
 
 * r 6 F (ikc)' 
 
 This expression will apply to any particular canal if we take U to 
 denote the normal velocity at the sphere's surface for that particular 
 canal ; and therefore to obtain an expression applicable at once 
 to all the canals, we have merely to write U for U . To facilitate 
 a comparison with (5) and (6), I shall, however, write %U n for U. 
 We have then, 
 
 _ _ L ik(at-r+c) 
 
 F (ikc)~ 
 
 It must be remembered that this is merely an expression appli- 
 cable at once to all the canals, the motion in each of which takes 
 place wholly along the radius vector, and accordingly the expres- 
 sion is not to be differentiated with respect to 6 or co with the 
 view of finding the transverse velocities. 
 
 " On comparing (7) with the expression for the function yjr in 
 the actual motion at a great distance from the sphere (6), we see 
 that the two are identical with the exception that U n is divided 
 by two different constants, namely F (ikc) in the former case and 
 F n (ike) in the latter. The same will be true of the leading terms 
 (or those of the order r~ l ) in the expressions for the condensation 
 and velocity. Hence if the mode of vibration of the sphere be 
 such that the normal velocity of its surface is expressed by a 
 Laplace's function of any one order, the disturbance at a great 
 distance from the sphere will vary from one direction to another 
 according to the same law as if lateral motions had been pre- 
 vented, the amplitude of excursion at a given distance from the 
 centre varying in both cases as the amplitude of excursion, in a 
 normal direction, of the surface of the sphere itself. The only 
 difference is that expressed by the symbolic ratio F n (ikc) : F (ike). 
 If we suppose F n (ike) reduced to the form /j, n (cos a n + i sin a n ), 
 the amplitude of vibration in the actual case will be to that in the 
 supposed case as ^ to p n , and the phases in the two cases will 
 differ by a n . 
 
 " If the normal velocity of the surface of the sphere be not 
 expressible by a single Laplace's Function, but only by a series, 
 finite or infinite, of such functions, the disturbance at a given 
 R. ii. 16 
 
242 
 
 EFFECT OF LATERAL MOTION. 
 
 [324. 
 
 great distance from the centre will no longer vary from one direc- 
 tion to another according to the same law as the normal velocity 
 of the surface of the sphere, since the modulus p n and likewise 
 the amplitude ctn of the imaginary quantity F n (ike) vary with the 
 order of the function. 
 
 " Let us now suppose the disturbance expressed by a Laplace's 
 function of some one order, and seek the numerical value of the 
 alteration of intensity at a distance, produced by the lateral 
 motion which actually exists. 
 
 "The intensity will be measured by the vis viva produced in a 
 given time, and consequently will vary as the density multiplied 
 by the velocity of propagation multiplied by the square of the 
 amplitude of vibration. It is the last factor alone that is different 
 from what it would have been if there had been no lateral motion. 
 The amplitude is altered in the proportion of /* to p, n , so that if 
 pj : ^ 2 = I n , I n is the quantity by which the intensity that would 
 have existed if the fluid had been hindered from lateral motion 
 has to be divided. 
 
 " If X be the length of the sound-wave corresponding to the 
 period of the vibration, k = 2?r/X, so that kc is the ratio of the 
 circumference of the sphere to the length of a wave. If we sup- 
 pose the gas to be air and \ to be 2 feet, which would correspond 
 to about 550 vibrations in a second, and the circumference 2?rc to 
 be 1 foot (a size and pitch which would correspond with the case 
 of a common house-bell), we shall have kc = . The following 
 table gives the values of the squares of the modulus and of the 
 
 kc 
 
 w = 
 
 n = l 
 
 71=2 
 
 w = 3 
 
 71 = 4 
 
 
 4 
 
 17 
 
 16-25 
 
 14-879 
 
 13-848 
 
 20-177 
 
 g 
 
 2 
 
 5 
 
 5 
 
 9-3125 
 
 80 
 
 1495-8 
 
 3 
 (D 
 
 1 
 
 2 
 
 5 
 
 89 
 
 3965 
 
 300137 
 
 00 
 
 0-5 
 
 1-25 
 
 16-25 
 
 1330-2 
 
 236191 
 
 72086371 
 
 g, 
 
 0-25 
 
 1-0625 
 
 64-062 
 
 20878 
 
 14837899 
 
 18160 x 10 6 
 
 
 
 4 
 
 1 
 
 0-95588 
 
 0-87523 
 
 0-81459 
 
 1-1869 
 
 t 
 
 2 
 
 1 
 
 1 
 
 1-8625 
 
 16 
 
 299-16 
 
 
 1 
 
 1 
 
 2-5 
 
 44-5 
 
 1982-5 
 
 150068 
 
 OQ 
 
 0-5 
 
 1 
 
 13 
 
 1064-2 
 
 188953 
 
 57669097 
 
 
 0-25 
 
 1 
 
 60-294 
 
 19650 
 
 13965 x 10* 
 
 17092 x 10 6 
 
 S ^ 
 
 ratio I n for the functions F n (ike) of the first five orders, for each 
 of the values 4, 2, 1, , and J of kc. It will presently appear why 
 
324.] STOKES' INVESTIGATION. 243 
 
 the table has been extended further in the direction of values 
 greater than J than it has in the opposite direction. Five signi- 
 ficant figures at least are retained. 
 
 " When kc = oo we get from the analytical expressions I n = 1. 
 We see from the table that when kc is somewhat large I n is liable 
 to be a little less than 1, and consequently the sound to be a little 
 more intense than if lateral motion had been prevented. The 
 possibility of that is explained by considering that the waves of 
 condensation spreading from those compartments of the sphere 
 which at a given moment are vibrating positively, i.e. outwards, 
 after the lapse of a half period may have spread over the neigh- 
 bouring compartments, which are now in their turn vibrating 
 positively, so that these latter compartments in their outward 
 motion work against a somewhat greater pressure than if such 
 compartment had opposite to it only the vibration of the gas 
 which it had itself occasioned ; and the same explanation applies 
 mutatis 'mutandis to the waves of rarefaction. However, the in- 
 crease of sound thus occasioned by the existence of lateral motion 
 is but small in any case, whereas when kc is somewhat small I n 
 increases enormously, and the sound becomes a mere nothing 
 compared with what it would have been had lateral motion been 
 prevented. 
 
 "The higher be the order of the function, the greater will be the 
 number of compartments, alternately positive and negative as to 
 their mode of vibration at a given moment, into which the surface 
 of the sphere will be divided. We see from the table that for a 
 given periodic time as well as radius the value of I n becomes con- 
 siderable when n is somewhat high. However practically vibra- 
 tions of this kind are produced when the elastic sphere executes, 
 not its principal, but one of its subordinate vibrations, the pitch 
 corresponding to which rises with the order of vibration, so that k 
 increases with that order. It was for this reason that the table 
 was extended from kc = 0'5 further in the direction of high pitch 
 than low pitch, namely, to three octaves higher and only one octave 
 lower. 
 
 " When the sphere vibrates symmetrically about the centre, i.e. 
 so that any two opposite points of the surface are at a given 
 moment moving with equal velocities in opposite directions, or 
 more generally when the mode of vibration is such that there is 
 no change of position of the centre of gravity of the volume, there 
 
 162 
 
244 LESLIE'S EXPERIMENT. [324. 
 
 is no term of order 1. For a sphere vibrating in the manner of a 
 bell the principal vibration is that expressed by a term of the 
 order 2, to which I shall now more particularly attend. 
 
 " Putting, for shortness, k?c z = q, we have 
 tf = q + 1, ^ = fa* + 9?-*) 2 + (4 - 9<r7 = q - 2 + 9^ + 810-*, 
 
 _q 3 - 2q* + 9q 4- 81 
 
 ? 2 (<? + !) 
 " The minimum value of 7 2 is determined by 
 
 q * - 6^ _ 84^ - 54 = 0, 
 giving approximately, 
 
 2 = 12-859, kc = 3-586, /tf = 13-859, /i 2 2 =12'049, 
 / 2 = -86941 ; 
 
 so that the utmost increase of sound produced by lateral motion 
 amounts to about 15 per cent. 
 
 "I now come more particularly to Leslie's experiments. Nothing 
 is stated as to the form, size, or pitch of his bell; and even if these 
 had been accurately described, there would have been a good deal 
 of guess-work in fixing on the size of the sphere which should be 
 considered the best representative of the bell. Hence all we can 
 do is to choose such values for k and c as are comparable with the 
 probable conditions of the experiment. 
 
 " I possess a bell, belonging to an old bell-in-air apparatus, 
 which may probably be somewhat similar to that used by Leslie. 
 It is nearly hemispherical, the diameter is 1'96 inch, and the pitch 
 an octave above the middle c of a piano. Taking the number of 
 vibrations 1056 per second, and the velocity of sound in air 1100 
 feet per second, we have X = 12'5 inches. To represent the bell by 
 a sphere of the same radius would be very greatly to underrate the 
 influence of local circulation, since near the mouth the gas has but 
 a little way to get round from the outside to the inside or the 
 reverse. To represent it by a sphere of half the radius would still 
 apparently be to underrate the effect. Nevertheless for the sake 
 of rather under-estimating than exaggerating the influence of the 
 cause here investigated, I will make these two suppositions suc- 
 cessively, giving respectively c = '98 and c = '49, kc = '4926, and 
 kc = -2463 for air. 
 
324.] 
 
 NUMERICAL RESULTS. 
 
 245 
 
 " If it were not for lateral motion the intensity would vary from 
 gas to gas in the proportion of the density into the velocity of 
 propagation, and therefore as the pressure into the square root of 
 the density under a standard pressure, if we take the factor de- 
 pending on the development of heat as sensibly the same for the 
 gases and gaseous mixtures with which we have to deal. In the 
 following Table the first column gives the gas, the second the 
 
 
 So? 
 
 <o 
 
 t- 00 t~ 
 
 CO i-l O 
 
 9 9 
 
 
 
 O O >O i-H 
 
 8 
 
 O O3 CS <M 
 
 9^9^ 
 
 8 3 
 
 o o o 10 o 
 
 2 - 
 
 1 
 
246 DEFICIENCY OF TERM OF ZERO ORDER. [324. 
 
 pressure p, in atmospheres, the third the density D under the 
 pressure p, referred to the density of the air at the atmospheric 
 pressure as unity, the fourth, Q r , what would have been the inten- 
 sity had the motion been wholly radial, referred to the intensity 
 in air at atmospheric pressure as unity, or, in other words, a 
 quantity varying as p x (the density at pressure 1)*. Then follow 
 the values of q, 7 2 , and Q, the last being the actual intensity 
 referred to air as before. 
 
 "An inspection of the numbers contained in the columns headed 
 Q will shew that the cause here investigated is amply sufficient to 
 account for the facts mentioned by Leslie." 
 
 The importance of the subject, and the masterly manner in 
 which it has been treated by Prof. Stokes, will probably be thought 
 sufficient to justify this long quotation. The simplicity of the true 
 explanation contrasts remarkably with conjectures that had pre- 
 viously been advanced. Sir J. Herschel, for example, thought 
 that the mixture of two gases tending to propagate sound with 
 different velocities might produce a confusion resulting in a rapid 
 stifling of the sound. 
 
 [The subject now under consideration may be still more simply 
 illustrated by the problems of 268, 301. The former, for in- 
 stance, may be regarded as the extreme case of the present, in 
 which the spherical surface is reduced to a plane vibrating in 
 rectangular segments. If we suppose the size of these segments, 
 determined by p and q, to be given, and trace the effect of gradu- 
 ally increasing frequency, we see that it is only when the frequency 
 attains a certain value that sensible vibrations are propagated to 
 infinity, the law of diminution with distance being exponential 
 in its form. On the other hand vibrations whose frequency 
 exceeds the critical value are propagated without loss, escaping 
 the attenuation to which spherical waves must of necessity 
 submit.] 
 
 325. The term of zero order 
 
 where S is a complex constant, corresponds to the potential of a 
 simple source of arbitrary intensity and phase, situated at the 
 centre of the sphere ( 279). If, as often happens in practice, the 
 
325.] REACTION ON RIGID VIBRATING SPHERE. 247 
 
 source of sound be a solid body vibrating without much change of 
 volume, this term is relatively deficient. In the case of a rigid 
 sphere vibrating about a position of equilibrium, the deficiency is 
 absolute 1 , inasmuch as the whole motion will then be represented 
 by a term of order 1 ; and whenever the body is very small in 
 comparison with the wave-length, the term of zero order must 
 be insignificant. For if we integrate the equation of motion, 
 V 2 Ajr + & 2 <\Jr = 0, over the small volume included between the body 
 and a sphere closely surrounding it, we see that the whole quan- 
 tity of fluid which enters and leaves this space is small, and that 
 therefore there is but little total flow across the surface of the 
 sphere. 
 
 Putting n = 1, we get for the term of the first order 
 
 and $! is proportional to the cosine of the angle between the 
 direction considered and some fixed axis. This expression is of 
 the same form as the potential of a double source ( 294), situated 
 at the centre, and composed of two equal and opposite simple 
 sources lying on the axis in question, whose distance apart is 
 infinitely small, and intensities such that the product of the 
 intensities and distance is finite. For, if x be the axis, and the 
 cosine of the angle between x and r be /JL, it is evident that the 
 potential of the double source is proportional to 
 
 d fe~ ikr \ d fe~ ikr \ .. per** L 1 ) 
 
 =/A-J- [- -I -tfc- - -a +7^-4. 
 
 eto\r/ r ar\r/ r { ikr) 
 
 It appears then that the disturbance due to the vibration of a 
 sphere as a rigid body is the same as that corresponding to a 
 double source at the centre whose axis coincides with the line of 
 the sphere's vibration. 
 
 The reaction of the air on a small sphere vibrating as a rigid 
 body with a harmonic motion, may be readily calculated from 
 preceding formulae. If f denote the velocity of the sphere at 
 time t, 
 
 U ie *<* = & ........................... (3), 
 
 and therefore for the value of ty at the surface of the sphere, we 
 have from (5) 324, 
 
 The centre of the sphere being the origin of coordinates. 
 
248 INCREASE OF EFFECTIVE INERTIA. [325. 
 
 The force B due to aerial pressures accelerating the motion is 
 given by 
 
 = p jjfi-jrdS 
 
 If we write 
 
 then H = p.$7rpc s .% - qka.%7rpc 3 .% ............... (6), 
 
 inasmuch as f = ika %. 
 
 The operation of the air is therefore to increase the effective 
 inertia of the sphere by p times the inertia of the air displaced, 
 and to retard the motion by a force proportional to the velocity, 
 and equal to | Trpc 3 . qka%, these effects being in general functions 
 of the frequency of vibration. By introduction of the values of /i 
 and F l we find 
 
 ! (ike) 
 F,(ikc) 
 
 2 + & 2 c 2 k*c 3 
 
 ^ = 4TF?' *-4+S* .................. (8) ' 
 
 When kc is small, we have approximately p = ^, q = %&(?. 
 Hence the effective inertia of a small sphere is increased by one- 
 half of that of the air displaced a quantity independent of the 
 frequency and the same as if the fluid were incompressible. The 
 dissipative term, which corresponds to the energy emitted, is of 
 high order in kc, and therefore (the effects of viscosity being 
 disregarded) the vibrations of a small sphere are but slowly 
 damped. 
 
 The motion of an ellipsoid through an incompressible fluid has 
 been investigated by Green 1 , and his result is applicable to the 
 calculation of the increase of effective inertia due to a compressible 
 fluid, provided the dimensions of the body be small in comparison 
 with the wave-length of the vibration. For a small circular disc 
 vibrating at right angles to its plane, the increase of effective 
 inertia is to the mass of a sphere of fluid, whose radius is equal to 
 
 1 Edinburgh Transactions, Dec. 16, 1833. Also Green's Mathematical Papers, 
 edited by Ferrers. Macmillan & Co., 1871. 
 
325.] MULTIPLE SOURCES. 249 
 
 that of the disc, as 2 to TT. The result for the case of a sphere 
 given above was obtained by Poisson 1 , a short time before the 
 publication of Green's paper. 
 
 It has been proved by Maxwell 2 that the various terms of the 
 harmonic expansion of the common potential may be regarded as 
 due to multiple points of corresponding degrees of complexity. 
 
 Thus Vi is proportional to ^ ^ - =y- (-), where there are i 
 
 dh^dh* ...dhi \rj 
 
 differentiations of r~ l with respect to the axes h lt h 2 , &c., any 
 number of which may in particular cases coincide. It might 
 perhaps have been expected that a similar law would hold for the 
 velocity potential with the substitution of r~ l e~ ikr for r~\ This 
 however is not the case ; it may be shewn that the potential of a 
 
 quadruple source, denoted by -^, . -- , corresponds in general 
 
 Qikr 
 
 not to the term of the second order simply, viz., S 2 f 2 (ikr), 
 
 but to a combination of this with a term of zero order. The 
 analogy therefore holds only in the single instance of the double 
 point or source, though of course the function r~ l e~ ikr after any 
 number of differentiations continues to satisfy the fundamental 
 equation 
 
 It is perhaps worth notice that the disturbance outside any 
 imaginary sphere which completely encloses the origin of sound 
 may be represented as due to the normal motion of the surface of 
 any smaller concentric sphere, or, as a particular case when the 
 radius of the sphere is infinitely small, as due to a source concen- 
 trated in one point at the centre. This source will in general be 
 composed of a combination of multiple sources of all orders of 
 complexity. 
 
 326. When the origin of the disturbance is the vibration of a 
 rigid body parallel to its axis of revolution, the various spherical 
 harmonics S n reduce to simple multiples of the zonal harmonic 
 P n (IJL), which may be defined as the coefficient of e n in the expan- 
 sion of {1 2e/4-f e 2 }~* in rising powers of e. [For the forms of 
 these functions see 334.] And whenever the solid, besides being 
 
 1 Memoir es de V Academic des Sciences, Tom. xi. p. 521. 
 
 2 Maxwell's Electricity and Magnetism, Ch. ix. 
 
250 CASES OF SYMMETRY. [326. 
 
 symmetrical about an axis, is also symmetrical with respect to an 
 equatorial plane (whose intersection with the axis is taken as 
 origin of co-ordinates), the expansion of the resulting disturbance 
 in spherical harmonics will contain terms of odd order only. For. 
 example, if the vibrating body were a circular disc moving perpen- 
 dicularly to its plane, the expansion of -v/r would contain terms 
 proportional to P l (/*), P 3 (/u,), P 5 (/*), &c. In the case of the sphere, 
 as we have seen, the series reduces absolutely to its first term, and 
 this term will generally be preponderant. 
 
 On the other hand we may have a vibrating system symmetri- 
 cal about an axis and with respect to an equatorial plane, but in 
 such a manner that the motions of the parts on the two sides of 
 the plane are opposed. Under this head comes the ideal tuning- 
 fork, composed of equal spheres or parallel circular discs, whose 
 distance apart varies periodically. Symmetry shews that the 
 velocity-potential, being the same at any point and at its image in 
 the plane of symmetry, must be an even function of //., and there- 
 fore expressible by a series containing only the even functions 
 PO(P), Pz(fJ>), &c. The second function P. 2 (fJ<) would usually 
 preponderate, though in particular cases, as for example if the 
 body were composed of two discs very close together in comparison 
 with their diameter, the symmetrical term of zero order might 
 become important. A comparison with the known solution for the 
 sphere whose surface vibrates according to any law, will in most 
 cases furnish material for an estimate as to the relative importance 
 of the various terms. 
 
 [The accompanying table, p. 251, giving P n as a function of 
 6, or cos" 1 fji, is abbreviated from that of Perry 1 .] 
 
 327. The total emission of energy by a vibrating sphere is 
 found by multiplying the variable part of the pressure (proportional 
 to ijr) by the normal velocity and integrating over the surface 
 ( 245). In virtue of the conjugate property the various spherical 
 harmonic terms may be taken separately without loss of generality. 
 We have (323) 
 
 .Cf p ik(at-r) \ 
 
 * = *' -- fn(ikr)\ 
 
 Phil. Mag. vol. xxxn., p. 516, 1891. 
 
327.] 
 
 TABLE OF ZONAL HARMONICS. 
 
 251 
 
 Table of Zonal Spherical Harmonics. 
 
 e 
 
 PI 
 
 P 2 
 
 P 3 
 
 p 
 
 P 5 
 
 P 6 
 
 *7 
 
 o 
 
 
 1-0000 
 
 1-0000 
 
 1-0000 
 
 1-0000 
 
 1-0000 
 
 1-0000 
 
 1-0000 
 
 2 
 
 9994 
 
 9982 
 
 9963 
 
 9939 
 
 9909 
 
 9872 
 
 9829 
 
 4 
 
 9976 
 
 9927 
 
 9854 
 
 9758 
 
 9638 
 
 9495 
 
 9329 
 
 6 
 
 9945 
 
 9836 
 
 9674 
 
 9459 
 
 9194 
 
 8881 
 
 8522 
 
 8 
 
 9903 
 
 9709 
 
 9423 
 
 9048 
 
 8589 
 
 8053 
 
 7448 
 
 10 
 
 9848 
 
 9548 
 
 9106 
 
 8532 
 
 7840 
 
 7045 
 
 6164 
 
 12 
 
 9781 
 
 9352 
 
 8724 
 
 7920 
 
 6966 
 
 5892 
 
 4732 
 
 14 
 
 9703 
 
 9122 
 
 8283 
 
 7224 
 
 5990 
 
 4635 
 
 3219 
 
 16 
 
 9613 
 
 8860 
 
 7787 
 
 6454 
 
 4937 
 
 3322 
 
 1699 
 
 18 
 
 9511 
 
 8568 
 
 7240 
 
 5624 
 
 3836 
 
 2002 
 
 0289 
 
 20 
 
 9397 
 
 8245 
 
 6649 
 
 4750 
 
 2715 
 
 0719 
 
 - -1072 
 
 22 
 
 9272 
 
 7895 
 
 6019 
 
 3845 
 
 1602 
 
 - -0481 
 
 - -2201 
 
 24 
 
 9135 
 
 7518 
 
 5357 
 
 2926 
 
 0525 
 
 - -1559 
 
 - -3095 
 
 26 
 
 8988 
 
 7117 
 
 4670 
 
 2007 
 
 - -0489 
 
 - -2478 
 
 -3717 
 
 28 
 
 8829 
 
 6694 
 
 3964 
 
 1105 
 
 -1415 
 
 - -3211 
 
 - -4052 
 
 30 
 
 8660 
 
 6250 
 
 3248 
 
 0234 
 
 - -2233 
 
 -3740 
 
 - -4101 
 
 32 
 
 8480 
 
 5788 
 
 2527 
 
 - -0591 
 
 - -2923 
 
 - -4052 
 
 - -3876 
 
 34 
 
 8290 
 
 5310 
 
 1809 
 
 -1357 
 
 - -3473 
 
 - -4148 
 
 - -3409 
 
 36 
 
 8090 
 
 4818 
 
 1102 
 
 - -2052 
 
 - -3871 
 
 - -4031 
 
 - -2738 
 
 38 
 
 7880 
 
 4314 
 
 0413 
 
 - -2666 
 
 -4112 
 
 -3719 
 
 -1918 
 
 40 
 
 7660 
 
 3802 
 
 - -0252 
 
 - -3190 
 
 - -4197 
 
 - -3234 
 
 - -1003 
 
 42 
 
 7431 
 
 3284 
 
 - -0887 
 
 - -3616 
 
 - -4128 
 
 - -2611 
 
 - -0065 
 
 44 
 
 7193 
 
 2762 
 
 - -1485 
 
 - -3940 
 
 - -3914 
 
 - -1878 
 
 0846 
 
 46 
 
 6947 
 
 2238 
 
 - -2040 
 
 - -4158 
 
 - -3568 
 
 - -1079 
 
 1666 
 
 48 
 
 6691 
 
 1716 
 
 - -2547 
 
 - -4270 
 
 - -3105 
 
 - -0251 
 
 2349 
 
 50 
 
 6428 
 
 1198 
 
 - -3002 
 
 - -4275 
 
 - -2545 
 
 + -0563 
 
 2854 
 
 52 
 
 6157 
 
 0686 
 
 - -3401 
 
 - -4178 
 
 -1910 
 
 + 1326 
 
 3153 
 
 54 
 
 5878 
 
 0182 
 
 - -3740 
 
 - -3984 
 
 - -1223 
 
 + 2002 
 
 3234 
 
 56 
 
 5592 
 
 - -0310 
 
 - -4016 
 
 - -3698 
 
 - -0510 
 
 + 2559 
 
 3095 
 
 58 
 
 5299 
 
 - -0788 
 
 - -4229 
 
 - -3331 
 
 0206 
 
 + -2976 
 
 2752 
 
 60 
 
 5000 
 
 - -1250 
 
 - -4375 
 
 - -2891 
 
 0898 
 
 + 3232 
 
 2231 
 
 62 
 
 4695 
 
 - -1694 
 
 - -4455 
 
 - -2390 
 
 1545 
 
 + 3321 
 
 1571 
 
 64 
 
 4384 
 
 - -2117 
 
 - -4470 
 
 - -1841 
 
 2123 
 
 + -3240 
 
 0818 
 
 66 
 
 4067 
 
 - -2518 
 
 - -4419 
 
 - -1256 
 
 2615 
 
 + 2996 
 
 0021 
 
 68 
 
 3746 
 
 - -2896 
 
 - -4305 
 
 - -0650 
 
 3005 
 
 + 2605 
 
 - -0763 
 
 70 
 
 3420 
 
 - -3245 
 
 - -4130 
 
 - -0038 
 
 3281 
 
 + -2089 
 
 - -1485 
 
 72 
 
 3090 
 
 - -3568 
 
 - -3898 
 
 0568 
 
 3434 
 
 + 1472 
 
 - -2099 
 
 74 
 
 2756 
 
 - -3860 
 
 -3611 
 
 1153 
 
 3461 
 
 + -0795 
 
 - -2559 
 
 76 
 
 2419 
 
 - -4112 
 
 -3275 
 
 1705 
 
 3362 
 
 + 0076 
 
 - -2848 
 
 78 
 
 2079 
 
 - -4352 
 
 - -2894 
 
 2211 
 
 3143 
 
 - -0644 
 
 - -2943 
 
 80 
 
 1736 
 
 - -4548 
 
 - -2474 
 
 2659 
 
 2810 
 
 -1321 
 
 - -2835 
 
 82 
 
 1392 
 
 - -4709 
 
 - -2020 
 
 3040 
 
 2378 
 
 - -1926 
 
 - -2536 
 
 84 
 
 1045 
 
 - -4836 
 
 - -1539 
 
 3345 
 
 1861 
 
 - -2431 
 
 - -2067 
 
 86 
 
 0698 
 
 - -4927 
 
 - -1038 
 
 3569 
 
 1278 
 
 - -2811 
 
 - -1460 
 
 88 
 
 0349 
 
 - -4982 
 
 - -0522 
 
 3704 
 
 0651 
 
 - -3045 
 
 - -0735 
 
 90 
 
 0000 
 
 - -5000 
 
 - -0000 
 
 3750 
 
 0000 
 
 - -3125 
 
 0000 
 
252 ENERGY EMITTED [327. 
 
 or on rejecting the imaginary part 
 
 - [ff cos & ( at -r) + a sin k (at r)} 
 
 ...... (2), 
 
 where F=aL + i/3, f=a! + iff .................. (3). 
 
 THUS 
 
 cos 2 & (a - r) - a'/3 sin 2 A; (a - r) 
 
 + (aaf ftff) sin k (at r) cos k (at ?)}. 
 
 When this is integrated over a long range of time, the periodic 
 terms may be omitted, and thus 
 
 ^] n dS.dt = ^( a /3'-a'/3) UMr (4). 
 
 (JLi 1(* J J 
 
 Now, since there can be on the whole no accumulation of 
 energy in the space included between two concentric spherical 
 surfaces, the rates of transmission of energy across these surfaces 
 must be the same, that is to say r" 1 (a'/3 ffa) must be independent 
 of r. In order to determine the constant value, we may take the 
 particular case of r indefinitely great, when 
 
 F n (ikr) = ikr, a = 0, ff = kr, 
 f n (ikr) = l, a' = l, ' = 0. 
 Thus a'/3 - ffa. = kr, identically (5). 
 
 It may be observed that the left-hand member of (5) when 
 multiplied by i is the imaginary part of (a + iff) (a' iff) or of 
 F n (ikr)f n ( ikr), so that our result may be expressed by saying 
 that the imaginary part of F n (ikr) f n ( ikr) is ikr, or 
 
 F n (ikr) f n (- ikr) - F n (- ikr) f n (ikr) = 2ikr (6). 
 
 In this form we shall have occasion presently to make use of it. 
 
 The same conclusion may be arrived at somewhat more directly 
 by an application of Helmholtz's theorem ( 294), i.e. that if two 
 functions u and v satisfy through a closed space S the equation 
 (V 2 + k 2 ) u = 0, then 
 
 dv c 
 
327.] FROM A VIBRATING SPHERICAL SURFACE. 253 
 
 If we take for S the space between two concentric spheres, 
 
 making 
 
 we find that r~ l {F n (ikr)f n (-ikr) - F n (-ikr)f n (ikr)} must be 
 independent of r. 
 
 We have therefore 
 
 so that the expression for the energy emitted in time t is (since 
 
 2 d(T ..................... (8). 
 
 It will be more instructive to exhibit W as a function of the 
 normal motion at the surface of a sphere of radius c. From (2) 
 
 7 i O 
 
 -^ W = -- " [cos kat (a cos kc + ft sin kc) 
 
 + sin kat (a. sin kc ft cos kc)], 
 
 so that, if the amplitude of d^ n /dr be U n , we have as the relation 
 between S n and U n 
 
 ..................... (9). 
 
 Thus F-Gr.'*r .................. (10). 
 
 This formula may be verified for the particular cases n = and 
 71 = 1, treated in 280, 325 respectively. 
 
 328. If the source of disturbance be a normal motion of a 
 small part of the surface of the sphere (r = c) in the immediate 
 neighbourhood of the point /JL 1, we must take in the general 
 solution applicable to divergent waves, viz. 
 
 -I 
 
 U n = !(2 + 1) P n O) . l7P B ( A t) dp 
 
 (2); 
 
254 SOURCE SITUATED [328. 
 
 for where U is sensible, P n (p) = 1. Thus 
 
 *--^^- s <* +1 > p -<^ ...... < 3 >- 
 
 In this formula 1 1 UdS measures the intensity of the source. 
 If ike be very small, 
 
 .. 
 F (ike) F l (ike) \ ikr 
 
 so that ultimately 
 
 and the waves diverge as from a simple source of equal magnitude. 
 
 We will now examine the problem when kc is not very small, 
 taking for simplicity the case where T/T is required at a great 
 distance only, so that f n (ikr) = l. The factor on which the rela- 
 tive intensities in various directions depend is 
 
 s (2n + l) f.Qi) 
 
 ~T~ 3W 
 
 and a complete solution of the question would involve a discussion 
 of this series as a function of p and kc. 
 
 Thus, if 
 
 T ............... 6 
 
 ^ = - JL JJraS . {.F 2 + G 2 ]* . e*<-^ e + ......... (7), 
 
 where tan0 = : F ....................... (8). 
 
 The intensity of the vibrations in the various directions is thus 
 measured by J' 2 + 6r 2 . If, as before, F n = ot 
 
 2 a 2 
 
 2 
 
 The following table gives the means of calculating F and G 
 for any value of /*, when &c = , 1, or 2. In the last case it is 
 necessary to go as far as n = 7 to get a tolerably accurate result, and 
 for larger values of kc the calculation would soon become very 
 
328.] 
 
 ON THE SURFACE OF A SPHERE. 
 
 255 
 
 laborious. In all problems of this sort the harmonic analysis seems 
 to lose its power when the waves are very small in comparison 
 with the dimensions of bodies. 
 
 n 
 
 2a 
 
 2/9 
 
 (n + toa+(a* + p) 
 
 (n+l)p+(a* + (P) 
 
 
 
 + 2 
 
 + 1 
 
 + 4 
 
 + 2 
 
 1 
 
 + 4 
 
 7 
 
 + 1846153 
 
 - -3230768 
 
 2 
 
 64 
 
 35 
 
 - -0001391 
 
 - -0328885 
 
 3 
 
 - 466 
 
 + 853 
 
 - -0034527 
 
 + -0063201 
 
 4 
 
 + 14902 
 
 + 8141 
 
 + -0004653 
 
 + -0002542 
 
 5 
 
 + 175592 
 
 - 321419 
 
 + -0000144 
 
 - -0000264 
 
 11 
 
 a 
 
 ft 
 
 (u + i)a-(a 2 + /3 2 ) 
 
 (n + i)/3+(a 2 + /3 2 ) 
 
 
 
 + 1 
 
 + 1 
 
 + 25 
 
 + 25 
 
 1 
 
 + 2 
 
 1 
 
 + 6 
 
 -3 
 
 2 
 
 5 
 
 8 
 
 - -140449 
 
 -224719 
 
 3 
 
 53 
 
 + 34 
 
 - -046784 
 
 + -030013 
 
 4 
 
 + 296 
 
 + 461 
 
 + -004438 
 
 + -006912 
 
 5 
 
 + 4951 
 
 - 3179 
 
 + -000787 
 
 - -000505 
 
 6 
 
 - 40613 
 
 - 63251 
 
 - -000047 
 
 - -000073 
 
 7 
 
 - 936340 
 
 + 601217 - -000006 
 
 + -000004 
 
 Arc =2. 
 
 n 
 
 a 
 
 ft 
 
 (tt + |)a-h(a 2 + /3 2 ) 
 
 (w + i)/3--(a 2 + /3 2 ) 
 
 
 
 + 1 
 
 + 2 
 
 + -1 
 
 + 2 
 
 1 
 
 + 2 
 
 + 1 
 
 + 6 
 
 + 3 
 
 2 
 
 + 1-75 
 
 - 2-5 
 
 + -46980 
 
 - -67114 
 
 3 
 
 8 
 
 4 
 
 -35 
 
 -175 
 
 4 
 
 - 16-1875 
 
 + 35-125 
 
 - -04870 
 
 + '10567 
 
 5 
 
 + 186-625 
 
 + 85-4375 
 
 + -02436 
 
 + -01115 
 
 6 
 
 + 538-80 
 
 -1177-3 
 
 + -00209 
 
 - -00456 
 
 7 
 
 -8621-7 
 
 -3945-8 
 
 - -00072 
 
 - -00033 
 
 The most interesting question on which this analysis informs 
 us is the influence which a rigid sphere, situated close to the 
 source, has on the intensity of sound in different directions. 
 By the principle of reciprocity ( 294) the source and the place of 
 observation may be interchanged. When therefore we know the 
 
256 
 
 NUMERICAL RESULTS. 
 
 [328. 
 
 relative intensities at two distant points B, B', due to a source A 
 on the surface of the sphere, we have also the relative intensities 
 (measured by potential) at the point A, due to distant sources at 
 B and B'. On this account the problem has a double interest. 
 
 As a numerical example I have calculated the values of F + iG 
 and F~ -f 6r 2 for the above values of kc, when //, = !,//, = !,//, = (), 
 that is, looking from the centre of the sphere, in the direction of 
 the source, in the opposite direction, and laterally. 
 
 When kc is zero, the value of F 2 + 6r 2 is '25, which therefore 
 represents on the same scale as in the table the intensity due to 
 an unobstructed source of equal magnitude. We may interpret kc 
 as the ratio of the circumference of the sphere to the wave-length 
 of the sound. 
 
 kc 
 
 /* 
 
 F + iG 
 
 F2+G 2 
 
 
 1 
 
 521503 + '149417i 
 
 294291 
 
 I 
 
 -1 
 
 159149 - -484149* 
 
 259729 
 
 
 
 
 430244 - -216539i -231999 
 
 
 1 
 
 667938 + -238369i 
 
 502961 
 
 1 
 
 -1 
 
 - -440055 - -302609* 
 
 285220 
 
 
 
 
 + -321903 - -364974* 
 
 236828 
 
 
 1 
 
 79683 +-2342H 
 
 6898 
 
 2 
 
 -1 
 
 24954 +-50586i 
 
 3182 
 
 
 1 
 
 - -15381 - -57662i 
 
 3562 
 
 In looking at these figures the first point which attracts 
 attention is the comparatively slight deviation from uniformity 
 in the intensities in different directions. Even when the circum- 
 ference of the sphere amounts to twice the wave-length, there is 
 scarcely anything to be called a sound shadow. But what is 
 perhaps still more unexpected is that in the first two cases the 
 intensity behind the sphere exceeds that in a transverse direction. 
 This result depends mainly on the preponderance of the term of 
 the first order, which vanishes with p. The order of the more 
 important terms increases with kc; when kc is 2, the principal 
 term is that of the second order. 
 
 Up to a certain point the augmentation of the sphere will 
 increase, the total energy emitted, because a simple source emits 
 
328.] NUMERICAL RESULTS. 257 
 
 twice as much energy when close to a rigid plane as when entirely 
 in the open. Within the limits of the table this effect masks the 
 obstruction due to an increasing sphere, so that when /z = 1, 
 the intensity is greater when the circumference is twice the wave- 
 length than when it is half the wave-length, the source itself 
 remaining constant. 
 
 If the source be not simple harmonic with respect to time, the 
 relative proportions of the various constituents will vary to some 
 extent both with the size of the sphere and with the direction 
 of the point of observation, illustrating the fundamental character 
 of the analysis into simple harmonics. 
 
 When kc is decidedly less than one-half, the calculation may 
 be conducted with sufficient approximation algebraically. The 
 result is 
 
 4- terms in k*c 6 ..................... (10). 
 
 It appears that so far as the term in & 2 c 2 , the intensity is an 
 even function of p, viz. the same at any two points diametrically 
 opposed. For the principal directions yu,= 1, or 0, the numerical 
 calculation of the coefficient of & 4 c 4 is easy on account of the simple 
 values then assumed by the functions P. Thus 
 
 (fi=l) t ^ J + 2 = i + T f A; 2 c 2 + -77755 ^c 4 + ...... 
 
 (fi = - 1 ), F- + G 2 = J + T f k 2 c 2 + -02 7 5 5 & 4 c 4 + ...... 
 
 19534 k*c*+ ...... 
 
 When A, 4 c 4 can be neglected, the intensity is less in a lateral 
 direction than immediately in front of or behind the sphere. Or, 
 by the reciprocal property, a source at a distance will give a greater 
 intensity on the surface of a small sphere at the point furthest 
 from the source than in a lateral position. 
 
 If we apply these formulae to the case of kc = , we get 
 
 which agree pretty closely with the results of the more complete 
 calculation. 
 
 /? 17 
 
 f( UN IV 
 
258 EFFECT OF SMALL SPHERE. [328. 
 
 For other values of /*, the coefficient of k*c 4 in (10) might be 
 calculated with the aid of tables of Legend re's functions, or from 
 the following algebraic expression in terms of JJL I , 
 
 1 + f /* + ST P 2 + If P * - */^ + TT5 P 4 
 
 = '78138 + 1-5 /A + '85938 /z 2 - "03056 p*. 
 
 The difference of intensities in the directions //, = + ! and 
 //, = 1 may be very simply expressed. Thus 
 
 If Arc= 
 
 If kc = $, 
 
 At the same time the total value of F 2 + 6r 2 approximates to 
 25, when &c is small. 
 
 These numbers have an interesting bearing on the explanation 
 of the part played by the two ears in the perception of the quarter 
 from which a sound proceeds. 
 
 It should be observed that the variations of intensity in different 
 directions about which we have been speaking are due to the 
 presence of the sphere as an obstacle, and not to the fact that 
 the source is on the circumference of the sphere instead of at 
 the centre. At a great distance a small displacement of a 
 source of sound will affect the phase but not the intensity in any 
 direction. 
 
 In order to find the alteration of phase we have for a small 
 sphere 
 
 tan0 = #:JP=&c(-l+f /*), or = fcc(- 1 +f ft) nearly. 
 Thus in (7) e ik < a< - r+c > + i9 = e ik < a '- r+ ^c) } 
 
 from which we may infer that the phase at a distance is the same 
 as if the source had been situated at the point //, = 1, T = \G 
 (instead of r = c), and there had been no obstacle. 
 
 329. The functional symbols f and F may be expressed in 
 terms of P. It is known 2 that 
 
 P (u\= n n + l l -^ , tt(" -1) (" + !)(" + 2) (1-A*) 8 
 1*1 2 1.2 1.2 2 2 
 
 1 For the forms of the functions P, see 334. 
 
 2 Thomson and Tait's Nat. Phil. 782 (quoted from Murphy). 
 
329.] ANALYTICAL EXPRESSIONS. 259 
 
 or, on changing /M into 1 //,, 
 
 p ri-.irt-1-S n + 1 A* , ^O-l) (n + l)(rc + 2) /4 2 
 
 rHT'l 4 1.2 ~T72 2~ 2 ----( 1 )- 
 
 Consider now the symbolic operator P n M J , and let it 
 operate on y~ l . 
 
 . = (_!)(_ 2 ) 
 
 A comparison with (9) 323 now shews that 
 
 i .................. (2), 
 
 from whicji we deduce by a known formula, 
 
 In like manner, 
 
 ty/(-2/) = -F 
 If we now identify y with ikr, we see that the general solution, 
 (12) 323, may be written 
 
 - . . ...(4) 
 
 tHjT* I 7 n y 
 
 from which the second term is to be omitted, if no part of the 
 disturbance be propagated inwards. 
 
 Again from (14) 323 we see that 
 
 Fn (y} . 
 
 y y 
 whence F n W -tfP n (l -|) (l -|) .1 ............ (5), 
 
 and .f ............... (6) . 
 
 2/ 2 * \dy)dy y 
 
 , -f . 
 f \dy) dy y 
 
 172 
 
 Similarly, ^^|^ = - P , (-f ) -f . ^ ............... (7). . 
 
 f \dy) dy y 
 
260 ANALYTICAL EXPRESSIONS. [329. 
 
 Using these expressions in (13) 323, we get 
 
 / A \ rl >-ikr 
 
 In P. 
 
 d.ikr/ d.ikr' ikr 
 
 -k*S n 'P n 
 
 330. We have already considered in some detail the form 
 assumed by our general expressions when there is no source at 
 infinity. An equally important class of cases is defined by the 
 condition that there be no source at the origin. We shall now 
 investigate what restriction is thereby imposed on our general 
 expressions. 
 
 Reversing the series for/ rt , we have 
 1.3. 5...2w-l 
 
 n (ikr)- n 
 
 + (-l) n S n 'e +ikr (I-ikr +...)}, 
 
 shewing that, as r diminishes without limit, rifr n approximates to 
 
 1.3.5 ...(27i-l) M 
 r + n = ~ (ikr)- -i& + <- !)"&} 
 
 In order therefore that -^r n may be finite at the origin, 
 
 & + (-!) S n ' = ......... .................. (1) 
 
 is a necessary condition ; that it is sufficient we shall see later. 
 Accordingly (12) 323 becomes 
 
 r+n = S n {e-^f n (ikr) - (- 1) e^fn (- ikr)} ...... (2). 
 
 If, separating the real and imaginary parts of f nt we write (as 
 before) 
 
 / = ' + # ........................... (3), 
 
 (2) may be put into the form 
 r^ n = - 2i+ I S n [a! sin (kr + i HTT) - /3 7 cos (kr + \ mr)} ...... (4). 
 
 Another form may be derived from (4) 329. We have 
 
 d \ e +ikr -e- ikr 
 
 2ikr 
 d \ sin kr 
 
330.] MOTION CONTINUOUS THROUGH POLE. 261 
 
 Since the function P n is either wholly odd or wholly even, the 
 expression for ty. n is wholly real or wholly imaginary. 
 
 In order to prove that the value of ty n in (5) remains finite 
 when r vanishes, we begin by observing that 
 
 so that 2P n (* = P n - ** ^ 
 
 \d . ikrl kr 
 
 (7), 
 
 as is obvious when it is considered that the effect of differentiating 
 e ikm. an y num ber of times with respect to ikr is to multiply it by 
 the corresponding power of jj,. It remains to expand the expres- 
 sion on the right in ascending powers of r. We have 
 +i r+i 
 
 , (t&r) n 
 I.2...n'* 
 
 Now any positive integral power of //,, such as /JL P , can be 
 expanded in a terminating series of the functions P, the function 
 of highest order being P p . It follows that, if p < n, 
 
 *+i 
 
 by known properties of these functions ; so that the lowest power 
 
 f.+l 
 of ikr in I P n (//.) eP*+ dp is (ikr) n . Retaining only the leading 
 
 term, we may write 
 
 From the expression for P n (p) in terms of JJL, viz. 
 
 n(n-l)(n-2)(n~3) 
 2.4.(27i-l)(2n-3) p 
 we see that 
 
 ' ' ' ' ' ' 
 
 r - =-r- 
 
 P n (fj,) + terms in p of lower order than yu- n ; 
 
262 ANALYTICAL EXPRESSIONS [330. 
 
 and therefore 
 
 f +1 1.2. 3. ..n /+* 
 
 p* P n (p) dp = T -^ fi- =y [P n 0*)? dp 
 
 J _! 1 . 6. o ... {An 1) J _i 
 
 1 A O / yj O 
 
 Accordingly, by (5) and (7) 
 
 which shews that i/r n vanishes with r, except when ?i = 0. 
 
 The complete series for ^r n) when there is no source at the 
 pole, is more conveniently obtained by the aid of the theory of 
 Bessel's functions. The differential equations (4) 200, satisfied 
 by these functions, viz. 
 
 z^+:ira.|.n __jy (H^ 
 
 may also be written in the form 
 
 l/m2 _ 1\ 
 
 g* = (12). 
 
 It is known ( 200) that the solution of (11) subject to the 
 condition of finiteness when z = 0, is y = A J m (z), where 
 
 J = 
 
 m 
 
 2r(m+) 2 . (2m + 2) 
 
 is the Bessel's function of order m. 
 
 When m is integral, r(m + l) = 1.2.3...m; but here we have 
 to do with m fractional and of the form n + J , n being integral. 
 In this case 
 
 Referring now to (12), we see that the solution of 
 
 under the same condition of finiteness when z = 0, is 
 
 z) ........................ (16). 
 
330.] FOR VELOCITY-POTENTIAL. 263 
 
 Now the function ^ n , with which we are at present concerned, 
 satisfies (4) 323, viz. 
 
 <"> 
 
 which is of the same form as (15), if ra = n 4- ^ ; so that the solu- 
 tion is 
 
 i _ 
 
 
 _ 
 1 . 3 ... (2n + 1) VT 2 . (2n + 3) 
 
 ..(18). 
 
 Determining the constant by a comparison with (10), we find 
 
 ..,., ,x.o W" 
 
 3) 
 
 + ... 
 
 (19), 
 
 as the complete expression for ^ n in rising powers of r. 
 
 Comparing the different expressions (5) and (19) for ty n) we 
 obtain 
 
 .(20). 
 
 If F = a + if}, the corresponding expressions for d^r n /dr, are 
 (*r) - (- 1)" 
 
 = ~ 2 ^ { sin (kr + % mr) - cos (lar + J mr)} 
 
 , - - - 
 
 ^ ikrj d . kr kr 
 L n + 2 , 
 
264 PARTICULAR CASES. [330. 
 
 It will be convenient to write down for reference the forms of 
 tjr and dty/dr for the first three orders. 
 
 /I// 
 
 7i = 4 , , n .j a , . 7 
 
 ch/r 2io (sin AT , ( 
 
 -f- = 5 J C os kn . 
 
 \ dr r [ AT 
 
 2$! ( , sin h 
 K=- ^ cos AT = 
 
 7 / 7 I * 
 
 (~dr~~ "T 2 "^ V ? ~&r/ 
 
 2i& (A, 3 \ 3 
 
 - -I 1 TX-T, sin kr + y- cos 
 r (V fftrj kr 
 
 d& 2iS z (( 9 \ . /, 9 \ ) 
 
 --r- = - \ 4 7 sin wr ^r -,- cos A;r> . 
 
 dr r 2 (V /rr*/ \ AT/ j 
 
 331. One of the most interesting applications of these results 
 is to the investigation of the motion of a gas within a rigid 
 spherical envelope. To determine the free periods we have only 
 to suppose that dty/dr vanishes, when r is equal to the radius of 
 the envelope. Thus in the case of the symmetrical vibrations, we 
 have to determine k, 
 
 tan kr = kr (1), 
 
 an equation which we have already considered in the chapter 
 on membranes, 207. The first finite root (kr = 1*4303 TT) corre- 
 sponds to the symmetrical vibration of lowest pitch. In the case 
 of a higher root, the vibration in question has spherical nodes, 
 whose radii correspond to the inferior roots. 
 
 Any cone, whose vertex is at the origin, may be made rigid 
 without affecting the conditions of the question. 
 
 The loops, or places of no pressure variation, are given by 
 (kr)~ l sin kr = 0, or kr = m?r, where m is any integer, except 
 zero. 
 
 The case of n = 1 , when the vibrations may be called dia- 
 metral, is perhaps the most interesting. 8 lt being a harmonic 
 of order 1, is proportional to cos where 6 is the angle between r 
 and some fixed direction of reference. Since dfa/dO vanishes only 
 
331.] DIAMETRAL VIBRATIONS. 265 
 
 at the poles, there are no conical nodes 1 with vertex at the centre. 
 Any meridianal plane, however, is nodal, and may be supposed 
 rigid. Along any specified radius vector, ^ and d^/dd vanish, 
 and change sign, with cos kr (kr)~ l sin kr, viz. when tan kr = kr. 
 
 To find the spherical nodes, we have 
 
 The first root is kr = 0. Calculating from Trigonometrical 
 Tables by trial and error, I find for the next root, which cor- 
 responds to the vibration of most importance within a sphere, 
 kr = 119-26 x Tr/180 ; so that r :\ = '3:313. 
 
 The air sways from side to side in much the same manner as 
 in a doubly closed pipe. Without analysis we might anticipate 
 that the pitch would be higher for the sphere than for a closed 
 pipe of equal length, because the sphere may be derived from the 
 cylinder with closed ends, by filling up part of the latter with 
 obstructing material, the effect of which must be to sharpen the 
 spring, while the mass to be moved remains but little changed. 
 In fact, for a closed pipe of length 2r, 
 
 The sphere is thus higher in pitch than the cylinder by about 
 a Fourth. 
 
 The vibration now under consideration is the gravest of which 
 the sphere is capable ; it is more than an octave graver than the 
 gravest radial vibration. The next vibration of this type is such 
 that kr = 340'35 ir/180, or 
 
 r : X =s -9454, 
 
 and is therefore higher than the first radial. 
 
 When kr is great, the roots of (2) may be conveniently calcu- 
 lated by means of a series. If kr = cnr y, [where cr is an integer,] 
 then 
 
 2 (CTTT - y) 
 y * 
 
 from which we find 
 
 tan y , . 
 
 (0-7T - 7/) 2 - 2 ' 
 
 1 A node is a surface which might be supposed rigid, viz. one across which there 
 is no motion. 
 
266 VIBRATIONS OF SECOND ORDER. [331. 
 
 When n 2, the general expression for S n is 
 
 $ 2 = A (cos'0 - i) + (A 1 cos o> + B! sin G>) sin 6 cos 6 
 
 2 cos 2o> + J5 2 sin 2o>) sin 2 0. . ..(4), 
 
 from which we may select for special consideration the following 
 notable cases: 
 
 (a) the zonal harmonic, 
 
 Here d-fyJdQ is proportional to sin 20, and therefore vanishes 
 when 6 = ^TT. This shews that the equatorial plane is a nodal 
 surface, so that the same motion might take place within a closed 
 hemisphere. Also since $ 2 does not involve co, any meridianal plane 
 may be regarded as rigid. 
 
 (/3) the sectorial harmonic 
 
 2 = ^. 2 cos2ft>sin 2 ........................ (5). 
 
 Here again dfa/d6 varies as sin 26, and the equatorial plane is 
 nodal. But dfa/dco varies as sin 2o>, and therefore does not vanish 
 independently of 6, except when sin 2 = 0. It appears accordingly 
 that two, and but two, meridianal planes are nodal, and that these 
 are at right angles to one another. 
 
 (7) the tesseral harmonic, 
 
 $ 2 = -4iCos o> sin 0cos 6 ..................... (6). 
 
 In this case dfa/dO vanishes independently of o> with cos 20, 
 that is, when = JTT, or f TT, which gives a nodal cone of revolution 
 whose vertical angle is a right angle, dfyjda) varies as sin o>, and 
 thus there is one meridianal nodal plane, and but one l . 
 
 The spherical nodes are given by 
 
 - 9kr 
 
 of which the first finite solution is 
 
 AT = 3-3422, 
 giving a tone graver than any of the radial group. 
 
 In the case of the general harmonic, the equation giving the 
 
 1 [I owe to Prof. Lamb the remark that the difference between (/3) and (7) is 
 only in relation to the axes of reference.] 
 
331.] 
 
 WAVE-LENGTHS OF VIBRATIONS. 
 
 267 
 
 tones possible within a sphere of radius r may be written (21) 
 330 
 
 p *(Ar}A-^ = <>-- ..(9), 
 
 \d. ifcr/ d.fcr fcr 
 or again, 
 
 O Z*/v j , (n V i T~" / i (rC r i\ I 1 i 
 
 [For the roots of 
 
 T-jir* /(*))() (11), 
 
 a 
 
 equivalent to (10), Prof. M c Mahon gives * 
 w _ Rf _ m + 7 _ 4 (7m 2 + 
 
 Zv ~^ '~zfT' ~T 
 
 32 (83m 3 4- 3535m 2 + 3561m + 6133) 
 
 15(8/3') 5 
 
 where m = 4z/ 2 , and 
 If w = 1, so that v = f, 
 
 and (12) gives a result in harmony with (3).] 
 
 Table A shews the values of X for a sphere of radius unity, 
 corresponding to the more important modes of vibration. In B is 
 exhibited the frequency of the various vibrations referred to the 
 gravest of the whole system. The Table is extended far enough 
 to include two octaves. 
 
 TABLE A, 
 
 Giving the values of X for a sphere of unit radius. 
 Order of Harmonic. 
 
 
 1-3983 
 81334 
 57622 
 44670 
 36485 
 30833 
 
 3-0186 
 1-0577 
 68251 
 50653 
 40330 
 33523 
 
 1-8800 
 86195 
 59208 
 45380 
 
 1-392 
 7320 
 5248 
 
 1-113 
 
 6385 
 
 9300 
 
 8002 
 
 1 Annals of Mathematics, vol. ix. no. 1. 
 
268 
 
 WAVE-LENGTHS OF VIBKATIONS. 
 
 [331 
 
 TABLE B. 
 
 Pitch of each 
 tone, referred 
 to gravest. 
 
 Order 
 of 
 Harmonic. 
 
 Number 
 of internal 
 spherical 
 nodes. 
 
 Pitch of each 
 tone, referred 
 to gravest. 
 
 Order 
 of 
 Harmonic. 
 
 Number 
 of internal 
 spherical 
 nodes. 
 
 1-0000 
 
 1 
 
 
 
 28540 
 
 1 
 
 1 
 
 1-6056 
 
 2 
 
 
 
 3-2458 
 
 5 
 
 
 
 2-1588 
 
 
 
 
 
 3-5021 
 
 2 
 
 1 
 
 2-169 
 
 3 
 
 
 
 3-7114 
 
 
 
 1 
 
 2-712 
 
 4 
 
 
 
 3-772 
 
 6 
 
 
 
 332. If we drop unnecessary constants, the particular solu- 
 tion for the vibrations of gas within a spherical case of radius 
 unity is represented by 
 
 ^ n = S n (kr)-* J n+ i (kr) cos (kat -6} ............ (1), 
 
 where k is a root of 
 
 (2). 
 
 In generalising this, we must remember that S n may be com- 
 posed of several terms, corresponding to each of which there may 
 exist a vibration of arbitrary amplitude and phase. Further, each 
 term in S n may be associated with any, or all, of the values of k, 
 determined by (2). For example, under the head of n = 2, we 
 might have 
 
 ^ 2 = J. (cos 2 <9 - i) (A?!?-)"* J^+j (k,r) cos (k.at + #0 
 
 + B cos 2o> sin 2 (k 2 r)-$ J n+ (k z r) cos (k 2 at + 0,,), 
 &j and & 2 being different roots of 
 
 Any two of the constituents of ^ are conjugate, i.e. will vanish 
 when multiplied together and integrated over the volume of the 
 sphere. This follows from the property of the spherical harmonics, 
 wherever the two terms considered correspond to different values of 
 n, or to two different constituents of S n l . The only case remaining 
 for consideration requires us to shew that 
 
 = ...... (3), 
 
 1 Thomson and Tait's Nat. Phil. p. 151. 
 
332.] UNIFORM INITIAL VELOCITY. 269 
 
 where fa and fa are different roots of 
 
 2kJ f n ^(k) = Jn + ^(k) ..................... (4), 
 
 and this is an immediate consequence of a fundamental property 
 of these functions ( 203). There is therefore no difficulty in 
 adapting the general solution to prescribed initial circumstances. 
 
 In order to illustrate this subject we will take the case where 
 initially the gas is in its position of equilibrium but is moving 
 with constant velocity parallel to x. This condition of things 
 would be approximately realised, if the case, having been pre- 
 viously in uniform motion, were suddenly stopped. 
 
 Since there is no initial condensation or rarefaction, all the 
 quantities 6 n vanish. If dty/dx be initially unity, we have 
 TJr = as = r/j,, which shews that the solution contains only terms of 
 the first order in spherical harmonics. The solution is therefore 
 of the form 
 
 -v/r = A l (far)"* J| (far) //, cos faat 
 
 + Az(far)-*J%(far)fjicosfaat + .................. (5), 
 
 where fa, fa, &c. are roots of 
 
 2kJj(k) = Jt(k) ........................... (6). 
 
 To determine the coefficients, we have initially for values of r 
 from to 1, 
 
 r = A, (far)-* Jf (far) + A 2 (far)-* J(k,r) + ......... (7). 
 
 Multiplying by r% J$ (kr) and integrating with respect to r from 
 to 1, we find 
 
 (8), 
 
 the other terms on the right vanishing in virtue of the conjugate 
 property. Now by (16), 203, 
 
 [J. (kr)frdr = [/,' 
 
 ..................... (9), 
 
 by (6). 
 
 The evaluation of I r* J^ (kr) dr may be effected by the aid of 
 
270 CASE OF UNIFORM [332. 
 
 a general theorem relating to these functions. By the fundamental 
 differential equation 
 
 f r , , fl d f dJ n (kr)\ f,. ri*\ T n N ~| , 
 r n+i _ ( r 5LV > } + (k 2 }J n (kr) dr = 0, 
 
 I T rlT \ fl'T I V V* I 
 
 .'o [_rar \ ar / \ / / j 
 
 whence by integration by parts we obtain, 
 
 dr 
 or, if we make r 1, 
 
 i (kr) dr = nJ n (k) kJ n '(k) (11). 
 
 o 
 Thus in the case, with which we are here concerned, 
 
 Equation (8) therefore takes the form 
 
 A - *& 
 
 ~(k*" 
 and the final solution is 
 
 where the summation is to be extended to all the admissible 
 values of k. 
 
 When t = 0, and r = 1, we must have ty = //., and accordingly 
 
 It will be remembered that the higher values of k are approxi- 
 mately, (3) 331, 
 
 & = cr7r- .............................. (15). 
 
 <77T 
 
 The first value of & is 2'0815, and the second 5'9402, whence 
 ^2-85742, ^ = 06009, 
 
 shewing that the first term in the series for ^ is by far the most 
 important. 
 
332.] INITIAL VELOCITY. 271 
 
 It may be well to recall here that 
 
 Equation (14) may be verified thus : the quantities k are the 
 roots of 
 
 or, if $ = z~% J| (z\ the roots of <' = 0, where (f> satisfies 
 
 ..................... (17). 
 
 Now, since the leading term in the expansion of <j> in ascending 
 powers of z is independent of z t we may write 
 
 !>' = const. ! 1 j- t 
 
 AY 
 
 whence, by taking the logarithms and differentiating, 
 
 _ = _^_ 2 * 
 <' AY* - 2 A; 2 2 - z* 
 
 If we now put z* = 2, we get by (17), 
 
 333. In a similar manner we may treat the problem of the 
 vibrations of air included between rigid concentric spherical 
 surfaces, whose radii are n and r 2 . For by (13) 323, if d^ n /dr 
 vanish for these values of r, 
 
 n 
 
 F n (+ikr,} 
 whence 
 
 where as before 
 
 = a + i ........................ (2). 
 
 When the difference between rj arid r 2 is very small compared with 
 either, the problem identifies itself with that of the vibration of a 
 spherical sheet of air, and is best solved independently. In (1) 
 
272 SPHERICAL SHELL. [333. 
 
 323, if ^Jr be independent of r, as it is evident that it must 
 approximately be in the case supposed, we have 
 
 ( sin \ + + ft.^ = . . . (3), 
 
 6 dO \ dd) si&6 da 2 
 
 sm 
 whose solution is simply 
 
 ^n = <8f ................................. (4), 
 
 while the admissible values of & 2 are given by 
 
 (5). 
 
 The interval between the gravest tone (n I) and the next is such 
 that two of them would make a twelfth (octave + fifth). The 
 problem of the spherical sheet of gas will be further considered in 
 the following chapter. [For a derivation of (5) from the funda- 
 mental determinant, equivalent to (1), the reader may be referred 
 to a short paper 1 by Mr Chree.] 
 
 334. The next application that we shall make of the spherical 
 harmonic analysis is to investigate the disturbance which ensues 
 when plane waves of sound impinge on an obstructing sphere. 
 Taking the centre of the sphere as origin of polar co-ordinates, and 
 the direction from which the waves come as the axis of /*, let <f> 
 be the potential of the unobstructed plane waves. Then, leaving 
 out an unnecessary complex coefficient, we have 
 
 = e ik tt+x } = & ikat ^ 
 
 and the solution of the problem requires the expansion of e ikr ^ in 
 spherical harmonics. On account of the symmetry the harmonics 
 reduce themselves to Legendre's functions P n (/JL), so that we may 
 take 
 
 e* r " = A + A 1 P 1 +...+A n P n + ............ (2), 
 
 where A ... are functions of r, but not of /-t. From what has 
 been already proved we may anticipate that A nt considered as a 
 function of r, must vary as 
 
 sin kr 
 
 but the same result may easily be obtained directly. Multiplying 
 
 1 Messenger of Mathematics, vol. xv. p. 20, 1886. 
 
334.] PLANE WAVES. 273 
 
 (2) by P n (p), and integrating with respect to //, from //, = 1 to 
 //. = + 1, we find 
 
 r+i r+i <2A 
 
 I ^(rie^dn^ 
 
 and, as in 330, 
 
 so that finally 
 
 In the problem in hand the whole' motion outside the sphere 
 may be divided into two parts ; the first, that represented by <f> 
 and corresponding to undisturbed plane waves, and the second 
 a disturbance due to the presence of the sphere, and radiating 
 outwards from it. If the potential of the latter part be >Jr, we 
 have (2) 324 on replacing the general harmonic S n by a n P n (fji), 
 
 r f n (ikr) \ 
 
 The velocity-potential of the whole motion is found by addition 
 of < and ty, the constants a n being determined by the boundary 
 conditions, whose form depends upon the character of the obstruc- 
 tion presented by the sphere. The simplest case is that of a rigid 
 and fixed sphere, and then the condition to be satisfied when r = c 
 is that 
 
 - 
 dr dr 
 
 a relation which must of course hold good for each harmonic 
 element separately. For the element of order n, we get 
 
 kc 2 e ikc - / d \ d sinkc 
 
 -ET-7-.-r-xn i / -JT' i 
 
 ' F n (ike) \d .ike/ d . kc kc 
 
 Corresponding to the plane waves < = e ik(at+x) , the disturbance 
 due to the presence of the sphere is expressed by 
 
 r 
 
 ^ W =QO 2w+l . . / d \ d sin kc 
 
 R. II. 
 
 18 
 
274 DISTURBANCE DUE TO [334. 
 
 At a sufficient distance from the source of disturbance we may 
 take f n (ikr) = 1. In order to pass to the solution of a real 
 problem, we may separate the real and imaginary parts, and 
 throw away the latter. On this supposition the plane waves are 
 represented by 
 
 [</>]= cos k (at + x) (9). 
 
 Confining ourselves for simplicity's sake to parts of space at a 
 great distance from the sphere, where f n (ikr) = l, we proceed to 
 extract the real part of (8). Since the functions P are wholly 
 even or wholly odd, 
 
 p / d \ d smkc 
 n \d . ike) d.kc' kc 
 
 is wholly real or wholly imaginary, so that this factor presents no 
 difficulty. {F n (ikc)}~~\ however, is complex, and since F n (ikc)=a+i/3, 
 
 {Fn 
 
 where tan 7 = ff/a. [If the positive value of V( 2 + /3 2 ) be taken 
 in all cases, 7 must be so chosen that cos 7 has the same sign as a.] 
 
 Thus 
 -vjr = 2 (2n + 1) - 
 
 When therefore n is even, 
 
 Jcr z 
 
 0/r] = (2n+ 1)^ cos {k (at-r + c) + 7} 
 
 /DQ) a. r> ( ^ \ d sin kc 
 x^ + W-tp^^j. 
 
 while, if n be odd, 
 
 Z, r 2 
 
 (2n + 1) tsin {k(at - r + c) + 7} 
 
 As examples we may write down the terms in [T/T], in- 
 volving harmonics of orders 0, 1, 2. The following table of the 
 functions P n (p) will be useful. 
 
334.1 A RIGID SPHERICAL OBSTACLE. 275 
 
 We have, 
 
 
 ..1, 
 
 , 3fc 2 4 -J d 2 sin fa; 
 
 f 4 )-J d 2 sin fa; 
 
 f ' 2 + Fc2 } a -^ -;- / - n {A (a* - r + c) + ,} 
 
 ........................... (14); 
 
 . 
 
 The solution of the problem here obtained, though analytically 
 quite general, is hardly of practical use except when kc is a small 
 quantity. In this case we may advantageously expand our results 
 in rising powers of kc. 
 
 __(! _ 
 
 7o}- (16). 
 
 x jj, . sin \k (at r 4- c) + 71} (1^)> 
 
 - 1) cos (A? (a* - r + c) + 72} ....... (18). 
 
 It appears that while [^ ] and [^J are of the same order in 
 the small quantity kc, [^ 2 ] is two orders higher. We shall find 
 presently that the higher harmonic components in [>/r] depend upon 
 
 182 
 
276 SPHERICAL OBSTACLE. [334. 
 
 still more elevated powers of kc. For a first approximation, then, 
 we may confine ourselves to the elements of order and 1. 
 
 Although [^o] contains a cosine, and [i/rj a sine, they never- 
 theless differ in phase by a small quantity only. Comparing two 
 of the values of d^ n /dr in (21) 330 we see that 
 
 a sin (kc + J ntr) - ft cos (kc + J rnr) 
 
 n (kc\ n+l 
 = -(- 1) lT g7 5 _( 2n + 1} + hi S her P owers of ke 
 
 identically. Dividing by a cos (kc + ^UTT), we get ultimately 
 
 (-l) n n(kc) n+l 
 
 tan (kc + *rwr) -- = -- 77 , --- ; . ^ ^ ~ /0 . -. v 
 a a cos (kc + ^HTT) 1 . 3 . o ... (2n + 1) 
 
 When ft is even, this equation becomes on substitution for a of 
 its leading term from (16) 323, 
 
 fe,_ 
 
 ~ 
 
 . 
 
 a ~ (n + 1)~(2 + 1) {1 . 3 . 5 . . . (2n - I)} 2 ' ' 
 For example, if ?i = 2, 
 
 7 f/3\ 
 tan^c- - = 
 
 ....... 
 
 3 3 .5 
 
 When 7i is at all high, the expressions tan kc and /3/a become 
 very nearly identical for moderate values of kc. 
 
 When n is odd, we get in a nearly similar manner, 
 o n (kc)- n ~ l 
 
 oot * + ; - (OTH2+-i) {i :3T5...(2-i)} + ...... (2 
 
 [From (19) we see that when n is even tan 7, or /3/a, is 
 approximately equal to tan kc, and from (20) when n is odd that 
 cot 7 = tan &c. In the first case, by (16) 323, a has the sign of 
 i~ n or of ( l)* n ; and in the second case a has the sign of i~ n+1 or 
 of ( l)*^- 1 ). In both cases the approximate solution may be 
 expressed 
 
 The velocity-potential of the disturbance due to a small rigid 
 and fixed sphere is therefore approximately, 
 
 - (1 + 1/*) <=os k (at - r) 
 
 rp 
 
 (l+ijft)ooft*(at-.r) ...... (21), 
 
 7*A 
 
 1 This emendation and others consequential to it are due to Dr Burton. 
 
334.] INTENSITY OF SECONDARY WAVES. 277 
 
 if T denote the volume of the obstacle, the corresponding direct 
 wave being 
 
 [</>] = cos k (at + as) (22). 
 
 For a given obstacle and a given distance the ratio of the 
 amplitudes of the scattered and the direct waves is in general pro- j 
 portional to the inverse square of the wave-length, and the ratio of 
 intensities is proportional to the inverse fourth power ( 296). 
 
 In order to compare the intensities of the primary and 
 scattered sounds, we may suppose the former to originate in a 
 simple source, provided it be sufficiently distant (R) from T. 
 Thus, if 
 
 cos k (at R) 
 
 (1+Wco8 * (a *" r ? ........... (24); 
 
 so that at equal distances from their sources the secondary and 
 the primary waves are in the ratio 
 
 The intensities are therefore in the ratio 
 
 + ttf ...................... < 26 >> 
 
 (27). 
 
 which, in the case of //, = + 1, gives approximately 
 
 61-72 T* 
 
 It must be well understood that in order that this result may 
 apply, \ must be great compared with the linear dimension of T, 
 and R must be great compared with \. 
 
 To find the leading term in the expression for ^ n , when kc is 
 small, we have in the first place, 
 
 d \ d sin kc 
 
 1.3.5...(2n-l) ( 2. n. (2^ + 3) 
 
 ^ 
 
278 FURTHER APPROXIMATION. [334. 
 
 Again, 
 
 F n (ike) x F n (- ike) 
 
 (29); 
 
 so that 
 
 (kc) n 
 
 i- _ ^ _ ^^-^ __ \- 
 *.(,-H)(2^l) 
 
 ........................ (30). 
 
 Hence, from (10), 
 ~- 
 
 When n is even, [since 7 = kc + ^mr approximately,] 
 
 while if n be odd, we have merely to replace i n by i n+l [and cos by 
 sin], the result being then still real. 
 
 By means of (31) we may verify the first two terms in the 
 expressions for fyj, [i|rj, in (17), (18). To the case of n = 0, (31) 
 does not apply. 
 
 Again, by (31), 
 
 7 
 
 I/* 3 - f /} sin (i- (a* - r + c) + 7,) . . -(33), 
 
 Combining (17), (18), (33), (34), we have the value of 
 complete as far as the terms which are of the order & 6 c 6 compared 
 with the two leading terms given in (21). In compounding the 
 partial expressions, it is as necessary to be exact with respect to 
 the phases of the components as with respect to their amplitudes ; 
 but for purposes requiring only one harmonic element at a time, 
 
334.] PRESSURES ON OBSTACLE. 279 
 
 the phase is often of subordinate importance. In such cases we 
 may take 
 
 From (31) or (32) it appears that the leading term in ty n rises 
 two orders in kc with each step in the order of the harmonic ; and 
 that ty n is itself expressed by a series containing only even, or only 
 odd, powers of kc. But besides being of higher order in kc, the 
 leading term becomes rapidly smaller as n increases, on account of 
 the other factors which it contains. This is evident, because for 
 all values of n and //,, P n (//,)<!; the same is true of n/(n + I); 
 while i n only affects the phase. 
 
 In particular cases any one of the harmonic elements of 
 may vanish. From (11), (12), since (a 2 + /3 2 )~* cannot vanish, we 
 have in such a case 
 
 p / d \ d sin&c_ ft 
 ^(d.ikc)d.ke kc ' 
 
 the same equation as that which gives the periods of the vibrations 
 of order n in a closed sphere of radius c. A little consideration 
 will shew that this result might have been expected. The table 
 of 331 is applicable to this question and shews, among other 
 things, that when kc is small, no harmonic element in [^J can 
 vanish. 
 
 In consequence of the aerial pressures the sphere is acted on 
 by a force parallel to the axis of //,, whose tendency is to set the 
 sphere into vibration. The magnitude of this force, if cr be the 
 density of the fluid, is given by 
 
 in which, by the conjugate property of Legendre's functions, only 
 the term of the first order affects the result of the integration. 
 Now, when r = c, 
 
 ... d sinkc 
 
 ' = 3e '' 
 
 __ 
 .ikc'd.kc- kc ' 
 
 where 
 
 /, (ike) = 1 + gL , F> (ike) = ike + 2 + ~ . 
 
280 SOURCE AT FINITE DISTANCE. [334. 
 
 In order that the force may vanish, it would be necessary that 
 
 d smkc , fi(ikc) d? smkc _~ 
 '~~~ CC ~ 
 
 which cannot be satisfied by any real value of kc. We conclude 
 that, if the sphere be free to move, it will always be set into 
 vibration. 
 
 If instead of being absolutely plane, the primary waves have 
 their origin in a unit source at a great, though finite, distance R 
 from the centre of the sphere, we have 
 
 n f d \ d Sin&C ,na\ 
 
 XP W U rf-} -= = = (36). 
 
 \d . ike d.kc kc 
 
 On the sphere itself r = c, so that the value of the total poten 
 tial at any point at the surface is 
 
 [p / d \ sin kc , f n (ike) p / _ d^ \ d sin A:cl 
 ^ n (d^ikc) ~lcc~ * C F n (ikc) n \d7ikc) dTkc ~kc~ J ' 
 
 This expression may be simplified. We have 
 sin kc 1 
 
 d. 
 
 and thus the quantity within square brackets may be written 
 
 f F n (ikc)f n (- ike) - F n (- ikc)f n (Me) 
 
 u * f " F n (ikc) 
 
 which by (6) 327 is identical with e ikc [F n (ikc)]~\ Thus 
 
 which is the same as if the source had been on the sphere, and 
 the point at which the potential is required at a great distance 
 ( 328), and is an example of the general Principle of Reciprocity. 
 
334.] SYMMETRICAL EXPRESSION. 281 
 
 By assuming the principle, and making use of the result (3) of 
 328, we see that if the source of the primary waves be at a finite 
 distance R, the value of the total potential at any point on the 
 sphere is 
 
 If A and B be any two points external to the sphere, a unit 
 source at A will give the same total potential at B y as a unit 
 source at B would give at A. In either case the total potential is 
 made up of two parts, of which the first is the same as if there 
 were no obstacle to the free propagation of the waves, and the 
 second represents the disturbance due to the obstacle. Of these 
 two parts the first is obviously the same, whichever of the two 
 points be regarded as source, and therefore the other parts must 
 also be equal, that is the value of >|r at B when A is a source is 
 equal to the value of ty at A when B is an equal source. Now 
 when the source A is at a great distance R, the value of -^ at a 
 point B whose angular distance from A is cos" 1 //,, and linear 
 distance from the centre is r, is (36) 
 
 = - J*L ik ( at-R-r + c } V 
 
 F n (ikc) 
 
 d \ d smkc 
 t&)d.kc' kc ' 
 
 and accordingly this is also the value of \/r at a great distance R, 
 when the source is at B. But since ty is a disturbance radiating 
 outwards from the sphere, its value at any finite distance R may 
 be inferred from that at an infinite distance by introducing into 
 each harmonic term the factor f n (ikR). We thus obtain the 
 following symmetrical expression 
 
 (39), 
 
 which gives this part of the potential at either point, when the 
 other is a unit source. 
 
 It should be observed that the general part of the argument 
 does not depend upon the obstacle being either spherical or rigid. 
 
282 INVESTIGATION FOB THE CASE [334. 
 
 From the expansion of e ikr{J - in spherical harmonics, we may 
 deduce that of the potential of waves issuing from a unit simple 
 source A finitely distant (r) from the origin of co-ordinates. The 
 potential at a point B at an infinite distance R from the origin, 
 and in a direction making an angle cos" 1 p with r, will be 
 
 ik(Rnr) 
 
 the time factor being omitted. 
 Hence by the expansion of e ikr>J - 
 
 e 
 
 -ikR 
 
 from which we pass to the case of a finite R by the simple intro- 
 duction of the factor f n (ikR). 
 
 Thus the potential at a finitely distant point B of a unit source 
 at A is 
 
 335. Having considered at some length the case of a rigid 
 spherical obstacle, we will now sketch briefly the course of the 
 investigation when the obstacle is gaseous. Although in all 
 natural gases the compressibility is nearly the same, we will 
 suppose for the sake of generality that the matter occupying the 
 sphere differs in compressibility, as well as in density, from the 
 medium in which the plane waves advance. 
 
 Exterior to the sphere, < is the same exactly, and ty is of 
 the same form as before. For the motion inside the sphere, if 
 k f = 2-7T/V be the internal wave-length, (2) 330, 
 
 f n (- ik'r)}, 
 
 satisfying the condition of continuity through the centre. 
 
 If <7, <r' be the natural densities, m, m the compressibilities, 
 //yA^o-'/o-.m/m' ........................ (1); 
 
335.] OF A GASEOUS OBSTACLE. 283 
 
 and the conditions, to be satisfied by each harmonic element 
 separately, are 
 
 d(f>ldr + d^/dr (outside) = d-^/dr (inside) ......... (2), 
 
 orfy + ty (outside)} = crS/r (inside) ............ (3), 
 
 expressing respectively the equalities of the normal motions and 
 of the pressures on the two sides of the bounding surface. From 
 these equations the complete solution may be worked out ; but 
 we will here confine ourselves to finding the value of the leading 
 terms, when kc, k'c are very small. 
 
 In this case, when r = c, 
 
 (inside) = - 2^V ) 
 (inside) =f ik' 3 ca ' )" 
 
 1 1 ,-. 
 
 2 " 
 
 (outside) = a /c 
 (outside) = -a /c* 
 
 Using these in (2), (3), and eliminating a ', retaining only the 
 principal term, we find 
 
 m m 
 
 In like manner for the term of first order, 
 
 ^ (inside) = fa/ A/ 2 
 r (inside) = - Ja/A/^ 
 
 <P! = ikcfi ) ,QX 
 
 d(pi/dr = ik/ju } 
 
 ^ (outside) = a,i/ikc* . p 
 r (outside) = 
 which give 
 
 At a distance from the sphere the disturbance due to it is 
 expressed by 
 
284 GASEOUS OBSTACLE. [335. 
 
 If we introduce the relations 
 
 and throw away the imaginary part, we obtain 
 
 as the expression for the most important part of the disturb- 
 ance, corresponding to (21) 334 for a fixed rigid sphere. It 
 appears, as might have been expected, that the term of zero 
 order is due to the variation of compressibility, and that of 
 order one to the variation of density. 
 
 From (13) we may fall back on the case of a rigid fixed sphere, 
 by making both &' and m' infinite. It is not sufficient to make a' 
 by itself infinite, apparently because, if m' at the same time 
 remained finite, k'c would not be small, as the investigation has 
 assumed. 
 
 When m' m, <r' & are small, (13) becomes equivalent to 
 
 fi [ cos k (at r), 
 
 AT/ ^ 'lib CT ) 
 
 corresponding to </> = cos kat at the centre of the sphere. This 
 agrees with the result (13) of 296, in which the obstacle may be 
 of any form. 
 
 In actual gases mf = m, and the term of zero order disappears. 
 If the gas occupying the spherical space be incomparably lighter 
 than the other gas, <r' = 0, and 
 
 TrT 
 
 ty = 3 - LL cos k (at r) (14), 
 
 \ 2 r 
 
 so that in the term of order one, the effect is twice that of a rigid 
 body, and has the reverse sign. 
 
 The greater part of this chapter is taken from two papers by 
 the author " On the vibrations of a gas contained within a rigid 
 spherical envelope," and an " Investigation of the disturbance pro- 
 duced by a spherical obstacle on the waves of sound 1 ," and from 
 the paper by Professor Stokes already referred to. 
 
 1 Math. Society's Proceeding*, March 14, 1872 ; Nov. 14, 1872. 
 
CHAPTER XVIII. 
 
 SPHERICAL SHEETS OF AIE. MOTION IN TWO DIMENSIONS. 
 
 336. IN a former chapter ( 135), we saw that a proof of 
 Fourier's theorem might be obtained by considering the mechanics 
 of a vibrating string. A similar treatment of the problem of 
 a spherical sheet of air will lead us to a proof of Laplace's 
 expansion for a function which is arbitrary at every point of 
 a spherical surface. 
 
 As in 333, if -^ is the velocity-potential, the equation of 
 continuity, referred to the ordinary polar co-ordinates 6, &>, takes 
 the form, 
 
 I d 
 
 Whatever may be the character of the free motion, it can 
 be analysed into a series of simple harmonic vibrations, the 
 nature of which is determined by the corresponding functions 
 A^T, considered as dependent on space. Thus, if ^oce 1 * ^, the 
 equation to determine i/r as a function of 9 and &> is 
 
 -. 
 
 sin 6 d0 \ d6 J sin 2 6 dc 
 
 Again, whatever function ^ may be, it can be expanded by 
 Fourier's theorem 1 in a series of sines and cosines of the multiples 
 of ft). Thus 
 
 ^ = ^o + ^i cos ft) + A/T/ sin a) + -^2 cos 2ft) -I- ^/ sin 2&) 
 
 cos sco + ^ s ' sin 5&) + ......... (2), 
 
 1 We here introduce the condition that $ recurs after one revolution round the 
 sphere. 
 
286 GENERAL DIFFERENTIAL EQUATION. [336. 
 
 where the coefficients ^r , ^ ... ^/, fa' ... are functions of d only ; 
 and by the conjugate property of the circular functions, each 
 term of the series must satisfy the equation independently. 
 Accordingly, 
 
 is the equation from which the character of -\Jr g or >|r/ is to be 
 determined. This equation may be written in various ways. 
 
 In terms of /JL (= cos 0), 
 
 or, if v = sin 6, 
 
 where h? is written for & 2 c 2 . 
 
 When the original function ^r is symmetrical with respect 
 to the pole, that is, depends upon latitude only, s vanishes, and 
 the equations simplify. This case we may conveniently take 
 first. In terms of //,, 
 
 The solution of this equation involves two arbitrary constants, 
 multiplying two definite functions of //,, and may be obtained 
 in the ordinary way by assuming an ascending series and de- 
 termining the exponents and coefficients by substitution. Thus 
 
 1.2.3.4.5.6 
 
 -4. 5) . 
 
 in which A and B are arbitrary constants. 
 
 Let us now further suppose that >/r besides being symmetrical 
 round the pole is also symmetrical with respect to the equator 
 (which is accordingly nodal), or in other words that ty is an 
 
336.] CONDITION TO BE SATISFIED AT POLES. 287 
 
 even function of the sine of the latitude (//,). Under these circum- 
 stances it is clear that B must vanish, and the value of i/r be 
 expressed simply by the first series, multiplied by the arbitrary 
 constant A. This value of the velocity-potential is the logical 
 consequence of the original differential equation and of the two 
 restrictions as to symmetry. The value of h? might appear 
 to be arbitrary, but from what we know of the mechanics of the 
 problem, it is certain beforehand that h? is really limited to a 
 series of particular values. The condition, which yet remains 
 to be introduced and by which h is determined, is that the 
 original equation is satisfied at the pole itself, or in other words 
 that the pole is not a source; and this requires us to consider 
 the value of the series when //, = !. Since the series is an 
 even function of /*, if the pole /* = + 1 be not a source, neither 
 will be the pole /A = 1. It is evident at once that if h 2 be of 
 the form n(n + l), where n is an even integer, the series termi- 
 nates, and therefore remains finite when ^ = 1 ; but what we 
 now want to prove is that, if the series remain finite for /* = !, 
 h z is necessarily of the above-mentioned form. By the ordinary 
 rule it appears at once that, whatever be the value of h 2 , 
 the ratio of successive terms tends to the limit /i 2 , and there- 
 fore the series is convergent for all values of JJL less than unity. 
 But for the extreme value //.= !, a higher method of discrimi- 
 nation is necessary. 
 
 It is known 1 that the infinite hypergeometrical series 
 
 ab a(a + l)b(b + l) a(a + l)(a + 2)6(6 + l)(6 + 2) 
 "*" cd + c(c + l)d(d + 1) c(c + l)(c + 2) d(d + l)(d + 2) + 
 
 is convergent, if c + d a b be greater than 1, and divergent 
 if c + d a b be equal to, or less than 1. In the latter case 
 the value of c + d a b affords a criterion of the degree of 
 divergency. Of two divergent series of the above form, for 
 which the values of c + d a b are different, that one is relatively 
 infinite for which the value of c + d a b is the smaller. 
 
 Our present series (7) may be reduced to the standard form 
 by taking & 2 = w(n + l), where n is not assumed to be integral. 
 Thus 
 
 1 Boole's Finite Differences, p. 79. 
 
288 CRITERION OF DIVERGENCY. [336. 
 
 l)(n-2)(ro + 3) 
 1.2 ^ 1.2.3.4 
 
 , L . i a , 
 
 14 1.2.J.f 
 
 + ............. (9), 
 
 which is of the standard form, if 
 
 a = -$n, b = ^n + , c = -J-, d-l. 
 
 Accordingly, since c + d a 6 = 1, the series is divergent for 
 /A = 1, unless it terminate ; and it terminates only when n is an 
 even integer. We are thus led to the conclusion that when 
 the pole is not a source, and >/r is an even function of /z,, h* must 
 be of the form n(n + 1), where n is an even integer. 
 
 In like manner, we may prove that when ^r is an odd function 
 of p, and the poles are not sources, A = 0, and A 2 must be of the 
 form n(n+ I), n being an odd integer. 
 
 If n be fractional, both series are divergent for /JL = 1, and 
 although a combination of them may be found which remains 
 finite at one or other pole, there can be no combination which 
 remains finite at both poles. If therefore it be a condition that 
 no point on the surface of the sphere is a source, we have no 
 alternative but to make n integral, and even then we do not 
 secure finiteness at the poles unless we further suppose A = 0, 
 when n is odd, and B = 0, when n is even. We conclude that 
 for a complete spherical layer, the only admissible values of -fy, 
 which are functions of latitude only, and proportional to harmonic 
 functions of the time, are included under 
 
 where P W (/A) is Legendre's function, and n is any odd or even 
 integer. The possibility of expanding an arbitrary function of 
 latitude in a series of Legendre's functions is a necessary con- 
 sequence of what has now been proved. Any possible motion 
 of the layer of gas is represented by the series 
 
 = A. + PM A, cos + Bl gin i + . . 
 
 C C / 
 
 co.ite:^ 
 
336.] TRANSITION TO TWO DIMENSIONS. 289 
 
 When * = 0, 
 
 n ( f i) + ......... (11), 
 
 and the value of t/r when t = is an arbitrary function of latitude. 
 
 The method that we have here followed has also the advantage 
 of proving the conjugate property, 
 
 (12), 
 
 where n and m are different integers. For the functions P(/u.) 
 are the normal functions ( 94) for the vibrating system under 
 consideration, and accordingly the expression for the kinetic 
 energy can only involve the squares of the generalized velocities. 
 If (12) do not hold good, the products also of the velocities must 
 enter. 
 
 The value of i/r appropriate to a plane layer of vibrating gas 
 can of course be deduced as a particular case of the general solu- 
 tion applicable to a spherical layer. Confining ourselves to the 
 case where there is no source at the pole (//,= !), we have to in- 
 vestigate the limiting form of ty = CP n (^), where n(n + l) = & 2 c 2 , 
 when c 2 and n 2 are infinite. At the same time //, 1 and v are 
 infinitesimal, and cv passes into the plane polar radius (r), so 
 that nv = kr. For this purpose the most convenient form of P n (ji) 
 is that of Murphy 1 : 
 
 - ............... (13). 
 
 The limit is evidently 
 
 {^,2^2 !/*** 
 
 l ~y+X^ 
 
 shewing that the Bessel's function of zero order is an extreme case 
 of Legendre's functions. 
 
 When the spherical layer is not complete, the problem re- 
 quires a different treatment. Thus, if the gas be bounded by walls 
 stretching along two parallels of latitude, the complete integral 
 involving two arbitrary constants will in general be necessary. 
 
 1 Thomson and Tait's Nat. Phil. 782. [t=sin 2 J0, not 4 sin 2 J0.] Todhunter's 
 Laplace's Functions, 19. 
 
 B. II. 19 
 
290 VIBRATIONS OF A SPHERICAL SHEET [336. 
 
 The ratio of the constants and the admissible values of h? are to be 
 determined by the two boundary conditions expressing that at the 
 parallels in question the motion is wholly in longitude. The value 
 of //, being throughout numerically less than unity, the series are 
 always convergent. 
 
 If the portion of the surface occupied by gas be that included 
 between two parallels of latitude at equal distances from the 
 equator, the question becomes simpler, since then one or other of 
 the constants A and B in (7) vanishes in the case of each normal 
 function. 
 
 337. When the spherical area contemplated includes a pole, 
 we have, as in the case of the complete sphere, to introduce the 
 condition that the pole is not a source. For this purpose the solu- 
 tion in terms of v, i.e. sin 0, will be more convenient. 
 
 If we restrict ourselves for the present to the case of symmetry, 
 we have, putting s = in (5) 336, 
 
 + ^,to=o ......... (i). 
 
 One solution of this equation is readily obtained in the ordinary 
 way by assuming an ascending series and substituting in the 
 differential equation to determine the exponents and coefficients. 
 We get 1 
 
 This value of ^ is the most general solution of (1), subject to 
 the condition of finiteness when v = 0. The complete solution 
 involving two arbitrary constants provides for a source of arbitrary 
 intensity at the pole, in which case the value of -t/r is infinite when 
 v = 0. Any solution which remains finite when v = and involves 
 one arbitrary constant, is therefore the most general possible under 
 the restriction that the pole be not a source. Accordingly it is 
 unnecessary for our purpose to complete the solution. The nature 
 of the second function (involving a logarithm of v) will be illus- 
 trated in the particular case of a plane layer to be considered 
 presently. 
 
 1 Heine's Kugelfumtionen, 28. 
 
337.J BOUNDED BY A SMALL CIRCLE. 291 
 
 By writing n (n + 1) for h? the series within brackets becomes 
 
 2 
 
 22 2 2 .4 2 ~ ~" 
 
 or, when reduced to the standard hypergeometrical form, 
 
 . . .(-!*)(** + 
 1.1 
 
 corresponding to 
 
 . . . . , , 
 
 1.1 1.2.1.2 + '"' 
 
 , c = , c=l. 
 
 Since c + cZ a 6 = f, the series converges for all values of v 
 from to 1 inclusive. To values of 6 (= sin" 1 v) greater than JTT 
 the solution is inapplicable. 
 
 When n is an integer, the series becomes identical with 
 Legendre's function P n (^). If the integer be even, the series 
 terminates, but otherwise remains infinite. Thus, when n = 1, the 
 series is identical with the expansion of yu,, viz. ^(1 z/ 2 ), in powers 
 of v. 
 
 The expression for ty in terms of v may be conveniently applied 
 to the investigation of the free symmetrical vibrations of a spheri- 
 cal layer of air, bounded by a small circle, whose radius is less than 
 the quadrant. The condition to be satisfied is simply dtyjdv = 0, 
 an equation by which the possible values of A 2 , or & 2 c 2 , are con- 
 nected with the given boundary value of v. 
 
 Certain particular cases of this problem may be treated by 
 means of Legendre's functions. Suppose, for example, that n = 6, 
 so that A 2 = & 2 c 2 = 42. The corresponding solution is ty = AP 6 (/jL). 
 The greatest value of //, for which difr/djJL= is yu,= '8302, corre- 
 sponding to 6 = 33 53' = -59137 radians 1 . 
 
 If we take c& = r, so that r is the radius of the small circle 
 measured along the sphere, we get 
 
 kr = V(42) x -59137 = 3'8325, 
 
 which is the equation connecting the value of k (= 2?r/X) with the 
 curved radius r, in the case of a small circle, whose angular radius 
 is 33 53 7 . If .the layer were plane ( 339), the value of kr would 
 be 3'8317 ; so that it makes no perceptible difference in the pitch 
 of the gravest tone whether the radius (r) of given length be 
 
 1 The radian is the unit of circular measure. 
 
 UNIVERSITY 
 
292 UNSYMMETBICAL MOTION. [337. 
 
 straight, or be curved to an arc of 33. The result of the com- 
 parison would, however, be materially different, if we were to take 
 the length of the circumference as the same in the two cases, that 
 is, replace c6 = r by cv = r. 
 
 In order to deduce the symmetrical solution for a plane layer, 
 it is only necessary to make c infinite, while cv remains finite. On 
 account of the infinite value of A 2 , the solution assumes the simple 
 form 
 
 or, if we write cv = r, where r is the polar radius in two dimensions, 
 
 {^2 r 2 JMr 4 ") 
 
 ^iF + Si- ......... J-4Jo(*r) ......... (5), 
 
 as in (14) 336. 
 
 The differential equation for i/r in terms of v, when c is infinite 
 and cv = r, becomes 
 
 ** + i + **-0 ..................... (6). 
 
 ar 5 r dr 
 
 An independent investigation and solution for the plane problem 
 will be given presently. 
 
 338. When s is different from zero, the differential equation 
 satisfied by the coefficients of sin 5o>, cos sco, is 
 
 and the solution, subject to the condition of finiteness when v = O 1 , 
 is easily found to be 
 
 >-fr 2 (5 + 2) (5 + 3) -A 2 | 
 
 2(25+2) 4(25 + 4) -]' 
 
 or, if we put h? = n (n + 1), 
 
 4 
 
 " ' 
 
 2. 4. (25 + 2) (25 + 4) 
 
 1 The solution may be completed by the addition of a second function derived 
 from (2) by changing the sign of , which occurs in (1) only as s 2 , but a modification 
 is necessary, when is a positive integer. The method of procedure will be 
 exemplified presently in the case of the plane layer. 
 
338.] UNSYMMETRICAL MOTION. 293 
 
 We have here the complete solution of the problem of the 
 vibrations of a spherical layer of gas bounded by a small circle 
 whose radius is less than the quadrant. For each value of s, there 
 are a series of possible values of n, determined by the condition 
 d^Jdv^Q] with any of these values of n the function on the 
 right-hand side of (2), when multiplied by cossw or sinso>, is a 
 normal function of the system. The aggregate of all the normal 
 functions corresponding to every admissible value of s and n, with 
 an arbitrary coefficient prefixed to each, gives an .expression 
 capable of being identified with the initial value of A|T, i.e. with a 
 function given arbitrarily over the area of the small circle. 
 
 When the radius of the sphere c is infinitely great, h? is infinite. 
 If cv = r, h* v 2 = k*r 2 , and (2) becomes 
 
 k 2 T 2 
 
 l ~ 
 
 2(2+2) 24(2s + 2) (2. + 4) 
 a function of r proportional to J s (kr). 
 
 In terms of p, the differential equation satisfied by the co- 
 efficient of cos sco, or sin sco, is 
 
 Assuming ^ = (1 ^ 2 )**< s , we find as the equation for </> s 
 
 = <>.. ..(5), 
 
 which will be more easily dealt with. 
 To solve it, let 
 
 and substitute in (5). The coefficient of the lowest power of 
 fji is a(a-l); so that a = 0, or a = l. The relation between 
 i, and a 2m , found by equating to zero the coefficient of /A a+2m/ , is 
 
 (cc + 2m + s - n) (a+ 2m +s + n +1) 
 -a ( a + 2wi + l)(a + 2m + 2) 
 
 where ?t (n + 1) = M 
 
294 CONDITIONS TO BE SATISFIED [338. 
 
 The complete value of (f> 8 is accordingly given by 
 
 . . (s-n)(s-n+2)(s+n+l)(s+n+3) 
 
 172 -- ** T.2.3.4 
 
 1.2.3.4.5.6 
 
 
 ) 
 
 where J. and 5 are arbitrary constants ; 
 
 and ^ S = (1- M 2 )^ ........................ (7). 
 
 We have now to prove that the condition that neither pole is 
 a source requires that n s be a positive integer, in which case 
 one or other of the series in the expression for $> s terminates. 
 For this purpose it will not be enough to shew that the series 
 (unless terminating) are infinite when //, = + 1 ; it will be necessary 
 to prove that they remain divergent after multiplication by 
 (1 yu, 2 )^, or as we may put it more conveniently, that they are 
 infinite when /tt= + l in comparison with (1 //. 2 )~H It will be 
 sufficient to consider in detail the case of the first series. 
 
 We have 
 
 (s-n)(s + n + 1) (s-n)(s-n+2)(s+n+l)(s+n+3) 
 
 1.2 1.2.3.4 * ...... 
 
 = 1 , 
 
 . 
 
 which is of the standard form (8) 336 
 
 if 
 
 ab q( 
 
 "*" erf" 1 " C (c+i)d(d+i 
 
 The degree of divergency is determined by the value of a + 6 - c - d, 
 which is here equal to s 1. 
 
338.] WHEN THE POLES ARE NOT SOURCES. 295 
 
 On the other hand, the binomial theorem gives for the ex- 
 pansion Of (1 /JL 2 )~$ S 
 
 which is of the standard form, if 
 
 a = ^s, c = l, b = d, and makes a + 
 
 Since s l>^s 1, it appears that the series in the expression 
 for <t> 8 are infinities of a higher order than (1 /t 2 )"**, and there- 
 fore remain infinite after multiplication by (1 /* 2 )**. Accordingly 
 tyg cannot be finite at both poles unless one or other of the series 
 terminate, which can only happen when n s is zero, or a positive 
 integer. If the integer be even, we have still to suppose B ; 
 and if the integer be odd, A = 0, in order to secure finiteness at 
 the poles. 
 
 In either case the value of (f> s for the complete sphere may be 
 put into the form 
 
 where the constant multiplier is omitted. The complete expres- 
 sion for that part of ifr which contains cos so) or sin so) as a factor 
 is therefore 
 
 . cos so) r x , _ d* D , v , a ^ 
 
 A ** P *M ........................ < 9 
 
 where A n is constant with respect to //, and to, but as a function 
 of the time will vary as 
 
 
 For most purposes, however, it is more convenient to group 
 the terms for which n is the same, rather than those for which s 
 is the same. Thus for any value of n 
 
 where every coefficient A s , B s may be regarded as containing a 
 time factor of the form (10). 
 
 Initially i/r is an arbitrary function of JJL and co, and therefore 
 any such function is capable of being represented in the form 
 
296 FORMULA OF DERIVATION. [338. 
 
 n=oo s=n fJs p / ,.\ 
 
 ^ = 2 2 v 8 - ^P(4/cos*a> + JV* sin say)... (12), 
 
 which is Laplace's expansion in spherical surface harmonics. 
 
 From the differential equation (5), or from its general solution 
 (6), it is easy to prove that (f> 8 is of the same form as dc^gi/dp, so 
 that we may write 
 
 (in which no connection between the arbitrary constants is as 
 serted), or in terms of T|T by (7), 
 
 Equation (13) is a generalization of the property of Laplace's 
 functions used in (8). 
 
 The corresponding relations for the plane problem may be 
 deduced, as before, by attaching an infinite value to n, which 
 in (13), (14) is arbitrary, and writing nv = kr. Since /u, 2 + v 2 = 1, 
 
 ijr being regarded as a function of v. In the limit //, (even 
 though subject to differentiation) may be identified with unity, 
 and thus we may take 
 
 When the pole is not a source, ^ is proportional to J s (kr). 
 The constant coefficient, left undetermined by (15), may be 
 readily found by a comparison of the leading terms. It thus 
 appears that 
 
 J Q (kr) ............ (16), 
 
 a well-known problem of Bessel's functions 1 . 
 
 The vibrations of a plane layer of gas are of course more 
 easily dealt with, than those of a layer of finite curvature, but 
 I have preferred to exhibit the indirect as well as the direct 
 method of investigation, both for the sake of the spherical problem 
 
 1 Todhunter's Laplace's Functions, 390. 
 
338.] VIBRATION IN TWO DIMENSIONS. 297 
 
 itself with the corresponding Laplace's expansion 1 , and because 
 the connection between Bessel's and Laplace's functions appears 
 not to be generally understood. We may now, however, proceed 
 to the independent treatment of the plane problem. 
 
 339. If in the general equation of simple aerial vibrations 
 
 we assume that ty is independent of z, and introduce plane polar 
 coordinates, we get ( 241) 
 
 or, if ^r be expanded in Fourier's series 
 
 ^=^0 + ^1+...+^ + ..................... (2), 
 
 where ty n is of the form A n cos n6 -f B n sin n6, 
 
 _ _ ............... 
 
 dr* r dr \ r 2 / T 
 
 This equation is of the same form as that with which we had to 
 deal in treating of circular membranes ( 200) ; the principal 
 mathematical difference between the two questions lies in the 
 fact that while in the case of membranes the condition to be 
 satisfied at the boundary is -^ = 0, in the present case interest 
 attaches itself rather to the boundary condition dty/dr = 0, corre- 
 sponding to the confinement of the gas by a rigid cylindrical 
 envelope 3 . 
 
 The pole not being a source, the solution of (3) is 
 
 + n = AJ n (kr) ........................ (4), 
 
 and the equation giving the possible periods of vibration within 
 a cylinder of radius r, is 
 
 J n '(kr) = ........................... (5). 
 
 The lower values of kr satisfying (5) are given in the following 
 table 4 , which was calculated from Han sen's tables of the functions 
 
 1 I have been much assisted by Heine's Handbuch der Kugelfunctionen, Berlin, 
 1861, and by Sir W. Thomson's papers on Laplace's Theory of the Tides, Phil. 
 Mag. Vol. L. 1875. 
 
 2 I here recur to the usual notation, but the reader will understand that n cor- 
 responds to the s of preceding sections. The n of Laplace's functions is now infinite. 
 
 3 [The symmetrical vibrations within a cylindrical boundary, corresponding to 
 w = 0, were considered by Duhamel (Liouville Journ. Math. Vol. 14, p. 69, 1849).] 
 
 4 Notes on Bessel's Functions. Phil. Mag. Nov. 1872. 
 
298 
 
 RIGID CIRCULAR BOUNDARY. 
 
 [339. 
 
 J by means of the relations allowing J n to be expressed in terms 
 of J Q and /!. 
 
 Number of in- 
 ternal circu- 
 lar nodes. 
 
 = o 
 
 n = l 
 
 n = 2 
 
 n = 3 
 
 
 
 3-832 
 
 1-841 
 
 3-054 
 
 4-201 
 
 1 
 
 7-015 
 
 5-332 
 
 6-705 
 
 8-015 
 
 2 
 
 10-174 
 
 8-536 
 
 9-965 
 
 11-344 
 
 3 
 
 13-324 
 
 11-706 
 
 
 
 4 
 
 16-471 
 
 14-864 
 
 
 
 5 
 
 19-616 
 
 18-016 
 
 
 
 [For the roots of the equation J n '(z) = 0, Prof. McMahon 1 finds 
 
 4 (7m 2 + 82m -9) 
 3(8/37 
 
 32 (83m 3 + 2075m 2 - 3039m + 3527) 
 15 
 
 
 where m = 4m*, and ff = ITT (2n -f 4s + 1). It will be found that 
 n = in (6 a) gives the same result as n = 1 in (4) 206, in 
 accordance with the identity JJ(z) = - Ji(z).] 
 The particular solution may be written 
 
 ty n = (A cos n6 + B sin nO) J n (kr) cos kat 
 
 + (Ccosn0 + Dsmn6)J n (kr)smkat ............ (6), 
 
 where A, B, C, D are arbitrary for every admissible value of 
 n and k. As in the corresponding problems for the sphere and 
 circular membrane, the sum of all the particular solutions must 
 be general enough to represent, when t 0, arbitrary values of 
 i|r and -Jr. 
 
 As an example of compound vibrations we may suppose, as 
 in 332, that the initial condition of the gas is that defined by 
 
 -^ = 0, i|r = x = r cos 6. 
 Under these circumstances (6) reduces to 
 
 i|r = A l cos 6 J l (k^) cos hat 4- A 2 cos 6 J l (k 2 r) cos k^at + . . .(7), 
 
 and, if we suppose the radius of the cylinder to-be unity, the 
 admissible values of k are the roots of 
 
 Ji'(A;) = ........................... (8). 
 
 1 Annah of Mathematics, Vol. ix. No. 1. 
 
339.] CASE OF COMPOUND VIBRATIONS. 299 
 
 The condition to determine the coefficients A is that for all values 
 of r from r = to r = 1, 
 
 r = A l J l (k 1 r) + AJ 1 (k t r) + ............... (9), 
 
 whence, as in 332, 
 
 The complete solution is therefore 
 
 where the summation extends to all the values of k determined 
 
 by (8). 
 
 If we put t = and r = 1, we get from (9) and (10) 
 
 2 
 
 (12), 
 
 an equation which may be verified numerically, or by an analy- 
 tical process similar to that applied in the case of (14) 332. 
 We may prove that 
 
 (Z 2 \ 
 1 ~p) 
 
 whence by differentiation 
 
 From this (12) is derived by putting z = 1, and having regard 
 to the fundamental differential equation satisfied by J lt which 
 shews that 
 
 //'(!) :</i'(l) = -l. 
 [More generally, if J n '(k) = 0, 
 
 v 2" _! i 
 
 F^'~ 
 
 Hitherto we have supposed the cylinder complete, so that 
 i/r recurs after each revolution, which requires that n be integral ; 
 but if instead of the complete cylinder we take the sector included 
 between 6 = and 6 = /5, fractional values of n will in general pre- 
 sent themselves. Since dty/d6 vanishes at both limits of 6, ty 
 must be of the form 
 
 i/r = A cos (kat + e) cos n6 J n (kr) (13), 
 
 where n = virjp, v being integral. If /3 be an aliquot part of 
 TT (or TT itself), the complete solution involves only integral values 
 
300 ANALOGOUS PROBLEMS FOB WATER WAVES. [339. 
 
 of n t as might have been foreseen ; but, in general, functions of 
 fractional order must be introduced. 
 
 An interesting example occurs when /8 = 2-Tr, which corre- 
 sponds to the case of a cylinder, traversed by a rigid wall 
 stretching from the centre to the circumference (compare 207). 
 The effect of the wall is to render possible a difference of pressure 
 on its two sides; but when no such difference occurs, the wall 
 may be removed, and the vibrations are included under the 
 theory of a complete cylinder. This state of things occurs 
 when v is even. But when v is odd, n is of the form (integer -I- J), 
 and the pressures on the two sides of the wall are different. In 
 the latter case J n is expressible in finite terms. The gravest 
 tone is obtained by taking v = 1, or n = J, when 
 
 (14), 
 
 and the admissible values of k are the roots of tan k = 2k. The 
 first root (after k0) is k = 1*1655, corresponding to a tone 
 decidedly graver than any of which the complete cylinder is 
 capable. 
 
 The preceding analysis has an interesting application to 
 the mathematically analogous problem of the vibrations of water 
 in a cylindrical vessel of uniform depth. The reader may 
 consult a paper on waves by the author in the Philosophical 
 Magazine for April, 1876, and papers by Prof. Guthrie to which 
 reference is there made. The observation of the periodic time 
 is very easy, and in this way may be obtained an experimental 
 solution of problems, whose theoretical treatment is far beyond 
 the power of known methods. 
 
 340. Returning to the complete cylinder, let us suppose it 
 closed by rigid transverse walls at z=Q, and z = I, and remove 
 the restriction that the motion is to be the same in all transverse 
 sections. The general differential equation ( 241) is 
 
 < 
 
 Let ty be expanded by Fourier's theorem in the series 
 
 27TZ 
 
 TT TT 1*2 TT 27TZ / TrZ\ 
 
 ^ = H Q + H l CQs-j-+ HZ cos -j- + ... + H p cos Ip -j-J + ...(2), 
 where the coefficients H p may be functions of r and 6. This form 
 
340.] VIBRATIONS IN A CLOSED CYLINDER'. 301 
 
 secures the fulfilment of the boundary conditions, when z 0, = I, 
 and each term must satisfy the differential equation separately. 
 Thus 
 
 ~~ P 2 ~n }Hp = ( 3 )' 
 
 r dr 
 
 which is of the same form as when the motion is independent of 
 z, k 2 being replaced by k' 2 p' 2 7T 2 l~ 2 . The particular solution may 
 therefore be written 
 
 ^ = (A n cos n6 + B n sin nd) .cosp-^-. J n (Jk 2 -p 2 7r 2 l~ 2 . r) cos kat 
 
 7TZ 
 
 + (G n cos n6 + D n sin nd) cosp-^-. J n (J& ~p 2 7r 2 l~ 2 . r) sin kat. . .(4), 
 
 which must be generalized by a triple summation, with respect to 
 all integral values of p and n, and also with respect to all the 
 values of k, determined by the equation, 
 
 p 2 7r 2 l- 2 .r) = .......................... (5). 
 
 If r = 1, and K denote the values of k given in the table ( 339), 
 corresponding to purely transverse vibrations, we have 
 
 k 2 = K 2 +p 2 7T 2 /l 2 ........................ (6). 
 
 The purely axial vibrations correspond to a zero value of K, 
 not included in the table. 
 
 341. The complete integral of the equation 
 
 dr 2 r dr 
 
 when there is no limitation as to the absence of a source at the 
 pole, involves a second function of r, which may be denoted by 
 J- n (kr). Thus, omitting unnecessary constant multipliers, we may 
 take ( 200) 
 
 + 
 
 but the second series requires modification, if n be integral. When 
 fi = 0, the two series become identical, and thus the immediate 
 result of supposing n in (2) lacks the necessary generality. The 
 
302 GENERAL SOLUTION. [341. 
 
 required solution may, however, be obtained by the ordinary rule 
 applicable to such cases. Denoting the coefficients of A and B 
 in (2) by f(n), /( n\ we have 
 
 by Maclaurin's theorem. Hence, taking new arbitrary constants, 
 we may write as the limiting form of (2), 
 
 In this equation /(O) is J (&r) ; to fmd/'(0) we have 
 1 - 
 
 d 
 
 dn } 2 . 2 + 2n 2 . 4 . 2 + 2n . 4 
 
 If % denote the general term (involving r 2m ) of the series within 
 brackets, taken without regard to sign, 
 
 ldu_d\ogu_ 2 2 2 
 
 M cn dn 2 + 2?i 4 + 2n 2m + 2n ' 
 
 //Y?/\ 
 
 iso that [-T-J = u n =oSm) 
 
 if S m = 
 
 Thus 
 
 l " ' + 
 
 (2 2 2 2 .4 2 '2 2 .4 2 .6 2 
 and the complete integral for the case n =0 is 
 
 k22 j^4^4 
 
 1- * + --- 
 
 For the general integral value of n the corresponding ex- 
 pression may be derived by means of (15) 338 
 
341.] EXPRESSION BY DESCENDING SERIES. 303 
 
 The formula of derivation (5) may be obtained directly from 
 the differential equation (1). Writing z for kr and putting 
 
 ^n = *"*n .............................. (6), 
 
 we find in place of (1) 
 
 Again (7) may be put into the form 
 
 from which it follows at once that 
 
 <#>-] ........................... (9); 
 
 so that * = ~*> ........................ (10) ' 
 
 or by (6) + ..................... (11), 
 
 which is equivalent to (5), since the constants in T|T O are arbitrary 
 in both equations. 
 
 The serial expressions for ^r n thus obtained are convergent for 
 all values of the argument, but are practically useless when the 
 argument is great. In such cases we must have recourse to semi- 
 convergent series corresponding to that of (10) 200. 
 
 Equation (1) may be put into the form 
 
 whence by 323 (4), (12), we find as the general solution of (1) 
 
 -4T2 2 ) 
 
 (I 2 - 4w 2 ) (3 2 - 4w a ) (5 2 - 4?i 2 ) 
 
 tt')(3-4n)(5-4n) 1 
 1.2.3. (8tfcr) 
 
304 DIVERGENT WAVE. [341. 
 
 When n is integral, these series are infinite and ultimately 
 divergent, but ( 200, 802) this circumstance does not interfere 
 with their practical utility. 
 
 The most important application of the complete integral of (1) 
 is to represent a disturbance diverging from the pole, a problem 
 which has been treated by Stokes in his memoir on the communi- 
 cation of vibrations to a gas. The condition that the disturbance 
 represented by (13) shall be exclusively divergent is simply 
 D = 0, as appears immediately on introduction of the time factor 
 e ikat by supposing r to be very great ; the principal difficulty of 
 the question consists in discovering what relation between the 
 coefficients of the ascending series corresponds to this condition, 
 for which purpose Stokes employs the solution of (1) in the form 
 of a definite integral. We shall attain the same object, perhaps 
 more simply, by using the results of 302. 
 
 By (22), (24) 302 
 
 1 s I 2 . 3 2 
 
 2W I 1 . 8w . 1 . 2 . (Biz] 
 
 and thus the question reduces itself to the determination of the 
 form of the right-hand member of (14) when z is small. By (5) 
 302 and (5) 200 we have 
 
 JTT [K(z) + i J (z)} = z + ^ITT + higher terms in z ...... (15), 
 
 so that all that remains is to find the form of the definite integral 
 in (14), when z is small. Putting \/(/3 2 + z 9 ) =y /3, we have 
 
 y z y 
 
 When z is small, 2 2 /2y is also small throughout the range of 
 integration, and thus we may write 
 
 The first integral on the right is 
 
 1 De Morgan's Differential and Integral Calculus, p. 653. 
 
341.] DIVERGENT WAVE. 305 
 
 where 7 is Eider's constant ('5772...); and, as we may easily 
 satisfy ourselves by integration by parts, the other integrals do not 
 contribute anything to the leading terms. Thus, when z is very 
 small, 
 
 l.Sfcr 
 
 
 ) 
 
 "} 
 
 (17). 
 
 Replacing z by kr, and comparing with the form assumed by (4) 
 when r is small, we see that in order to make the series identical 
 we must take 
 
 so that a series of waves diverging from the pole, whose expression 
 in descending series is 
 
 is represented also by the ascending series 
 
 In applying the formula of derivation (11) to the descending 
 series, the parts containing e~ ikr and e +ikr as factors will evidently 
 remain distinct, and the complete integral for the general value 
 of 11, subject to the condition that the part containing e +ikr shall 
 not appear, will be got by differentiation from the complete 
 integral for n = subject to the same condition. Thus, since 
 
 1.2.3. (Sikry 
 R. II. 20 
 
306 SOUNDING BOARDS. [341, 
 
 or, in terms of the ascending series, 
 
 l ~ kr\ 2 2 + 2 2 .4 2 
 
 ikr\{kr A^r 3 
 log T J|2--^r 4 + 2 2 
 
 These expressions are applied by Prof. Stokes to shew how feebly 
 the vibrations of a string, (corresponding to the term of order 
 one), are communicated to the surrounding gas. For this purpose 
 he makes a comparison between the actual sound, and what would 
 have been emitted in the same direction, were the lateral motion 
 of the gas in the neighbourhood of the string prevented. For a 
 piano string corresponding to the middle C, the radius of the 
 wire may be about '02 inch, and X is about 25 inches ; and it 
 appears that the sound is nearly 40,000 times weaker than it would 
 have been if the motion of the particles of air had taken place in 
 planes passing through the axis of the string. " This shews the 
 vital importance of sounding-boards in stringed instruments. 
 Although the amplitude of vibration of the particles of the sound- 
 ing-board is extremely small compared with that of the particles 
 of the string, yet as it presents a broad surface to the air it is able 
 to excite loud sonorous vibrations, whereas were the string 
 supported in an absolutely rigid manner, the vibrations which it 
 could excite directly in the air would be so small as to be almost 
 or altogether inaudible." 
 
 Fig. 64. 
 
 " The increase of sound produced by the stoppage of lateral 
 motion may be prettily exhibited by a very simple experiment. 
 Take a tuning-fork, and holding it in the fingers after it has been 
 
341.] SYMMETRICAL DIVERGENT WAVES. 307 
 
 made to vibrate, place a sheet of paper, or the blade of a broad 
 knife, with its edge parallel to the axis of the fork, and as near to 
 the fork as conveniently may be without touching. If the plane of 
 the obstacle coincide with either of the planes of symmetry of the 
 fork, as represented in section at A or B, no effect is produced; 
 but if it be placed in an intermediate position, such as C, the 
 sound becomes much stronger 1 ." 
 
 342. The real expression for the velocity-potential of sym- 
 metrical waves diverging in two dimensions is obtained from (18) 
 341 after introduction of the time factor e ikat by rejecting the 
 imaginary part ; it is 
 
 \i f I 2 3 2 
 
 -J cosfc(a*-r-^X)|l- 1 ^ 2 ^ 8 ^ )2 +... 
 
 7T \* . , f I 2 1 2 .3 2 .5 2 ) 
 
 ^ )sm*(o-r-iX) r^-l7270^ + "' (1) ' 
 
 in which, as usual, two arbitrary constants may be inserted, one as 
 a multiplier of the whole expression and the other as an addition 
 to the time. 
 
 The problem of a linear source of uniform intensity may also 
 be treated by the general method applicable in three dimensions. 
 Thus by (3) 277, if p be the distance of any element dx from 0, 
 the point at which the potential is to be estimated, and r be the 
 smallest value of p, so that p z = r 2 + # 2 , we may take 
 
 which must be of the same form as (1). Taking y ^ r y we 
 may write in place of (2) 
 
 e -ikr e -Hcy dy 
 
 from which the various expressions follow as in (14) 341. When 
 kr is great, an approximate value of the integral may be obtained 
 by neglecting the variation of \/(2r + ?/), since on account of the 
 rapid fluctuation of sign caused by the factor e~ iky we need attend 
 
 1 Phil. Trans, vol. 158, p. 447, 1868. 
 
 202 
 
308 LINEAR SOURCE. [342. 
 
 only to small values of y. Now 
 
 [cosxdx_ rsmxdx_ / (ir\ 
 
 Jo V* ~Jo "V* ~V V2/" 
 
 so that #-a-')' r * erHW .......... (5) - 
 
 Introducing the factor e ikat , and rejecting the imaginary part 
 of the expression, we have finally 
 
 as the value of the velocity-potential at a great distance. A 
 similar argument is applicable to shew that (1) is also the expres- 
 sion for the velocity-potential on one side of an infinite plane 
 ( 278) due to the uniform normal motion of an infinitesimal strip 
 bounded by parallel lines. 
 
 In like manner we may regard the term of the first order 
 (20) 341 as the expression of the velocity-potential due to double 
 sources uniformly distributed along an infinite straight line. 
 
 From the point of view of the present section we see the 
 significance of the retardation of JX, which appears in (1) and in 
 the results of the following section (16), (17). In the ordinary 
 integration for surface distributions by Fresnel's zones ( 283) 
 the whole effect is the half of that of the first zone, and the phase 
 of the effect of the first zone is midway between the phases due 
 to its extreme parts, i.e. JX behind the phase due to the central 
 point. In the present case the retardation of the resultant 
 relatively to the central element is less, on account of the pre- 
 ponderance of the central parts. 
 
 [From the formulae of the present section for the velocity- 
 potential of a linear source we may obtain by integration a 
 corresponding expression for a source which is uniformly distributed 
 over a plane. The waves issuing from this latter are necessarily 
 plane waves, of which the velocity-potential can at once be written 
 down, and the comparison of results leads to the evaluation of 
 certain definite integrals relating to Bessel's and allied functions 1 .] 
 
 1 On Point-, Line-, and Plane-Sources of Sound. Proc. London Math. Soc. 
 Vol. xix. p. 504, 1888. 
 
343.] CYLINDRICAL OBSTACLE. 309 
 
 343. In illustration of the formulae of 341 we may take 
 the problem of the disturbance of plane waves of sound by a 
 cylindrical obstacle, whose radius is small in comparison with 
 the length of the waves, and whose axis is parallel to their plane. 
 (Compare 335.) 
 
 Let the plane waves be represented by 
 
 J. __ gik (at+x) aikat aikr cos 9 / J\ 
 
 The general expansion of <f> in Fourier's series may be readily 
 effected, the coefficients of the various terms being, as might 
 be anticipated, simply the BessePs functions of corresponding 
 orders. [Thus, as in (12) 272 a, 
 
 gib- cue = J 9 (kr) + 2iJ,(kr) cos d + . . . + 2i n J n (kr) cos nd + ....] 
 
 But, as we confine ourselves here to the case where c the radius of 
 the cylinder is small, we will at once expand in powers of r. 
 
 Thus, when r = c, if e ikat be omitted, 
 
 = 1-1&V 2 + ike. cos 0+ ................. (2), 
 
 .Qs0 + ..................... (3). 
 
 The amount and even the law of the disturbance depends upon 
 the character of the obstacle. We will begin by supposing the 
 material of the cylinder to be a gas of density er' and compressi- 
 bility ra'; the solution of the problem for a rigid obstacle may 
 finally be derived by suitable suppositions with respect to a ', m . 
 If k' be the internal value of k, we have inside the cylinder by the 
 condition that the axis is not a source ( 339), 
 
 so that, when r = c, 
 
 ^ (inside) = A (1 - i&V) + A,c (1 - JfcV) . cos . . .(4), 
 
 ^ (inside) = -iA^c 4- ^(1-ffcV) cos <9 ........... (5). 
 
 CX?" 
 
 Outside the cylinder, when r = c > we have by (19), (21) 341, 
 
 COS ^ 
 
310 CYLINDRICAL OBSTACLE. [343. 
 
 The conditions to be satisfied at the surface of separation 
 are thus 
 
 -A&V = -& 2 c 2 + 2J9 ..................... (8), 
 
 (7 + log *) ...... (9), 
 
 B l 
 
 ................ (11), 
 
 from which by eliminating A , A l we get approximately 
 
 (13). 
 
 Thus at a distance from the cylinder we have by (18) and 
 (20) 341, 
 
 27T.7TC 2 ... ,, . fm 7 m a' a /1 . x 
 
 J-Y- e-^^+i^ -^ r + -7 cos 0^ . . .(14). 
 r*X* 2m -- 
 
 Hence, corresponding to the primary wave 
 
 2 
 ^008^((rf + ?) ..................... (15), 
 
 A. 
 
 the scattered wave is approximately 
 
 2?r . 7TC 2 fm' m cr' cr J 2?r . , 
 
 cos * (erf - r - 
 
 The fact that -^ varies inversely as X^ might have been 
 anticipated by the method of dimensions, as in the corresponding 
 problem for the sphere ( 296). As in that case, the symmetrical 
 part of the divergent wave depends upon the variation of com- 
 pressibility, and would disappear in the application to an actual 
 
343.] PASSAGE OF SOUND THROUGH FABRICS. 311 
 
 gas ; and the term of the first order depends upon the variation of 
 density. 
 
 By supposing a and in' to become infinite, in such a manner 
 that their ratio remains finite, we obtain the solution corresponding 
 to a rigid and immoveable obstacle, 
 
 )co8(ai-r-iX) ...... (17). 
 
 The exceeding smallness of the obstruction offered by fine 
 wires or fibres to the passage of sound is strikingly illustrated 
 in some of Tyndall's experiments. A piece of stiff' felt half an 
 inch in thickness allows much more sound to pass than a wetted 
 pocket-handkerchief, which in consequence of the closing of 
 its pores behaves rather as a thin lamina. For the same reason 
 fogs, and even rain and snow, interfere but little with the free 
 propagation of sounds of moderate wave-length. In the case 
 of a hiss, or other very acute sound, the effect would perhaps 
 be apparent. 
 
 [The partial reflections from sheets of muslin may be utilized 
 to illustrate an important principle. If a pure tone of high 
 (inaudible) pitch be reflected from a single sheet so as to impinge 
 upon a sensitive flame, the intensity will probably be insufficient 
 to produce a visible effect. If, however, a moderate number of 
 such sheets be placed parallel to one another and at such equal 
 distances apart that the partial reflections agree in phase, then 
 the flame may be powerfully affected. The parallelism and 
 equidistance of the sheets may be maintained mechanically by 
 a lazy-tongs arrangement, which nevertheless allows the common 
 distance to be varied. It is then easy to trace the dependence of 
 the action upon the accommodation of the interval to the wave 
 length of the sound. Thus, if the incidence were perpendicular, 
 the flame would be most powerfully influenced when the interval 
 between adjacent sheets was equal to the half wave length ; 
 and although the exigencies of experiment make it necessary 
 to introduce obliquity, allowance for this is readily made 1 .] 
 
 1 Iridescent Crystals, Proc. Roy. Inst. April 1889. See also Phil. Mag. vol. xxiv. 
 p. 145, 1887 ; vol. xxvi. p. 256, 1888. 
 
CHAPTER XIX. 
 
 FRICTION AND HEAT CONDUCTION. 
 
 344. THE equations of Chapter XL and the consequences that 
 we have deduced from them are based upon the assumption ( 236), 
 that the mutual action between any two portions of fluid separated 
 by an imaginary surface is normal to that surface. Actual fluids 
 however do not come up to this ideal ; in many phenomena the 
 defect of fluidity, usually called viscosity or fluid friction, plays an 
 important and even a preponderating part. It will therefore be 
 proper to inquire whether the laws of aerial vibrations are sensibly 
 influenced by the viscosity of air, and if so in what manner. 
 
 In order to understand clearly the nature of viscosity, let us 
 conceive a fluid divided into parallel strata in such a manner that 
 while each stratum moves in its own plane with uniform velocity, 
 a change of velocity occurs in passing from one stratum to another. 
 The simplest supposition which we can make is that the velocities 
 of all the strata are in the same direction, but increase uniformly 
 in magnitude as we pass along a line perpendicular to the planes 
 of stratification. Under these circumstances a tangential force 
 between contiguous strata is called into play, in the direction of 
 the relative motion, and of magnitude proportional to the rate at 
 which the velocity changes, and to a coefficient of viscosity, com- 
 monly denoted by the letter p. Thus, if the strata be parallel to 
 xy and the direction of their motion be parallel to y, the tangential 
 force, reckoned (like a pressure) per unit of area, is 
 
 dv 
 
 The dimensions of /*, are 
 
 The examination of the origin of the tangential force belongs 
 to molecular science. It has been explained by Maxwell in ac- 
 
344.1 FLUID FRICTION. 313 
 
 x^ 
 
 cordance with the kinetic theory of gases as resulting from inter- 
 change of molecules between the strata, giving rise to diffusion of 
 momentum. Both by theory and experiment the remarkable 
 conclusion has been established that within wide limits the force 
 is independent of the density of the gas. For air at Centigrade 
 Maxwell l found 
 
 ^ = 0001878(1 + '00366(9) ..................... (2), 
 
 the centimetre, gramme, and second being units. 
 
 345. The investigation of the equations of fluid motion in 
 which regard is paid to viscous forces can scarcely be considered 
 to belong to the subject of this work, but it may be of service 
 to some readers to point out its close connection with the more 
 generally known theory of solid elasticity. 
 
 The potential energy of unit of volume of uniformly strained 
 isotropic matter may be expressed 2 
 
 V= |raS 2 + \n (e- +f 2 +g* - 2fg - 2ge - 2ef+ a? + 6 2 + c 2 ) 
 = ^^ 3 + i^(2e 2 + 2/ 2 + 2^ 2 -|S 2 + a 2 + 6 2 + c 2 ) ......... (1), 
 
 in which B(= e+f+g) is the dilatation, e,f, g, a, b, c are the six 
 components of strain, connected with the actual displacements 
 a, ft, 7 by the equations 
 
 da. , d/3 dy 
 
 - + , b= + , c = + . ...(3), 
 
 dz dy dx dz dy dx 
 
 and m, n, K are constants of elasticity, connected by the equation 
 K m ^n ................................. (4), 
 
 of which n measures the rigidity, or resistance to shearing, and K 
 measures the resistance to change of volume. The components of 
 stress P, Q, R, S, T, U. corresponding respectively to e,f, g, a, b, c, 
 are found from V by simple differentiation with respect to those 
 quantities ; thus 
 
 P = rc8 + 2n(e-%S), &c ................... (5), 
 
 S = na, &c ..................................... (6). 
 
 1 On the Viscosity or Internal Friction of Air and other Gases. Phil. Trans. 
 vol. 156, p. 249, 1866. 
 
 2 Thomson and Tait's Natural Philosophy. Appendix C. 
 
314 EQUATIONS OF MOTION. [345. 
 
 If X, Y, Z be the components of the applied force reckoned per 
 unit of volume, the equations of equilibrium are of the form 
 
 dP dU dT 
 
 -T-+- r - + -T- + A r = 0, &c ................ (7), 
 
 dx dy dz 
 
 from which the equations of motion are immediately obtainable 
 by means of D'Alembert's principle. In terms of the displace- 
 ments a, 0, 7, these equations become 
 
 (8), 
 
 ~ da. d6 dy 
 
 where fr- + p+ ' 
 
 dx dy dz 
 
 In the ordinary theory of fluid friction no forces of restitution 
 are included, but on the other hand we have to consider viscous 
 forces whose relation to the velocities (u, v, w) of the fluid elements 
 is of precisely the same character as that of the forces of restitution 
 to the displacements (a, /3, 7) of an isotropic solid. Thus if ' be 
 the velocity of dilatation, so that 
 
 s , = d p dw_ 
 dx dy dz 
 
 the force parallel to x due to viscosity is, as in (8), 
 
 So far K and n are arbitrary constants ; but it has been argued 
 with great force by Prof. Stokes, that there is no reason why a 
 motion of dilatation uniform in all directions should give rise to 
 viscous force, or cause the pressure to differ from the statical pres- 
 sure corresponding to the actual density. In accordance with this 
 argument we are to put K = ; and, as appears from (6), n coincides 
 with the quantity previously denoted by //,. The factional terms 
 are therefore 
 
 , d (du dv dw\\ s 
 u + J -y- ( -^ + -=- + -=- )\ , &c. ; 
 dx\dx dy dz /] 
 
 and ( 237) the equations of motion take the form 
 
 Du v \ dp _ d /du dv dw\ _ 
 
 -f--X) +-f--f A V*u-$u,- r -(- r + T- + -," =0 ...... (12); 
 
 Dt J dx ^dx\dx dy dz J 
 
345.] PLANE WAVES. 315 
 
 or, if there be no applied forces and the square of the motion be 
 neglected, 
 
 du dp _., d (du dv dw\ 
 
 Po -JI + ^T-^ ~u-u, -j- [-J- + -j-+ r - = ......... (13). 
 
 dt dx r dx\dx dy dz) 
 
 We may observe that the dissipative forces here considered 
 correspond to a dissipation function, whose form is the same with 
 respect to u, v, w as that of V with respect to a, /3, 7, in the theory 
 of isotropic solids. Thus putting /c = 0, we have from (1) 
 
 du dv dwY 
 
 ..... (14), 
 
 dy J \dx dz] \dy dx) J 
 
 in agreement with Prof. Stokes' calculation 1 . The theory of friction 
 for the case of a compressible fluid was first given by Poisson 2 . 
 
 346. We will now apply the differential equations to the in- 
 vestigation of plane waves of sound. Supposing that v and w are 
 zero and that u, p, &c. are functions of x only, we obtain from 
 (13) 345 
 
 du dp 4pd*u_ 
 P dt+dx T3^~ 
 
 The equation of continuity (3) 238 is in this case 
 
 and the relation between the variable part of the pressure $p and 
 the condensation s is as usual ( 244) 
 
 Bp = a-p s ........................... (3). 
 
 Thus, eliminating &p and s between (1), (2), (3), we obtain 
 
 _, _ 
 
 dt* d& 3p da,*dt~ 
 
 which is the equation given by Stokes 3 . 
 
 Let us now inquire how a train of harmonic waves of wave- 
 length X, which are maintained at the origin (x = 0), fade away 
 
 1 Cambridge Transactions, vol. ix. 49, 1851. 
 
 2 Journal de VEcole Poly technique, t. xin. cah. 20, p. 139. 
 8 Cambridge Transactions, vol. vin. p. 287, 1845. 
 
316 EFFECTS OF FRICTION. [346. 
 
 as x increases. Assuming that u varies as e int , we find as in 
 148, 
 
 y = Ae~** cos (nt fix) ..................... (5), 
 
 
 In the application to air at ordinary pressures fjt, may be con- 
 sidered to be a very small quantity and its square may be 
 neglected. Thus 
 
 It appears that to this order of approximation the velocity of 
 sound is unaffected by fluid friction. If we replace n by 27ra\~\ 
 the expression for the coefficient of decay becomes 
 
 8?r> ,. 
 
 = 3V^i .................. 
 
 shewing that the influence of viscosity is greatest on the waves of 
 short wave-length. The amplitude is diminished in the ratio 
 e : 1, when x = ar l . In c. G. s. measure we may take 
 
 p = -0013, M= -00019, a = 33200; 
 whence ^=8800X 2 ........................... (9). 
 
 Thus the amplitude of waves of one centimetre wave-length is 
 diminished in the ratio e : 1 after travelling a distance of 88 
 metres. A wave-length of 10 centimetres would correspond nearly 
 to g lv ; for this case x = 8800 metres. It appears therefore that at 
 atmospheric pressures the influence of friction is not likely to be 
 sensible to ordinary observation, except near the upper limit of the 
 musical scale. The mellowing of sounds by distance, as observed in 
 mountainous countries, is perhaps to be attributed to friction, by 
 the operation of which the higher and harsher components are 
 gradually eliminated. It must often have been noticed that the 
 sound s is scarcely, if at all, returned by echos, and I have found ] 
 that at a distance of 200 metres a powerful hiss loses its character, 
 even when there is no reflection. Probably this effect also is due 
 to viscosity. 
 
 1 Acoustical Observations, Phil. Mag. vol. in. p. 456, 1877. 
 
346.] TRANSVERSE VIBRATIONS. 317 
 
 In highly rarefied air the value of a as given in (8) is much 
 increased, //, being constant. Sounds even of grave pitch may then 
 be affected within moderate distances. 
 
 From the observations of Colladon in the lake of Geneva it 
 would appear that in water grave sounds are more rapidly damped 
 than acute sounds. At a moderate distance from a bell, struck 
 under water, he found the sound short and sharp, without musical 
 character. 
 
 347. The effect of viscosity in modifying the motion of air in 
 contact with vibrating solids will be best understood from the solu- 
 tion of the problem for a very simple case given by Stokes. Let us 
 suppose that an infinite plane (yz) executes harmonic vibrations in 
 a direction (y) parallel to itself. The motion being in parallel 
 strata, u and w vanish, and the variable quantities are func- 
 tions of x only. The first of equations (13) 345 shews that the 
 pressure is constant ; the corresponding equation in v takes the 
 
 form 
 
 dv_ptfv 
 
 dt~ p da?" 
 
 similar to the equation for the linear conduction of heat. If we 
 now suppose that v is proportional to e int , the resulting equation 
 in x is 
 
 and its general solution 
 
 (3), 
 
 where in = 
 
 If the gas be on the positive side of the vibrating plane the motion 
 is to vanish when x = + oo . Hence B = 0, and the value of v 
 becomes on rejection of the imaginary part 
 
 (5), 
 corresponding to the motion 
 
 V=Acoant ........................... (6) 
 
 at x = 0. The velocity of the fluid in contact with the plane is 
 usually assumed to be the same as that of the plane itself on the 
 
318 PROPAGATION OF SOUND [347. 
 
 apparently sufficient ground that the contrary would imply an 
 infinitely greater smoothness of the fluid with respect to the solid 
 than with respect to itself. On this supposition (5) expresses the 
 motion of the fluid on the positive side due to a motion of the 
 plane given by (6). 
 
 The tangential force per unit area acting on the plane is 
 
 or ^ 
 
 if A = 1. The first term represents a dissipative force tending to 
 stop the motion ; the second represents a force equivalent to an 
 increase in the inertia of the vibrating body. The magnitude of 
 both forces depends upon the frequency of the vibration. 
 
 We will apply this result to calculate approximately the velocity 
 of sound in tubes so narrow that the viscosity of air exercises a 
 sensible influence. As in 265, let X denote the total transfer of 
 fluid across the section of the tube at the point x. The force, 
 due to hydrostatic pressure, acting on the slice between x and 
 x + Bx is, as usual, 
 
 The force due to viscosity may be inferred from the investigation 
 for a vibrating plane, provided that the thickness of the layer of 
 air adhering to the walls of the tube be small in comparison with 
 the diameter. Thus, if P be the perimeter of the tube, and V be 
 the velocity of the current at a distance from the walls of the 
 tube, the tangential force on the slice, whose volume is SSa, is 
 
 by (7) 
 
 7 ~y 
 
 or on replacing V by -=- -H S 
 dt 
 
 (9). 
 
 The equation of motion for this period is therefore 
 
 PBx/dX ld*X\ , d* 
 
347.] IN NARROW TUBES. 319 
 
 dt 2 { S\ 
 The velocity of sound is approximately 
 
 or in the case of a circular tube of radius r, 
 
 The result expressed in (12) was first obtained by Helmholtz. 
 
 348 *. In the investigation of Kirchhoff 2 , to which we now 
 proceed, account is taken not only of viscosity but of the equally 
 important effects arising from the generation of heat and its 
 communication by conduction to and from the solid walls of a 
 narrow tube. 
 
 The square of the motion being neglected, the "equation of 
 continuity " (3) 237 is 
 
 ds du dv dw . 
 
 -r:+-7-+;T- + -i- = .................. (1); 
 
 dt dx dy dz 
 
 so that the dynamical equations (13) 345 may be written in the 
 form 
 
 ^ 1^_/, A _^_ 
 
 dt^ptdx p. ^3p dxdt" 
 
 The thermal questions involved have already been considered 
 in 247. By equation (4) 
 
 where v is a constant representing the thermometric conductivity. 
 
 By (3) 247 
 
 p/p Q = 6 2 (1 + s + a0) ..................... (4), 
 
 in which b denotes Newton's value of the velocity of sound, viz. 
 V(PO//OO)- If we denote Laplace's value for the velocity by a, 
 
 aY& 2 = 7 = l + a/3 ........................ (5), 
 
 so that = (a 2 - 6 2 )/6 2 a ........................ (6). 
 
 1 This and the following appear for the first time in the second edition. 
 The first edition closed with 348, there devoted to the question of dynamical 
 similarity. 
 
 2 Pogg. Ann. vol. cxxxiv., p. 177, 1868. 
 
320 KIRCHHOFF'S INVESTIGATION. [348. 
 
 It will simplify the equations if we introduce a new symbol & in 
 place of 6, connected with it by the relation 6' = 0/0. Thus (3) 
 becomes 
 
 and the typical equation (2) may be written 
 
 where // is equal to fi/p . p" represents a second constant, whose 
 value according to Stokes' theory is ^//. This relation is in 
 accordance with Maxwell's kinetic theory, which on the intro- 
 duction of more special suppositions further gives 
 
 " = 4/ .............................. (9). 
 
 In any case //, ///', v may be regarded as being of the same order 
 of magnitude. 
 
 We will now, following Kirchhoff closely, introduce the suppo- 
 sition that the variables u, v, w, s, 6' are functions of the time on 
 account only of the factor ^ f , where h is a constant to be after- 
 wards taken as imaginary. Differentiations with respect to t are 
 then represented by the insertion of the factor h, and the equations 
 become 
 
 = Q ............ (10), 
 
 hv - p'V-v = - dPjdy .................. (11), 
 
 P = (b* + hv,")s + (a*-b>)6' ............... (12), 
 
 8 = ff -(v/h)?*? ..................... (13). 
 
 By (13), if s be eliminated, (12) and (10) become 
 
 V*0' ......... (14), 
 
 + + + /,0'_^' = 0.. ,.(15). 
 
 dx dy dz 
 
 By differentiation of equations (11) with respect to x, y, z, 
 with subsequent addition and use of (14), (15), we find as the 
 equation in & 
 
 h*ff - a * + h ' + "+ v V'0' + &+ h 
 
348.] EFFECTS OF HEAT CONDUCTION. 321 
 
 A solution of (16) may be obtained in the form 
 
 0' = A<3i + A& ..................... (17), 
 
 where Q lt Q- 2 are functions satisfying respectively 
 
 V'Q^X^, V*Q 2 = X 2 Q 2 ............ (18), 
 
 X 1} X 2 being the roots of 
 
 # _ [ a * + h (/*' + /*" + P)} X 4- {6 2 + h (/*' + /*")} X 2 = 0. . .(19), 
 
 while A lt A 2 denote arbitrary constants. 
 
 In correspondence with this value of 0', particular solutions of 
 equations (11) are obtained by equating u, v, w to the differential 
 coefficients of 
 
 AQi + 5Ci, 
 
 taken with respect to x, y, z. The relation of the constants B l , B< 2 
 to A lt A 2 appears at once from (15), which gives 
 
 so that by (18) 
 
 More general solutions may be obtained by addition to u, v, w 
 respectively of u', v', w', where u', v', w' satisfy 
 
 W = A u ' V 2 !;' = 4 tf, VV = -, w' . . . (21). 
 ppf* 
 
 Thus 
 
 u = u' +B l dQ l jdx + BtdQJdx \ 
 
 ............ (22), 
 
 w = w' + BtdQJdz + B 2 dQ 2 /dz ) 
 where B lt B z have the values above given. 
 
 By substitution in (15) of the values of u, v, w specified in (22) 
 it appears that 
 
 ^' + *' + ^' = ..................... (23). 
 
 dx dy dz 
 
 349. These results are first applied by Kirchhoff to the case 
 of plane waves, supposed to be propagated in infinite space in 
 the direction of +#. Thus v' and w' vanish, while u', Q l} Q 2 are 
 independent of y and z. It follows from (23) 348 that u' also 
 vanishes. The equations for Q l and Q 2 are 
 
 dQ 1 /cfa? = \ 1 Q 1 , d*QJdtf = \ 2 Q z ......... (1); 
 
 R. H. 21 
 
322 PLANE WAVES. [349. 
 
 so that we may take 
 
 ft --*', ft --*' ............... (2), 
 
 where the signs of the square roots are to be so chosen that the 
 real parts are positive. Accordingly 
 
 "* ...... (3), 
 
 (4), 
 
 in which the constants A lt A 2 may be regarded as determined by 
 the values of u and & when x 0. 
 
 The solution, as expressed by (3), (4), is too general for our 
 present purpose, providing as it does for arbitrary communication 
 of heat at x = 0. From the quadratic in X, (19) 348, we see that 
 if ft, p", v be regarded as small quantities, one of the values of X, 
 say X 1? is approximately equal to A 2 /a 2 , while the other \z is very 
 great. The solution which we require is that corresponding to Xi 
 simply. The second approximation to X x is by (19) 348 
 
 _ _ _ J^ _ yfcV _ h? ( _ h(ji'+fi,"+v)} vfrh* 
 l ~a? + h(fjL' + n" + v)* ha z ~a 2 } a 2 j" 1 a 6 
 
 so that VXx = - - ^ W + p" + v(l- 6 2 /a 2 )} ......... (5). 
 
 a ACL 
 
 If we now write in for h, we see that the typical solution is 
 
 r * e in(t-x/a) ........................ g 
 
 where 
 
 In (6) an arbitrary multiplier and an arbitrary addition to t 
 may, as usual, be introduced ; and, if desired, the solution may be 
 realized by omitting the imaginary part. 
 
 These results are in harmony with those already obtained for 
 particular cases. Thus, if v = 0, (7) gives 
 
 in agreement with (7) 346, where 
 
 ' On the other hand if viscosity be left out of account, so that 
 p = p" = 0, we fall back upon (18) 247. It is unnecessary to add 
 anything to the discussions already given. 
 
349.] SYMMETRY BOUND AN AXIS. 323 
 
 In the case of spherical waves, propagated in the direction of 
 + r, Kirchhoff finds in like manner as the expression for the radial 
 velocity 
 
 where m' has the same value (7) as before. 
 
 350. We will now pass on to the more important problem and 
 suppose that the air is contained in a cylindrical tube of circular 
 section, and that the motion is symmetrical with respect to the 
 axis of x. If y* + z* = r 2 , and 
 
 v = q . yjr, w = q. z\r, 
 
 v'=q'.ylr, w' = q.zfr y 
 
 then u, u', q, q, Q 1 , Q 2 are to be regarded as functions of x and r. 
 We suppose further that as functions of x these quantities are 
 proportional to e mx , where m is a complex constant to be deter- 
 mined. The equations (18) 348 for Q lt Q 2 become 
 
 < 
 
 For u' y q' equations (21), (23) give 
 
 d?u' , Idu' fh \ , 
 
 + _ = _ _ m s \ u f ............... (3) 
 
 dr 2 r dr \/j, ) 
 
 + = ....................... (5). 
 
 dr r 
 
 These three equations are satisfied if u' be determined by 
 means of the first, and q' is chosen so that 
 
 ,_ m du!_ , fi , 
 
 q = ~' 2 " 
 
 a, relation obtained by subtracting from (4) the result of differen- 
 tiating (5) with respect to r. The solution of (3) may be written 
 u' = A Q, in which A is a constant, and Q a function of r satisfying 
 
 J -(?-) .................. <" 
 
 212 
 
324 CIRCULAR TUBE. [350. 
 
 Thus, by (20), (22) 348, 
 
 u = AQ-A l m^p)Q l -A,m^-^Q, ...... (8), 
 
 m dQ . h \dQ, h \dQ, 
 
 0'^ft + ^ft ..................... (10). 
 
 On the walls of the tube u, q, 6' must satisfy certain conditions. 
 It will here be supposed that there is neither motion of the gas 
 nor change of temperature ; so that when r has a value equal to 
 the radius of the tube, u, q, 6' vanish. The condition of which \ve 
 are in search is thus expressed by the evanescence of the determi- 
 nant of (8), (9), (10), viz. : 
 
 _m?h^ /!__ lA 
 
 A/p'-'Ut " 
 
 _ \ d log ft 
 
 ~~ v ) ~w 
 
 .o ...... (ID. 
 
 dr 
 
 The three functions ft ft, ft, which are required to remain finite 
 when r = 0, are Bessel's functions of order zero ( 200), so that we 
 may write in the usual notation 
 
 ) 
 
 (12). 
 
 In equation (11) the values of \ lt X 2 are independent of ?% 
 being determined by (19) 348. In the application to air under 
 normal conditions yu/, p", v may be regarded as small, and we have 
 approximately 
 
 X 1 = /t 2 /a 2 , \ 2 = /ia 2 /^ 2 ............... (13). 
 
 A second approximation to the value of Xj has already been given 
 in (5) 349. It is here assumed that the velocity of propagation 
 of viscous effects of the pitch in question, viz. ^(2firi), 347, is 
 small compared with that of sound, so that tn^t'/a*, or hfjf/a 2 , is a 
 small quantity. 
 
 In interpreting the solution there are two extreme cases 
 worthy of special notice. The first of these, which is that 
 considered by Kirchhoff, arises when //, /u,", v are treated as very 
 small, so small that the layer of gas immediately affected by the 
 walls of the tube is but an insignificant fraction of the whole 
 contents. When // &c. vanish, we have 
 
350.] VISCOSITY SMALL. 325 
 
 so that r\/(m 2 X]) is here to be regarded as small. On the other 
 hand r*J(m* h/fi), r*J(m* \^) are large. 
 
 The value of J (z\ when z is small, is given by the ascending 
 series (5) 200 ; from which it follows at once that 
 
 d log JQ (z)jdz = \z. 
 
 When z is very large and such that its imaginary part is positive, 
 (10) 200 gives 
 
 d log JQ (z)jdz = tan (z JTT) = i. 
 Thus, if we retain only the terms of highest order, 
 
 d\ogQ l ldr = r(\ 1 -m*) ............... (14). 
 
 dlog Q. 2 /dr = J(ha?lvb 2 ) J 
 
 Using these in (11) with the approximate values of X l5 X 2 from 
 (13), we find 
 
 ..................... <> 
 
 where y^*/p' + (a/b b/a)/v .................. (16), 
 
 and the sign of \Jh is to be so chosen that the real part is positive. 
 
 We now write 
 
 h = ni ..................... . ........ (17), 
 
 so that the frequency is n/^w. Thus 
 
 Jh = ^n).(l+i) ...................... (18); 
 
 and m=(m / + im // ) ......................... (19), 
 
 where by (15) 
 
 ' = ^, bt"= = + ^ ............ (20). 
 
 V2 . ar a V2 . ar 
 
 If we restore the hitherto suppressed factors dependent upon as 
 and t, we have 
 
 u = BRe ht+mx , q = BR^ t+mx , & = BR"e ht+mx , 
 
 where B is an arbitrary constant, and R, R, R" are certain 
 functions of r, which vanish when r is equated to the radius of 
 the tube, and which for points lying at a finite distance from the 
 walls assume the values 
 
326 VELOCITY OF SOUND. [350. 
 
 The realized solution for u, applicable at points which lie at a 
 finite distance from the walls, may be written 
 
 u == Citf*'* sin (nt + m"x + &) + C. 2 e~ m ' x sin (nt - m"x + &>). . .(21), 
 
 where C lt (7 2 , 81, 8 2 denote four real arbitrary constants. Ac- 
 cordingly m' determines the attenuation which the waves suffer 
 in their progress, and m" determines the velocity of propagation. 
 This velocity is 
 
 ^"=4-vofc} (22) ' 
 
 in harmony with (12) 347. 
 
 The diminution of the velocity of sound in narrow tubes, as 
 indicated by the wave-length of stationary vibrations, was observed 
 by Kundt ( 260), and has been specially investigated by Schnee- 
 beli 1 and A. Seebeck 2 . From their experiments it appears that 
 the diminution of velocity varies as r~ 1 , in accordance with (22), 
 but that, when n varies, it is proportional rather to n~% than to 
 ?rt Since //, is independent of the density (/>), the effect would 
 be increased in rarefied air. 
 
 We will now turn to the consideration of another extreme case 
 of equation (11). This arises when the tube is such that the 
 layer immediately affected by the friction, instead of merely 
 forming a thin coating to the walls, extends itself over the whole 
 section, as must inevitably happen if the diameter be sufficiently 
 reduced. Under these circumstances /ir 2 /// is a small, and not, as 
 in the case treated by Kirchhoff, a large quantity, and the argu- 
 ments of all the three functions in (12) are to be regarded as 
 small. 
 
 One result of the investigation may be foreseen. When the 
 diameter of the tube is very much reduced, the conduction of heat 
 from the centre to the circumference of the column of air becomes 
 more and more free. In the limit the temperature of the solid 
 walls controls that of the included gas, and the expansions and 
 rarefactions take place isothermally. Under these circumstances 
 there is no dissipation due to conduction, and everything is the 
 same as if no heat were developed at all. Consequently the 
 coefficient of heat-conduction will not appear in the result, which 
 
 1 Fogg. Ann. vol. cxxxvi. p. 296, 1869. 
 
 2 Pogg. Ann. vol. cxxxix. p. 104, 1870. 
 
350.] EXCEEDINGLY NARROW TUBES. 327 
 
 will involve, moreover, the Newtonian value (6) of the velocity of 
 sound, and not that of Laplace (a). 
 
 When z is small, 
 
 so that approximately 
 
 d log J" (z)/dz = - \z (1 + \z 1 } (23). 
 
 When the results of the application of (23) to Q, Q lf Q 2 are 
 introduced into (11), the equation may be divided by Jr, and the 
 left-hand member will then consist of two parts, of which the first 
 is independent of r and the second is proportional to r 2 . The first 
 part reduces itself without further approximations to v (X 2 ~ ^i). 
 For the second part the leading terms only need be retained. 
 Thus with use of (13) 
 
 , . a 2 r 2 Cm 2 h 2 (a 2 b 2 )\ 
 
 \^ / 
 
 whence ra 2 
 
 The ratio of the second term to the first is of the order hr 2 /v, by 
 supposition a small quantity, so that we are to take simply 
 
 as the solution applicable under the supposed conditions. 
 
 Before leaving this question it may be worth while to consider 
 briefly the corresponding problem in two dimensions, although it 
 is of less importance than that of the circular tube treated by 
 Kirchhoff. The analysis is a little simpler; but, as it follows 
 practically the same course, we may content ourselves with a mere 
 indication of the necessary changes. The motion is supposed to 
 be independent of z and to take place between parallel walls at 
 
 y = yi- 
 
 The equations (1) to (11) of the preceding investigation may 
 be regarded as still applicable in the present problem, if we write 
 v for q and y for r, with omission of the terms where r occurs in 
 the denominator. The general solution of the equations corre- 
 sponding to (1), (2), (7) contains two functions whose form is that 
 of sines and cosines of multiples of y. But from (8), (9), (10) it 
 is evident that the conditions of the problem at y = require the 
 
328 TWO DIMENSIONS. [351. 
 
 absence of the sine function, so that in (12) we are simply to 
 replace the function J" by the cosine. 
 
 In the case where /// &c. are regarded as infinitely small we 
 have as in (14), when y = y lt 
 
 d 
 
 but in place of the second of equations (14) 
 
 When these values are substituted in (11), the resulting equation 
 is unchanged, except that r is replaced by 2^. The same substi- 
 tution is to be made in (15), (20), (22). The latter gives for the 
 velocity of sound 
 
 (27). 
 
 It is worth notice that (27) is what (11) 347 becomes for 
 this case when we replace V/*' by 7' ; and we may perhaps infer 
 that the same change is sufficient to render that equation ap- 
 plicable to a section of any form when thermal effects are to be 
 taken into account. 
 
 In the second extreme case where the distance between the 
 walls (2y x ) is so small that hy^jv is to be neglected, we have in 
 place of (23) 
 
 (28). 
 
 The equations following are thus adapted to our present 
 purpose if we replace Jr 2 by Ji/j 2 . The analogue of (24) is ac- 
 cordingly 
 
 351. The results of 350 have an important bearing upon 
 the explanation of the behaviour of porous bodies in relation to 
 sound. Tyndall has shewn that in many cases sound penetrates 
 such bodies more freely than would have been expected, although 
 it is reflected from thin layers of continuous solid matter. On 
 the other hand a hay-stack seems to form a very perfect obstacle. 
 It is probable that porous walls give a diminished reflection, so 
 that within a building so bounded resonance is less prolonged 
 than if the walls were formed of continuous matter. 
 
351.] POROUS WALL. 329 
 
 When we inquire into the mechanical question, it is evident 
 that sound is not destroyed by obstacles as such. In the absence 
 of dissipative forces, what is not transmitted must be reflected. 
 Destruction depends upon viscosity and upon conduction of heat ; 
 but the influence of these agencies is enormously augmented by 
 the contact of solid matter exposing a large surface. At such a 
 surface the tangential as well as the normal motion is hindered, 
 and a passage of heat to and fro takes place, as the neighbouring 
 air is heated and cooled during its condensations and rarefactions. 
 With such rapidity of alternation as we are concerned with in the 
 case of audible sounds, these influences extend to only a very thin 
 layer of the air and of the solid, and are thus greatly favoured by 
 a fine state of division. 
 
 Let us conceive an otherwise continuous wall, presenting a 
 flat face, to be perforated by a great number of similar narrow 
 channels, uniformly distributed, and bounded by surfaces every- 
 where perpendicular to the face of the wall. If the channels be 
 sufficiently numerous, the transition, when sound impinges, from 
 simple plane waves on the outside to the state on the inside of 
 aerial vibration corresponding to the interior of a channel of 
 unlimited length, occupies a space which is small relatively to 
 the wave-length of the vibration, and then the connection between 
 the condition of things inside and outside admits of simple ex- 
 pression. 
 
 Considering first the interior of one of the channels, and 
 taking the axis of x parallel to the axis of the channel, we suppose 
 that as functions of x the velocity components u, v, w and the 
 condensation s are proportional to e ikx , while as functions of t 
 everything is proportional to e int , n being real. The relationship 
 between k and n depends upon the nature of the gas and upon 
 the size and form of the channel, and has been determined for 
 certain important cases in 350, ik being there denoted by m. 
 Supposing it to be known, we will go on to shew how the problem 
 of reflection is to be dealt with. 
 
 For this purpose consider the equation of continuity as 
 integrated over the cross-section a of the channel. Since the 
 walls of the channel are impenetrable, 
 
 so that nf/sd<T + kf/ud<r = ..................... (1). 
 
330 KEFLECTION OF SOUND [351. 
 
 This equation is applicable at points distant from the open end 
 more than several diameters of the channel. 
 
 Taking now the origin of x at the face of the wall, we have to 
 form corresponding expressions for the waves outside ; and we 
 may there neglect the effects of viscosity of conduction of heat. 
 If a be the velocity of sound in the open, and k -= n/a, we may 
 write for waves incident and reflected perpendicularly 
 
 s = (6*** + Be~ ik * x ) e ini ..................... (2), 
 
 e** ................. (3); 
 
 so that the incident wave is 
 
 or, on throwing away the imaginary part, 
 
 s = cos (nt + k x) ........................ (5). 
 
 These expressions are applicable when x exceeds a moderate 
 multiple of the distance between the channels. Close up to the 
 face the motion will be more complicated ; but we have no need 
 to investigate it in detail. The ratio of u and s at a place near 
 the wall- is given with sufficient accuracy by putting x = in (2) 
 and (3), 
 
 We now assume that a space, defined by parallel planes one 
 on either side of # = 0, may be taken so thin relatively to the 
 wave-length that the mean pressures are sensibly the same at the 
 two boundaries, and that the flow into the space at one boundary 
 is sensibly equal to the flow out of the space at the other boundary, 
 and yet broad enough relatively to the transverse dimensions of the 
 channels to allow the application of (6) at one bounding plane and 
 of (1) at the other bounding plane. The equality of flow does not 
 imply an equality of mean velocities, since the areas concerned are 
 different. The mean velocities will be inversely proportional to 
 the corresponding areas that is in the ratio a- : a + a-', if a' denote 
 the area of the unperforated part of the wall corresponding to each 
 channel. By (1) and (6) the connection between the inside and 
 outside motion is expressed by 
 
351.J AT THE FACE OF A POROUS WALL. 331 
 
 We will denote the ratio of the unperforated to the perforated 
 parts of the wall by g, so that g = o-'/o: Thus 
 
 * m 
 
 If $r = 0, k = k , that is, if the wall be abolished, or if it be 
 reduced to infinitely thin partitions between the channels while at 
 the same time the dissipative effects are neglected, there is no 
 reflection. If there are no perforations (g = cc), then B = l, 
 signifying total reflection. Generally in place of (7) we may write 
 
 Q 
 
 -- 
 
 which is the solution of the problem proposed. It is understood 
 that waves which have once entered the wall do not return. 
 When dissipative forces act, this condition may always be satisfied 
 by supposing the channels to be long enough. The necessary 
 length of channel, or thickness of wall, will depend upon the 
 properties of the gas and upon the size and shape of the channels. 
 Even in the absence of dissipative forces there must be reflection, 
 except in the extreme case g = 0. Putting k = k Q in (8), we 
 have 
 
 If g = l, that is if half the wall be cut away, = J, B- = ^ so 
 that the reflection is but small. If the channels be circular and 
 arranged in square order as close as possible to one another, 
 # = (4-7r)/7r, whence 5 = 121, .B 2 ='015, nearly all the motion 
 being transmitted. 
 
 If the channels be circular in section and so small that nr*/v 
 may be neglected, we have, (24) 350, 
 
 so that (21) the wave propagated into a channel is proportional to 
 
 +% l } ..................... (11), 
 
 > 
 
 where ra =m = .--^ -' = -*WLL ' .............. (12), 
 
 or ar 
 
 7 being the ratio of specific heats 246. 
 
 ^ 
 
 T^> CfTTV 
 
332 NUMERICAL EXAMPLES. [351. 
 
 To take a numerical example, suppose that the pitch is 256, 
 so that n = ZTT x 256. The value of /*' for air is '16 C.G.S., and that 
 of v is '256. If we take r = TI ^ cm., we find nr*/8v equal to about 
 -j-^j^. If r were ten times as great, the approximation in (10) 
 would perhaps still be sufficient. 
 
 From (12), if n = 2?r x 256, 
 
 m' = m" = '00115/r ......... . ........... (13); 
 
 so that, if r = T ^ r?r cm., m' = ri5. In this case the amplitude is 
 reduced in the ratio e : I in passing over the distance 1/m', that is 
 about one centimetre. The distance penetrated is proportional to 
 the radius of the channel. 
 
 The amplitude of the reflected wave is by (8) 
 m'( 
 
 "m' 
 
 or, as we may write it, 
 
 Ml iM 
 
 where M=(I+g)m'/k ..................... (15). 
 
 If / be the intensity of the reflected sound, that of the incident 
 
 sound being unity, 
 
 - 
 
 ..................... ( 
 
 The intensity of the intromitted sound is given by 
 By (12), (15) 
 
 If we suppose r = yJ^ cm., and g 1, we shall have a wall 
 of pretty close texture. In this case by (18), Jf=47'4 and 
 17 = '0412. A loss of 4 per cent, may not appear to be im- 
 portant; but we must remember that in prolonged resonance 
 we are concerned with the accumulated effect of a large number 
 of reflections, so that a comparatively small loss in a single re- 
 flection may well be material. The thickness of the porous layer 
 necessary to produce this effect is less than one centimetre. 
 
 Again, suppose r=y^cm.,^=l. We find M =4*74, I-/ ='342; 
 and the necessary thickness would be less than 10 centimetres. 
 
352.] RESONANCE OF BUILDINGS. 333 
 
 If r be much greater than -^ cm., the exchange of heat 
 between the air and the sides of the channel is no longer suffi- 
 ciently free to allow of the use of (24) 350. When the diameter 
 is so great that the thermal and viscous effects extend through 
 only a small fraction of it, we have the case discussed by Kirchhoff 
 (15) 350. Here 
 
 (19), 
 
 which value is to be substituted in (8). If for simplicity we put 
 g = 0, we find 
 
 ya-0 
 
 ~" 
 
 (21). 
 
 The supposition that g = is, however, inconsistent with the 
 circular section ; and it is therefore preferable to use the solution 
 corresponding to (27) 350, applicable when the channels assume 
 the form of narrow crevasses 1 . We have merely to replace r in 
 (19), (20), (21) by 2y lt 2^ being the width of a crevasse. The 
 incident sound is absorbed more and more completely as the width 
 of the channels increases ; but at the same time a greater length 
 of channel, or thickness of wall, becomes necessary in order to 
 prevent a return from the further side. If g = 0, there is no 
 theoretical limit to the absorption ; and, as we have seen, a 
 moderate value of g does not of itself entail more than a com- 
 paratively small reflection. A loosely compacted hay-stack would 
 seem to be as effective an absorbent of sound as anything likely to 
 be met with. 
 
 In large spaces bounded by non-porous walls, roof, and floor, 
 and with few windows, a prolonged resonance seems inevitable. 
 The mitigating influence of thick carpets in such cases is well 
 known. The application of similar material to the walls and to 
 the roof appears to offer the best chance of further improve- 
 ment. 
 
 352. One of the most curious consequences of viscosity is the 
 generation in certain cases of regular vortices. Of this an example, 
 discovered by Dvorak, has already been mentioned in 260. In 
 
 1 It may be remarked that even in the two-dimensional problem the sup- 
 position g = Q involves an infinite capacity for heat in the material composing 
 the partitions. 
 
334 TWO-DIMENSIONAL EQUATIONS [352. 
 
 a theoretically inviscid fluid no such effect could occur, 240 ; and, 
 even when viscosity enters, the phenomenon is one of the second 
 order, dependent, that is, upon the square of the motion. Three 
 problems of this kind have been treated by the author 1 on a 
 former occasion, but here we must limit ourselves to Dvorak's 
 phenomenon, further simplifying the question by taking the case 
 of two dimensions and by neglecting the terms dependent upon 
 the development and conduction of heat. 
 
 If we suppose that p = a 2 p, and write s for \og(p/p ), the 
 fundamental equations (12) 345 are 
 
 ds du du da ,__ d (du dv\ ._ x 
 
 a?- r = --r t -u- r -v- i - + p'V 2 u + // ' -j- ( -j- + -=- ).. .(1), 
 dx dt dx dy dx\dx dy) 
 
 with a corresponding equation for v, and the equation of continuity 
 238 
 
 du dv ds ds ds 
 
 j- + -3- + ^ + u j-+ v ^r=Q ( 2 )- 
 
 dx dy dt dx dy 
 Whatever may be the actual values of u and v, we may write 
 
 in which 
 
 _ du dv __ , du dv 
 
 V 2 ( = -j- -f -j- , V->Jr = - -=- (4). 
 
 dx dy dy dx 
 
 From (1), (2) 
 ( a *+ "d.}^ = - + 'V* u 
 
 du du d ( ds ds\ 
 
 u-j v -j JJL -j-r (u -j- + v -j-] (o), 
 
 dx dy dx \ dx dy) 
 
 f f f d\ds dv^ ,^ 2 
 
 \ dt) dy dt 
 
 dv dv fl d f ds ds\ 
 dx dy dy\ dx dy) 
 
 Again, from (5), (6), 
 
 ,d , ,,d\r, d^s d (_ds . ds' 
 
 a 
 
 ,,- t . 
 
 - " * " f<tt " V- 
 
 "\n-i ds\ d { du du\ d f dv dv 
 
 " V 2 U-- + V-- --T- j- + V j- -3- ( tt 'j- + T- 
 
 d ay/ dy \ dx dy 
 
 1 On the Circulation of Air observed in Kundt's Tubes, and on some allied 
 Acoustical Problems. ' Phil. Trans, vol. 175, p. 1, 1883. 
 
352.] TO A SECOND APPROXIMATION. 335 
 
 For the first approximation the terms of the second order in 
 u, v t s are to be omitted. If we assume that as functions of t all 
 the periodic quantities are proportional to e int , and write q for 
 a 2 + inp + infi", (7) becomes 
 
 qV 2 s + n z s = Q ........................ (8). 
 
 Now by (2), (4) V 2 < = - ins = i (q/n) V 2 s, 
 
 so that (j> = iqs/n ........................... (9) 1 , 
 
 iq ds d-dr iq ds dty 
 
 and u = -^ -J- + -T-, v = - --T- ~ j ......... ( 10 )- 
 
 n dx ay n dy dx 
 
 Substituting in (5), (6), with omission of the terms of the 
 second order, we get in view of (8), 
 
 whence (/a'V 8 - w)^r = ..................... (11). 
 
 If we eliminate s directly from equations (1), we get 
 
 dv\ 
 
 t 'j- 
 
 d) 
 
 . d A d du du\ d dv 
 - 
 
 j jj j 
 
 dt / T dy \ dx dy] dx\ dx dy 
 
 (12). 
 
 If we now assume that as functions of x the quantities s, ty, &c. 
 are proportional to e ikx , equations (8), (11) may be written 
 
 (d*/df-k"*)s = ..................... (13), 
 
 where k" 2 = k z n*/q, 
 
 (d*/df-k'*)^ = ..................... (14), 
 
 where k' 2 = k z + in/fj,'. 
 
 If the origin for y be in the middle between the two parallel 
 bounding planes, s must be an even function of y, and ty must be 
 an odd function. Thus we may write 
 
 s = A cosh Wy . e int . e ikx , ^ = B sinh k'y . ^ nt . e ikx . . .(15), 
 u = (- kq/n . A cosh k"y 4- k'B cosh k'y) e ini . e ikx 
 v = (iqk"ln . A sinh k"y - ikB sinh k'y) e ini . e ik * 
 
 1 It is unnecessary to add a complementary function <f>' satisfying vV = 0, for 
 the motion corresponding thereto may be regarded as covered by \^. 
 
336 MOTION BETWEEN PARALLEL WALLS. [352. 
 
 If the fixed walls are situated at y y l , u and v must vanish 
 for these values of y. Eliminating from (16) the ratio of A to B, 
 we get as the equation for determining k, 
 
 # tanh %! = #&" tanh Fy! (17), 
 
 where k', k" are the functions of k above defined. Equation (17) 
 may be regarded as a modified and simplified form of (11) 350, 
 modified on account of the change from symmetry about an axis 
 to two dimensions, and simplified by the omission of the thermal 
 terms represented by v. The comparison is readily made. Since 
 X 2 = oo , the third term in (11), involving Q 3 , disappears altogether, 
 and then X^ 1 divides out. In (11), (12) r is to be replaced by y, 
 and Jo by cosine, as has already been explained. Further, 
 
 ra 2 = k~, h = in. 
 
 We now introduce further approximations dependent upon the 
 assumption that the direct influence of viscosity extends through 
 a layer whose thickness is a small fraction only of y lm In this case 
 2 _ f^jo^ nearly, so that k"y is a small quantity and k'y^ is a large 
 quantity, and we may take 
 
 tanh k'y^ = 1, tanh k"y^ = lc"y^. 
 Equation (17) then becomes 
 
 te = k'k"* yi (18), 
 
 or, if we introduce the values of k', k" from (13), (14), 
 
 Thus approximately 
 k 
 
 in agreement with the result already indicated in 350. 
 
 In taking approximate forms for (16) we must specify which 
 half of the symmetrical motion we contemplate. If we choose 
 that for which y is negative, we replace coshAA/ and siuhk'y by 
 %e~ Vy . For cosh k"y we may write unity, and for sinh k"y simply 
 k"y. If we change the arbitrary multiplier so that the maximum 
 value of u is u and for the present take u equal to unity, we have 
 
 u = (- 1 + e-#(y+yi> e ikx e int 
 
 (20), 
 
 , c j 
 
 in which, of course, u and v vanish when y^y^. 
 
352.] FIRST APPROXIMATION. 337 
 
 If in (20) we change k into k and then take the mean, we 
 obtain 
 
 u = (-I+ e~ k> ( y + yJ ) cos kx e int \ 
 
 v = - kjk' . (yjy, + e - k>( y + yJ) sin kx e ini \ ......... 
 
 Although k is not absolutely a real quantity, we may consider it 
 to be so with sufficient approximation for our purpose. We may 
 also take in (14) 
 
 (22), 
 
 if (S = *J(n/2p'). Using this approximation in (21), we get in terms 
 of real quantities, 
 
 u = cos kx [ cos nt + e~& to + vJ cos [nt ft (y + y^ 
 
 &sin kx t 7 / 4 _ 
 
 L..(23). 
 
 ~l*-0( if *y$] i 
 
 It will shorten the expressions with which we have to deal if 
 we measure y from the wall (on the negative side) instead of, as 
 hitherto, from the plane of symmetry, for which purpose we must 
 write y for y + y lt Thus 
 
 U-L = cos kx [ cos nt + e~? y cos (?? /3?/)] 
 
 the subscripts indicating the order of the terms. 
 
 These are the values of the velocities when the square of the 
 motion is neglected. In proceeding to a second approximation we 
 require to form expressions for the right-hand members of (7) and 
 (12), which for the purposes of the first approximation were 
 neglected altogether. The additional terms dependent upon the 
 square of the motion are partly independent of the time and 
 partly of double frequency involving 2nt. The latter are not of 
 much interest, so that we shall confine ourselves to the non- 
 periodic part. Further simplifications are admissible in virtue 
 of the small thickness of the retarded layer in proportion to the 
 width of the channel (2^) and still more in proportion to the 
 wave-length (X). Thus k//3 is a small quantity and may usually 
 be neglected. 
 
 R. II. 22 
 
338 MOTION BETWEEN PARALLEL WALLS [352. 
 
 From (24) 
 
 V-\^ = /3V2 . cos kx e~to sin (nt - JTT - 0y) ...... (25), 
 
 dujdx + dvjdy = k sin kx cos nt ............ (26), 
 
 sn 6- - cos / 
 
 4- terms in 2?i ............... (27), 
 
 \ Ot5? O 
 
 + terms in 2?i# ............... (28). 
 
 Thus for the non-periodic part of \jr of the second order, we 
 have from (12) 
 
 V 4 ^ 2 = - , sin 2Ara? e-^ (sin 0y + 3 cos /Sy - 2e~^} . . . (29 ). 
 In this we identify V 4 with (d/dy) 4 , so that 
 
 sin + 8 cos 
 
 to which may be added a complementary function, satisfying 
 V 4 >/r 2 = 0, of the form 
 
 sinh 2 ^ (yi ~ y} + 5 (yi ~ y} cosh 2 ^(2/i-2/)}---(3i), 
 
 or, as we may take it approximately, if y 1 be small compared 
 with X, 
 
 l -yY} ....... (32). 
 
 Equations (30), (32) give the non-periodic part of ^ of the second 
 order. 
 
 The value of s to a second approximation would have to be 
 investigated by means of (7). It will be composed of two parts, 
 the first independent of t, the second a harmonic function of 2nt. 
 In calculating the part of d<f>/dx independent of t from 
 
 V 2 </> = ds/dt lids Ida; v ds/dy, 
 
 we shall obtain nothing from ds/dt. In the remaining terms on 
 the right-hand side it will be sufficient to employ the values u, v, s 
 of the first approximation. From 
 
 ds/dt du/dx dv/dy, 
 
 
352.] TO A SECOND APPROXIMATION. 339 
 
 in conjunction with (26), we get 
 
 s uja . sin kx sin nt, 
 whence d? <f> 2 /d (/3y) 2 = kul%af& . cos 2 kx e~ fty sin fty. 
 
 From this it is easily seen that the part of u 2 resulting from 
 d<p/dv in (3) is of order %?//& in comparison with the part (33) 
 resulting from i|r 2 , and may be omitted. Accordingly by (30), 
 with introduction of the value of ft and (in order to restore 
 homogeneity) of ?/ 2 , 
 
 , U " ^ "'^- - {4 sin ,% + 8008^ + ^, ...( 3 3), 
 
 , . 
 
 {sin y +3 cos 0y + \e~^\ . . .(34); 
 
 and from (32) 
 
 sn 
 
 cos 
 
 The complete value of the terms of the second order in u, v are 
 given by addition of (33), (35) and of (34), (36). The constants 
 A', B' are to be determined by the condition that these values 
 vanish when y = 0. We thus obtain as the complete expression of 
 the terms of the second order 
 
 cos y 
 
 y\ 
 
 ............... (37), 
 
 _ fe (gin ^ 
 
 Outside the thin film of air immediately influenced by the 
 friction we may put e~^ y = 0, and then 
 
 } 3(y 1 -2/ 
 
 ^7" 
 
 222 
 
340 VORTICES. [352. 
 
 From (39) we see that u. 2 changes sign as we pass from the 
 boundary y to the plane of symmetry y y\, the critical value 
 of y being ^ (1 - Vi), or '423^. 
 
 The value of i^ from (24) corresponding to (39) is 
 
 u l = u Q cos kx cos nt (41), 
 
 so that the loops correspond to kx = 0, TT, 2?r, . . . , and the nodes 
 correspond to kx= |TT, f TT, ... . 
 
 The steady motion represented by (39), (40) is of a very simple 
 character. It consists of a series of vortices periodic with respect 
 to x in the distance JX. From (40) it appears that v is positive 
 at the nodes and negative at the loops, vanishing of course in each 
 case both at the wall y = and at the plane of symmetry y = y^. 
 
 Fig. 65. 
 C- D 
 
 -B 
 
 ITT TT |TT 2r 
 
 In the figure AB represents the wall, CD the plane of symmetry, 
 and the directions of motion in the vortices are indicated by 
 arrows. It is especially to be remarked that the velocity of the 
 vortical motion is independent of /A', so that this effect is not to be 
 obviated by taking the viscosity infinitely small. In that way 
 the tendency to generate the vortices may indeed be diminished, 
 but in the same proportion the maintenance of the vortices is 
 facilitated, so that when the motion has reached a final state the 
 vortices are as important with a small as with a large viscosity. 
 The fact that when viscosity is neglected from the first no such 
 vortices make their appearance in the solution shews what extreme 
 care is required in dealing with problems relating to the be- 
 haviour of slightly viscous fluid in contact with solid bodies. 
 
 In estimating the mean motion to the second order there is 
 another point to be considered which has not yet been touched 
 upon. The values of u^ and v x in (24) are, it is true, strictly 
 periodic, but the same property does not attach to the motions 
 thereby defined of the particles of the fluid. In our notation u is 
 not the velocity of any individual particle of the fluid, but of the 
 particle, whichever it may be, that at the moment under conside- 
 
352.] CIRCULAR TUBE. 341 
 
 ration occupies the position x, y, ( 237). If x + f, y + 77 define 
 the actual position at time t of the particle whose mean position 
 during several vibrations is (a?, y\ then the actual velocities of the 
 particle at time t are, not u l} v lt but 
 
 , d^ du^ dv r .. dv l 
 
 u *+ i t+-zri' v i + j + j *?; 
 dx* dy dx* dy ' 
 
 and thus the mean velocity parallel to a? is not necessarily zero, 
 but is equal to the mean value of 
 
 fdMjdat+iidUi/dy ..................... (42), 
 
 in which again 
 
 ?=/M*, y^fadt ........... ....... (43). 
 
 In the present case the mean value of (42) is 
 
 - u<?/4>a . sin 2kx e~^ (e~M - cos &y) ........... (44), 
 
 which is to be regarded as an addition to (37). However, at a 
 short distance from the wall (44) may be neglected, so that (39) 
 remains adequate. 
 
 We have seen that the width of the direct current along the 
 wall y = is '423^, and that of the return current, measured up 
 to the plane of symmetry, is "5 7 7^. The ratio of these widths is 
 not altered by the inclusion of the second half of the channel 
 lying beyond the plane of symmetry ; so that the direct current is 
 distinctly narrower than the return current. This disproportion 
 will be increased in the case of a tube of circular section. The 
 point under consideration depends in fact only upon a comple- 
 mentary function analogous to (32), and is so simple that it may 
 be worth while briefly to indicate the steps of the calculation. 
 
 The equation for ^ is 1 
 
 but, if we suppose that the radius of the tube is small in compari- 
 son with X, k' 2 may be omitted. The general solution is 
 
 i|r 2 = [A + r 2 + B'r* log r + Or 4 } sin 2knc ....... (46), 
 
 so that 
 
 r = (2 + ff (2 log r + 1) + 4<7r 2 } sin 2kx . . . (47), 
 
 1 Stokes, Trans. Camb. Phil. Soc., vol. ix. 1856; Basset's Hydrodynamics, 
 485. 
 
342 DIRECT AND RETURN CURRENTS. [352. 
 
 whence B' = 0, by the condition at r = 0. Again, 
 
 v. 2 = - d^Jrdas = - 2k {Ar~ l + Br+ Cr 5 } cos 2kx . . .(48), 
 whence A = 0. 
 
 We may therefore take 
 
 (49). 
 
 If, as in (40), ?> 2 = 0, when r = R, B+ CR> = 0, and 
 
 (50). 
 Thus 2 vanishes, when 
 
 The direct current is thus limited to an annulus of thickness 
 293 R, the return current occupying the whole interior and having 
 therefore a diameter of 2 x '7Q7R, or T414.R. 
 
CHAPTER XX. 
 
 CAPILLARITY. 
 
 353. THE subject of the present chapter is the behaviour of 
 inviscid incompressible fluid vibrating under the action of gravity 
 and capillary force, more especially the latter. In virtue of the 
 first condition we may assume the existence of a velocity-potential 
 ($), which by the second condition must satisfy ( 241) the 
 equation 
 
 V 2 < = .............................. (1), 
 
 throughout the interior of the fluid. In terms of <f> the equation 
 for the pressure is ( 244) 
 
 Bp/p = R-d^/dt ......................... (2), 
 
 if we assume that the motion is so small that its square may be 
 neglected. The only impressed force, acting upon the interior of 
 the fluid, which we have occasion to consider is that due to gravity ; 
 so that, if z be measured vertically downwards, R gz, and (2) 
 becomes 
 
 fy/p**ff*-d<l>/dt ........................ (3). 
 
 Let us now consider the propagation of waves upon the hori- 
 zontal surface (z = 0) of water, or other liquid, of uniform depth I, 
 limiting our attention to the case of two dimensions, where the 
 motion is confined to the plane zx. The general solution of (1) 
 under this condition, and that the motion is proportional to 
 
 or, with regard to the condition that the vertical velocity must 
 vanish at the bottom where z=l, 
 
 (z-l).e lkx ..................... (4). 
 
344 WAVES ON WATER. [353. 
 
 If the motion be proportional also to e int , and we throw away the 
 imaginary part in (4), we get as the expression for waves propa- 
 gated in the negative direction 
 
 $ = cosh k (z 1} cos (nt + kx) ............... (5), 
 
 in which it remains to find the connection between n and k. 
 
 If h denote the elevation of the water surface at the point x, 
 and T the constant tension, the pressure at the surface due to 
 capillarity is TcPh/dx*, and (3) becomes 
 
 d<f> 
 
 or, if we differentiate with respect to t and remember that 
 dk/dt = - d<f>/dz, 
 
 T #* d*_#* (6) 
 
 pd&ds 9 d* dt*" 
 
 Applying this equation to (5) where z 0, we get for the velocity 
 of propagation 
 
 F 2 = ?i 2 / 2 = (gjk + Tk/p) tanh kl ............... (7) 1 , 
 
 where, as usual, 
 
 & = 27r/X .... ....................... (8). 
 
 In many cases the depth of liquid is sufficient to allow us to 
 take tanh kl = 1 ; and then 
 
 gives the relation between V and X. When X is great, the waves 
 move mainly under gravity and with velocity approximately equal 
 to *J(g\/%7r). On the other hand, when X is small, the influence 
 of capillarity becomes predominant and the expression for the 
 velocity assumes the form 
 
 F=V(27rr/pX) ........................ (10). 
 
 Since X = FT, the relation between wave-length and periodic 
 time corresponding to (10) is 
 
 \ 3 /r*=27rT/p ........................ (11). 
 
 Except as regards the numerical factor, the relations (10), (11) 
 can be deduced by considerations of dimensions from the fact that 
 the dimensions of T are those of a force divided by a line. 
 
 1 A more general formula for the velocity of propagation (w/&) at the interface 
 between two liquids is given in (7) 365. 
 
 2 Kelvin, Phil. Man. vol. XLII. p. 375, 1871. 
 
353.] 
 
 MINIMUM VELOCITY. 
 
 345 
 
 If we inquire what values of X correspond to a given value of 
 F, we obtain from the quadratic (9) 
 
 (12), 
 
 which shews that for no wave-length can F be less than F , 
 
 where 
 
 (13), 
 
 The values of X and of r corresponding to the minimum 
 velocity are given by 
 
 Xo = 27r(2y0p)*, T = 27r-(2VVV>)i ......... (14). 
 
 If we take in c.G.s. measure <7 = 981, and for water /o = l, 
 T=76, we have F = 23'l, X = 1'71, 1/T=13'6. 
 
 The accompanying table gives a few corresponding values of 
 wave-length, velocity, and frequency in the neighbourhood of the 
 critical point : 
 
 Wave-length 
 
 5 ! 1-0 
 
 1-7 
 
 2-5 
 
 3-0 
 
 5-0 
 
 Velocity 
 
 31-5 ! 24-7 
 
 23-1 
 
 23-9 
 
 24-9 
 
 29-5 
 
 Frequency 
 
 63-0 1 24-7 
 
 13-6 
 
 9-6 
 
 8-3 
 
 5-9 
 
 A comparison of Kelvin's formula (9) with observation has 
 been effected by Matthiessen 1 , the ripples being generated by 
 touching the surface of the various liquids with dippers attached 
 to vibrating forks of known pitch. Among the liquids tried were 
 water, mercury, alcohol, ether, bisulphide of carbon ; and the 
 agreement was found to be satisfactory. The observations include 
 frequencies as high as 1832, and wave-lengths as small as 
 '04 cm. 
 
 Somewhat similar experiments have been carried out by the 
 author 2 with the view of determining T by a method independent 
 of any assumption respecting angles of contact between fluid and 
 solid, and admitting of application to surfaces purified to the 
 utmost from grease. In order to see the waves well, the light 
 was made intermittent in a period equal to that of the waves 
 ( 42), and Foucault's optical method was employed for rendering 
 visible small departures from truth in plane or spherical reflecting 
 
 1 Wied. Ann. vol. xxxvm. p. 118, 1889. 
 
 2 On the Tension of Water Surfaces, clean and contaminated. Phil. Mag. 
 vol. xxx. p. 386, 1890. 
 
346 T DETERMINED BY RIPPLES. [_354. 
 
 surfaces. From the measured values of T and \, T may be deter- 
 mined by (11), corrected, if necessary, for any small effect of 
 gravity. The values thus found were for clean water 74'0 c.G.s., 
 for a surface greasy to the point where camphor motions nearly 
 cease 53'0, for a surface saturated with olive-oil 41 '0, and for one 
 saturated with oleate of soda 25'0. It should be remembered that 
 the tension of contaminated surfaces is liable to variations depen- 
 dent upon the extension which has taken place, or is taking 
 place ; but it is not necessary for the purposes of this work to 
 enter further upon the question of "superficial viscosity." 
 
 354. Another way of generating capillary waves, or crispa- 
 tions as they were termed by Faraday, depends upon the principle 
 discussed in 68 b. If a glass plate, held horizontally and made 
 to vibrate as for the production of Chladni's figures, be covered 
 with a thin layer of water or other mobile liquid, the phenomena 
 in question may be readily observed 1 . Over those parts of the 
 plate which vibrate sensibly the surface is ruffled by minute waves, 
 the degree of fineness increasing with the frequency of vibration. 
 The same crispations are observed upon the surface of liquid in a 
 large wine-glass or finger-glass which is caused to vibrate in the 
 usual manner by carrying the moistened finger round the circum- 
 ference ( 234). All that is essential to the production of 
 crispations is that a body of liquid with a free surface be 
 constrained to execute a vertical vibration. It is indifferent 
 whether the origin of the motion be at the bottom, as in the 
 first case, or, as in the second, be due to the alternate advance 
 and retreat of a lateral boundary, to accommodate itself to which 
 the neighbouring surface must rise and fall. 
 
 More than sixty years ago the nature of these vibrations was 
 examined by Faraday 2 with great ingenuity and success. The 
 conditions are simplest when the motion of the vibrating horizontal 
 plate on which the liquid is spread is a simple up and down 
 motion without rotation. To secure this Faraday attached the 
 plate to the centre of a strip of glass or lath of deal, supported 
 at the nodes, and caused to vibrate by friction. Still more con- 
 venient is a large iron bar, maintained in vibration electrically, to 
 which the plate may be attached by cement. 
 
 1 On the Crispations of Fluid resting upon a Vibrating Support. Phil. Mag. 
 vol. xvi. p. 50, 1883. 
 
 2 Phil. Trans. 1831, p. 290. 
 
354.} FARADAY'S CRISPATIONS. 347 
 
 The vibrating liquid standing upon the plate presents appear- 
 ances which at first are rather difficult to interpret, and which 
 vary a good deal with the nature of the liquid in respect of 
 transparency and opacity, and with the incidence of the light. 
 The vibrations are too quick to be followed by the eye ; and thus 
 the effect observed is an average, due to the superposition of an 
 indefinite number of elementary impressions corresponding to the 
 various phases. 
 
 If the plate be rectangular, the motion of the liquid consists of 
 two sets of stationary vibrations superposed, the ridges and furrows 
 of the two sets being perpendicular to one another and usually 
 parallel to the edges of the plate. Confining our attention for the 
 moment to one set of stationary waves, let us consider what 
 appearance it might be expected to present. At one moment 
 the ridges form a set of parallel and equidistant lines, the interval 
 being X. Midway between these are the lines which represent at 
 that moment the position of the furrows. After the lapse of a J 
 period the surface is flat; after another J period the ridges and 
 furrows are again at their maximum developement, but the 
 positions are exchanged. Now, since only an average effect can 
 be perceived, it is clear that no distinction is recognizable between 
 the ridges and the furrows, and that the observed effect must be 
 periodic within a distance equal to JX. If the liquid on the plate 
 be rendered moderately opaque by addition of aniline blue, and be 
 seen by diffused transmitted light, the lines of ridge and furrow 
 will appear bright in comparison with the intermediate nodal 
 lines where the normal depth is preserved throughout the vi- 
 bration. The gain of light when the thickness is small will, in 
 accordance with the law of absorption, outweigh the loss of light 
 which occurs half a period later when the furrow is replaced by a 
 ridge. 
 
 The actual phenomenon is more complicated in consequence of 
 the coexistence of the two sets of ridges and furrows in perpendi- 
 cular directions (x, y). In the adjoining figure (Fig. 66) the thick 
 lines represent the ridges, and the thin lines the furrows, of the 
 two systems at a moment of maximum excursion. One quarter 
 period later the surface is flat, and one half period later the ridges 
 and furrows are interchanged. The places of maximum elevation 
 and depression are the intersections of the thick lines with one 
 another and of the thin lines with one another, places not distin- 
 
348 
 
 RESOLUTION INTO PROGRESSIVE WAVES. 
 
 [354. 
 
 guishable by ordinary vision. They appear like holes in the sheet 
 of colour. The nodal lines where the normal depth of colour is 
 preserved throughout the vibration are shewn dotted ; they are 
 inclined at 45, and pass through the intersections of the thin 
 lines with the thick lines. The pattern is recurrent in the 
 
 Fig. 66. 
 
 directions both of x and y, and in each case with an interval 
 equal to the real wave-length (X). The distance between the 
 bright spots measured parallel to a? or y is thus X ; but the 
 shortest distance between these spots is in directions inclined at 
 45, and is equal to X/V2. 
 
 As in all similar cases, these stationary waves may be resolved 
 into their progressive components by a suitable motion of the eye. 
 Consider, for example, the simple set of waves represented by 
 
 2 cos kx cos nt = cos (nt + kx) + cos (nt kx). 
 
 This is with reference to an origin fixed in space. But let us 
 refer the phenomenon to an origin moving forward with the velocity 
 (n/k) of the waves, so as to obtain the impression that would be 
 produced upon the eye, or in a photographic camera, carried 
 forward in this manner. Writing kx' + nt for kx, we get 
 
 cos (kx + 2nt) + cos kx'. 
 
355.] LISSAJOUS' PHENOMENON. 349 
 
 Now the average effect of the first term is independent of a/, so 
 that what is seen is simply that set of progressive waves which 
 moves with the eye. 
 
 In order to see the progressive waves it is not necessary to 
 move the head as a whole, but only to turn the eye as when we 
 follow the motion of a real object. To do this without assistance 
 is not very easy at first, especially if the area of the plate be 
 somewhat small. By moving a pointer at various speeds until the 
 right one is found, the eye may be guided to do what is required 
 of it ; and after a few successes repetition becomes easy. 
 
 Faraday's assertion that the waves have a period double that of 
 the support has been disputed, but it may be verified in various 
 ways. Observation by stroboscopic methods is perhaps the most 
 satisfactory. The violence of the vibrations and the small depth 
 of the liquid interfere with an accurate calculation of frequency on 
 the basis of the observed wave-length. The theory of vibrations 
 in the sub-octave has already been considered ( 68 6). 
 
 355. Typical stationary waves are formed by the superposi- 
 tion of equal positive and negative progressive waves of like 
 frequency. If the one set be derived from the other by reflection, 
 the equality of frequencies is secured automatically; but if the two 
 sets of waves originate in different sources, the unison is a matter 
 of adjustment, and a question arises as to the effect of a slight 
 error. We may take as the expression for the two sets of 
 progressive waves of equal amplitude and of approximately equal 
 frequency 
 
 cos (kx nt) + cos (k'x + n't), 
 
 or, which is the same, 
 
 2 cos {} (k + k')x + (n r - n) t] x cos { J (k r -k)x + % (ri + n) t] 
 
 (1). 
 
 If n' = n, k f k, the waves are absolutely stationary ; but we 
 have now to interpret (1) when (n' n), (&' k) are merely small. 
 
 The position at any time t of the crests and hollows of the 
 nearly stationary waves represented by (1) is given by 
 
 %(k + k')x+ %(ri -ri)t = mir (2), 
 
 where m is an integer. The velocity of displacement U is 
 accordingly 
 
350 STANDING WAVES [356. 
 
 or approximately 
 
 ff = ( w -w')/2A? (3) 1 , 
 
 from vvhich it appears that in every case the shifting takes place 
 in the direction of waves of higher pitch, or towards the source of 
 graver pitch. If V be the velocity (n/k) of propagation of the 
 progressive waves, (3) may be written 
 
 U/V=(n-ri)/2n (4). 
 
 The slow travel under these circumstances of the places where 
 the maximum displacements occur is a general phenomenon, not 
 dependent upon the peculiarities of any particular kind of waves ; 
 but the most striking example is that afforded by capillary waves 
 and described by Lissajous 2 . In his experiment two nearly 
 unisonant forks touch the surface of water so as to form approxi- 
 mately stationary waves in the region between the points of 
 contact. Since the crests and troughs cannot be distinguished, 
 the pattern seen has an apparent wave-length half that of the real 
 waves, and it travels slowly towards the graver fork. A frequency 
 of about 50 will be found suitable for convenient observation. 
 
 If the waves be aerial, there is no difference of velocity ; but 
 (4) still holds good, and gives the rate at which the ear must 
 travel in order to remain continually in a loop or in a node. 
 
 356. One of the best opportunities for the examination of capil- 
 lary waves occurs when they are reduced to rest by a contrary 
 movement of the water. Waves of this kind are sometimes described 
 as standing waves, and they may usually be observed when the 
 uniform motion of a stream is disturbed by obstacles. Thus when 
 the surface is touched by a small rod, or by a fishing-line, or is 
 displaced by the impact of a gentle stream of air from a small 
 nozzle, a beautiful pattern is often displayed, stationary with 
 respect to the obstacle. This was described and figured by Scott 
 Russell 3 , who remarked that the purity of the water had much to 
 do with the extent and range of the phenomenon. On the 
 up-stream side of the obstacle the wave-length is short, and, as 
 was first clearly shewn by Kelvin, the force governing the vibra- 
 
 1 Phil. Mag. vol. xvi. p. 57, 1883. 
 
 2 Compt. Rend. vol. LXVII. p. 1187, 1868. 
 
 3 Brit. Ass. Rep. 1844, p. 375, Plate 57. See also Poncelet, Ann. d. Chim. 
 vol. XLVI. p. 5, 1831. 
 
357.] ON RUNNING WATER. 
 
 tions is principally cohesion. On the down-stream side the waves 
 are longer and are governed principally by gravity. Both sets of 
 waves move with the same velocity relatively to the water ( 353) ; 
 namely, that required in order that they may maintain a fixed 
 position relatively to the obstacle. The same condition governs 
 the velocity and therefore the wave-lengths of those parts of the 
 pattern where the fronts are oblique to the direction of motion. 
 If the angle between this direction and the normal to the wave- 
 front be called 0, the velocity of propagation must be equal to 
 v cos 0, where v represents the velocity of the water. 
 
 If VQ be less than 23 cm. per sec., no wave-pattern is possible, 
 for no waves can then move over the surface so slowly as to 
 maintain a stationary position with respect to the obstacle. When 
 VQ exceeds 23 cm. per sec., a pattern is formed ; but the angle has 
 a limit defined by V Q cos 6 = 23, and the curved wave-front has a 
 corresponding asymptote. 
 
 It would lead us too far to go further into the matter here, but 
 it may be mentioned that the problem in two dimensions admits 
 of analytical treatment 1 , and that the solution explains satis- 
 factorily one of the peculiar features of the case, namely, the 
 limitation of the smaller capillary waves to the up-stream side, 
 and of the larger (gravity) waves to the down-stream side of the 
 obstacle. 
 
 357. A large class of phenomena, interesting not only in 
 themselves but also as throwing light upon others yet more 
 obscure, depend for their explanation upon the transformations 
 undergone by a cylindrical body of liquid when slightly displaced 
 from its equilibrium configuration and then left to itself. Such a 
 cylinder is formed when liquid issues under pressure through a 
 circular orifice, at least when gravity may be neglected ; and the 
 behaviour of the jet, as studied experimentally by Savart, Magnus, 
 Plateau and others, is substantially independent of the forward 
 motion common to all its parts. It will save repetition and be 
 more in accordance with the general character of this work if we 
 commence our investigation with the theory of an infinite cylinder 
 of liquid, considered as a system in equilibrium under the action 
 
 1 On the form of Standing Waves on the Surface of Running Water. Proc. 
 Lond. Math. Soc. vol. xv. p. 69, 1883. 
 
352 LIQUID CYLINDER. [357. 
 
 of the capillary force. With a solution of this mechanical problem 
 most of the experimental results will easily be connected. 
 
 Taking cylindrical coordinates z, r, <j>, the equation of the 
 slightly disturbed surface may be written 
 
 in which f ($>, z) is always a small quantity. By Fourier's theorem 
 the arbitrary function /may be expanded in a series of terms of 
 the type a n cos n<j> cos kz ; and, as we shall see in the course of the 
 investigation, each of these terms may be considered independently 
 of the others. Either cosine may be replaced by a sine ; and the 
 summation extends to all positive values of k and to all positive 
 integral values of n, zero included. 
 
 During the motion the quantity a does not remain absolutely 
 constant ; its value must be determined by the condition that the 
 enclosed volume is invariable. Now for the surface 
 
 r = a + oin cos n<j> cos kz .................. (2), 
 
 we find 
 
 Volume = $ffr*d$dz = z (7ra 2 -f J?ra 7 f ) ; 
 
 so that, if a, denote the radius of the section of the undisturbed 
 cylinder, 
 
 a 2 = a 2 + ia n 2 , 
 
 whence approximately 
 
 a = a(l-ia, 2 /a 2 ) ..................... (3). 
 
 This holds good when ?i = l, 2, 3.... If ?? = 0, (2) gives in place 
 of (3) 
 
 > ..................... (4). 
 
 The potential energy of the system in any configuration, due to 
 the capillary force, is proportional simply to the surface. Now 
 in (2) 
 
 Surface = 
 
 so that by (3), if a- denote the surface corresponding upon the 
 average to unit of length, 
 
 r(fc 2 a 2 + tt 2 -l)a n 2 / a ............ (5). 
 
357.] POTENTIAL ENERGY. 353 
 
 The potential energy due to capillarity, estimated per unit 
 length and from the configuration of equilibrium, is accordingly 
 
 P = lTTT(k*a? + n*-l)oL n *la ............... (6), 
 
 T denoting, as usual, the superficial tension. 
 
 In (6) it is supposed that k and n are not zero. If k be zero, 
 (6) requires to be doubled in order to give the potential energy 
 corresponding to 
 
 r = a + a w cos n<f> ..................... (7); 
 
 and again, if n be zero, we are to take 
 
 P = i7rr(& 2 a 2 -l)a 2 /a .................. (8), 
 
 corresponding to 
 
 r = a + o cos kz ........................ (9). 
 
 From (6) it appears that when n is unity or any greater 
 integer, the value of P is positive, shewing that for all displace- 
 ments of these kinds the original equilibrium is stable. For the 
 case of displacements symmetrical about the axis (n = 0), we see 
 from (8) that the equilibrium is stable or unstable according as ka 
 is greater or less than unity, i.e. according as the wave-length 
 (2-7T/&) of the symmetrical deformation is less or greater than the 
 circumference of the cylinder, a proposition first established by 
 Plateau. 
 
 If the expression for r in (2) involve a number of terms with 
 various values of n and A 1 , and with arbitrary substitution of 
 sines for cosines, the corresponding expression for P is found by 
 simple addition of the expressions relating to the component 
 terms, and it contains the squares only (and not the products) of 
 the quantities a. 
 
 We have now to consider the kinetic energy of the motion. 
 Since the fluid is supposed to be inviscid, there is a velocity- 
 potential A/T, and this in virtue of the incompressibility satisfies 
 Laplace's equation. Thus, (4) 241, 
 
 Icty 1 d 2 ^ 
 
 rdr + r* d<f> 2 + dz* = 
 
 or, if in order to correspond with (2) we assume that the variable 
 part is proportional to cos 11$ cos kz, 
 
 >+ _ ............... 
 
 dr 1 r dr \r 2 J 1 
 R. n. 23 
 
354 KINETIC ENERGY. [357. 
 
 The solution of (10) under the condition that there is no 
 introduction or abstraction of fluid along the axis of symmetry 
 
 is 200 
 
 >|r = j3 n J n (ikr) cos n<J> cos kz ............... (11). 
 
 The constant ff n is to be found from the condition that the 
 radial velocity when r a coincides with that implied in (2). 
 
 Thus 
 
 ikp n j n '(ika)-da*/dt ................. ,(12). 
 
 If p be the density, the kinetic energy of the motion is by 
 Green's theorem (2) 242 
 
 i/ 3 S\fr<tyl&r\M* ad(f)dz = ^irpz.ika.J n (ika) J n '(ika).^ n 2 ; 
 so that by (12), if K denote the kinetic energy per unit of length, 
 
 J n (ika) (dot n \ 
 1 - r / / -7 \ -v. 
 ika.J n (ika) \ dt J 
 
 When n = 0, we must take in place of (13) 
 
 Jo (ika) f 
 
 . r//-/ x 
 ika. J Q (ika) \dt 
 
 The most general value of K is to be found by simple summa- 
 tion from the particular values expressed in (13), (14). Since the 
 expressions for P and K involve the squares only, and not the 
 products, of the quantities a, da/dt, and the corresponding quanti- 
 ties in which cosines are replaced by sines, it follows that the 
 motions represented by (2) take place in perfect independence of 
 one another, so long as the whole displacement is small. 
 
 For the free motion we get by Lagrange's method from 
 
 (6), (13) 
 
 d?a n T ika.Jn (ika} . 
 
 -j*+ T / 7 > -()i 2 + &a--l)a n = () ...... (15), 
 
 dt 2 pa 3 J n (ika) 
 
 which applies without change to the case n = 0. Thus, if On varies 
 as cos (pt e), 
 
 giving the frequency of vibration in the cases of stability. 
 
 If ?i = 0, and ka<\ 9 the solution changes its form. If we 
 suppose that oto varies as e* qt , 
 
 ............... 
 
 pa* J (ika) 
 Proc. Roy. Soc. vol. xxix. p. 94, 1879. 
 
357.] FREQUENCY EQUATION. 355 
 
 When n is greater than unity, the circumstances are usually 
 such that the motion is approximately in two dimensions only. 
 We may then advantageously introduce into (16) the supposition 
 that ka is small. In this way we get, (5) 200, 
 
 T f 
 
 a 2 )- 
 pa 3 I 
 
 ?i(2?i 
 
 (18), 
 
 or, if ka be neglected altogether, 
 
 f*\*-*ijf ..................... (19), 
 
 the two-dimensional formula. When n = 1 , there is no force of resti- 
 tution for a displacement purely in two dimensions. If X denote 
 the wave-length measured round the circumference, \ = 
 Thus in (19), if n and a are infinite, 
 
 in agreement with the theory of capillary waves upon a plane 
 surface. Compare (7) 353. A similar conclusion may be reached 
 by the consideration of waves whose length is measured axially. 
 Thus, if X = 2-7T/&, and a = oc , n = Q, (16) reduces to (20) in virtue 
 of the relation, 302, 350, 
 
 358. Many years ago Bidone investigated by experiment the 
 behaviour of jets of water issuing horizontally under considerable 
 pressure from orifices in thin plates. If the orifice be circular, the 
 section of the jet, though diminished in area, retains the circular 
 form. But if the orifice be not circular, curious transformations 
 ensue. The peculiarities of the orifice are exaggerated in the jet, 
 but in an inverted manner. Thus in the case of an elliptical 
 aperture, with major axis horizontal, the sections of the jet taken 
 at increasing distances gradually lose their ellipticity until at a 
 certain distance the section is circular. Further out the section 
 again assumes ellipticity, but now with major axis vertical, and 
 (in the circumstances of Bidone's experiments) the ellipticity 
 increases until the jet is reduced to a flat sheet in the vertical 
 plane, very broad and thin. This sheet preserves its continuity to 
 a considerable distance (e.g. six feet) from the orifice, where finally 
 it is penetrated by air. If the orifice be in the form of an equi- 
 
 232 
 
356 OBSERVATIONS BY BIDONE AND MAGNUS. [358. 
 
 lateral triangle, the jet resolves itself into three sheets disposed 
 symmetrically round the axis, the planes of the sheets being 
 perpendicular to the sides of the orifice; and in like manner if 
 the aperture be a regular polygon of any number of sides, there 
 are developed a corresponding number of sheets perpendicular to 
 the sides of the polygon. 
 
 Bidone explains the formation of these sheets by reference to 
 simpler cases of meeting streams. Thus equal jets, moving in the 
 same straight line with equal and opposite velocities, flatten them- 
 selves into a disc situated in the perpendicular plane. If the axes 
 of the jets intersect obliquely, a sheet is formed symmetrically in 
 the plane perpendicular to that of the impinging jets. Those 
 portions of a jet which proceed from the outlying parts of a single 
 unsymmetrical orifice are regarded as behaving in some degree 
 like independent meeting streams. 
 
 In many cases, especially when the orifices are small and the 
 pressures low, the extension of the sheets reaches a limit. Sections 
 taken at still greater distances from the orifice shew a gradual 
 gathering together of the sheets, until a compact form is regained 
 similar to that at the first contraction. Beyond this point, if the 
 jet retains its coherence, sheets are gradually thrown out again, 
 but in directions bisecting the angles between the directions of 
 the former sheets. These sheets may in their turn reach a limit 
 of developement, again contract, and so on. The forms assumed 
 in the case of orifices of various shapes including the rectangle, 
 the equilateral triangle, and the square, have been carefully 
 investigated and figured by Magnus. Phenomena of this kind 
 are of every day occurrence, and may generally be observed 
 whenever liquid falls from the lip of a moderately elevated 
 vessel. 
 
 As was first suggested by Magnus 1 and BufF' 2 , the cause of the 
 contraction of the sheets after their first developement is to be 
 found in the capillary force, in virtue of which the fluid behaves 
 as if enclosed in an envelope of constant tension; and the re- 
 current form of the jet is due to vibrations of the fluid column 
 about the circular figure of equilibrium, superposed upon the 
 general progressive motion. Since the phase of the vibration, 
 initiated during passage through the aperture, depends upon the 
 
 1 Hydraulische Untersuchungen, Pogg. Ann. vol. xcv, p. 1, 1855. 
 
 2 Pogg. Ann. vol. c, p. 168, 1857. 
 
358.] FURTHER EXPERIMENTS. 357 
 
 time elapsed, it is always the same at the same point in space, 
 and thus the motion is steady in the hydrodynamical sense, and the 
 boundary of the jet is a fixed surface. Relatively to the water the 
 waves here concerned are progressive, such as may be compounded 
 of two stationary systems, and they move up stream with a velocity 
 equal to that of the water so as to maintain a fixed position rela- 
 tively to external objects, 356. 
 
 If the departure from the circular form be small, the vibrations 
 are those considered in 357, of which the frequency is determined 
 by equations (16), (18), (19). The distance between consecutive 
 corresponding points of the recurrent figure, or, as it may be called, 
 the wave-length of the figure, is the space travelled over by the 
 stream during one vibration. Thence results a relation between 
 wave-length and period. If the circumference of the jet be small 
 in comparison with the wave-length, so that (19) 357 is appli- 
 cable, the periodic time is independent of the wave-length ; and 
 then the wave-length is directly proportional to the velocity of 
 the jet, or to the square root of the pressure. The elongation of 
 wave-length with increasing pressure was remarked by Bidone and 
 by Magnus, but no definite law was arrived at. 
 
 In the experiments of the author 1 upon elliptical, triangular, 
 and square apertures, the jets were caused to issue horizontally in 
 order to avoid the complications due to gravity ; and, if the pressure 
 were not too high, the law above stated was found to be verified. 
 At higher pressures the observed wave-lengths had a marked 
 tendency to increase more rapidly than the velocity of the jet. 
 This result points to a departure from the law of isochronous 
 vibration. Strict isochronism is only to be expected when vibra- 
 tions are infinitely small, that is when the section of the jet never 
 deviates more than infinitesimally from the circular form. Under 
 the high pressures in question the departures from circularity were 
 very considerable, and there is no reason for expecting that such 
 vibrations will be executed in precisely the same time as vibrations 
 of infinitely small amplitude. 
 
 The increase of amplitude under high pressure is easily ex- 
 plained, inasmuch as the lateral velocities to which the vibrations 
 are mainly due vary in direct proportion to the longitudinal 
 velocity of the jet. Consequently the amplitude varies approxi- 
 
 1 Proc. Roy. Soc. vol. xxix, p. 71, 1879. 
 
358 COMPARISON WITH THEORY. [358. 
 
 mately as the square root of the pressure, or as the wave-length. 
 In general, the periodic time of a vibration is an even function of 
 amplitude ( 67) ; and thus, if h represent the head of liquid, the 
 wave-length may be expected to be a function of h of the form 
 (M '+ Nh) /\/h, where M and N are constants for a given aperture. 
 It appears from experiment, and might perhaps have been ex- 
 pected, that N is here positive. 
 
 For a comparison with theory it is necessary to keep within the 
 range of the law of isochronism ; and it is convenient to employ in 
 the calculations the area of the section of the jet in place of the 
 mean radius. Thus, if A = 7ra 2 , (19) 357 may be written 
 
 (1), 
 
 in which A is to be determined by experiments upon the rate of 
 total discharge. For the case of water ( 353) we may take in 
 C.G.s. measure T=74, p=l; so that for the frequency of the 
 gravest vibration (n = 2) we get from (1) 
 
 2>/27r = 7-914-* ........................ (2). 
 
 For a sectional area of one square centimetre there are thus 
 about 8 vibrations per second. A pitch of 256 would correspond 
 to a diameter of about one millimetre. 
 
 For the general value of n, we have 
 
 jp/27r = 3-234-* VO 3 -) .................. (3). 
 
 If h be the head of water to which the velocity of the jet is due 
 and X the wave-length, 
 
 V(2^).4* 
 3-23V<y-n)'" 
 
 In one experiment with an elliptical aperture (n = 2) the 
 observed value of X was 3*95 while the value calculated from 
 (4) is 3-93. In the case of a triangular aperture (n 3) the 
 observed value of \ was 2*3 and the calculated was 2*1. Again, 
 the observed value for a square aperture (w = 4) was T85 and the 
 calculated T78. The excess of the observed over the calculated 
 values in the last two cases may perhaps have been due to exces- 
 sive departure from the circular figure. 
 
 The general theory, unrestricted to small amplitudes, would 
 doubtless involve great complications; but some information 
 
358.] LARGE AMPLITUDES. 359 
 
 respecting it may be obtained with facility by the method of 
 dimensions. If the shape of the orifice be given, X may be re- 
 garded as a function of T, p, A, and H the pressure under which 
 the jet escapes. Of these T is a force divided by a line, so that its 
 dimensions are 1 in mass, in length, and 2 in time ; p is of 
 dimensions 1 in mass, 3 in length, in time ; A is of dimensions 
 
 in mass, 2 in length, in time ; and finally H is of dimensions 
 
 1 in mass, 1 in length, and 2 in time. If we assume 
 
 X cc T x pvA z H u , 
 
 then x + y -f u = 0, 3y + Zz - u = 1, 2# - 2u = 0, 
 whence u = x, y = 0, z = \(\ x)\ 
 
 so that X x A* (TA-H~ l ) x . 
 
 The exponent x is here undetermined ; and, since any number 
 of terms with different values of x may occur simultaneously, all 
 that we can infer is that X is of the form 
 
 or, if we prefer it, 
 
 ) .................. (5), 
 
 where f and F are arbitrary functional symbols. Thus for a given 
 liquid and shape of orifice there is complete dynamical similarity 
 if the pressure be taken inversely proportional to the linear 
 dimension. The simple case previously considered where the 
 departures from circularity are small, and the vibrations take place 
 approximately in two dimensions, corresponds to F= constant. 
 
 The method of determining T by observations upon X is 
 scarcely delicate enough to compete with others that may be 
 employed for the same purpose when the tension is constant. 
 But the possibility of thus experimenting upon surfaces which 
 have been formed but a fraction of a second earlier is of consi- 
 derable interest. In this way it may be proved with great ease 
 that the tension of a soapy solution immediately after the forma- 
 tion of a free surface differs comparatively little from that of pure 
 water, whereas when a few seconds have elapsed the difference 
 becomes very great 1 . 
 
 1 On the Tension of Recently Formed Liquid Surfaces, Proc. Roy. Soc. vol. 
 XLVII, p. 281, 1890. 
 
360 SYMMETRICAL DISTURBANCES. [358. 
 
 Hitherto it has been supposed for the sake of simplicity that 
 the jet after its issue from the nozzle is withdrawn from the action 
 of gravity. If the direction of projection be vertically downwards, 
 as is often convenient, the velocity of flow (v) continually increases, 
 while at the same time the area of the section diminishes, the 
 relation being vA = constant. But, so far as regards X, the dis- 
 turbance which thus ensues is less than might have been expected, 
 for the changes in v and A compensate one another to a con- 
 siderable extent. By (1) 
 
 X oc v/p oc ui oc /i, 
 
 if h denote the whole difference of level between the surface of 
 liquid in the reservoir and the place where X is measured. 
 
 359. In 358 the motion of the liquid is regarded as steady, 
 every portion as in turn it passes the orifice being similarly 
 affected. Under these circumstances no term corresponding to 
 tt = can appear in the mathematical expressions; but it must 
 not be forgotten that for certain disturbances of this type the 
 cylindrical form is unstable and that therefore the jet cannot long 
 preserve its integrity. The minute disturbances required to bring 
 the instability into play are such as act differently at different 
 moments of time, and have their origin in eddying motions of the 
 fluid due to friction, and especially in vibration communicated to 
 the nozzle and of such a character as to render the rate of discharge 
 subject to a slight periodic variation. If v be the velocity of the 
 jet and r the period of the vibration, the cylindrical column issuing 
 from a circular orifice is launched subject to a disturbance of 
 wave-length (X) equal to vr. If this wave-length exceed the 
 circumference of the jet (2?ra), the disturbance grows exponential^ 
 until finally the column of liquid is divided into detached masses 
 separated by the common interval X, and passing a fixed point 
 with velocity v and frequency 1/r. Even though no regular 
 vibration has access to the nozzle, the instability cannot fail to 
 assert itself, and casual disturbances of a complex character will 
 bring about disintegration. It will be convenient to discuss in 
 the first place somewhat in detail the theory of the case of n = 
 in (16), (17) 357, and then to consider its application to the 
 beautiful phenomena described by Savart and to a large extent 
 explained by Plateau. 
 
359.] 
 
 INSTABILITY. 
 
 361 
 
 If ka = z, and we introduce the notation of 221 a, (17) 357 
 becomes 
 
 a, 
 
 In this equation Ii(z) and I (z) are both positive, so that as z 
 decreases (or as X increases) q first becomes real when z\. At 
 this point instability commences, and at first the degree of in- 
 stability is infinitely small. Also when z is very small, or X is 
 
 T z* 
 very great, ^""a?? 
 
 ultimately, so that q is again small. For some value of z 
 between and I, q is a maximum, and ,the investigation of this 
 value is a matter of importance, because, as has already been 
 shewn 87, the unstable equilibrium will give way by preference 
 in the mode so characterized. 
 
 The function to be made a maximum is 
 
 or, expanded in powers of z, 
 
 2 4 .3 2 U .3.5 
 
 Hence, to find the maximum, we obtain on differentiation 
 
 If the last terms be neglected, the quadratic gives 2 2 = '4914. If 
 this value be substituted in the small terms, the equation becomes 
 
 98928 -f* 2 
 whence 2 =-486, 
 
 The values of expression (2), or of its square root, to which q 
 is proportional, may be calculated from tables of / and I lt 221 a. 
 We have 
 
 z 
 
 {(2)}* 
 
 z 
 
 {(2)}* 
 
 o-o 
 
 0000 
 
 0-6 
 
 3321 
 
 0-1 
 
 0703 
 
 0-7 
 
 3433 
 
 0-2 
 
 1382 
 
 0-8 
 
 3269 
 
 0-3 
 
 2012 
 
 0-9 
 
 2647 
 
 0-4 
 
 2567 
 
 1-0 
 
 0000 
 
 0-5 
 
 3015 
 
 
 
 1 On the Instability of Jets, Proc. Lond. Math. Soc. vol. x, p. 7, 1878. 
 
362 MAXIMUM INSTABILITY. [359. 
 
 From these values we find for the maximum by Lagrange's 
 interpolation formula z '696, corresponding to 
 
 X = 27ra/^ = 4r51 x 2a ..................... (3). 
 
 Hence the maximum instability occurs when the wave-length 
 of disturbance is about half as great again as that at which 
 instability first commences. 
 
 Taking for water in c.G.s. units T = 73, /o = l, we get for the 
 case of maximum instability 
 
 This is the time in which the disturbance is multiplied in the 
 ratio e : 1. Thus in the case of a diameter of one centimetre the 
 disturbance is multiplied 2'7 times in about J second. If the 
 disturbance be multiplied 1000 fold in time t, qt = 3 \og e 10 = 6'9, 
 so that t = '828 (2a)*. For example, if the diameter be one milli- 
 metre, the disturbance is multiplied 1000 fold in about ^ second. 
 In view of these estimates the rapid disintegration of a jet of water 
 will not cause surprise. 
 
 The above theory of the instability of a cylindrical surface 
 separating liquid from gas may be extended to meet the case 
 where the liquid is outside and the gas, whose inertia is neglected, 
 is inside the surface. This represents a jet of gas discharged 
 under liquid ; and it appears that the degree of maximum in- 
 stability is even higher than before, and that it occurs when 
 X=6'48x2a 1 . But it is scarcely necessary for our purpose to 
 pursue this part of the subject further. 
 
 360. The application of our mathematical results to actual 
 jets presents no great difficulty. The disturbances, by which 
 equilibrium is upset, are impressed upon the fluid as it leaves 
 the aperture, and the continuous portion of the jet represents the 
 distance travelled over during the time necessary to produce 
 disintegration. Thus the length of the continuous portion neces- 
 sarily depends upon the character of the disturbances in respect of 
 amplitude and wave-length. It may be increased considerably, as 
 Savart shewed 2 , by a suitable isolation of the reservoir from 
 
 1 On the Instability of Cylindrical Fluid Surfaces, Phil. Mag. vol. xxxiv, p. 177, 
 1892. 
 
 2 Ann. de Chimie, LIII, p. 337, 1833. 
 
360.] SAVART'S LAWS. 363 
 
 tremors, whether due to external sources or to the impact of the 
 jet itself in the vessel placed to receive it. Nevertheless it does 
 not appear possible to carry the prolongation very far. Whether 
 the residual disturbances are of external origin or are due to 
 friction, or to some peculiarity of the fluid motion within the 
 reservoir, has not been satisfactorily determined. On this point 
 Plateau's explanations are not very clear, and he sometimes 
 expresses himself as if the time of disintegration depended 
 only upon the capillary tension without reference to initial dis- 
 turbances at all. 
 
 Two laws were formulated by Savart with respect to the length 
 of the continuous portion of a jet, and have been to a certain 
 extent explained by Plateau 1 . For a given fluid and a given 
 orifice the length is approximately proportional to the square root 
 of the head. This follows at once from theory, if it can be assumed 
 that the disturbances remain always of the same character, so that 
 the time of disintegration is constant. When the head is given, 
 Savart found the length to be proportional to the diameter of the 
 orifice. From (4) 359 it appears that the time in which a small 
 disturbance is multiplied in a given ratio varies not as a, but as a*. 
 Again, when the fluid is changed, the time varies as p*T~*. But 
 it may well be doubted whether the length of the continuous 
 portion obeys any very simple laws, even when external disturb- 
 ances are avoided as far as possible. 
 
 When a jet falls vertically downwards, the circumstances upon 
 which its stability or instability depend are continually changing, 
 more especially if the initial velocity be very small. The kind of 
 disturbance to which the jet is most sensitive as it leaves the 
 nozzle is one which impresses upon it undulations of length equal 
 to about 4J times the initial diameter. But as the jet falls, its 
 velocity increases, with consequent lengthening of the undulations, 
 and its diameter diminishes, so that the degree of instability soon 
 becomes much reduced. On the other hand, the kind of disturb- 
 ance which will be effective in a later stage is altogether ineffective 
 in the earlier stages. The change of conditions during fall has 
 thus a protective influence, and the continuous part tends to 
 become longer than would be the case were the velocity constant, 
 the initial disturbances being unaltered. 
 
 1 Statique experimentale et- theorique des Liquides soumis aux seules forces 
 moleculaires, Paris, 1873. 
 
 
364 PLATEAU'S THEORY. [360. 
 
 When the circumstances are such that the reservoir is 
 influenced by the shocks due to the impact of the jet, the 
 disintegration often assumes a complete regularity and is attended 
 by a musical note (Savart). The impact of the regular series of 
 drops, which at any moment strike the receiving vessel, determines 
 the rupture into similar drops of the portion of the jet at the same 
 moment passing the orifice. The pitch of the note, though not 
 definite, cannot differ greatly from that which corresponds to the 
 division of the column into wave-lengths of maximum instability ; 
 and in fact Savart found that the frequency was directly as the 
 square root of the head, inversely as the diameter of the orifice, 
 and independent of the nature of the fluid laws which follow 
 immediately from Plateau's theory. 
 
 From the observed pitch of the note due to a jet of given 
 diameter, and issuing under a given head, the wave-length of the 
 nascent divisions can be at once deduced. Reasoning from some 
 observations of Savart, Plateau found in this way 4'38 as the ratio 
 of the length of a division to the diameter of the jet. Now that 
 the length of a division can be estimated a priori, it is preferable 
 to reverse Plateau's calculation and to exhibit the frequency of 
 vibration in terms of the other data of the problem. Thus 
 
 frequency - ..................... (1), 
 
 and in many cases, where the jet is not too fine, v may be replaced 
 by \/(2gh) with sufficient accuracy. 
 
 But the most certain method of attaining complete regularity 
 of resolution is to bring the reservoir under the influence of an 
 external vibrator, whose pitch is approximately the same as that 
 proper to the jet. Magnus 1 employed a Neef 's hammer, attached 
 to the frame which supported the reservoir. Perhaps an electrically 
 maintained tuning-fork is still better. Magnus shewed that the 
 most important part of the effect is due to the forced vibration of 
 that side of the vessel which contains the orifice, and that but little 
 of it is propagated through the air. With respect to the limits of 
 pitch, Savart found that the note might be a fifth above, and more 
 than an octave below, that proper to the jet. According to theory 
 there is no well defined lower limit ; while, on the other side the 
 external vibration cannot be efficient if it tends to produce divisions 
 
 1 Pogg. Ann. vol. cvi, p. 1, 1859. 
 
360.] VIBRATIONS OF LOW FREQUENCY. 365 
 
 whose length is less than the circumference of the jet. This gives 
 for the interval denning the upper limit TT : 4'51, or about a fifth. 
 In the case of Plateau's numbers (TT : 4*38) the discrepancy is a 
 little greater. 
 
 361. The question of the influence of vibrations of low 
 frequency is difficult to treat experimentally in consequence of 
 the complications which arise from the almost universal presence 
 of harmonic overtones. It is evident that the octave, for example, 
 of the principal tone, though present in a very subordinate degree,, 
 may nevertheless be the more important agent of the two in 
 determining the behaviour of the jet, if its pitch happen to lie 
 in the neighbourhood of that of maximum instability. In my own 
 experiments 1 tuning-forks were employed as sources of vibration, 
 and in every case the behaviour of the jet on its horizontal course 
 was examined not only by direct inspection, but also by the 
 method of intermittent illumination ( 42) so arranged that 
 there was one view for each complete period of the phenomenon 
 to be observed. Except when it was important to eliminate the 
 octave as far as possible, the vibration was communicated to the 
 reservoir through the table on which it stood. The forks were 
 either screwed to the table and vibrated by a bow, or maintained 
 electrically, the former method being adequate when only one fork 
 was required at a time. The circumstances of the jet were such 
 that the pitch of maximum sensitiveness, as determined by calcu- 
 lation, was 259, and that forming the transition between stability 
 and instability 372. 
 
 With pitches varying downwards from 370 to about 180, the 
 observed phenomena agreed perfectly with the unambiguous pre- 
 dictions of theory. From the point decidedly below 370 at 
 which a regular effect was first observed, there was always one 
 drop for each complete vibration of the fork, and a single stream, 
 each drop breaking away under precisely the same conditions as 
 its predecessor. After passing 180 it becomes a question whether 
 the octave of the fork's note may not produce an effect as well as 
 the prime. If this effect be sufficient, the number of drops is 
 doubled, and when the prime is very subordinate indeed, there is 
 a double stream, alternate drops breaking away under different 
 conditions and (under the action of gravity) taking sensibly 
 
 1 Proc. Roy. Soc. vol. xxxiv, p. 133, 1882. 
 
366 FILES OF DROPS [361. 
 
 different courses. In these experiments the influence of the 
 prime was usually sufficient to determine the number of drops, 
 even in the neighbourhood of pitch 128. Sometimes, however, 
 the octave became predominant and doubled the number of drops. 
 When the octave is not strong enough actually to double the 
 drops, it often produces an effect which is very apparent to an 
 observer examining the transformation through the revolving 
 holes. " On one occasion a vigorous bowing of the fork, which 
 favours the octave, gave at first a double stream, but this after 
 a few seconds passed into a single one. Near the point of 
 resolution those consecutive drops which ultimately coalesce as 
 the fork dies down are connected by a ligament. If the octave 
 is strong enough, this ligament subsequently breaks, and the 
 drops are separated ; otherwise the ligament draws the half-formed 
 drops together, and the stream becomes single. The transition 
 from the one state of things to the other could be watched with 
 facility. 
 
 In order to get rid entirely of the influence of the octave a 
 different arrangement was necessary. It was found that the 
 desired result could be arrived at by holding a 128 fork in the 
 hand over a resonator of the same pitch resting upon the table. 
 The transformation was now quite similar in appearance to that 
 effected by a fork of frequency 256, the only differences being that 
 the drops were bigger and twice as widely spaced, and that the 
 spherule, which results from the gathering together of the liga- 
 ment, was much larger. We may conclude that the cause of the 
 doubling of a jet by the sub-octave of the note natural to it is to be 
 found in the presence of the second component from which hardly 
 any musical notes are free. 
 
 When two forks of pitches 128 and 256 were sounded together, 
 the single or double stream could be obtained at pleasure by 
 varying the relative intensities. Any imperfection in the tuning 
 is rendered very evident by the behaviour of the jet, which per- 
 forms evolutions synchronous with the audible beats. This 
 observation, which does not require the aid of the stroboscopic 
 disc, suggests that the effect depends in some degree upon the 
 relative phases of the two tones, as might be expected a priori. 
 In some cases the influence of the sub-octave is shewn more in 
 making the alternate drops unequal in magnitude than in pro- 
 jecting them into very different paths. 
 
361.] DOUBLED BY OVERTONES. 367 
 
 Returning now to the case of a single fork screwed to the table, 
 it was found that as the pitch was lowered below 128, the double 
 stream was regularly established. The action of the twelfth (85^) 
 below the principal note demands special attention. At this pitch 
 we might expect the first three components of a compound note to 
 influence the result. If the third component were pretty strong, 
 it would determine the number of drops, and the result would be 
 a three-fold stream. In the case of a fork screwed to the table the 
 third component of the note must be extremely weak if not alto- 
 gether missing ; but the second (octave) component is fairly strong, 
 and in fact determined the number of drops (190f). At the same 
 time the influence of the prime (85J) is sufficient to cause the 
 alternate drops to pursue different paths, so that a double stream 
 is observed. 
 
 By the addition of a 256 fork there was no difficulty in 
 obtaining a triple stream ; but it was of more interest to examine 
 whether it were possible to reduce the double stream to a single 
 one with only 85| drops per second. In order to secure as strong 
 and as pure a fundamental tone as possible and to cause it to act 
 upon the jet in the most favourable manner, the air space in the 
 reservoir (an aspirator bottle) above the water was tuned to the 
 note of the fork by sliding a plate of glass over the neck so as 
 partially to cover it ( 805). When the fork was held over the 
 resonator thus formed, the pressure which expels the jet was 
 rendered variable with a frequency of 85^, and overtones were 
 excluded as far as possible. To the unaided eye, however, the jet 
 still appeared double, though on more attentive examination one 
 set of drops was seen to be decidedly smaller than the other. 
 With the revolving disc, giving about 85 views per second, the 
 real state of the case was made clear. The smaller drops were the 
 spherules, and the stream was single in the same sense as the 
 streams given by pure tones of frequencies 128 and 256. The 
 increased size of the spherule is of course to be attributed to the 
 greater length of the ligament, the principal drops being now three 
 times as widely spaced as when the jet is under the influence of 
 the 256 fork. 
 
 With still graver forks screwed to the table the number of 
 drops continued to correspond to the second component of the 
 note. The double octave of the principal note (64) gave 128 drops 
 per second, and the influence of the prime was so feeble that the 
 
368 BELL'S EXPERIMENTS. [361. 
 
 duplicity of the stream was only just recognisable. Below 64 the 
 observations were not carried, and even at this pitch attempts to 
 attain a single stream of drops were unsuccessful. 
 
 362. Savart's experiments upon this subject have been further 
 developed by Mr C. A. Bell, who shewed that a jet maybe made to 
 play the part of a telephonic receiver 1 . The external vibrations 
 may be conveyed to the nozzle through a string telephone ( 156 a). 
 An india rubber membrane, stretched over the upper end of a 
 metal tube, receives the jet and communicates the vibration due 
 to the varying impact to the cavity behind, with which the ear 
 may be connected. The diameter and velocity of the jet require 
 to be accommodated to the general character as to pitch of the 
 sounds to be dealt with. " When the membrane is held close 
 under the jet orifice, no sound will be audible in the ear-piece ; but 
 as the receiving tube is gradually withdrawn along the jet path, a 
 sound will be heard corresponding in pitch and quality to the dis- 
 turbing sound provided, of course, that the jet is at such pressure 
 as to be capable of responding to all the higher tones to which the 
 disturbing sound may owe its timbre. The intensity of this sound 
 grows as the distance between jet orifice and membrane is in- 
 creased. Finally, while the jet is still continuous above the 
 membrane, a point of maximum intensity and purity of tone will 
 be reached ; and if the membrane be carried beyond this point the 
 sound heard will at first increase in loudness, becoming harsh in 
 character at the same time, and at a still lower point will de- 
 generate into an unmusical roar. In the latter case the jet will be 
 seen to break above the membrane." 
 
 From the fact that small jets travelling at high speeds respond 
 equally to sounds whose pitch varies over a wide range Mr Bell 
 argues that Plateau's theory is inadequate, and he looks rather to 
 vortex motion, dependent upon unequal velocity at the centre and 
 at the exterior of the column, as the real cause of the phenomena 
 presented by these jets. 
 
 As an example of a jet self-excited, the interrupter of 235 r 
 may be referred to. In this case the machinery by which the 
 effect is carried back to the nozzle is electric. But ordinary 
 mechanical devices answer the purpose equally well. The intro- 
 duction of a resonator, such as the fork of 235 r, or the telephone 
 
 1 Phil. Trans, vol. 177, p. 383, 1886. 
 
362.] COLLISIONS BETWEEN DROPS. 369 
 
 plate which may be made to take its place, if the telephone be 
 brought in contact with the nozzle, gives greater regularity to the 
 process, and usually allows also of a greater latitude in respect of 
 pitch. It should not be forgotten that in all these cases of self- 
 excitation a certain condition as to phase needs to be satisfied. 
 If for instance in the interrupter of 235 r, supposed to be working 
 well, the platinum points be displaced through half the interval 
 between consecutive drops, it is evident that the action will cease 
 until some fresh accommodation is brought about. 
 
 363. When a small jet is projected upwards in a nearly 
 vertical direction, there are complications dependent upon the 
 collisions of the drops with one another. Such collisions are 
 inevitable in consequence of the different velocities acquired by 
 the drops as they break away irregularly from the continuous 
 portion of the column. Even when the resolution is regularized 
 by the action of external vibrations of suitable frequency, the 
 drops must still come into contact before they reach the summit 
 of their parabolic path. In the case of a continuous jet the 
 "equation of continuity" shews that as the jet loses velocity in 
 ascending, it must increase in section. When the stream consists 
 of drops following the same path in single file, no such increase 
 of section is possible ; and then the constancy of the total stream 
 demands a gradual approximation of the drops, which in the case 
 of a nearly vertical direction of motion cannot stop short of actual 
 contact. Regular vibration has, however, the effect of postponing 
 the collisions and consequent scattering of the drops, and in the 
 case of a direction of motion less nearly vertical may prevent them 
 altogether. 
 
 The behaviour of a nearly vertical fountain is influenced in an 
 extraordinary manner by the neighbourhood of an electrified body. 
 The experiment may be tried with a jet from a nozzle of 1 mm. 
 diameter rising about 50 centims. In its normal state the jet resolves 
 itself into drops, which even before passing the summit, and still 
 more after passing it, are scattered through a considerable width. 
 W^hen a feebly electrified body is presented to it, the jet undergoes 
 a remarkable transformation, and appears to become coherent; 
 but under more powerful electrical action the scattering becomes 
 even greater than at first. The second effect is readily attributed 
 to the mutual repulsion of the electrified drops, but the action of 
 R. n. 24 
 
370 INFLUENCE OF ELECTRICITY. [363. 
 
 feeble electricity in producing apparent coherence depends upon a 
 different principle. 
 
 It has been shewn by Beetz 1 that the coherence is apparent 
 only, and that the place where the jet breaks into drops is not 
 perceptibly shifted by the electricity. By screening various parts 
 with metallic plates connected to earth, Beetz further proved that, 
 contrary to the opinion of earlier observers, the seat of sensitive- 
 ness is not at the root of the jet where it leaves the orifice, but at 
 the place of resolution into drops. As in Lord Kelvin's water- 
 dropping apparatus for atmospheric electricity, the drops carry 
 away with them an electric charge, which may be collected by 
 receiving them in an insulated vessel. 
 
 It may be proved by instantaneous illumination that the 
 normal scattering is due to the rebound of the drops when they 
 come into collision. Under moderate electrical influence there is 
 no material change in the resolution into drops nor in the subse- 
 quent motion of the drops up to the moment of collision. The 
 difference begins here. Instead of rebounding after collision, as 
 the unelectrified drops of clean water generally do, the electrified 
 drops coalesce, and thus the jet is no longer scattered about 2 . An 
 elaborate discussion of this subject would be out of place here. 
 It must suffice to say that the effect depends upon a difference of 
 potential between the drops at the moment of collision, and that 
 when this difference is too small to cause coalescence there is 
 complete electrical insulation between the contiguous masses. 
 
 When the jet is projected upwards at a moderate obliquity, 
 the scattering is confined to the vertical plane. Under these 
 circumstances there are few or no collisions, as the drops have 
 room to clear one another, and moderate electrical influence is 
 without effect. At a higher obliquity the drops begin to be 
 scattered out of the vertical plane, which is a sign that collisions 
 are taking place. Moderate electrical influence will reduce the 
 scattering to the vertical plane by causing coalescence of drops 
 which come into contact. 
 
 If, as in Savart's beautiful experiments, the resolution into 
 drops is regularized by external vibrations of suitable frequency, 
 
 1 Pogg. Ann. vol. CXLIV. p. 443, 1872. 
 
 2 The influence of Electricity on Colliding Water Drops, Proc. Roy. Soc. vol. 
 xxvni. p. 406, 1879. 
 
363.] SAVART'S OBSERVATIONS. 371 
 
 the principal drops follow the same course, and unless the 
 projection is nearly vertical there are no collisions between them. 
 But it sometimes happens that the spherules are thrown out 
 laterally in a distinct stream, making a considerable angle with 
 the main stream. This is the result of collisions between the 
 spherules and the principal drops. It may even happen that the 
 former are reflected backwards and forwards several times until at 
 last they escape laterally. In all cases the behaviour under 
 feeble electrical influence is a criterion of the occurrence of 
 collisions. 
 
 In an experiment, due to Magnus 1 , the spherules are diverted 
 from the main stream without collisions by electrical attraction. 
 Advantage may be taken of this to obtain a regular procession 
 of drops finer than would otherwise be possible. 
 
 364. The detached masses of liquid into which a jet is 
 resolved do not at once assume and retain a spherical figure, but 
 execute a series of vibrations, being alternately compressed and 
 elongated in the direction of the axis of symmetry. When the 
 resolution is effected in a perfectly periodic manner, each drop is 
 in the same phase of its vibration as it passes through a given 
 point of space; and thence arises the remarkable appearance of 
 alternate swellings and contractions described by Savart. The 
 interval from one swelling to the next is the space described by 
 the drop during one complete vibration about its figure of equi- 
 librium, and is therefore, as Plateau shewed, proportional cceteris 
 paribus to the square root of the head. 
 
 The time of vibration is of course itself a function of the 
 nature of the fluid (T, p) and of the size of the drop, to the 
 calculation of which we now proceed. It may be remarked that 
 the argument from dimensions is sufficient to shew that the 
 time (T) of an infinitely small vibration of any type is proportional 
 to *J(pVjT\), where Fis the volume of the drop. 
 
 In the mathematical investigation of the small vibrations of a 
 liquid mass about its spherical figure of equilibrium, we will 
 confine ourselves to modes of vibration symmetrical about an axis, 
 which suffice for the problem in hand. These modes require for 
 their expression only Legendre's functions P n ; the more general 
 
 1 Pogg. Ann. vol. cvi. p. 27, 1859. 
 
 242 
 
372 MATHEMATICAL THEORY OF [364. 
 
 problem, involving Laplace's functions, may be treated in the 
 same way and leads to the same results. 
 
 The radius r of the surface bounding the liquid may be 
 expanded at any time t in the series ( 336) 
 
 r = a + o 1 P 1 0)+... + a n P n GO + ............ (1), 
 
 where Oj, a z ... are small quantities relatively to a , and //, repre- 
 sents, as usual, the cosine of the colatitude (6). 
 
 For the volume (V) included within the surface (1) we have 
 
 [1 + 32 (2n + l)- 1 ^ 2 /^ 2 ] ....(2), 
 
 the summation commencing at n = 1. Thus, if a be the radius of 
 the sphere of equilibrium, 
 
 a = a [l+2(2^ + l)- 1 a n 2 /a 2 ] ............... (3). 
 
 The potential energy of capillarity is the product of the 
 tension T and of the surface S. To calculate S we have 
 
 ^Yi* d0 = 2-rr flr- + i f^ 
 at// j J du 
 
 8 = 27T ( r sin r 2 4- d0 = 2-rr r- + sin 
 
 J 
 
 For the first part 
 
 f +1 r^/x = 2a 2 + 22 (2rc + I)- 1 
 For the second part 
 
 The value of the quantity on the right may be found with the 
 aid of the formula 
 
 in which m is an integer equal to or different from n. Thus 
 r+i 
 
 */>- 
 
 a n 2 f +1 P n 
 
 J -i 
 
 Accordingly 
 
 S = 47ra 2 + 2?r 2 (2w + I)' 1 (w 8 4- w + 2) a n 2 
 = 4?ra 2 + 2w2(w - l)(w + 2)(2n + l)' 1 ^ 2 ...... (4) 
 
 by (3). 
 
364.] VIBRATIONS OF DROPS. 373 
 
 Thus, if T be the cohesive tension, the potential energy (P) 
 corresponding thereto may be taken to be 
 
 l)-*an> ......... (5). 
 
 We have now to calculate the kinetic energy of the motion. 
 The velocity-potential ty may be expanded in the series 
 
 + = & + /3 1 rP 1 (fi)+...+p n rP n (iA) + ......... (6); 
 
 and thus for the kinetic energy we get 
 
 K = $p J[ d^/dr . a? d(f> dp 
 
 = Zirpa* . 2 (2n + I)- 1 na m -^ n \ 
 But by comparison of the value of d^jr/dr from (6) with (1) we find 
 
 na n ~ l p n = da n /dt ; 
 and thus 
 
 K = 2-n-pa B . 2 (2n 2 + n)~ l (da n ldt) z ............ (7). 
 
 Since the products of the quantities a n and da n /dt do not occur 
 in the expressions for P and K, the motions represented by the 
 various terms take place independently of one another. The 
 equation for a n is by Lagrange's method ( 87) 
 
 ^bl + (n_l)( + 2)^ 1 a 1 ,-0 ............ (8); 
 
 so that, if a n x cos (pt + e), 
 
 p* = n(n-V)(n + 2). } .................. (9) 1 . 
 
 The periodic time is equal to 2-Tr/p, so that in terms of V (equal 
 to f Tra 3 ) 
 
 ~ 
 
 or in the particular case of n equal to 2 
 
 r = \/{3>7rpV/8T} ...................... (11). 
 
 To find the radius of the sphere of water which vibrates 
 seconds, we put in (9) p = 27r } T=74, p = 1, n = 2. Thus 
 a = 2'47 centims., or a little less than one inch. 
 
 An attempt to compare (11) with the phenomena observed in 
 a jet did not bring out a good agreement. A stream of 19'7 cub. 
 
 1 Proc. Roy. Soc. vol. xxix. p. 97, 1879 ; Webb, Mess, of Math. vol. ix. p. 177, 1880. 
 
374 INSTABILITY DUE TO ELECTRICITY. [364. 
 
 cent, per second was broken up under the action of a fork making 
 128 vibrations per second. Neglecting the mass of the small 
 spherules, we may take for the volume of each principal drop 
 19-7/128, or -154 cub. cent. Thence by (11), putting p = 1, 
 T = 74, we have r = '0494 second. This is the calculated value. 
 By observation of the vibrating jet the distance between the first 
 and second swellings, corresponding to the maximum oblateness of 
 the drops, was 16'5 centims. The level of the contraction midway 
 between the two swellings was 36 '8 centims. below the surface of 
 the liquid in the reservoir, corresponding to a velocity of 269 
 centims. per second. These data give for the time of vibration 
 
 T = 16-5/269 = -0612 second. 
 
 The discrepancy between the two values of T is probably attribu- 
 table to excessive amplitude, entailing a departure from the 
 law of isochronism. Observations upon the vibrations of drops 
 delivered singly from pipettes have been made by Lenard 1 . 
 
 The tendency of the capillary force is always towards the 
 restoration of the spherical figure of equilibrium. By electrifying 
 the drop we may introduce a force operative in the opposite direc- 
 tion. It may be proved 2 that if Q be the charge of electricity in 
 electrostatic measure, the formula corresponding to (9) is 
 
 ^. _ 
 pa? i" 47ra 3 ) 
 
 If T > Q*/167ra s , the spherical form is stable for all displace- 
 ments. When Q is great, the spherical form becomes unstable for 
 all values of n below a certain limit, the maximum instability 
 corresponding to a great, but still finite, value of n. Under these 
 circumstances the liquid is thrown out in fine jets, whose fineness, 
 however, has a limit. 
 
 Observations upon the swellings and contractions of a regularly 
 resolved jet may be made stroboscopically, one view corresponding 
 to each complete period of the vibrator ; or photographs may be 
 taken by the instantaneous illumination furnished by a powerful 
 electric spark 3 . 
 
 1 Wied. Ann. vol. xxx. p. 209, 1887. 
 
 2 Phil. Mag. vol. xiv. p. 184, 1882. 
 
 3 Some Applications of Photography, Proc. Roy. Soc. Inst. vol. xm. p. 261, 
 1891 ; Nature, vol. XLIV. p. 249, 1891. 
 
364.] THREADS OF VISCOUS FLUID. 375 
 
 In the mathematical investigations of this chapter no account 
 has been taken of viscosity. Plateau held the opinion that the 
 difference between the wave-length of spontaneous division of a 
 jet (4*5 x 2a) and the critical wave-length (TT x 2a) was an effect 
 of viscosity ; but we have seen that it is sufficiently accounted for 
 by inertia. The inclusion of viscosity considerably complicates 
 the mathematical problem 1 , and it will not here be attempted. 
 The result is to shew that, when viscosity is paramount, long 
 threads do not tend to divide themselves into drops at mutual 
 distances comparable with the diameter of the thread, but rather 
 to give way by attenuation at few and distant places. This 
 appears to be in agreement with the observed behaviour of highly 
 viscous threads of glass, or treacle, when supported only at the 
 terminals. A separation into numerous drops, or a varicosity 
 pointing to such a resolution, may thus be taken as evidence that 
 the fluidity has been sufficient to bring inertia into play. 
 
 A still more general investigation, in which the influence of 
 electrification is considered, has been given by Basset 2 . 
 
 1 Phil. Mag. vol. xxxiv. p. 145, 1892. 
 
 2 Amer. Journ. of Math. vol. xvi. No. 1. 
 
CHAPTER XXI. 
 
 VORTEX MOTION AND SENSITIVE JETS. 
 
 365. A LARGE and important group of acoustical phenomena 
 have their origin in the instability of certain fluid motions of the 
 kind classified in hydrodynamics as steady. A motion, the same 
 at all times, satisfies the dynamical conditions, and is thus in a 
 sense possible; but the smallest departure from the ideal so 
 defined tends spontaneously to increase, and usually with great 
 rapidity according to the law of compound interest. Examples of 
 such instability are afforded by sensitive jets and flames, seolian 
 tones, and by the flute pipes of the organ. These phenomena are 
 still very imperfectly understood ; but their importance is such as 
 to demand all the consideration that we can give them. 
 
 So long as we regard the fluid as absolutely inviscid there is 
 nothing to forbid a finite slip at the surface where two masses 
 come into contact. At such a surface the vorticity ( 239) is 
 infinite, and the surface may be called a vortex sheet. The 
 existence of a vortex sheet is compatible with the dynamical 
 conditions for steady motion ; but, as was remarked at an early 
 date by v. Helmholtz 1 , the steady motion is unstable. The 
 simplest case occurs when a plane vortex sheet separates two 
 masses of fluid which move with different velocities, but without 
 internal relative motion a problem considered by Lord Kelvin in 
 his investigation of the influence of wind upon waves 2 . In the 
 following discussion the method of Lord Kelvin is applied to 
 determine the law of falling away from steady motion in some of 
 the simpler cases of a plane surface of separation. 
 
 1 Phil. Mag. vol. xxxvi. p. 337, 1868. 
 
 2 Phil. Mag. vol. XLII. p. 368, 1871. See also Proc. Math. Soc. vol. x. p. 4, 
 1878; Basset's Hydrodynamics, 391, 1888; Lamb's Hydrodynamics, 224, 1895. 
 
365.] FINITE SLIP. 377 
 
 Let us suppose that below the plane z = the fluid is of 
 constant density p and moves parallel to x with velocity V, and 
 that above that plane the density is p and the velocity V. As 
 in 353, let z be measured downwards, and let there be rigid 
 walls bounding the lower fluid at z = I and the upper fluid at 
 z l r . The disturbance is supposed to involve x and t only 
 through the factors e ikx , e lnt . The velocity potential ( Vx + <) in 
 the lower fluid satisfies Laplace's equation, and thus </> by the 
 condition at z = I takes the form 
 
 (1) ; 
 and a similar expression, 
 
 <t>' = C'coshk(z + l').e i{nt+k v ............... (2), 
 
 applies to the lower fluid, if the whole velocity-potential be there 
 (F'# + <'). The connection between <f> and the elevation (h) at 
 the common surface is 
 
 so that, if h 
 
 kCsmhkl = i(n + kV)H ............... (4). 
 
 In like manner, -kC' smh kl' = i (n + kV) H ............... (5). 
 
 We have now to express the condition relating to pressures at 
 z = 0. The general equation (2), 244, gives for the lower fluid 
 
 = gh in<f) ikV<f>, 
 
 squares of small quantities being neglected. In like manner for 
 the upper fluid at z = 
 
 If there be no capillary tension, Bp and 8p' are equal. If the 
 capillary tension be T, the difference is 
 
 so that 
 
 ......... (6). 
 
378 GRAVITY AND CAPILLARITY. [365. 
 
 When the values of <, <' at z - are introduced from (1), (2), 
 (4), (5), the condition becomes 
 
 g ( p _ p ') + jfcay = ty, ( F + n/Jc)* coth M + kp' ( V + n/&) 2 coth M' 
 
 This is the equation which determines the values of n/k. If 
 the roots of the quadratic are real, waves are propagated with the 
 corresponding real velocities ; if on the other hand the roots are 
 imaginary, exponential functions of the time enter into the 
 solution, indicating that the steady motion is unstable. The 
 criterion of stability is accordingly 
 
 (p coth kl + p' coth kV) {g (p - //) + Tk*} 
 
 - kpp coth kl coth kl' ( V- F') 2 > ...... (8). 
 
 If g and T both vanish, the motion is unstable for all disturb- 
 ances, that is, whatever may be the value of k. If T vanish, the 
 operation of gravity may be to secure stability for certain values 
 of k, but it cannot render the steady motion stable on the whole. 
 For when k is infinitely great, that is, when the corrugations are 
 infinitely fine, coth&/ = cothkl' = 1, and the term in g disappears 
 from the criterion. In spite of the impressed forces tending to 
 stability the motion is necessarily unstable for waves of infini- 
 tesimal length; and this conclusion may be extended to vortex 
 sheets of any form and to impressed forces of any kind. 
 
 If T be finite, then on the contrary there is of necessity 
 stability for waves of infinitesimal length, although there may be 
 instability for waves of finite length. 
 
 For further examination we may take the simpler conditions 
 which arise when I and I' are infinite. The criterion of stability 
 then becomes 
 
 and the critical case is determined by equating the left-hand 
 member to zero. This gives a quadratic in k. If the roots of the 
 quadratic are imaginary, the criterion (9) is satisfied for all inter- 
 mediate values of k, as well as for the infinitely small and in- 
 finitely large values by which it is satisfied in all cases, provided 
 that p > p. The condition of complete stability is thus 
 
365.] CHARACTER OF INSTABILITY. 379 
 
 Let W denote the minimum velocity ( 353) of waves when 
 7=0, 7 = 0. Then by (7) 
 
 and (10) may be written 
 
 If (7 V) do not exceed the value thus determined, the 
 steady motion is stable for all disturbances ; otherwise there will 
 be some finite wave-lengths for which disturbances increase ex- 
 ponentially. 
 
 If we now omit the terms in (7) dependent upon gravity and 
 upon capillarity, the equation becomes 
 
 p (n + kV)* coth kl + p (n + kV'} z coth kl' = ...... (13). 
 
 When I = I', or when both these quantities are infinite, we 
 have simply 
 
 p'(n + kV')* = ............... (14), 
 
 n_ . 
 
 f 
 
 k 
 
 We see from (15) that, as was to be expected, a motion 
 common to both parts of the liquid has no dynamical significance. 
 An equal addition to 7 and 7' is equivalent to a deduction of 
 like amount from n/k. Ifp=p', (15) becomes 
 
 nlk = -\(V+V')^(V-V) ............ (16). 
 
 The essential features of the case are brought out by the 
 simple case where V V, so that the steady motions of the two 
 masses of fluid are equal and opposite. We have then 
 
 n/k=iV ........................ (17); 
 
 and for the elevation, 
 
 h = He krt cos(kx+e) .................. (18), 
 
 corresponding to h = H cos (nx + e) ..................... (19), 
 
 initially. 
 
 If when t = 0, dh/dt = 0, 
 
 e) ............... (20), 
 
 indicating that the waves upon the surface of separation are 
 stationary, and increase in amplitude with the time according to 
 
380 INSTABILITY OF JETS [365. 
 
 the law of the hyperbolic cosine. The rate of increase of the term 
 with the positive exponent is extremely rapid. Since k = 2?r/X, 
 the amplitude is multiplied by e*,' or about 23, in the time 
 occupied by either stream in passing over a distance X. 
 
 If V = V, the roots (16) are equal, but the general solution 
 may be obtained by the usual method. Thus, if we put 
 
 where a is ultimately to vanish, 
 
 n/k=- V 
 and h = e ik(x - vt} {Ad*-* + Be-? ikvt - a }, 
 
 where A, B are arbitrary constants. Passing now to the limit 
 where a = ; and taking new arbitrary constants, we get 
 
 or in real quantities, 
 
 h = {C + Dt] cos k (x - Vt + e). 
 If initially h = cos kx, dh/dt = 0, 
 
 x) ......... (21). 
 
 The peculiarity of this case is that previous to the displacement 
 there is no real surface of separation at all. 
 
 The general solution involving I and I' may be adapted to 
 represent certain cases of disturbance of a two-dimensional jet of 
 width 21 playing into stationary fluid. For if the disturbance be 
 symmetrical, so that the median plane is a plane of symmetry, the 
 conditions are the same as if a fixed wall were there introduced. 
 If the surrounding fluid be unlimited, ' = oo , coth M 1 ; and 
 the equation determining n becomes, if V = 0, p' = p, 
 
 (n + kV) 2 cothkl + ri* = ............... (22), 
 
 of which the solution is 
 
 n - 
 
 kV = 1 + tonhM .................. ( " 
 
 Thus h = He** cos k Ix - r -^L n l . . . . (24), 
 
 I l + tanh&Z) 
 
 where v/(tanhM) 
 
 M 1+tanhH 
 
365.] WITH FINITE SLIP. 381 
 
 This represents the progression of symmetrical disturbances in a 
 jet of width 21 playing into a stationary environment of the 
 same density. 
 
 If kl be very small, so that the wave-length is large in com- 
 parison with the thickness of the jet, 
 
 h = He lf M- tYt cosk{a;-Vt} ............... (26). 
 
 The investigation of the asymmetrical disturbance of a jet 
 requires the solution of the problem of a single vortex sheet when 
 the condition to be satisfied at z = I is $ = 0, instead of as hitherto 
 dtj>/dz = 0. The value of <f> is 
 
 (27); 
 
 from which, if as before d<f>'jdz = when z = l', 
 
 p(n + kV)*teuhkl+p'(n + &F') 2 coth kl' = ... (28). 
 in'^00,/^/5, F'=0, 
 
 (?i + &F) 2 tanh&Z + ?i 2 =0 ............... (29). 
 
 This is applicable to a jet of width 21, moving with velocity V 
 in still fluid and displaced in such a manner that the sinuosities 
 of its two surfaces are parallel. 
 
 When kl is small, we have approximately 
 
 h = He "W' kVt cosk(x-kl.Vi) ............ (30). 
 
 By a combination of the solutions represented by (26), (30), we 
 may determine the consequences of any displacements in two 
 dimensions of the two surfaces of a thin jet moving with velocity 
 V in still fluid of its own density. 
 
 366. The investigations of 365 may be considered to afford 
 an adequate general explanation of the sensitiveness of jets. In 
 the ideal case of abrupt transitions of velocity, constituting vortex 
 sheets, in frictionless fluid, the motion is always unstable, and the 
 degree of instability increases as the wave-length of the disturb- 
 ance diminishes. 
 
 The direct application of this result to actual jets would lead 
 us to the conclusion that their sensitiveness increases indefinitely 
 with pitch. It is true that, in the case of certain flames, the 
 pitch of the most efficient sounds is very high, not far from the 
 
382 TENDENCY OF VISCOSITY. [366. 
 
 upper limit of human hearing; but there are other kinds of 
 sensitive jets on which these high sounds are without effect, and 
 which require for their excitation a moderate or even a grave 
 pitch. 
 
 A probable explanation of the discrepancy readily suggests 
 itself. The calculations are founded upon the supposition that 
 the changes of velocity are discontinuous a supposition that 
 cannot possibly agree with reality. In consequence of fluid 
 friction a surface of discontinuity, even if it could ever be formed, 
 would instantaneously disappear, the transition from the one 
 velocity to the other becoming more and more gradual, until the 
 layer of transition attained a sensible width. When this width is 
 comparable with the wave-length of a sinuous disturbance, the 
 solution for an abrupt transition ceases to be applicable, and we 
 have no reason for supposing that the instability would increase 
 for much shorter wave-lengths. 
 
 A general idea of the influence of viscosity in broadening a 
 jet may be obtained from Fourier's solution of the problem where 
 the initial width is supposed to be infinitesimal. Thus, if in the 
 general equations v and w vanish, while u is a function of y only, 
 the equation satisfied by u is (as in 347) 
 
 du _ fju d 2 u . 
 
 dt~pdy*" 
 
 The solution of this equation for the case where u is initially 
 sensible only at y = is 
 
 where *> = /*//>, and f/i denotes the initial value of Judy. When 
 2/ 2 = 4>vt, the value of u is less than that to be found at the same 
 time at 7/ = in the ratio e : 1. For air y = '16 C.G.S., and thus 
 after a time t the thickness (2y) of the jet is comparable in 
 magnitude with 1-6 V* 5 for example, after one second it may be 
 considered to be about 1 cm. 
 
 There is therefore ample foundation for the suspicion that the 
 phenomena of sensitive jets may be greatly influenced by fluid 
 friction, and deviate materially from the results of calculations 
 based upon the supposition of discontinuous changes of velocity. 
 Under these circumstances it becomes important to investigate 
 
366.] GENERAL EQUATION. 383 
 
 the character of the equilibrium of stratified motion in cases more 
 nearly approaching what is met with in practice. A complete 
 investigation which should take account of all the effects of 
 viscosity would encounter many formidable difficulties. For the 
 present purpose we shall treat the fluid as frictionless and be 
 content to obtain solutions for laws of stratification which are free 
 from discontinuity. For the undisturbed motion the component 
 velocities v, w are zero, and u is a function of y only, which we 
 will denote by U. A curve in which U is ordinate and y is 
 abscissa represents the law of stratification, and may be called for 
 brevity the velocity curve. The vorticity Z ( 239) of the steady 
 motion is equal to ^dU/dy. 
 
 If in the disturbed motion, assumed to be in two dimensions, 
 the velocities be denoted by U -f u, v, and the vorticity by Z + f, 
 the general equation (4), 239, takes the form 
 
 ) 
 
 at dx dy 
 
 in which dZ/dt = 0, dZ/das = 0. 
 
 Thus, if the square of the disturbances be neglected, the 
 equation may be written 
 
 $+U$ + ,-0 ..................... (3); 
 
 dt dx dy 
 
 and the equation of continuity for an incompressible fluid gives 
 
 + = ........................... <*> 
 
 If the values of Z and f in terms of the velocities be sub- 
 stituted in (3), 
 
 fd TJ d\(du dv\ d 2 U 
 
 l-iZU-j-ii-j -- jr}+9 ;rT = ............ ()- 
 
 \dt dx) \dy dx) dy* 
 
 We now introduce the supposition that as functions of x and t, 
 u and v are proportional to e tnt . e ikx . From (4) 
 
 iku + dv/dy = Q ............. ........... (6); 
 
 and if this value of u be substituted in (5), we obtain 
 n \d 2 v \ d 2 U 
 
 1 Proc. Math. Soc. vol. xi. p. 68, 1880. 
 
384 CASE OF STABILITY. [366. 
 
 In (7) k may be regarded as real, and in any particular 
 problem that may be proposed the principal object is to determine 
 the corresponding value of n, and especially whether it is real or 
 imaginary. One general proposition of importance relates to the 
 case where d*U/dy 2 is of one sign, so that the velocity curve is 
 wholly convex, or wholly concave, throughout the entire space 
 between two fixed walls at which the condition v = is satisfied. 
 Let n/k = p + iq, v = OL + i/3, where p, q, a, are real. Substituting 
 in (7) we get 
 
 or, on equating separately to zero the real and imaginary parts, 
 
 (9). 
 
 If- (p + UY + q 2 
 Multiplying (8) by {3, (9) by a, and subtracting, we get 
 d 2 a d 2 8 d /^da. 
 
 d 2 a d 2 = d( da dfr\ = <FU q(* + V 2 ) (w} 
 dy* dy 2 dy V dy dy) dy 2 (p + U) 2 + q 2 '" 
 
 At the limits v, and therefore both a and ft are by hypothesis 
 zero. Hence integrating (10) between the limits, we see that q 
 must be zero, if d 2 U/dy 2 is of one sign throughout the range of 
 integration. Accordingly n is real, and the motion, if not abso- 
 lutely stable, is at any rate not exponentially unstable. 
 
 Another general conclusion worthy of notice can be deduced 
 from (7). Writing it in the form 
 
 ufc 
 
 we see that, if n be real, v cannot pass from one zero value to 
 another zero value, unless d? U/dy* and (n + kU) be somewhere of 
 contrary signs. Thus if we suppose that U is positive and 
 d- U/dy* negative throughout, and that V is the greatest value of 
 U, we find that n + kV must be positive. 
 
 367. A class of problems admitting of fairly simple solution 
 is obtained by supposing the vorticity Z to be constant through- 
 out layers of finite thickness and to change its value only in 
 
367.] LAYERS OF UNIFORM VORTICITY. 385 
 
 passing a limited number of planes, for each of which y is constant. 
 In such cases the velocity curve is composed of portions of straight 
 lines which meet one another at finite angles. This state of things 
 is supposed to be disturbed by bending the surfaces of transition. 
 
 Throughout any layer of constant vorticity d? U/dy 2 = 0, and 
 thus by (7), 366, wherever n+kU is not equal to zero, 
 
 of which the solution is 
 
 v = AeP + Ber*y ........................ (2). 
 
 If there are several layers in each of which Z is constant, the 
 various solutions of the form (2) are to be fitted together, the 
 arbitrary constants being so chosen as to satisfy certain boundary 
 conditions. The first of these conditions is evidently the conti- 
 nuity of v, or as it may be expressed, 
 
 Av = .............................. (3). 
 
 The other necessary condition may be obtained by integrating 
 (7), 366, across the surface of transition. Thus 
 
 idjy/ V dy 
 
 These are the conditions that the velocity shall be continuous 
 at the places where dUjdy changes its value. 
 
 In the problems which we shall consider the fluid is either 
 bounded by a fixed plane at which y is constant, or else extends 
 to infinity. For the former the condition is simply v = 0. If 
 there be a layer extending to infinity in the positive direction, A 
 must vanish in the expression (2) applicable to this layer; if a 
 layer extend to infinity in the negative direction, the correspond- 
 ing B must vanish. 
 
 Under the first head we will consider a problem of some 
 generality, where the stratified steady motion Uf ^ 
 
 takes place between fixed walls at 2/ = and at 
 
 The vorticity is constant throughout each of 
 the three layers bounded by y = 0, y \ ; y b t , 
 y = &! + b' ; y ^-^-b', y = &! + &' -f & 2 (Fig. 67). 
 There are thus two internal surfaces where the 
 vorticity changes. The values of U at these 
 surfaces may be denoted by C/i, U a . Fig. 67. 
 
 R. II. 25 
 
386 LAYERS OF UNIFORM VORTICITY. [367. 
 
 In conformity with (3) and with the condition that v = when 
 y = 0, we may take in the first layer 
 
 v = v 1 = sinh ky ........................ (5) ; 
 
 in the second layer 
 
 v = v 2 = v 1 + H! sinh k (y b^ ............... (6) ; 
 
 in the third layer 
 
 v = v s = v% + M 2 sinh k (y bi - b') ............ (7). 
 
 The condition that v = 0, when y = b 1 + b' + b z , now gives 
 
 = M z sinh kb a + M l sinh k (6 2 + V) + sinh k (b z + b' + b,). . .(8). 
 
 We have still to express the other two conditions (4) at the 
 surfaces of transition. At the first surface 
 
 v sinh kb it & (dv/dy) = kM t ; 
 
 at the second surface 
 
 v Mi sinh kb f + sinh k (bi + b'), A (dv/dy) = kM 2 . 
 
 If we denote the values of A (d U/dy) at the two surfaces 
 respectively by A 1} A 2 , our conditions become 
 
 (n + kUJMi- A! sinh &&! = () ............... (9), 
 
 (n + kU 2 ) M a - A 2 {Mi sinh W + sinh k (b, + b')} = O...(10). 
 By (8), (9), (10) the values of M lt M Z) n are determined. 
 
 The equation for n is found by equating to zero the determi- 
 nant of the three equations. It may be written 
 
 An* + Bn+C=Q ..................... (11), 
 
 where 
 
 6 1 ) ............................................. (12), 
 
 + A 2 sinh kbz sinh k (&, + b') + Aj sinh kb sinh k (6 2 + b'). . . (13), 
 
 C = k*Ui U 2 sinh k (6, + b' + b,) 
 
 + kUi& z sinh kb 2 sinh k (6 X + 6') + A; ^Ai sinh kbi sinh A; (6 a + b') 
 + A^a sinh kbi sinh A;6 2 sinh A?6' .............................. (14). 
 
 To find the character of the roots we have to form the expression 
 for J5 2 4 AC. On reduction we get 
 
 B^-^AC= {k ( U, - U,) sinh k (b, + b'+ b,) 
 
 + AX sinh kbi sinh k (6 2 + 6') - A 2 sinh kb 2 sinh k (6 X + b')} 2 
 
 + 4AA sinh 2 kbi sinh 2 ^6 2 ....................................... (15). 
 
367.J FIXED WALLS. 387 
 
 Hence if A 1? A 2 have the same sign, that is, if the velocity 
 curve ( 366) be of one curvature throughout, B 2 4<AC is positive, 
 and the two values of n are real. Under these circumstances the 
 disturbed motion is stable. 
 
 We will now suppose that the surfaces at which the vorticity 
 changes are symmetrically situated, so that 6 X = 6 2 = 6. 
 
 In this case we find 
 
 4 = sinhfr(2& + 6') (16), 
 
 B =&(/!+ / 2 )sinh&(26+6 / )+(A 1 +A 2 )sinh&6 sinh&(6-f 6')...(17), 
 
 C = k* U^ U z sinh k (26 + b') + k ( ETA + ETA) sirm ^ sinh & (& + &') 
 + A A sinh 2 kb sinh kb' (18), 
 
 4A A sinh 4 kb 
 
 Under this head there are two sub-cases which may be 
 especially noted. The first is that in which the 
 values of U are the same on both sides of the 
 median plane, so that the middle layer is a 
 region of constant velocity without vorticity, 
 and the velocity curve is that shewn in Fig. 68. g 
 We may suppose that U= V in the middle 
 layer, and that U= at the walls, without loss 
 of generality, since any constant velocity (ET ) 
 superposed upon this system merely alters n by Fi g- 68 - 
 
 the corresponding quantity kU 0) as is evident from (7), 366. 
 
 and B?-*AC = 4 A 2 sinh 4 kb. 
 
 Hence n + *F- ** ^W + V ^ ** (20). 
 
 b sinh k (26 -f 6 ) 
 
 As was to be expected, since the curvature of the velocity 
 curve is of one sign, the values of n in (20) are real. It is easy 
 from the symmetry to see that the two normal disturbances are 
 such that the values of v at the surfaces of separation are either 
 equal or opposite for a given value of x. In the first case the 
 surfaces are bent towards the same side, and (as may be found 
 from the equations or inferred from the particular case presently 
 to be mentioned) the corresponding value of n in (20) has the 
 
 252 
 
388 
 
 LAYERS OF UNIFORM VORTICITY. 
 
 [367. 
 
 upper sign. In the second case the motion is symmetrical with 
 respect to the median plane which behaves as a fixed wall. 
 
 If the middle layer be absent (V = 0), one value of n, that 
 corresponding to the symmetrical motion, vanishes. The remain- 
 ing value is given by 
 
 2sinh 2 &6 Ftanh&fc 
 
 n+kV= . , Q77 =- - (21). 
 
 smh 2kb b 
 
 The other case which we shall consider is that in which the 
 velocities U on the two sides of the median plane are opposite to 
 one another ; so that 
 
 U^-U^V, A^-A^-^F (22). 
 
 Here = 0, and 
 
 C = - k z F 2 sinh k (26 + b') - 2k fi F 2 sinh kb sinh k(b + V) 
 -yu, 2 F 2 sinh 2 A$ sinh kb'. 
 
 For the sake of brevity we will write kb = ft, kb' ft' ; so that 
 the equation for n becomes 
 
 <tf_ _ & 2 sinh(2/3+/3')+2A^sinh sinh (/3+/3')+/* 2 sinh 2 ft sinh 2 /3' 
 FF 2 ~ k z sinh (2/8 + ft') 
 
 _ {//. sinh/3 sinh ft'+k sinh (ft + /3QJ 2 - k* sinh 2 ft 
 & 2 sinh ' sinh (2/3 + /3') 
 
 Here the two values of n are equal and opposite; and, since 
 A 1} A 2 are of opposite signs, the question is 
 open as to whether n is real or imaginary. 
 
 It is at once evident that n is real if /* be 
 positive, that is, if A : and F are of the same 
 sign as in Fig. 69. 
 
 Even when //, is negative, n 2 is necessarily 
 positive for great values of k, that is, for small 
 wave-lengths. For we have ultimately from 
 (23) n=kV. 
 
 We may now inquire for what values of //, n 2 may be negative 
 when k is very small, that is, when the wave-length is very great. 
 Equating the numerator of (23) to zero, and expanding the 
 hyperbolic sines, we get as a quadratic in /x, 
 
 yu,W + 2/^6 (b + 6') + 26 + b' = 0, 
 whence /*=!/&, or ^ = -1/6-2/6' (24). 
 
367.] 
 
 FIXED WALLS. 
 
 389 
 
 When /JL lies between these limits (and then only), in? is nega- 
 tive, and the disturbance (of great wave-length) increases expo- 
 nentially with the time. 
 
 We may express these results by means of the velocity V at 
 the wall where y 0. We have 
 
 
 The limiting values of V are therefore bVfyb' and 0. The 
 velocity curve corresponding to the first limit is shewn in Fig. 70 
 by the line QPOP'Q', the point Q being found by drawing a line 
 AQ parallel to OP to meet the wall in Q. If 6' = 26, QP is 
 parallel to OA, or the velocity is constant in each of the extreme 
 layers. 
 
 At the second limit F = 0, and the velocity curve is that 
 shewn in Fig. 71. 
 
 i ! 
 
 
 i 
 
 
 *H^ 
 
 Q' 
 
 Q 
 
 Fig. 70. 
 
 A 
 
 \ 
 Fig. 71. 
 
 It is important to notice that motions represented by velocity 
 curves intermediate between these limits are unstable in a manner 
 not possible to motions in which the velocity curve, as in Fig. 68, 
 is of one curvature throughout. 
 
 According to the first approximation, the motion of Fig. 71 
 is on the border-line between stability and instability for disturb- 
 ances of great wave-length ; but, if we pursue the calculation, 
 we find that it is really unstable. Taking in (23) 
 
 -!/& -2/6', 
 
 we get, after reduction, 
 
 indicating instability. 
 
 2 
 
 (25), 
 
390 LAYERS OF UNIFORM VORTICITY. [367. 
 
 From the second form of (23) we see that, whatever may be 
 the value of k, it is possible so to determine /JL that the disturb- 
 ance shall be unstable. The condition is simply that p must be 
 
 between the limits 
 
 , sinh k (b -f V) sinh kb 
 sinh kb sinh kb' 
 
 or - k {coth kb + coth %W}, - k {coth kb + tanh $kb'} . . .(26), 
 of which the first corresponds to the superior limit to the numeri- 
 cal value of /z. 
 
 When k is very large, the limits are very great and very close. 
 When k is small, they become 
 
 -1/6-2/6' and -1/6, 
 
 as has already been proved. As k increases from to oo , the 
 numerical value of the upper limit increases continuously from 
 1/6 + 2/6' to oo , and in like manner that of the inferior limit from 
 1/6 to oo . The motion therefore cannot be stable for all values of 
 k, if p (being negative) exceed numerically 1/6. The final condi- 
 tion of complete stability is therefore that algebraically 
 
 /*>-!/& ......................... (27). 
 
 In the transition case 
 
 (28); 
 
 it is that represented in Fig. 70. If PQ be bent more downwards 
 than is there shewn, as for example in Fig. 71, the steady motion 
 is certainly unstable. 
 
 Reverting to the general equations (11), (12), (13), (14), (15), 
 let us suppose that A 2 = 0, amounting to the abolition of the 
 corresponding surface of discontinuity. We get 
 
 B = k(U^ + U t ) sinh k (6 2 + 6' + 6 X ) + A x sinh kb, sinh k (6 2 + 6'), 
 B* - 4AC = {k ( U, - U a ) sinh k (6 2 + b' + 6 X ) 
 
 + A 1 sinh&6 1 sinh&(6 2 + 6')} 2 ; 
 so that n = -kU 2 ........................... (29), 
 
 or -kU ii sn 
 
 -" ......... (S 
 
 The latter is the general solution for two layers of constant 
 vorticity of breadths b, and 6'+ 6 2 . An equivalent result may be 
 obtained by supposing in (11) &c. that 6' = 0, or that 6 X = 0. 
 
367.] FIXED WALLS. 391 
 
 The occurrence of (29) suggests that any value of kU is 
 admissible as a value of ?^, and the meaning of this is apparent 
 from the fundamental equation (7), 366. For, at the place where 
 n + kU=Q, (1) need not be satisfied, that is, the arbitrary con- 
 stants in (2) may change their values. It is evident that, with 
 the prescribed values of n and k, a solution may be found satisfy- 
 ing the required conditions at the walls and at the surfaces where 
 dU/dy changes value, as well as equation (3) at the plane where 
 n + kU=0. In this motion an additional vorticity is supposed to 
 be communicated to the fluid at the plane in question, and it 
 moves with the fluid at velocity U. 
 
 We may inquire what occurs at a second place in the fluid 
 where the velocity happens to be the same as at the first place of 
 added vorticity. The second place may be either within a layer of 
 originally uniform vorticity, or upon a surface of transition. In 
 the first case nothing very special presents itself. If there be no 
 new vorticity at the second place, the value of v is definite as 
 usual, save as to one arbitrary multiplyer. But, consistently with 
 the given value of n, there may be new vorticity at the second as 
 well as at the first place, and then the complete value of v for the 
 given n may be regarded as composed of two parts, each propor- 
 tional to one of the new vorticities and each affected by an 
 arbitrary multiplyer. 
 
 If the second place lie upon a surface of transition, it follows 
 from (4) that v = 0, since A (d Ujdy) is finite. From this fact we 
 might be tempted to infer that the surface in question behaves 
 like a fixed wall, but a closer examination shews that the inference 
 would be unwarranted. In order to understand this, it may be 
 well to investigate the relation between v and the displacement of 
 the surface, supposed also to be proportional to e int . e ikx . Thus, if 
 the equation of the surface be 
 
 F=y-he int+ikx = ..................... (31), 
 
 the condition to be satisfied is 1 
 
 <& +U M M = ..................... (32), 
 
 at dx dy 
 
 so that -ih(n + kU) + v = ..................... (33) 
 
 1 Lamb's Hydrodynamics, % 10. 
 
392 LAYERS OF UNIFORM VORTICITY. [367. 
 
 is the required relation. A finite h is thus consistent with an 
 evanescent v. 
 
 368. In the problems of 367 the fluid is bounded by fixed 
 walls ; in those to which we now proceed, it will be considered to 
 be unlimited. As a first example, let us suppose that on the 
 upper side of a layer of thickness b the undisturbed velocity U is 
 equal to + V, and on the lower side to V t while inside the layer 
 
 Fig. 72. Fig. 73. Fig. 74. 
 
 it changes uniformly, Fig. 72. The vorticity within the layer is 
 V/b, and outside the layer it is zero. 
 
 The most straightforward method of attacking this problem is 
 perhaps on the lines of 367. From y = oo to y 0, we should 
 assume an expression of the form Vi = e^ y , satisfying the necessary 
 condition when y = oo . Then from y to y = b, 
 
 Vz = v l + M l sinh ky ; 
 and from = bto= + oo 
 
 sinh k(y b). 
 But by the conditions at + oo , v s must be of the form e~ ky , so that 
 
 The two other conditions may then be formed as in 367, and the 
 two constants M lt M 2 eliminated, giving finally an equation for n. 
 But it will be more appropriate and instructive to follow a 
 different course, suggested by vortex theory. 
 
 If we write the fundamental equation 
 
 in the form 
 
 we see that, if F= from y = - oo to y = + oo , then v = 0. Any 
 value that v may have may thus be regarded as dependent upon 
 F, and further, in virtue of the linearity, as compounded by simple 
 addition of the values corresponding to the partial values of F. 
 
368.] INFINITELY EXTENDED FLUID. 393 
 
 In the applications which we have in view Y vanishes, except at 
 certain definite places the surfaces of discontinuity where alone 
 d*U/dy 2 differs from zero. The complete value of v may thus be 
 found by summation of partial values, each corresponding to a 
 single surface of discontinuity. 
 
 To find the partial value corresponding to a surface of dis- 
 continuity situate at y = y l} we have to suppose in (2) that Y 
 vanishes at all other places, while v vanishes at +00. Thus, 
 when y>y ly v must be proportional to e~ k(y ~ yi) , and when y<y^ 
 v must be proportional to e +k(y ~y^. Moreover, since v itself must 
 be continuous at y = y\, the coefficients of the exponentials must 
 be equal, so that the value may be written 
 
 i;=(7e**(y-yi> (3), 
 
 when C is some constant. 
 
 In the particular problem above proposed there are two 
 surfaces of discontinuity, at y = and at y = b ; and accordingly 
 the complete value of v may be written in the form 
 
 v^AeW + Be***-* (4). 
 
 We have now to satisfy at each surface the equation of condi- 
 tion (4), 367. When y = 0, we have from (4) 
 
 v = A + Be~ kb , A (dv/dy) = - 2kA, 
 
 while U = -V, A (dU/dy) = + 2F/6 ; 
 
 and when y b, 
 
 v b = A e~ kb -{-B, A (dv/dy) b = - 2kB, 
 
 while U = + V, A (d U/dy) = - 2 V/b. 
 
 The conditions to be satisfied by B : A and n are thus 
 
 A{n-kV-+V/b}+B{Ve-* b /b} = (5), 
 
 A{Ve- kb /b\-B{n + kV-V/b} = (6); 
 
 from which by elimination of B : A, 
 
 n*-^' {<-!)*-*) (7). 
 
 When kb is small, that is, when the wave-length is great in 
 comparison with b, the case approximates to that of a sudden 
 transition from the velocity V to the velocity + V. Then 
 
 from (7) 
 
 w9sB -&>F 8 (8), 
 
394 
 
 LAYERS OF UNIFORM VOBTICITY. 
 
 [368. 
 
 in agreement with the value already found (17), 365. In this 
 case the steady motion is unstable. On the other hand, when kb 
 is great, we find from (7) 
 
 ^2_jL2TT2 /O\ . 
 
 n K> y \J}, 
 
 and, since the two values of n are real, the motion is stable. It 
 appears, therefore, that so far from the instability increasing 
 indefinitely with vanishing wave-length, as happens when the 
 transition from V to + V is sudden, a diminution of wave-length 
 below a certain value is accompanied by an instability which 
 gradually decreases, and is finally exchanged for actual stability. 
 The following table exhibits more in detail the progress of b*n*/V 2 
 as a function of kb : 
 
 kb 
 
 &V/F 2 
 
 kb 
 
 &% 2 /F 2 
 
 2 
 
 - -03032 
 
 1-0 
 
 - -13534 
 
 4 
 
 - -08933 
 
 1-2 
 
 - -05072 
 
 6 
 
 - -14120 
 
 1-3 
 
 + -01573 
 
 8 
 
 - -16190 
 
 2-0 
 
 + -98168 
 
 We see that the instability is greatest when kb = '8 nearly, 
 that is, when X = 8&; and that the passage from instability to 
 stability takes place when kb = 1*3 nearly, or X = 56. 
 
 Corresponding with the two values of n, there are two ratios 
 of B : A determined by (5) or (6), each of which gives a normal 
 mode of disturbance, and by means of these normal modes arbi- 
 trary initial circumstances may be represented. It will be seen 
 that for the stable disturbances the ratio B : A is real, indicating 
 that the sinuosities of the two surfaces are at every moment in 
 the same phase. 
 
 We may next take an example from a jet of thickness 26 
 moving in still fluid, supposing that the velocity in the middle of 
 the jet is V, and that it falls uniformly to zero on either side, 
 (Fig. 73). Taking the origin of y in the middle line, we may write 
 
 U=V(l+y/b) (10), 
 
 in which the sign applies to the upper, and the + sign to the 
 lower half of the jet (Fig. 73). There are now three surfaces 
 y = b, y = 0, y -f b, at which the form of v suffers discontinuity. 
 As in (4) we may take 
 
 (11); 
 
368.] INFINITELY EXTENDED FLUID. 395 
 
 so that, when 
 
 y = -b, 17=0, A(dC7dy)=F/6, 
 
 v = A + Be~ kb + Ce~* b , A (cfo/cfy) = - 2k A ; 
 
 when y = 0, Z7 = F, &(dU/dy) = -2V/b, 
 
 v = Ae~ kb + B + Ce~ kb , A (dv/dy) = - 2M? ; 
 
 when y = 6, Z7=0, A (d U/dy) = F/6, 
 
 w = 4 e-*& + 5e~* 6 + 0, A (dv/dy) = - 2A?C. 
 
 The introduction of these values into the equations of condition 
 (4), 367 gives 
 
 (12), 
 Q ............ (13), 
 
 (14), 
 which are the equations determining A : B : C and n. 
 
 By the symmetries of the case, or by inspection of (12), (13), 
 (14), we see that one of the normal disturbances is defined by 
 
 5 = 0, A + C=0 ..................... (15), 
 
 and that the corresponding value of m is 7 2 . Thus for the 
 symmetrical disturbance 
 
 (16), 
 
 indicating stability, so far as this mode is concerned. 
 
 The general determinant of the system of three equations may 
 be put into the form 
 
 (m - 7 2 ) {m 2 + ( 7 2 + 2kb - 3) m + f 2 (1 + 2kb)} = 0. . .(17), 
 
 in which the first factor corresponds to the symmetrical disturb- 
 ance already considered. The two remaining values of n are 
 real, if 
 
 (7 + ?tt-.*)P-47{l + *tt)>6 ............ (18), 
 
 but not otherwise. When kb is infinite, 7 = 0, and (18) is satis- 
 fied; so that the motion is stable when the wave-length of 
 disturbance is small in comparison with the thickness (2 b) of the 
 jet. On th.e other hand, as may be proved without difficulty by 
 expanding 7, or e~ kb , in (18), the motion is unstable, when the 
 wave-length is great in comparison with the thickness of the jet. 
 
396 
 
 LAYERS OF UNIFORM VORTICITY. 
 
 [368. 
 
 The values of the left-hand member of (18) can be more easily 
 computed when it is thrown into the form 
 
 (19). 
 
 Some corresponding values of (19) and 2kb are tabulated below : 
 
 2kb 
 
 (19) 
 
 2fc6 
 
 (19) 
 
 -5 
 
 - -054 
 
 2-5 
 
 -975 
 
 1-0 
 
 -279 
 
 3-0 
 
 -794 
 
 1-5 
 
 -599 
 
 3-5 
 
 -263 
 
 2-0 
 
 -876 
 
 4-0 
 
 + 671 
 
 The imaginary part of n, when such exists, is proportional to 
 the square root of (19). The wave-length of maximum instability 
 is thus determined approximately by 2&6=2'5, or X = 2'5x26. 
 The critical wave-length is given by 2kb = 3*5 nearly, or X = 1*8 x 26, 
 smaller wave-lengths than this leading to stability, and greater 
 wave-lengths to instability. In these respects there is a fairly 
 close analogy with cylindrical columns of liquid under capillary 
 force ( 357), although the nature of the equilibrium itself and the 
 manner in which it is departed from are so entirely different. 
 
 One more step in the direction of generality may be taken by 
 supposing the maximum velocity V to extend through a layer of 
 finite thickness b' in the middle of the jet (Fig. 74). In this layer 
 accordingly there is no vorticity, while in the adjacent layers of 
 thickness b the vorticity and velocity remain as before. 
 
 Taking, as in (11), four constants A, B, G, D to represent the 
 discontinuities at the four surfaces considered in order, and 
 writing 7 = e~ kb , 7' = e~ kb ', we have at the first surface 
 
 A (dv/dy) 
 
 at the second surface 
 
 U=V, 
 v = y A + B + yc + yy'D, 
 
 at the third surface 
 
 A (dv/dy) = -'2kB; 
 
 'A + y 'B+C + yD, 
 
 A (dv/dy) = -2kC; 
 
368.] INFINITELY EXTENDED FLUID. 397 
 
 at the fourth surface 
 
 7=0, 
 v = v 2 y'A + jy'B + yC + D, A (dv/dy) = - 2kD. 
 
 Using these values in (4) 367, we get 
 
 A{I + 2bn/ V] + yB + yy'C + <fy'D = ...... (20), 
 
 + n/V)}+yC + r/D = ...... (21), 
 
 - 26 (k + n/V)} + yD = ...... (22), 
 
 + yy'B + yC + D (1 + 26n/F| =0 ...... (23). 
 
 The elimination of the ratios A : B : C : D would give a bi- 
 quadratic in n, which, however, may be split into two quadratics, 
 one relating to symmetrical disturbances for which A + D = 0, 
 B + C = ; and the other to disturbances for which A D = 0, 
 B C 0. The resulting equation in n may be written 
 
 f - 1 + 2kb + 7 2 (1 7' 2&&y') = ...... (24). 
 
 In (24) the upper signs of the ambiguities correspond to the 
 symmetrical disturbances. The roots are real, and the correspond- 
 ing disturbances are stable, if 
 
 be positive. 
 
 In what follows we will limit our attention to the symmetrical 
 disturbances, that is, to the upper signs in (25), and to terms of 
 orders not higher than the first in 6'. The expression (25) may 
 then be reduced to 
 
 (1 _ 7 2 _ 2M) 2 + 2kb' (1 + 7 2 ) (1 - 7 2 - 2kb) ...... (26). 
 
 If kb be very small, this becomes 
 
 4,k'b*-$kb'.k*b 2 ..................... (27). 
 
 If b' is zero (27) is positive, and the disturbance is stable, as we 
 found before ; but, if b and b' be of the same order of magnitude 
 and both small compared with X, it follows from (27) that the 
 disturbance is unstable, although it be symmetrical. 
 
 If in (24) we suppose that b' = 0, we fall back upon the suppo- 
 
398 INFINITIES OCCURRING [368. 
 
 sitions of the previous problem. For the symmetrical disturbances, 
 putting 7' = 1 in (24), we get 
 
 shewing that the values of 2bn/V are 7 2 1 and - 2kb. The 
 former agrees with (16), and the latter gives n-\-kV=Q. We 
 have already seen that any value of kU is a possible solution 
 for n. 
 
 If on the other hand we suppose that b 0, we fall back upon 
 the case of a jet of uniform velocity V and thickness b' moving in 
 still fluid. The equation for n becomes, after division by 6 2 , 
 
 n" + (1 7') kV. n + i (1 7) & 2 F 2 = 0, 
 
 or (?i + A;F) 2 7 + n 2 = .................. (28). 
 
 In (28) = coth $W, = tanh 
 
 17 1+7 
 
 so that the result is in harmony with (22), (29), 365, where I 
 corresponds with &'. 
 
 Another particular case of (24), comparable with previous 
 results, is obtained by supposing b' to be infinite. 
 
 369. When d?U/dy* is finite, we must fall back upon the 
 general equation 366 
 
 ............... , 
 
 df 
 
 from which the curve representing v as a function of y can 
 theoretically be constructed when n (being real) is known. In fact 
 we may regard (1) as determining the curvature with which we 
 are to proceed in tracing the curve through any point. At a 
 place when n + kU vanishes, that is, where the stream-velocity is 
 equal to the wave- velocity, the curvature becomes infinite, unless 
 v vanishes. The character of the infinity at such a place (suppose 
 y = 0) would be most satisfactorily investigated by means of the 
 complete solution of some particular case. It is, however, sufficient 
 to examine the form of solution in the neighbourhood of y = 0, and 
 for this purpose the differential equation may be simplified. Thus, 
 when y is small, ?i + &Z7may be treated as proportional to y, and 
 
369.] WHEN n + kU=Q. 399 
 
 d 2 Ujdy- as approximately constant. In comparison with the large 
 term, k'-v 2 may be neglected, and it suffices to consider 
 
 d 2 v/dy* + y~ l v = ........................ (2), 
 
 a known constant multiplying y being omitted for the sake of 
 brevity. This falls under the head of Ricati's equation 
 
 d?v/df + y"v = ........................ (3), 
 
 of which the solution is in general (771 fractional) 1 
 
 v = V2/. (4 J(f ) + ./_. (0) ................ (4), 
 
 where w =!/(//. + 2), f = 2my 1/2TO .................. (5). 
 
 When, as in the present case, m is integral, J_ m (f ) is to be replaced 
 ( 341) by the function of the second kind Y m (f). The general 
 solution of (2) is accordingly 
 
 (6). 
 
 In passing through zero y changes sign and with it the 
 character of the functions. If we regard (6) as applicable on the 
 positive side, then on the negative side we may write 
 
 (7), 
 the argument of the functions in (7) being pure imaginaries. 
 
 From the known forms of the functions ( 341) we may deduce, 
 as applicable when y is small, 
 
 v A {y -| y"} 
 + B$(\-y + W)-\QgWy).(y-W) + y-%f} ...... (8); 
 
 so that ultimately 
 
 v remaining finite in any case. 
 
 We will now shew that any value of kU is an admissible 
 value of n in (1). The place where n + kU = is taken as origin 
 of y ; and in the first instance we will suppose that n + k U 
 vanishes nowhere else. In the immediate neighbourhood of y = 
 the solutions applicable upon the two sides are (6), (7), and they are 
 subject to the condition that v shall be continuous. Hence by 
 
 1 Lommel, Studien fiber die Bessel'schen Functionen 31, Leipzig, 1868 ; Gray 
 and Matthews' Bessel Functions, p. 233, 1895. 
 
400 SENSITIVE FLAMES. [369. 
 
 (9), B = D, leaving three constants arbitrary. The manner in 
 which the functions start from y = being thus ascertained, their 
 further progress is subject to the original equation (1), which 
 completely defines them when the three arbitraries are known. In 
 the present case two relations are given by the conditions to be 
 satisfied at the fixed walls or other boundaries of the fluid, and 
 thus is determined the entire form of v t save as to a constant 
 multiply er. If B and D are finite, there is infinite vorticity at the 
 origin. 
 
 Any other places at which n + kU=Q may be treated in a 
 similar manner, and the most general solution will contain as 
 many arbitrary constants as there are places of infinite vorticity. 
 But the vorticity need not be infinite merely because n + k U = ; 
 and in fact a particular solution may be obtained with only one 
 infinite vorticity. At any other of the critical places, such for 
 example as we may now suppose the origin to be, B and D may 
 vanish, so that v = 0, d*v/dy 2 = A,orC. 
 
 From this discussion it would seem that the infinities which 
 present themselves when n + kU= do not seriously interfere with 
 the application of the general theory, so long as the square of the 
 disturbance from steady motion is neglected. 
 
 A large part of the preceding paragraphs is taken from certain 
 papers by the author 1 . The reader should also consult Lord 
 Kelvin's writings 2 in which the effects of viscosity are dealt with. 
 
 370. It remains to describe the phenomena of sensitive 
 flames and to indicate, so far as can be done, the application of 
 theoretical principles. In a sense the combination of flame and 
 resonator described in 322 h may be called sensitive, but in this 
 case it is rather the resonator to which the name attaches, the 
 office of the flame being to maintain by a periodic supply of heat 
 the vibration of the resonator when once started. Following 
 Tyndall, we may conveniently limit the term to naked flames and 
 jets, where the origin of the sensitiveness is undoubtedly to be 
 found in the instability which accompanies vortex motion. 
 
 The earliest observation upon this subject was that of Prof. 
 
 1 Proc. Math. Soc., vol. xi. p. 57, 1880; vol. xix. p. 67, 1887. It is hoped shortly 
 to communicate a supplement. 
 
 2 Phil. Mag. vol. xxiv. pp. 188, 272, 1887. 
 
370.] EARLY OBSERVATIONS. 401 
 
 Leconte 1 , who noticed the jumping of the flame from an ordinary 
 fishtail burner in response to certain notes of a violoncello. The 
 sensitive condition demanded that in the absence of sound the 
 flame should be on the point of flaring. When the pressure of 
 gas was reduced, the sensitiveness was lost. 
 
 An independent observation of the same nature drew the 
 attention of Prof. Barrett to sensitive flames ; and he investigated 
 the kind of burner best suited to work with the ordinary pressure 
 of the gas mains 2 . " It is formed of glass tubing about | of an 
 inch (1 cm.) in diameter, contracted to an orifice ^ of an inch 
 (*16 cm.) in diameter. It is very essential that this orifice should 
 be slightly V-shaped.... Nothing is easier than to form such a 
 burner ; it is only necessary to draw out a piece of glass tubing in 
 a gas flame, and with a pair of scissors snip the contraction into 
 the shape indicated." 
 
 But the most striking by far is the high-pressure flame 
 employed by Tyndall. The gas is supplied from a special holder 
 under a pressure of say 25 cm. of water to a pinhole steatite 
 burner, and the flame rises to a height of about 40 cm. Under the 
 influence of a sound of suitable (very high) pitch the flame roars, 
 and drops down to perhaps half its original height 3 . Tyndall 
 shewed that the seat of sensitiveness is at the root of the flame. 
 Sound coming along a tube is ineffective when presented to the 
 flame a little higher up, and also when caused to impinge upon 
 the burner below the place of issue. 
 
 It is to Tyndall that we owe also the demonstration that it is 
 not to the flame as such that these extraordinary effects are to be 
 ascribed. Phenomena substantially the same are obtained when 
 a jet of unigriited gas, of carbonic acid, hydrogen, or even air 
 itself, issues from an orifice under proper pressure. They may be 
 rendered visible in two ways. By association with smoke the 
 whole course of the jet may be made apparent ; and it is found 
 that suitable smoke jets can surpass even flames in delicacy. 
 "The notes here effective are of much lower pitch than those 
 which are most efficient in the case of flames." Another way of 
 making the sensitiveness of an air-jet visible to the eye is to cause 
 
 1 On the Influence of Musical Sounds on the Flame of a Jet of Coal-gas. Phil. 
 Mag. vol. xv. p. 235, 1858. 
 
 - Phil. Mag. vol. xxxm. p. 216, 1867. 
 
 3 Phil. Mag. vol. xxxm. pp. 92, 375, 1867 ; Sound, 3rd ed. ch. vi. 
 
 R. ii. 26 
 
402 VARICOSE OR SINUOUS? [370. 
 
 it to impinge upon a flame, such as a candle flame, which plays 
 merely the part of an indicator. 
 
 In the sensitive flame of Prof. Govi * and of Mr Barry 2 the gas 
 is unignited at the burner, but catches fire on the further side of 
 wire-gauze held at a suitable distance. On the same principle is 
 an arrangement employed by the author 3 . A jet of coal gas from 
 a pinhole burner rises vertically in the interior of a cavity from 
 which air is excluded. It then passes into a brass tube a few 
 inches long, and on reaching the top burns in the open. The 
 front wall of the cavity is formed of a flexible membrane of tissue- 
 paper, through which external sounds can reach the burner. In 
 these cases the sensitive agent is the unignited part of the jet. 
 Used in this way a given burner requires a much less pressure of 
 gas than is necessary when the flame is allowed to reach it, and 
 the sounds which have the most influence are graver. 
 
 Struck by the analogy between these phenomena and those 
 of water-jets investigated by Savart and Plateau, the earlier ob- 
 servers seem to have leaped to the conclusion that the manner of 
 disintegration was also similar symmetrical, that is, about the 
 axis ; and Prof. Leconte went so far as to deduce the existence of 
 a cohesive force in gases. A surface tension, however, requires a 
 very abrupt transition between the properties of the matter on 
 the two sides, such as could have only a momentary existence 
 when there is a tendency to mix, so that it appears extremely 
 unlikely that capillarity plays here any sensible part. 
 
 The question of the manner of disintegration, whether it be by 
 gradually increasing varicosity or by gradually increasing sinuosity, 
 is of the greatest importance, and the answer is still, perhaps, in 
 some cases open to doubt. But that the latter is predominant in 
 general follows from a variety of arguments. The necessity, as 
 remarked by Barrett, for an unsymmetrical orifice points strongly 
 in this direction. The same conclusion is drawn by Ridout 4 from 
 the results of some ingenious experiments. The latter observer 
 found further that fishtail flames, formed by the union at a small 
 angle of jets from two perfectly similar glass nozzles, shewed a 
 
 1 Torino, Atti Acad. Sci. vol. v. p. 396, 1869. 
 8 Tyndall's Sound, 3rd edition, p. 240. 
 
 3 Camb. Phil. Soc. Proc. vol. iv. p. 17, 1883. 
 
 4 Nature, vol. xvm. p. 604, 1878. 
 
370.] NODES AND LOOPS. 403 
 
 sensitiveness dependent upon the direction of the sound. If this 
 direction lie in the plane of symmetry containing the flame (that 
 perpendicular to the plane of the nozzles), there is no response. 
 
 Even in the case of the tall high-pressure flame from a pin- 
 hole burner, where to all appearance both the nozzle and the 
 flame (when undisturbed) are perfectly symmetrical, there is 
 reason to believe that the manner of disintegration is sinuous, or 
 unsymmetrical. Perhaps the easiest road to this conclusion is by 
 examining the behaviour of the flame when exposed to stationary 
 sonorous waves, such as may be derived by superposing upon direct 
 waves from a source giving a pure tone the waves reflected perpen- 
 dicularly from a flat obstacle, e.g. a sheet of glass. According to 
 the analogy with capillary jets, an analogy pushed further than it 
 will bear by most writers upon this subject, the flame should be 
 excited when the nozzle is situated at a node, where the pressure 
 varies most, and remain unaffected at a loop where the pressure 
 does not vary at all. There was no difficulty in proving experi- 
 mentally J that the facts are precisely the opposite. The source 
 of sound was a bird-call ( 371), and the observations were made 
 by moving the burner to and fro in front of the reflector until 
 the positions were found in which the flame was least disturbed. 
 These positions were very well defined, and the measurements 
 shewed distances from the reflector proportional to the series 
 of numbers 1, 2, o, &c., and therefore corresponding to nodes. 
 If the positions had coincided with loops, the distances would 
 have formed a series proportional to the odd numbers 1, 3, 5, &c. 
 The wave-length of the sound, determined by the doubled 
 interval between consecutive minima, was 31'2 mm., corresponding 
 to pitch f$. 
 
 A few observations were made at the same time on the 
 positions of the silences as estimated by the ear listening through 
 a tube. As was to be expected, they coincided with the loops, 
 bisecting the intervals given by the flame. When the flame was 
 in a position of minimum effect, and the free end of the tube was 
 held close to the burner at an equal distance from the reflector, 
 the sound heard was a maximum, and diminished when the 
 end of the tube was displaced a little in either direction. It was 
 thus established that the flame is affected where the ear would 
 not be affected, and vice versa. 
 
 1 Phil. Mag. vol. vn. p. 153, 1879. 
 
 262 
 
404 OBSTRUCTION PREJUDICIAL. [370. 
 
 Flames from pinhole burners, which perform well in other 
 respects, seem always to shew a marked difference according to 
 the direction in which the sound arrives. If, while a bird-call is 
 in operation, the burner be turned steadily round its axis, two 
 positions differing by 180 are found, in which there is little or no 
 response. This peculiarity may sometimes be turned to account 
 in experiment 1 . Thus after such an adjustment has been made 
 that the direct sound has no effect, vigorous flaring may yet 
 result from the impact of sound from the same source after 
 reflection from a small pane of glass, the pane being held so that 
 the direction of arrival is at 90 to that of the direct sound, 
 and this although the distance travelled by the reflected sound is 
 the greater. 
 
 Tyndall 2 lays it down as an essential condition of complete 
 success in the more delicate experiments with these flames, " that 
 a free way should be open for the transmission of the vibrations 
 from the flame, backwards, through the gaspipe which feeds it. 
 The orifices of the stopcocks near the flame ought to be as wide 
 as possible." The recommendation is probably better justified 
 than the reason given for it. Prof. Barrett 3 attributes the evil 
 effect of a partially opened stopcock to the irregular flow and 
 consequent ricochetting of the current of gas from side to side of 
 the pipe. In some experiments of my own 4 the introduction of a 
 glass nozzle into the supply pipe, making the flow of gas in 
 the highest degree irregular, did not interfere, nor did other 
 obstructions unless attended by hissing sounds. The prejudicial 
 action of a partially opened stopcock was thus naturally attri- 
 buted to the production of internal sounds of the kind to which 
 the flame is sensitive, and this view of the matter was confirmed 
 by some observations of the pressure of the gas in the neighbour- 
 hood of the burner. " In the path of the gas there were inserted 
 two stopcocks, one only a little way behind the manometer 
 junction, the other separated from it by a long length of india- 
 rubber tubing. When the first cock was fully open, and the 
 flame was brought near the flaring-point by adjustment of the 
 distant cock, the sensitiveness to external sounds was great, 
 
 1 Proc. Roy. Inst. vol. xn. p. 192, 1888 ; Nature, vol. xxxvin. p. 208, 1888. 
 
 2 Phil. Mag. vol. xxxm. p. 99, 1867. 
 
 3 Phil. Mag. vol. xxxm. p. 288, 1867. 
 
 4 Phil. Mag. vol. xin. p. 345, 1882. 
 
370.] PERIODIC VIEW. 405 
 
 and the manometer indicated a pressure of 10 inches (25'4 cm.) 
 of water. But when the distant cock stood fully open and the 
 adjustment was effected at the other, high sensitiveness could not 
 be obtained ; and the reason was obvious, because the flame 
 flared without external excitation while the pressure was still an 
 inch (2*54 cm.) short of that which had been borne without 
 flinching in the former arrangement. On opening again the 
 neighbouring cock to its full extent, and adjusting the distant 
 one until the pressure at the manometer measured 9 inches 
 (22*9 cm.), the flame was found comparatively insensitive." 
 
 The most direct and satisfactory evidence as to the manner of 
 disintegration is of course that of actual observation. Using a jet 
 of phosphorus smoke from a glass nozzle and a stroboscopic disc, 
 I was able (in 1879) to see the sinuosities when the jet was 
 disturbed by a fork of pitch 256 vibrating in its neighbourhood 1 . 
 Moreover by placing the nozzle exactly in the plane of symmetry 
 between the prongs of the tuning-fork, it could be verified that 
 the disturbance required is motion transverse to the jet. In this 
 position there was but little effect ; but the slightest displacement 
 led to an early rupture of the jet. 
 
 " In order to exalt the sensitiveness of jets to notes of mode- 
 rate pitch, I found the use of resonators advantageous. These 
 may be of Helmholtz's pattern ; but suitably selected wide-mouth 
 bottles answer the purpose. What is essential is that the jet 
 should issue from the nozzle in the region of rapid reciprocating 
 motion at the mouth of the resonator, and in a transverse direction. 
 
 " Good results were obtained at a pitch of 256. When two 
 forks of about this pitch, and slightly out of tune with one another, 
 were allowed to sound simultaneously, the evolutions of the smoke- 
 jet in correspondence with the audible beats were very remarkable. 
 By gradually raising the pressure at which the smoke is supplied, 
 in the manner usual in these experiments, a high degree of 
 sensitiveness may be attained, either with a drawn-out glass 
 nozzle or with the steatite pinhole burner used by Tyndall. In 
 some cases (even at pitch 256) the combination of jet and resona- 
 tor' proved almost as sensitive to sound as the ear itself. 
 
 " The behaviour of the sensitive jet does not depend upon the 
 smoke-particles, whose office is merely to render the effects more 
 
 1 Phil. Mag. vol. xvn. p. 188, 1884. 
 
406 SENSITIVE JETS OF [370. 
 
 easily visible. I have repeated these observations without smoke 
 by simply causing air-jets from the same nozzles to impinge upon 
 the flame of a candle placed at a suitable distance. In such cases, 
 as has been pointed out by Tyndall, the flame acts merely as an 
 indicator of the condition of the otherwise invisible jet. Even 
 without a resonator the sensitiveness of such jets to hissing 
 sounds may be taken advantage of to form a pretty experiment. 
 
 " The combination of jet, resonator, and flame shows sometimes 
 a tendency to speak on its own account ; but I did not succeed in 
 getting a well-sustained sound. Such as it is, the effect probably 
 corresponds to one observed by Savart and Plateau with water-jets 
 breaking up under the operation of the capillary tension and, when 
 resolved into drops, impinging upon a solid obstacle, such as the 
 bottom of a sink, in mechanical connection with the nozzle from 
 which the jet originally issues. In virtue of the connexion, any 
 regular cycle in the mode of disintegration is able, as it were, to 
 propagate itself." 
 
 "In the hope of being able to make better observations 
 upon the transformations of unstable jets, I next had recourse to 
 coloured water issuing under water. In this form the experiment 
 is more manageable than in the case of smoke-jets, which are 
 difficult to light, and liable to be disturbed by the slightest 
 draught. Permanganate of potash was preferred as a colouring 
 agent, and the colour may be discharged by mixing with the 
 general mass of liquid a little acid ferrous sulphate. The jets 
 were usually projected downwards into a large beaker or tank 
 of glass, and were lighted from behind through a piece of ground 
 glass. 
 
 " The notes of maximum sensitiveness of these liquid jets were 
 found to be far graver than for smoke-jets or for flames. Forks 
 vibrating from 20 to 50 times per second appeared to produce the 
 maximum effect, to observe which it is only necessary to bring the 
 stalk of the fork into contact with the table supporting the appa- 
 ratus. The general behaviour of the jet could be observed without 
 stroboscopic appliances by causing the liquid in the beaker to 
 vibrate from side to side under the action of gravity. The line of 
 colour proceeding from the nozzle is seen to become gradually more 
 and more sinuous, and a little further down presents the appear- 
 ance of a rope bent backwards and forwards upon itself. I have 
 followed the process of disintegration with gradually increasing 
 
370.] LIQUID IN LIQUID. 
 
 frequencies of vibrational disturbance from 1 or 2 per second up 
 to about 24 per second, using electro-magnetic interrupters to 
 send intermittent currents through an electro-magnet which acted 
 upon a soft-iron armature attached to the nozzle. At each stage 
 the pressure at which the jet is supplied should be adjusted so as 
 to give the right degree of sensitiveness. If the pressure be too 
 great, the jet flares independently of the imposed vibration, and 
 the transformations become irregular : in the contrary case the 
 phenomena, though usually observable, are not so well marked as 
 when a suitable adjustment is made. After a little practice it is 
 possible to interpret pretty well what is seen directly; but in 
 order to have before the eye an image of what is really going on, 
 we must have recourse to intermittent vision. The best results 
 are obtained with two forks slightly out of tune, one of which is 
 used to effect the disintegration of the jet, and the other (by means 
 of perforated plates attached to its prongs) to give an intermittent 
 view. The difference of frequencies should be about one per 
 second. When the means of obtaining uniform rotation are at 
 hand, a stroboscopic disk may be substituted for the second fork 1 . 
 
 " The carrying out of these observations, especially when it is 
 desired to make a drawing, is difficult unless we can control the 
 plane of the bendings. In order to see the phases properly it is 
 necessary that the plane of bendings should be perpendicular to 
 the line of vision ; but with a symmetrical nozzle this would occur 
 only by accident. The difficulty may be got over by slightly 
 nicking the end of the drawn-out glass nozzle at two opposite 
 points (Barrett). In this way the plane of bending is usually 
 rendered determinate, being that which includes the nicks, so that 
 by turning the nozzle round its axis the sinuosities of the jet may 
 be properly presented to the eye, 
 
 " Occasionally the jet appears to divide itself into two parts 
 imperfectly connected by a sort of sheet. This seems to corre- 
 spond to the duplication of flames and smoke-jets under powerful 
 sonorous action, and to be due to what we may regard as the 
 broken waves taking alternately different courses." 
 
 " It has already been noticed that the notes appropriate to 
 water -jets are far graver than for air-jets from the same nozzles. 
 
 1 In the original paper (Phil. Hog. vol. xvn. p. 188, 1884) drawings by Mrs 
 Sidgwick are given. See also Proc. Roy. Inst. vol. xin. p. 261, 1891, for repro- 
 ductions of instantaneous photographs. 
 
408 VARYING VISCOSITY. [370. 
 
 Moreover, the velocities suitable in the former case are much less 
 than in the latter. This difference relates not, as might perhaps 
 be at first supposed, to the greater density, but to the smaller 
 viscosity of the water, measured of course kinematically. It is 
 not difficult to see that the density, presumed to be the same for 
 the jet and surrounding fluid, is immaterial, except of course in so 
 far as a denser fluid requires a greater pressure to give it an 
 assigned velocity. The influence of fluid viscosity upon these 
 phenomena is explained in a former paper on the Stability or 
 Instability of certain Fluid Motions 1 ; and the laws of dynamical 
 similarity with regard to fluid friction, laid down by Prof. Stokes 2 , 
 allow us to compare the behaviour of one fluid with another. The 
 dimensions of the kinematic coefficient of viscosity are those of an 
 area divided by a time. If we use the same nozzle in both cases, 
 we must keep the same standard of length ; and .thus the times 
 must be taken inversely, and the velocities directly, as the co- 
 efficients of viscosity. In passing from air to water the pitch and 
 velocity are to be reduced some ten times. But, in spite of the 
 smaller velocity, the water-jet will require the greater pressure 
 behind it, inasmuch as the densities differ in a ratio exceeding 
 100 : 1." 
 
 Guided by these considerations, I made experiments to try 
 whether the jets would behave differently in warm (less viscous) 
 water, and as to the effect of substituting for water a mixture of 
 alcohol and water in equal parts, a fluid known to be more viscous 
 than either of its constituents. The effect of varying the viscosity 
 was found to be very distinct. A jet which would not bear 
 a pressure of more than J inch ('63 cm.) of water without flaring 
 when the liquid was water at a temperature under the boiling- 
 point required about 25 inches (63 cm.) pressure to make it flare 
 when the alcoholic mixture was substituted. The importance 
 of viscosity in these phenomena was thus abundantly established. 
 
 The manner in which viscosity operates is probably as follows. 
 At the root of the jet, just after it issues from the nozzle, there is 
 a near approach to discontinuous motion, and a high degree 
 of instability. If a disturbance of sufficient intensity and of 
 
 1 Math. Soc. Proc. Feb. 12, 1880. See 366. . 
 
 - Camb. Phil. Tram. 1850, " On the Effect of Internal Friction of Fluids 011 the 
 Motion of Pendulums," 5. See also Helmholtz, Wied. Ann. Bd. vii. p. 337 (1879) 
 or Reprint, vol. i. p. 891. 
 
370.] BELL'S EXPERIMENT. 409 
 
 suitable period have access, the regular motion is lost and cannot 
 afterwards be recovered. But the instability has a very short 
 time in which to produce its effect. Under the influence of viscosity 
 the changes of velocity become more gradual, and the instability 
 decreases rapidly if it does not disappear altogether. Thus if the 
 disturbance be insufficient to cause disintegration during the 
 brief period of instability, the jet may behave very much as 
 though it had not been disturbed at all, and may reach the full 
 developement observed in long flames and smoke-jets. This 
 temporary character of the instability is a second feature differ- 
 entiating strongly these jets from those of Savart, in which 
 capillarity has an unlimited time of action. 
 
 When a flame is lighted at the burner, there are further 
 complications of which it is difficult to give an adequate 
 explanation. The high temperature leads indeed to increased 
 viscosity, and this tends to explain the higher pressure then 
 admissible, and the graver notes which then become operative. 
 But it is probable that the change due to ignition is of a still 
 more fundamental nature. 
 
 An ingenious method of observation, due to Mr C. Bell J , may 
 be applied so as to give valuable information with regard to 
 the disintegration of jets ; but the results obtained by the author 
 are not in harmony with the views of Mr Bell, who favours 
 the symmetrical theory. In this method a second similar nozzle 
 faces directly the nozzle from which the air issues, and is con- 
 nected with the ear of the observer by means of rubber tubing. 
 Suitable means are provided whereby the position of the hearing 
 nozzle may be adjusted with accuracy, both longitudinally and 
 laterally. When the distance is properly chosen, small disturb- 
 ances acting upon the jet are perceived upon a magnified scale. 
 Thus a fork vibrating feebly and presented to the jet is loudly 
 heard ; and that the effect is due to the peculiar properties of 
 the jet is proved at once by cutting off the supply of air, when 
 the sound becomes feeble, if not inaudible. Mr Bell proved that 
 the efficacy of the arrangement requires a small area in the 
 hearing nozzle ; if the latter be large enough to receive the whole 
 stream of air accompanying the jet, comparatively little is heard. 
 
 In the following experiments an air-jet from a well-regulated 
 bellows issued from a glass nozzle and impinged upon a similar 
 
 1 Phil. Trait*, vol. CLXXVII. p. 383, 1886. 
 
410 BIRD-CALLS. [370. 
 
 hearing nozzle. It was excited by forks (c' or c") held in the 
 neighbourhood. 
 
 If the position of the fork was such that the plane of its 
 prongs was perpendicular to the jet, and that the prolongation of 
 the axis of the stalk intersected the delivery end of the nozzle, 
 the sound perceived was much less than when the fork was 
 displaced laterally in its own plane so as to bring the nozzle 
 nearer to one prong. This appears to prove that here again the 
 effect is due, not to variation of pressure, but to transverse motion, 
 causing the jet to become sinuous. 
 
 Confirmatory evidence may be drawn from observations upon 
 the effect of slight movements of the hearing nozzle. When this 
 is adjusted axially, but little is perceived of the fundamental tone 
 of a fork presented laterally to the jet nozzle, but the octave tone 
 is heard and often very strongly. When, however, the hearing 
 nozzle is displaced laterally, the fundamental tone of the fork 
 comes in loudly. 
 
 371. In that very convenient source of sounds of high pitch, 
 the "bird-call," a stream of air issuing from a circular hole in 
 a thin plate impinges centrically upon a similar hole in a parallel 
 plate held at a little distance. The circumstances upon which 
 the pitch depends have been investigated by Sondhauss 1 , but 
 much remains obscure as regards the manner in which the 
 vibrations are excited. 
 
 According to Sondhauss the pitch is comparatively inde- 
 pendent of the size and shape of the plates, varying directly 
 as the velocity (v) of the jet and inversely as the distance (d) 
 between the plates. If we assume independence of other 
 elements, and that the frequency (n) is a function only of v, d, and 
 b the diameter of the jet, it follows from dynamical similarity 
 that 
 
 n = v/d.f(b/d) (1), 
 
 where / is an arbitrary function. Thus, if b/d be constant, 
 Sondhauss' law must hold. From the very small dimensions 
 employed it might fairly be argued that the action must be nearly 
 independent of the velocity of sound, and therefore (v being given) 
 of the density of the gas; but the question arises whether 
 viscosity may not be an element of importance. If we suppose 
 
 1 Pogg. Ann. vol. xci. p. 126, 1854. 
 
371.] EXTREME PITCH. 411 
 
 that geometrical similarity is maintained (b proportional to d), 
 the theoretical form, when viscosity is retained, is 
 
 n=v/d . F(v/vd) (2), 
 
 v being the kinematic coefficient of viscosity, of dimensions 2 
 in space and 1 in time. But when we take a numerical example, 
 it appears improbable that the degree of viscosity can play much 
 part in determining the pitch. In C.G.S. measure v='16 for air; 
 and if the pressure propelling the jet be 1 cm. of mercury, v = 
 4000 (cm./sec.). Thus, if we take d = '1 cm., we have v/vd = '0004, 
 so that F (v/vd) could hardly differ much from F(Q). 
 
 Bird-calls are very easily made. The first plate, of 1 or 2 cm. 
 in diameter, is cemented, or soldered, to the end of a short supply 
 tube. The second plate may conveniently be made triangular, 
 the turned down corners being soldered to the first plate. For 
 calls of medium pitch the holes may be made in tin plate, 
 but when it is desired to attain a very high pitch thin brass, 
 or sheet silver, is more suitable. The holes may then be as small 
 as J mm. in diameter, and the distance between them as little as 
 1 mm. In any case the edges of the holes should be sharp 
 and clean \ 
 
 In order to test a bird-call it should be connected with a well- 
 regulated supply of wind and with a manometer by which the 
 operative pressure can be measured with precision. When it 
 is found to speak well, the pressure and corresponding wave- 
 length should be recorded. If the tones are high or inaudible, a 
 high-pressure sensitive flame is required, the wave-length being 
 deduced from the interval between the positions in which 
 a reflector must be held in order that the flame may shew the 
 least disturbance ( 370). There is no difficulty in obtaining 
 wave-lengths (complete) as low as 1 cm., and with care wave- 
 lengths of *6 cm. may be reached, corresponding to about 50,000 
 vibrations per second. In experimenting upon minimum wave- 
 lengths, the distance between the call and the flame should 
 not exceed 50 cm., and the flame should be adjusted to the verge 
 of flaring. 
 
 In many cases a bird-call, which otherwise will not speak, may 
 be made to do so by a reflecting plate held at a short distance in 
 front. In practice the reflector is with advantage reduced to a 
 
 1 Prof. A. M. Mayer has constructed beautifully finished bird-calls in which the 
 distance between the plates is adjustable by a screw motion. 
 
412 AEOLIAN TONES. [371. 
 
 strip of metal, e.g. 1 cm. wide ; and, when this assistance is required, 
 the right distance is an (even or odd) multiple of the half wave- 
 length. In some cases the necessary position of the strip is very 
 sharply defined. 
 
 On the question whether the disturbance of the jet accom- 
 panying the production of the sound is varicose or sinuous, some 
 evidence may be derived from observations upon the manner in 
 which the sound radiates. Upon the latter view we might expect 
 that the sound would fall off, or even disappear altogether, in the 
 axial direction, as happens, for example, in the case of the sound 
 radiated from a bell ( 282). But, so far as I have been able to 
 observe, the sound emitted from a bird-call, speaking without the 
 aid of a reflecting strip, is uniform through a wide angle ; and this 
 fact may be regarded as telling strongly in favour of the view that 
 the disturbance is here symmetrical, or varicose, in character. 
 Other evidence tending in the same direction is afforded by the 
 behaviour of resonating pipes made to speak with the aid of bird- 
 calls. The pair of perforated plates is mounted symmetrically at 
 one end of a pipe 40 or 50 cm. long. The other end of the pipe is 
 acoustically open, and a gentle stream of air is made to pass the 
 bird-call, most easily with the aid of a very narrow tube inserted 
 into the open end and supplied from the mouth. By careful regu- 
 lation of the force of the blast, the pipe may be made to speak in 
 various harmonics, and the fact that it speaks at all seems to shew 
 that the issue of air through the bird-call is variable. 
 
 The manner of action is perhaps somewhat as follows. When 
 a symmetrical excrescence reaches the second plate, it is unable to 
 pass the hole with freedom, and the disturbance is thrown back, 
 probably with the velocity of sound, to the first plate, where it 
 gives rise to a further disturbance, to grow in its turn during the 
 progress of the jet. But the elucidation of this and many kindred 
 phenomena remains still to be effected. 
 
 372. ^Eolian tones, as in the aBolian harp, are generated when 
 wind plays upon a stretched wire capable of vibration at various 
 speeds, and their production also is doubtless connected with the 
 instability of vortex sheets. It is not essential, however, that the 
 wire should partake in the vibration, and the general phenomenon 
 has been investigated by Strouhal 1 , under the name of reibungstone. 
 
 1 Wied. Ann. vol. v. p. 216, 1878. See also W. Koblrausch, Wisd. Ann. vol. 
 xin. p. 545, 1881. 
 
372.] AEOLIAN HARP. 413 
 
 In Strouhal's experiments a vertical wire attached to a suitable 
 frame was caused to revolve with uniform velocity about a parallel 
 axis. The pitch of the seolian tone generated by the relative 
 motion of the wire and of the air was found to be independent of 
 the length and of the tension of the wire, but to vary with the 
 diameter (d) and with the speed (v) of the relative motion. 
 Within certain limits the relation between the frequency (n) and 
 these data was expressible by 
 
 w = -185 v/d (1), 
 
 the centimetre and second being units. 
 
 When the speed is such that the aeolian tone coincides with 
 one of the proper tones of wife, supported so as to be capable of 
 free independent vibration, the sound is greatly reinforced, and 
 with this advantage Strouhal found it possible to extend the range 
 of the observations. Under the more extreme conditions then 
 practicable the observed pitch deviated sensibly from the value 
 given by (1). He shewed further that with a given diameter and 
 a given speed a rise of temperature was attended by a fall in pitch. 
 
 Observations 1 upon a string, vibrating after the manner of 
 the aeolian harp under the stimulus of a chimney draught, have 
 shewn that, contrary to the opinion generally expressed, the vi- 
 brations are effected in a plane perpendicular to the direction of 
 the wind. According to (1) the distance travelled over by the wind 
 during one complete vibration is about 6 times the diameter of the 
 wire. 
 
 If, as appears probable, the compressibility of the fluid may be 
 left out of account, we may regard n as a function of v, d, and v the 
 kinematic coefficient of viscosity. In this case n is necessarily of 
 the form 
 
 n = v/d.f(v/vd) (2), 
 
 where / represents an arbitrary function ; and there is dynamical 
 similarity, if v x v d. In observations upon air at one temperature 
 v is constant ; and, if d vary inversely as v, ndjv should be constant, 
 a result fairly in harmony with the observations of Strouhal. Again, 
 if the temperature rises, v increases, and in order to accord with 
 observation, we must suppose that the function f diminishes with 
 increasing argument. 
 
 1 Phil. Mag. vol. vn. p. 161, 1879. 
 
414 STROUHAL'S LAW. [372. 
 
 An examination of the actual values in Strouhal's experiments 
 shew that v/vd was always small ; and we are thus led to represent 
 / by a few terms of Mac Laurin's series. If we take 
 
 f(x) = a 4- bx 
 we get 
 
 If the third term in (3) may be neglected, the relation between 
 n and v is linear. This law was formulated by Strouhal, and his 
 diagrams shew that the coefficient b is negative, as is also required 
 to express the observed effect of a rise of temperature. Further 
 
 , dn ' cv z 
 
 so that d . dn/dv is very nearly constant, a result also given by 
 Strouhal on the basis of his measurements. 
 
 On the whole it would appear that the phenomena are satis- 
 factorily represented by (2) or (3), but a dynamical theory has yet 
 to be given. It would also be of interest to extend the experi- 
 ments to liquids. 
 
CHAPTER XXII. 
 
 VIBRATIONS OF SOLID BODIES. 
 
 373. IT is impossible in the present work to attempt anything 
 approaching to a full consideration of the problems suggested by 
 vibrating solid bodies ; and yet the simpler parts of the theory 
 seem to demand our notice. We shall limit ourselves entirely to 
 the case of isotropic matter. 
 
 The general equations of equilibrium have already been stated 
 in 345. If p be the density, and 
 
 a 2 = (* + |n)/p, P=n/p ............... (1), 
 
 we have (a a -&) + 6 2 v 2 a + Z' = J etc ............... (2), 
 
 where X', Y', Z' are the impressed forces reckoned per unit of 
 mass. 
 
 If from these we separate the reactions against acceleration, 
 we obtain by D'Alembert's principle 
 
 and two similar equations. In (3) B is the dilatation, related to 
 a, ft, 7 according to 
 
 (4). 
 
 If a, & 7, etc. be proportional to e ipt , d?a/dt* = p*a, and (3) 
 becomes 
 
416 PLANE WAVES [373. 
 
 Differentiating equation (3) and its companions with respect 
 to x, y, z, and adding, we obtain by (4) 
 
 <PS dX' dY' dZ' 
 
 &*<***+ W + -ft + to ............... (6) ' 
 
 Similar equations may be obtained for the rotations (compare 
 239), defined by 
 
 ^t_dft ' ?-^=2< f-^ = ^'" ..... (7). 
 dy dz dz dx dx dy 
 
 Thus, if we differentiate the third of equations (3) with respect to 
 y, the second with respect to z, and subtract, 
 
 .dZ' dY' 
 
 and there are two similar equations relative to OT 
 be observed that <&', &", *r"' are subject to the relation 
 
 dT* f /dx + dv"/dy + dv'"/dz = Q ............... (9). 
 
 We will now consider briefly certain cases of the propagation 
 of plane waves in the absence of impressed forces. In (6), if 
 X', Y', Z vanish, and 8 be a function of x only, 
 
 d^ldt^tfd^ldx* ...................... (10), 
 
 of which the solution is, as in 245, 
 
 8=f(x-at) + F(x + at) ................... (11). 
 
 In this wave 8 doLJdx, while ft and 7 vanish ; so that the case is 
 similar to that of the propagation of waves in a compressible 
 fluid. It should be observed, however, that by (1) the velocity 
 depends upon the constant of rigidity (n) as well as upon that of 
 compressibility (K). 
 
 In the dilatational wave (11) the rotations w', CT", r'" vanish, 
 as appears at once from their expressions in (7). We have now 
 to consider a wave of transverse vibration for which 8 vanishes. 
 If, for example, we suppose that a and ft vanish and that 7 is a 
 function of x (and t) only, we have 
 
 8 = 0, CT ' = CT '" = 0, 2r" = -dy/<fo. 
 
 The equation for -&" is 
 
 #w7dP = &dflr"/c& ..................... (12), 
 
 of the same form as (10); and the same equation obtains for 7. 
 The transverse vibration is thus propagated in plane waves with 
 velocity 6, a velocity less than that (a) of the dilatational waves. 
 
373.] LIMITED INITIAL DISTURBANCE. 417 
 
 The formation of stationary waves by superposition of positive 
 and negative progressive waves of like wave-length need not be 
 dwelt upon. If k = 2?r/X, where X is the wave-length, the super- 
 position of the positive wave 7 = F cos k (bt x) upon the negative 
 wave 7 = F cos k (bt + as) gives 
 
 j = 2Tcoskbt.coskx (13). 
 
 The second progressive wave may be the reflection of the first at a 
 bounding surface impenetrable to energy. This may be either 
 a free surface, or one at which 7 is prevented from varying. 
 
 374. The problem of the propagation in three dimensions of 
 a disturbance initially limited to a finite region of the solid was 
 first successfully considered by Poisson, and the whole subject has 
 been exhaustively treated by Stokes 1 . By (6), (8) 373 the dila- 
 tation and the rotations satisfy the equations 
 
 <2 2 S/eft 2 = a 2 V 2 S, d*v/dt 2 = b*V*v (1), 
 
 the solutions of which, applicable to the present purpose, have 
 already been fully discussed in 273, 274. It appears that 
 distinct waves of dilatation and distortion are propagated out- 
 wards with different velocities, so that at a sufficient distance 
 from the source they become separated. If we consider what 
 occurs at a distant point, we see that at first there is neither 
 dilatation nor distortion. When the wave of dilatation arrives 
 this effect commences, but there is no distortion. After a while 
 the wave of dilatation passes, and there is an interval of no 
 dilatation and no distortion. Then the wave of distortion arrives 
 and for a time produces its effect, after which there is never again 
 either dilatation or distortion. 
 
 The complete discussion requires the expressions for the dis- 
 placements in terms of 3, w^, vr 2 , vr 3 , for the derivation of which 
 we have not space. From these it may be proved that before the 
 arrival of the wave of dilatation and subsequently to the passage 
 of the wave of distortion, the medium remains at rest. Between 
 the two waves the medium is not absolutely undisturbed, although 
 there is neither dilatation nor distortion. 
 
 If the initial disturbances be of such a character that there is 
 no wave of distortion, the whole disturbance is confined to the 
 wave of dilatation. 
 
 1 "Dynamical Theory of Diffraction," Camb. Phil. Trans. Vol. ix. p. 1, 1849. 
 
 R. ii. 27 
 
418 HARMONIC FORCE [375. 
 
 375. The subject of 374 was the free propagation of waves 
 resulting from a disturbance initially given. A problem at least 
 equally important is that of divergent waves maintained by 
 harmonic forces operative in the neighbourhood of a given centre. 
 
 We may take first the case of a harmonic force of such a 
 character as to generate waves of dilatation. By equation (6) 
 373 we may suppose that at all points except the origin of 
 
 coordinates 
 
 a*V*S ........................ (1); 
 
 or, if 8 as a function of x, y, z depend upon r, or VO* + 2T + * 2 )> 
 only, and as a function of the time be proportional to e ipt , 241, 
 
 + W-0 ....................... (2), 
 
 r dr 
 
 where h =p/a. The solution of (2) is, as in 277, 
 
 ihr 
 
 In terms of real quantities 
 
 ~ A cos ( pt hr + e) 
 
 ~~r~ 
 in which A and e are arbitrary. 
 
 By transformation of (4) 373, the relation between & and the 
 radial displacement w may be shewn to be 
 
 S = r-*d(r*w)ldr ........................ (5), 
 
 or at a great distance from the origin simply 
 
 $ = dw/dr ........................... (6). 
 
 Thus, when r is great, corresponding to (4) 
 
 A 
 
 W J- sin (pt hr + e) .................. (7). 
 
 In these purely dilatational waves the motion is radial, that is, 
 parallel to the direction of propagation, and the distribution is 
 symmetrical with respect to the origin. 
 
 The theory of forced waves of distortion proceeding outwards 
 from a centre is of still greater interest. The simplest case is 
 when the waves are due to a periodic force, say Z', acting through 
 
375.] OPERATIVE AT A POINT. 419 
 
 a space T at the origin. If we suppose in (8) 373 that X, Y' 
 vanish, and that all the quantities are proportional to e ipt , we find 
 
 V 2 tzr" + &&" -^b--dZ'/dx = Q (9), 
 
 V 2 OT'" + & 2 OT'" =0 (10), 
 
 k being written for p/b. 
 
 These equations are solved as in 277. We get w'" = 0, and 
 
 l r r r rl ^7* s* UGT 
 
 *-M>) 1 1 -%******> 
 
 r denoting the distance between the element at x y y, z near the 
 origin (0) and the point (P) under consideration. If we integrate 
 partially with respect to y, we find 
 
 the integrated term vanishing in virtue of the condition that Z' is 
 finite only within the space T. Moreover, since the dimensions of 
 T are supposed to be very small in comparison with the wave- 
 length, d(r~ l e~ ikr )ldy may be removed from under the integral 
 sign. It will be convenient also to change the meaning of x, y, z, 
 so that they shall now represent the coordinates of P relatively to 
 0. Thus, if Z' now stand for the mean value of Z' throughout the 
 space T, 
 
 TZ' d (p-ikr\ . 
 
 iff' = + ^- (- -)X. ...(12). 
 
 8?r6 2 dr \ r J r 
 
 In like manner 
 
 TZ' d fe~ 
 
 and r'" = (14). 
 
 In virtue of the symmetry round the axis of z it suffices to 
 consider points which lie in the plane ZX. Then TS' vanishes, so 
 that the rotation takes place about an axis perpendicular both to 
 the direction of propagation (r) and to that of the force (z). If 6 
 denote the angle between these directions, the resultant rotation, 
 coincident with -cr", is 
 
 TZ' sin 6 d 
 
 (( UN IV 
 
420 SMALL OBSTACLE. [375. 
 
 If we confine our attention to points at a great distance, this 
 
 becomes simply 
 
 ikTZ'sm0e~ ikr 
 
 The displacement, corresponding to (16), is perpendicular to r and 
 in the plane zr. Its value is given by 
 
 4-7T& 2 r 
 
 or, if we restore the factor e ikbt , and reject the imaginary part of 
 the solution, 
 
 -> ............. (17). 
 
 r 
 If Z l cos kbt denote the whole force applied at the origin, 
 
 Z,= TZ'.p .......................... (18), 
 
 so that (17) may be written 
 
 ) ............... (19). 
 
 The amplitude of the vibration radiated outwards is thus inversely 
 as the distance, and directly as the sine of the angle between the 
 ray and the direction in which the force acts. In the latter 
 direction itself there is no transverse vibration propagated. 
 
 These expressions may be applied to find the secondary vibra- 
 tion dispersed in various directions when plane waves impinge 
 upon a small obstacle of density different from that of the rest of 
 the solid. We may suppose that the plane waves are expressed 
 
 by 
 
 y = Tcosk(bt-x) ..................... (20), 
 
 and that they impinge at the origin upon an obstacle of volume T 
 and density p. The additional inertia of the solid at this place would 
 be compensated by a force (/>' />)y, or (p p) & 2 6 2 F cos kbt, 
 acting throughout T\ and, if this force be actually applied, the 
 primary waves would proceed without interruption. The secon- 
 dary waves may thus be regarded as due to a force equal to the 
 opposite of this, acting at parallel .to Z. The whole amount of 
 the force is given by 
 
 Z l coskbt = (p-p)k 2 TTcoBkbt ............ (21); 
 
 so that by (19) the secondary displacement at a distant point 
 (r, 0) is 
 
 cosk(bt-r) 
 "~ 
 
375.] LINEAR SOURCE. 421 
 
 The intensity of the scattered vibration is thus inversely as the 
 fourth power of the wave-length (F being given), and as the 
 square of the sine of the angle between the scattered ray and the 
 direction of vibration in the primary waves. Thus, if the primary 
 ray be along x and the secondary ray along z, there are no 
 secondary vibrations if (as above supposed) the primary vibrations 
 are parallel to z ; but if the primary vibrations are parallel to y, 
 there are secondary vibrations of full amplitude (sin 6 = 1), and 
 these vibrations are themselves executed in a direction parallel 
 to y} 
 
 376. In 375 we have examined the effect of a periodic force 
 Z l coskbt, localized at the origin. We now proceed to consider 
 the case of a force uniformly distributed along an infinite line. 
 
 Of this there are two principal sub-cases : the first where the 
 force, itself always parallel to z, is distributed along the axis of z, 
 the second where the distribution is along the axis of y. In the 
 first, with which we commence, the entire motion is in two 
 dimensions, symmetrical with respect to OZ, and further is such 
 that a and /3 vanish, while 7 is a function of (x 2 + y' 2 ) only. If, as 
 suffices, we limit ourselves to points situated along OX, CT', -or"' 
 vanish, and we have only to find or". 
 
 The simplest course to this end is by integration of the result 
 given in (16) 375. pTZ' will be replaced by Z n dz, the amount 
 of the force distributed on dz ; r denotes the distance between P 
 on OX and dz on OZ; 6 the angle between r and z. The rotation 
 sr" about an axis parallel to y arid due to this element of the force 
 is thus 
 
 '^.. ..a). 
 
 In the integration x is constant, and r 2 = x 2 + z 2 , so that we have 
 to consider 
 
 ie- ikr dr f ie- ikx e~ ikh dh 
 rVCr'-aJ*) ](x + h).4(Zx + h).*Jh '' 
 
 if we write r x = h. 
 
 1 "On the Light from the Sky, its Polarization and Colour." Phil. Mag. Vol. 
 XLI. pp. 107, 274, 1871 ; see also Phil. Mag. Vol. XLI. p. 447, 1871, for an investi- 
 gation of the case where the obstacle differs in elastic quality, as well as in density, 
 from the remainder of the medium. 
 
422 LINEAR SOURCE. [376. 
 
 From this integral a rigorous solution may be developed, but, 
 as in 342, we may content ourselves with the limiting form 
 assumed when kx is very great. Thus, as the equivalent of (2), 
 
 we get 
 
 * - -(*-) 
 
 so that as the integral of (1) 
 
 i(kx-fr) /A\ 
 
 From this 7 may be at once deduced. We have 
 
 or, if we restore the time-factor, and omit the imaginary part of 
 the solution, 
 
 This corresponds to the force Z u cos kbt per unit of length of the 
 axis of z. In virtue of the symmetry we may apply (6) to points 
 not situated upon the axis of x, if we replace x by \/(# 2 + 2/ 2 )- 
 That the value of 7 would be inversely as the square root of the 
 distance from the axis of z might have been anticipated from the 
 principle of energy. 
 
 The solution might also be investigated directly in terms of 7 
 without the aid of the rotations OT. 
 
 It now remains to consider the case in which the applied force, 
 still parallel to z, is distributed along OF, instead of along OZ. 
 The point P, at which the effect is required, may be supposed to 
 be situated in the plane ZX at a great distance R from and in 
 such a direction that the angle ZOP is 6. 
 
 In virtue of the two-dimensional character of the force, /3 = 0, 
 while a, 7 are independent of y. Hence ', is" vanish. But, 
 although these component rotations vanish as regards the resultant 
 effect, the action of a single element of the force Z u dy, situated 
 at y, would be more complicated. Into this, however, we need not 
 enter, because, as before, the effect in reality depends only upon 
 the elements in the neighbourhood of 0. Thus, in place of (1), 
 we may take 
 
 ikZ u dy . sin d e~ ikr 
 
376.] LINEAR OBSTACLE. 4 '2 3 
 
 r being the distance between dy and P, so that 
 
 Writing r - R = h, we get, as in (2), (3), (4), 
 
 and for the displacement, perpendicular to 
 
 Hence, corresponding to the force Z n cos kbt per unit of length of 
 the axis of y, we have the displacement perpendicular to R at the 
 point (R, d) 
 
 377. As in 375, we may employ the results of 376 to form 
 expressions for the secondary waves dispersed from a small 
 cylindrical obstacle, coincident with OZ and of density p', upon 
 which primary parallel waves impinge. If the expression for the 
 primary waves be (20) 375, we have 
 
 Z u =(p'-p)kW.ir*.r ..................... (1), 
 
 7TC 2 being the area of the cross section of the obstacle. Thus, if 
 we denote V(# 2 + 2/ 2 ) by r > we have from (6) 376 as the expression 
 of the secondary waves, 
 
 tr(p' -p).m-.r 2ir ., 
 
 - -cos-(6*-r-JX) ......... (2), 
 
 k being replaced by its equivalent (2ir/\). In this case the 
 secondary waves are symmetrical, and their intensity varies in- 
 versely as the distance and as the cube of the wave-length. 
 
 The solution expressed by (10) 376 shews that if primary 
 waves 
 
 P = Bcosk(bt-x) ........................ (3) 
 
 impinge upon the same small cylindrical obstacle, the displace- 
 ment perpendicular to the secondary ray, viz. r, will be 
 
 7T (p' p) . 7TC 2 . B . COS 6 2-7T ,, . 
 
 -eo._(i_ r -.jx) ......... (4), 
 
424 LINEAR OBSTACLE. [377. 
 
 6 denoting the angle between the direction of the primary ray (x) 
 and the secondary ray (r). In this case the secondary disturbance 
 vanishes in one direction, that is along a ray parallel to the 
 primary vibration. 
 
 Returning to the first case, in which a and /3 vanish through- 
 out, while 7 is a function of x and y only, let us suppose that 
 the material composing the cylindrical obstacle differs from its 
 surroundings in rigidity (ri) as well as in density (p f ). The 
 conditions to be satisfied at the cylindrical surface are 
 
 7 (inside) = 7 (outside), 
 ri<ty/dr (inside) = n dy/dr (outside). 
 In the exterior space 7 satisfies the equation ( 373) 
 
 d*y/dtf + d*y/dy z + k*y = 0, 
 
 where k = p/b ; and in the space interior to the cylinder 7 satisfies 
 
 0, 
 
 where k' =p/b' and b' denotes the velocity of transverse vibrations 
 in the material composing the cylinder. The investigation of the 
 secondary waves thrown off by the obstacle when primary plane 
 waves impinge upon it is then analogous to that of 343, and the 
 conclusion is that, corresponding to primary waves 
 
 (5), 
 
 the secondary waves thrown off by a small cylinder in a direction 
 making an angle 6 with x are given by 
 
 27T.7TC 2 . 
 
 which includes (2) as a particular case. 
 
 378. We now return to the fundamental problem, already 
 partially treated in 375, of the vibrations in an unlimited solid 
 due to the application of a periodic force at the origin of coordi- 
 nates. Equations (12), (13), (14) 375 give the solution so far as 
 to specify the values of the component rotations. If, as we shall 
 ultimately suppose, the solid be incompressible, we have in 
 addition 5 = 0. On this basis the solution might be completed, 
 but it may be more instructive to give an independent investi- 
 gation. 
 
378.] FORCE AT ONE POINT. 425 
 
 Since in the notation of 373 X' = T = 0, we have by (5) 
 
 ............... (1), 
 
 ............... (2), 
 
 -' ......... (3). 
 
 Let us assume 
 
 a = d*x/dx dz, fi = d*x/dy dz, j = d*x/dz 2 + w ...... (4), 
 
 and accordingly 
 
 (5). 
 The substitution of these values in (1) gives 
 
 so that (1) and (2) are satisfied if 
 
 a 2 V 2 x+j9 2 % + (a 2 -6 2 )w = ............... (6). 
 
 The same substitutions in (3) give 
 
 d? 
 
 fa f 2V2 % + P*X + O 2 ~ & 2 ) w ] + & V 2 ^ +p*w + Z' = 0, 
 
 or in virtue of (6) 
 
 & 2 V 2 w+^ 2 w + ^ = ..................... (7). 
 
 By this equation w is determined, and thence x by (6)- 
 
 In the notation of 375, k=p/b, h=p/a. Since Z' = at all 
 points other than the origin, (7) becomes 
 
 (V 2 + & 2 )w = ........................ (8), 
 
 whence by (6) (V 2 + A 2 ) (V 2 + & 2 ) x = ..................... (9) 
 
 is to be satisfied everywhere except at the origin. The solution 
 of (9) is 
 
 -ikr p -ihr 
 
 where A and B are constants. The corresponding values of w 
 and 8 are by (6) and (5) 
 
 ilcr fJ /0ihr\ 
 
 (11). 
 
 r z T 
 
 To connect A and B with Z' , we have from (7), as in 375, 
 
 1 /"/"/* e~ ikr Z e~ ikr 
 
 w = - r- \\\Z' dxdy dz 
 
 r TTP r 
 
 so that A = , ^ ........................... (12). 
 
426 COMPLETE SOLUTION. [378. 
 
 Again, by (6) 373, 
 
 V 2 S + A 2 S + a- 9 dZ'/dz = ; 
 so that, as in 375, 
 
 Z 1 d 
 
 Thus, by comparison with (11), 
 
 Z l e -ikr_ e -ihr 
 
 and Y 
 
 r 
 
 From the values of % and w thus fully determined a, /8, 7 are 
 found by simple differentiations, as indicated in (4). We have 
 
 m. 
 
 As the complete expressions are rather long, we will limit 
 ourselves to the case of incompressibility (/& = ()). Thus, if we 
 restore the time-factor (e ipt ) and throw away the imaginary part 
 of the solution, we get 
 
 3 \ 3 3 
 
 cos < -*- sm * ~ ^ r ~ 
 
 the value of /3 differing from that of a merely by the substitution 
 of y for x. The value of A is given in (12), and Z^cospt is the 
 whole force operative at the origin at time t. 
 
 At a great distance from the origin (17), (18) reduce to 
 Zi xzcos(pt-kr) 
 
 ~~ AKi~^& ~ ~ ~" V iy A 
 
 cos(pt-kr) ^ ^ 2Q ^ 
 
 in agreement with (19) 375. 
 
378.] COMPARISON WITH HERTZ. 427 
 
 W. Konig 1 has remarked upon the non-agreement of the 
 complete solution (17), (18), first given in a different form by 
 Stokes 2 , with the results of a somewhat similar investigation by 
 Hertz 3 , in which the terms involving cos pt, sin pt do not occur, 
 and he seems disposed to regard Stokes' results as affected by 
 error. But the fact is that the problems treated are essentially 
 different, that of Hertz having no relation to elastic solids. The 
 source of the discrepancy is in the first terms of (1) &c., which are 
 omitted by Hertz in his theory of the ether. But assuredly in a 
 theory of elastic solids these terms must be retained. Even when 
 the material is supposed to be incompressible, so that 8 vanishes, 
 the retention is still necessary, because, as was fully explained by 
 Stokes in the memoir referred to, the factor (a 2 6 2 ) is infinite at 
 the same time. 
 
 If we suppose in (17), (18) that p and k are very small, and 
 trace the limiting form, we obtain the solution of the statical 
 problem of the deformation of an incompressible solid by a force 
 localized at a point in its interior. 
 
 379. In 373 we saw that in a uniform medium plane waves 
 of transverse vibration 
 
 a = 0, = 0, y = Tcos(pt-kx) (1) 
 
 may be propagated without limit. We will now suppose that on 
 the positive side of the plane x the medium changes, so that 
 the density becomes p l instead of p, while the rigidity becomes n^ 
 instead of n. In the transmitted wave p remains the same, but k 
 is changed to k lt where 
 
 kf/fr^npjntf (2). 
 
 Assuming, as will be verified presently, that no change of phase 
 need be allowed for, we may take as the expressions for the 
 transmitted and reflected waves 
 
 7l = I\ cos (pt - &!#), 7 = F cos (pt + kx) (3), 
 
 so that altogether the value of 7 in the first medium is 
 
 7 = T cos(pt kx) + T' cos(pt + kx) (4), 
 
 and in the second 
 
 7 X = T 1 cos(pt-k l x) (5). 
 
 1 Wied. Ann. vol. xxxvu. p. 651, 1889. 
 
 2 Camb. Phil. Trans, vol. ix. p. 1, 1849; Collected Works, vol. n. p. 243. 
 
 3 Wied. Ann. vol. xxxvi. p. 1, 1889. 
 
428 REFLECTION OF PLANE WAVES. [379. 
 
 The conditions to be satisfied at the interface (x = 0), upon which 
 no external force acts, are 
 
 71 = 7, n^dyjdx^ndy/dx ............... (6); 
 
 so that rH-r=r 1? ^(r-r) = w 1 A: 1 r 1 ............... (7). 
 
 If, as can plainly be done, I", I\ be determined in accordance with 
 (7), the conditions are all satisfied. We have 
 
 ' nk - njc 
 
 nk + r^ &! \/(np) + V(% pi) ' ' 
 
 i>i)'' 
 
 by which the reflected and transmitted waves are determined. 
 The particular cases in which p l = p, or ^ == n, may be specially 
 noted. 
 
 When the incidence upon the plane separating the two bodies 
 is oblique, the problem becomes more complicated, and divides 
 itself into two parts according as the vibrations (always perpen- 
 dicular to the incident ray) are executed in the plane of incidence, 
 or in the perpendicular plane. Into these matters, which have 
 been much discussed from an optical point of view, we shall 
 not enter. The method of investigation, due mainly to Green, 
 is similar to that of 270. A full account with the necessary 
 references is given in Basset's Treatise on Physical Optics, 
 Ch. xii. 
 
 380. The vibrations of solid bodies bounded by free surfaces 
 which are plane, cylindrical, or spherical, can be investigated 
 without great difficulty, but the subject belongs rather to the 
 Theory of Elasticity. For an infinite plate of constant thickness 
 the functions of the coordinates required are merely circular 
 and exponential - 1 . The solution of the problem for an infinite 
 cylinder 2 depends upon Bessel's functions, and is of interest 
 as giving a more complete view of the longitudinal and flexural 
 vibrations of a thin rod. 
 
 The case of the sphere is important as of a body limited in 
 all directions. The symmetrical radial vibrations, purely dila- 
 tational in their character, were first investigated by Poisson and 
 
 1 Proc. Lond. Math. Soc. vol. xvn. p. 4, 1885 ; vol. xx. p. 225, 1889. 
 - Pochhammer, Crelle, vol. LXXXI. 1876 ; Chree, Quart. Journ. 1886. See also 
 Love's Theory of Elasticity, ch. xvn. 
 
380.] VIBRATIONS OF SOLID BODIES. 429 
 
 Clebsch \ The complete theory is due to Jaerisch 2 and especially 
 to Lamb 3 . An exposition of it will be found in Love's treatise 
 already cited. 
 
 The calculations of frequency are complicated by the existence 
 of two elastic constants K and n 373, or q and /u. 214. From 
 the principle of 88 we may infer, as Lamb has remarked, that 
 the frequency increases with any rise either of K or of n, for 
 as appears from (1) 345 either change increases the potential 
 energy of a given deformation. 
 
 381 4 . In the course of this work we have had frequent 
 occasion to notice the importance of the conclusions that may be 
 arrived at by the method of dimensions. Now that we are 
 in a position to draw illustrations from a greater variety of 
 acoustical phenomena relating to the vibrations of both solids and 
 fluids, it will be convenient to resume the subject, and to develope 
 somewhat in detail the principles upon which the method rests. 
 
 In the case of systems, such as bells or tuning-forks, formed of 
 uniform isotropic material, and vibrating in virtue of elasticity, the 
 acoustical elements are the shape, the linear dimension c, the 
 constants of elasticity q and n ( 149), and the density p. Hence,, 
 by the method of dimensions, the periodic time varies cceteris 
 paribus as the linear dimension, at least if the amplitude of vibra- 
 tion be in the same proportion ; and, if the law of isochronism 
 be assumed, the last-named restriction may be dispensed with. In 
 fact, since the dimensions of q and p are respectively [ML~ 1 T~'*\ 
 and [JfL~ 3 ], while /z is a mere number, the only combination 
 capable of representing a time is q~* . p* . c. 
 
 The argument which underlies this mathematical shorthand is 
 of the following nature. Conceive two geometrically similar bodies, 
 whose mechanical constitution at corresponding points is the 
 same, to execute similar movements in such a manner that the 
 corresponding changes occupy times 5 which are proportional to the 
 
 1 Theorie der Elasticitdt Fester Korper, Leipzig, 1862. 
 
 2 Crelle, vol. LXXXVIII. 1879. 
 
 3 Proc. Loud. Math. Soc. vol. xin. p. 189, 1882. 
 
 4 This section appeared in the First Edition as 348. 
 
 5 The conception of an alteration of scale in space has been made familiar by 
 the universal use of maps and models, but the corresponding conception for time 
 is often less distinct. Eeference to the case of a musical composition performed at 
 different speeds may assist the imagination of the student. 
 
430 PRINCIPLE OF [381. 
 
 linear dimensions in the ratio, say, of 1 : n. Then, if the one 
 movement be possible as a consequence of the elastic forces, the 
 other will be also. For the masses to be moved are as 1 : n 3 , the 
 accelerations as 1 : n~ l , and therefore the necessary forces are 
 as 1 : ft 2 ; and, since the strains are the same, this is in fact the 
 ratio of the elastic forces due to them when referred to corre- 
 sponding areas. If the elastic forces are competent to produce 
 the supposed motion in the first case, they are also competent to 
 produce the supposed motion in the second case. 
 
 The dynamical similarity is disturbed by the operation of a 
 force like gravity, proportional to the cubes, and not to the squares, 
 of corresponding lines; but in cases where gravity is the sole 
 motive power, dynamical similarity may be secured by a different 
 relation between corresponding spaces and corresponding times. 
 Thus if the ratio of corresponding spaces be 1 : n, and that of 
 corresponding times be 1 : n*, the accelerations are in both cases 
 the same, and may be the effects of forces in the ratio 1 : n s acting 
 on masses which are in the same ratio. As examples coming under 
 this head may be mentioned the common pendulum, sea-waves, 
 whose velocity varies as the square root of the wave-length, and 
 the whole theory of the comparison of ships and their models 
 by which Froude predicted the behaviour of ships from experi- 
 ments made on models of moderate dimensions. 
 
 The same comparison that we have employed above for elastic 
 solids applies also to aerial vibrations. The pressures in the cases 
 to be compared are the same, and therefore when acting over 
 areas in the ratio 1 : w 2 , give forces in the same ratio. These 
 forces operate on masses in the ratio 1 : n 3 , and therefore produce 
 accelerations in the ratio 1 : n~ l , which is the ratio of the actual 
 accelerations when both spaces and times are as 1 : n. Accordingly 
 the periodic times of similar resonant cavities, filled with the 
 same gas, are directly as the linear dimension a very important 
 law first formulated by Savart. 
 
 Since the same method of comparison applies both to elastic 
 solids and to elastic fluids, an extension may be made to systems 
 into which both kinds of vibration enter. For example, the scale 
 of a system compounded of a tuning-fork and of an air resonator 
 may be supposed to be altered without change in the motion other 
 than that involved in taking the times in the same ratio as 
 the linear dimensions. 
 
381.] DYNAMICAL SIMILARITY. 431 
 
 Hitherto the alteration of scale has been supposed to be 
 uniform in all dimensions, but there are cases, not coming under 
 this head, to which the principle of dynamical similarity may be 
 most usefully applied. Let us consider, for example, the flexural 
 vibrations of a system composed of a thin elastic lamina, plane or 
 curved. By 214, 215 we see that the thickness of the lamina b, 
 and the mechanical constants q and p, will occur only in the com- 
 binations qb s and bp, and thus a comparison may be made even 
 although the alteration of thickness be not in the same proportion 
 as for the other dimensions. If c be the linear dimension when 
 the thickness is disregarded, the times must vary cceteris paribus 
 as q~* . p* . c 2 . b~\ For a given material, thickness, and shape, the 
 times are therefore as the squares of the linear dimension. It must 
 not be forgotten, however, that results such as these, which involve 
 a law whose truth is only approximate, stand on a different level 
 from the more immediate consequences of the principle of 
 similarity. 
 
CHAPTER XXIII. 
 
 FACTS AND THEORIES OF AUDITION. 
 
 382. THE subject of the present chapter has especial relation 
 to the ear as the organ of hearing, but it can be considered only 
 from the physical side. The discussion of anatomical or physio- 
 logical questions would accord neither with the scope of this book 
 nor with the qualifications of the author. Constant reference to 
 the great work of Helmholtz is indispensable 1 . Although, as we 
 shall see, some of the positions taken by the author have been 
 relinquished, perhaps too hastily, by subsequent writers, the im- 
 portance of the observations and reasonings contained in it, as well 
 as the charm with which they are expounded, ensure its long 
 remaining the starting point of all discussions relating to sound 
 sensations. 
 
 383. The range of pitch over which the ear is capable of 
 perceiving sounds is very wide. Naturally neither limit is well 
 defined. From his experiments Helmholtz concluded that the 
 sensation of musical tone begins at about 30 vibrations per second, 
 but that a determinate musical pitch is not perceived till about 
 40 vibrations are performed in a second. Preyer 2 believes that he 
 heard pure tones as low as 15 per second, but it seems doubtful 
 whether the octave was absolutely excluded. On a recent review 
 of the evidence and in the light of some fresh experiments, Van 
 Schaik 3 sees no reason for departing greatly from Helmholtz's 
 estimate, and fixes the limit at about 24 vibrations per second. 
 
 1 Tonempjlndungen, 4th edition, 1877 ; Sensations of Tone, 2nd English edition 
 translated from the 4th German edition by A. J. Ellis. Citations will be made from 
 this English edition, which is further furnished by the translator with many valuable 
 notes. 
 
 2 Physiologische Abhandlungen, Jena, 1876. 
 
 3 Arch. N6erl. vol. xxix. p. 87, 1895. 
 
38 4.] ESTIMATION OF PITCH. 433 
 
 On the upper side the discrepancies are still greater. Much 
 no doubt depends upon the intensity of the vibrations. In experi- 
 ments with bird-calls ( 371) nothing is heard above 10,000, 
 although sensitive flames respond up to 50,000. But forks care- 
 fully bowed, or metal bars struck with a hammer, appear to give 
 rise to audible sounds of still higher frequencies. Preyer gives 
 20,000 as near the limit for normal ears. 
 
 In the case of very high sounds there is little or no appreciation 
 of pitch, so that for musical purposes nothing over 4000 need be 
 considered. 
 
 The next question is how accurately can we estimate pitch by 
 the ear only ? The sounds are here supposed to be heard in 
 succession, for ( 59) when two uniformly sustained notes are 
 sounded together there is no limit to the accuracy of comparison 
 attainable by the method of beats. From a series of elaborate 
 experiments Preyer 1 concludes that at no part of the scale can "20 
 vibration per second be distinguished with certainty. The sensi- 
 tiveness varies with pitch. In the neighbourhood of 120, '4 
 vibration per second can be just distinguished; at 500 about %3 
 vibration; and at 1000 about '5 vibration per second. In some 
 cases where a difference of pitch was recognised, the observer could 
 not decide which of the two sounds was the graver. 
 
 384. In determinations of the limits of pitch, or of the 
 perceptible differences of pitch, the sounds are to be chosen of 
 convenient intensity. But a further question remains behind as 
 to the degree of intensity at given pitch necessary for audibility. 
 The earliest estimate of the amplitude of but just audible sounds 
 appears to be that of Toepler and Boltzmann 2 . It depends upon an 
 ingenious application of v. Helmholtz's theory of the open organ - 
 pipe ( 313) to data relating to the maximum condensation within 
 the pipe, as obtained by the authors experimentally ( 322 d). 
 They conclude that plane waves, of pitch 181, in which the 
 maximum condensation (s) is 6'5 x 10~ 8 , are just audible. 
 
 It is evident that a superior limit to the amplitude of waves 
 giving an audible sound may be derived from a knowledge of the 
 energy which must be expended in a given time in order to 
 
 1 An account of Preyer's work was given by A. J. Ellis in the Proceedings of the 
 Musical Association, 3rd session, p. 1, 1877. 
 
 2 Pogg. Ann. vol. CXLI. p. 321, 1870. 
 
 R. II. 28 
 
434 MINIMUM AMPLITUDE [384. 
 
 generate them and of the extent of surface over which the waves 
 so generated are spread at the time of hearing. An estimate 
 founded on these data will necessarily be too high, both because 
 sound-waves must suffer some dissipation in their progress and 
 also because a part, and in some cases a large part, of the energy 
 expended never takes the form of sound-waves at all. 
 
 In the first application of the method 1 , the source of sound 
 was a whistle, mounted upon a Wolfe's bottle, in connection with 
 which was a siphon manometer for the purpose of measuring the 
 pressure of the wind. The apparatus was inflated from the lungs, 
 and with a little practice there was no difficulty in maintaining a 
 sufficiently constant blast of the requisite duration. The most 
 suitable pressure was determined by preliminary trials, and was 
 measured by a column of water 9 cm. high. 
 
 The first point to be determined was the distance from the 
 source to which the sound remained clearly audible. The experi- 
 ment was tried upon a still winter's day and it was ascertained 
 that the whistle could be heard without effort (in botn directions) 
 to a distance of 820 metres. 
 
 The only remaining datum necessary for the calculation is the 
 quantity of air which passes through the whistle in a given time. 
 This was determined by a laboratory experiment from which it 
 appeared that the consumption was 196 cub. cents, per second. 
 
 In working out the result it is most convenient to use con- 
 sistently the c. G. s. system. On this system of measurement the 
 pressure employed was 9-J x 981 dynes per sq. cent., and therefore 
 the work expended per second in generating the waves was 
 196 x 9J x 981 ergs 2 . 
 
 Now ( 245) the mechanical value of a series of progressive 
 waves is the same as the kinetic energy of the whole mass of air 
 concerned, supposed to be moving with the maximum velocity (v) 
 of vibration; so that, if S denote the area of the wave-front 
 considered, a the velocity of sound, p the density of air, the 
 mechanical value of the waves passing in a unit of time is 
 expressed by S.a.p. ^v 2 , in which the numerical value of a is 
 about 34100, and that of p about '0013. In the present applica- 
 tion 8 is the area of the surface of a hemisphere whose radius is 
 
 1 Proc. Roy. Soc. vol. xxvi. p. 248, 1877. 
 Nearly 2 x 10 6 ergs. 
 
384.] OF AUDIBLE SOUNDS. 435 
 
 82000 centimetres ; and thus, if the whole energy of the escaping 
 air were converted into sound and there were no dissipation on the 
 way, the value of v at a distance of 82000 centimetres would be 
 given by the equation 
 
 2 x 196 x 9J-X981 
 27r(82000) 2 x 34100 x -0013' 
 
 whence v = '0014 - , s = - = 4*1 x 10~ 8 . 
 sec. a 
 
 This result does not require a knowledge of the pitch of the 
 sound. If the period be r, the relation between the maximum 
 excursion x and the maximum velocity v is x = VT/%7r. In the 
 experiment under discussion the note was f ir , with a frequency of 
 about 2730. Hence 
 
 or the amplitude of the aerial particles was less than a ten- 
 millionth of a centimetre. It was estimated that under favourable 
 conditions an amplitude of 10~ 8 cm. would still have been audible. 
 
 It is an objection to the above method that when such large 
 distances are concerned it is difficult to feel sure that the disturb- 
 ing influence of atmospheric refraction is sufficiently excluded. 
 Subsequently experiments were attempted with pipes of lower 
 pitch which should be audible to a less distance, but these were 
 not successful, and ultimately recourse was had to tuning-forks. 
 
 " A fork of known dimensions, vibrating with a known ampli- 
 tude, may be regarded as a store of energy of which the amount 
 may readily be calculated. This energy is gradually consumed by 
 internal friction and by generation of sound. When a resonator 
 is employed the latter element is the more important, and in some 
 cases we may regard the dying down of the amplitude as sufficiently 
 accounted for by the emission of sound. Adopting this view for 
 the present, we may deduce the rate of emission of sonorous energy 
 from the observed amplitude of the fork at the moment in question 
 and from the rate at which the amplitude decreases. Thus if the 
 law of decrease be e~* kt for the amplitude of the fork, or e~ kt for 
 the energy, and if E be the total energy at time t, the rate at 
 which energy is emitted at that time is dEfdt, or IcE. The value 
 of k is deducible from observations of the rate of decay, e. g. of the 
 time during which the amplitude is halved. With these arrange- 
 
 282 
 
436 MINIMUM AMPLITUDE [384. 
 
 ments there is no difficulty in converting energy into sound upon 
 a small scale, and thus in reducing the distance of audibility to 
 such a figure as 30 metres. Under these circumstances the obser- 
 vations are much more manageable than when the operators are 
 separated by half a mile, and there is no reason to fear disturbance 
 from atmospheric refraction. 
 
 The fork, is mounted upon a stand to which is also firmly 
 attached the observing-microscope. Suitable points of light are 
 obtained from starch grains, and the line of light into which each 
 point is extended by the vibration is determined with the aid of 
 an eyepiece-micrometer. Each division of the micrometer-scale 
 represents '001 centim. The resonator, when in use, is situated in 
 the position of maximum effect, with its mouth under the free ends 
 of the vibrating prongs. 
 
 The course of an experiment was as follows : In the first place 
 the rates of dying down were observed, with and without the 
 resonator, the stand being situated upon the ground in the middle 
 of a lawn. The fork was set in vibration with a bow, and the time 
 required for the double amplitude to fall to half its original value 
 was determined. Thus in the case of a fork of frequency 256, the 
 time during which the vibration fell from 20 micrometer-divisions 
 to 10 micrometer-divisions was 16 s without the resonator, and 9 s 
 when the resonator was in position. These times of halving were, 
 as far as could be observed, independent of the initial amplitude. 
 To determine the minimum audible, one observer (myself) took up 
 a position 30 yards (27*4 metres) from the fork, and a second 
 (Mr Gordon) communicated a large vibration to the fork. At the 
 moment when the double amplitude measured 20 micrometer- 
 divisions the second observer gave a signal, and immediately 
 afterwards withdrew to a distance. The business of the first 
 observer was to estimate for how many seconds after the signal 
 the sound still remained audible. In the case referred to the 
 time was 12 s . When the distance was reduced to 15 yards (13'7 
 metres), an initial double amplitude of 10 micrometer-divisions was 
 audible for almost exactly the same time. 
 
 These estimates of audibility are not made without some diffi- 
 culty. There are usually 2 or 3 seconds during which the observer 
 is in doubt whether he hears or only imagines, and different 
 individuals decide the question in opposite ways. There is also 
 of course room for a real difference of hearing, but this has not 
 
384.] OF AUDIBLE SOUNDS. 437 
 
 obtruded itself much. A given observer on a given day will often 
 agree with himself surprisingly well, but the accuracy thus 
 suggested is, I think, illusory. Much depends upon freedom 
 from disturbing noises. The wind in the trees or the twittering 
 of birds embarrasses the observer, and interferes more or less with 
 the accuracy of results. 
 
 The equality of emission of sound in various horizontal direc- 
 tions was tested, but no difference could be found. The sound 
 issues almost entirely from the resonator, and this may be expected 
 to act as a simple source. 
 
 When the time of audibility is regarded as known, it is easy to 
 deduce the amplitude of the vibration of the fork at the moment 
 when the sound ceases to impress the observer. From this the 
 rate of emission of sonorous energy and the amplitude of the aerial 
 vibration as it reaches the observer are to be calculated. 
 
 The first step in the calculation is the expression of the total 
 energy of the fork as a function of the amplitude of vibration 
 measured at the extremity of one of the prongs. This problem is 
 considered in 164. If I be the length, p the density, and o> the 
 sectional area of a rod damped at one end and free at the other, 
 the kinetic energy T is connected with the displacement rj at the 
 free end by the equation (10) 
 
 At the moment of passage through the position of equilibrium 
 17 = and drj/dt has its maximum value, the whole energy being 
 then kinetic. The maximum value of drj/dt is connected with the 
 maximum value of 77 by the equation 
 
 so that if we now denote the double amplitude by 2?7, the whole 
 energy of the vibrating bar is ipo)l7r 2 /T y .(2r)) 2 , 
 
 or for the two bars composing the fork 
 
 ^ = i /0 ft)Z 7 r 2 /T 2 .(27;) 2 , ........................... (A) 
 
 where pcol is the mass of each prong. 
 
 The application of (A) to the 256-fork, vibrating with a double 
 amplitude of 20 micrometer-divisions, is as follows. We have 
 l = 14-0 cm., &> = -6x1-1 = -66 sq. cm., 
 
 l/r = 256, p = 7-8, 277 = '050 cm.; 
 and thus E = 4'06 x 10 3 ergs. 
 
438 MINIMUM AMPLITUDE [384. 
 
 This is the whole energy of the fork when the actual double 
 amplitude at the ends of the prongs is '050 centim. 
 
 As has already been shewn, the energy lost per second is kE, if 
 the amplitude vary as e~$ kt . For the present purpose k must be 
 regarded as made up of two parts, one & t representing the dissipa- 
 tion which occurs in the absence of the resonator, the other & 2 due 
 to the resonator. It is the latter part only which is effective 
 towards the production of sound. For when the resonator is out 
 of use the fork is practically silent ; and, indeed, even if it were 
 worth while to make a correction on account of the residual sound, 
 its phase would only accidentally agree with that of the sound 
 issuing from the resonator. 
 
 The values of k v and k are conveniently derived from the times, 
 ^ and t, during which the amplitude falls to one half. Thus 
 
 &=21og e 2./, ^ = 2105,2.^; 
 so that 
 
 k, = 2 log, 2 . (l/t - 1/0 = 1-386 (l/t - l/t,). 
 
 And the energy converted into sound per second is kJE. 
 
 We may now apply these formulae to the case, already quoted, 
 of the 256-fork, for which t = 9, ^ = 16. Thus t 9 , the time which 
 would be occupied in halving the amplitude were the dissipation 
 due entirely to the resonator, is 20'6; and k z = '0674. Accordingly, 
 
 k^E= 267 ergs per second, 
 
 corresponding to a double amplitude represented by 20 micrometer- 
 divisions. In the experiment quoted the duration of audibility 
 was 12 seconds, during which the amplitude would fall in the ratio 
 2 12/9 : 1, and the energy in the ratio 4 12 / 9 : 1. Hence at the moment 
 when the sound was just becoming inaudible the energy emitted 
 as sound was 42'1 ergs per second 1 . 
 
 1 It is of interest to compare with the energy-emission of a source of light. An 
 incandescent electric lamp of 200 candles absorbs about a horse-power, or say 10 10 
 ergs per second. Of the total radiation only about T fa part acts effectively upon 
 the eye ; so that radiation of suitable quality consuming 5 x 10 5 ergs per second 
 corresponds to a candle-power. This is about 10 4 times that emitted as sound by 
 the fork in the experiment described above. At a distance of 10 2 x30, or 3000 
 metres, the stream of energy from the ideal candle would be about equal to the 
 stream of energy just audible to the ear. It appears that the streams of energy 
 required to influence the eye and the ear are of the same order of magnitude, a 
 conclusion already drawn by Toepler and Boltzmann. 
 
384.] OF AUDIBLE SOUNDS. 439 
 
 The question now remains, What is the corresponding ampli- 
 tude or condensation in the progressive aerial waves at 27*4 metres 
 from the source ? If we suppose, as in my former calculations, 
 that the ground reflects well, we are to treat the waves as hemi- 
 spherical. On the whole this seems to be the best supposition to 
 make, although the reflexion is doubtless imperfect. The area 8 
 covered at the distance of the observer is thus 2?r x 2740 2 sq. 
 centim., and since ( 245) 
 
 S . %apv 2 = 8 . paV = 421, 
 
 -rir/i nTKi O^ _ , 
 
 TT x 2740* x -00125 x 34100 3 ' 
 and s = 6-0 x 10~ 9 . 
 
 The condensation s is here reckoned in atmospheres ; and the 
 result shews that the ear is able to recognize the addition and 
 subtraction of densities far less than those to be found in our 
 highest vacua. 
 
 The amplitude of aerial vibration is given by asr/Sir, where 
 l/r = 256, and is thus equal to T27 x 10~ 7 cm. 
 
 It is to be observed that the numbers thus obtained are still 
 somewhat of the nature of superior limits, for they depend upon 
 the assumption that all the dissipation due to the resonator repre- 
 sents production of sound. This may not be strictly the case even 
 with the moderate amplitudes here in question, but the uncertainty 
 under this head is far less than in the case of resonators or organ- 
 pipes caused to speak by wind. From the nature of the calculation 
 by which the amplitude or condensation in the aerial waves is 
 deduced, a considerable loss of energy does not largely influence 
 the final numbers. 
 
 Similar experiments have been tried at various times with forks 
 of pitch 384 and 512. The results were not quite so accordant as 
 was at first hoped might be the case, but they suffice to fix with 
 some approximation the condensation necessary for audibility. The 
 mean results are as follows : 
 
 " 9 
 
 c', frequency = 256, s = 6'0 x 10 
 g, =384, s = 4-6x!0" 9 , 
 
 c", =512, 5 = 4-6xKr 9 , 
 
 no reliable distinction appearing between the two last numbers. 
 Even the distinction between 6'0 and 4'6 should be accepted with 
 
440 BINAUKAL AUDITION. [384. 
 
 reserve ; so that the comparison must not be taken to prove much 
 more than that the condensation necessary for audibility varies but 
 slowly in the singly dashed octave 1 ." 
 
 Results of the same order of magnitude have been obtained 
 also by Wien 2 , who used an entirely different method. 
 
 385. For most purposes of experiment and for many of 
 ordinary life it makes but little difference whether we employ 
 one ear only, or both ; and yet there can be no doubt that we can 
 derive most important information from the simultaneous use of 
 the two ears. How this is effected still remains very obscure. 
 
 Although the utmost precautions be taken to ensure separate 
 action, it is certain that a sound led into one ear is capable of 
 giving beats with a second sound of slightly different pitch led 
 into the other ear. There is, of course, no approximation to such 
 silence as would occur at the moment of antagonism were the two 
 sounds conveyed to the same ear ; but the beats are perfectly 
 distinct, and remain so as the sounds die away so as to become 
 single all but inaudible 3 . It is found, however, that combination 
 tones ( 391) are not produced under these conditions 4 . Some 
 curious observations with the telephone are thus described by 
 Prof. S. P. Thompson 5 . "Almost all persons who have experi- 
 mented with the Bell telephone, when using a pair of instruments 
 to receive the sound, one applied to each ear, have at some time 
 or other noticed the apparent localization of the sounds of the 
 telephone at the back of the head. Few, however, seemed to be 
 aware that this was the result of either reversed order in the 
 connection of the terminals of the instruments with the circuit, or 
 reversed order in the polarity of the magnet of one of the receiving 
 instruments. When the two vibrating discs execute similar vi- 
 brations, both advancing or both receding at once, the sound is 
 heard as usual in the ears ; but if the action of one instrument be 
 reversed, so that when one disc advances the other recedes, and 
 the vibrations have opposite phases, the sound apparently changes 
 its place from the interior of the ear, and is heard as if proceeding 
 from the back of the head, or, as I would rather say, from the top 
 
 1 Phil. Mag. vol. xxxvm. p. 366, 1894. 
 
 2 Wied. Ann. vol. xxxvi. p. 834, 1889. 
 
 3 S. P. Thompson, Phil. Mag. vol. iv. p. 274, 1877. 
 
 4 See also Dove, Pogg. Ann. vol. cvn. p. 652, 1859. 
 
 5 Phil. Mag. vol. vi. p. 385, 1878. 
 
385.] BIN AURAL AUDITION. 441 
 
 of the cerebellum." "I arranged a Hughes's microphone with two 
 
 cells of a Fuller's battery and two Bell telephones, one of them 
 having a commutator under my control. Placing the telephones 
 to my ears, I requested my assistant to tap on the wooden support 
 of the microphone. The result was deafening. I felt as if simul- 
 taneous blows had been given to the tympana of my ears. But 
 on reversing the current through one telephone, I experienced a 
 sensation only to be described as of some one tapping with a hammer 
 on the back of the skull from the inside!' 
 
 In our estimation of the direction in which a sound comes to 
 us we are largely dependent upon the evidence afforded by bin- 
 aural audition. This is one of those familiar and instinctive 
 operations which often present peculiar difficulties to scientific 
 analysis. A blindfold observer in the open air is usually able to 
 indicate within a few degrees the direction of a sound, even though 
 it be of short duration, such as a single vowel or a clap of the 
 hands. The decision is made with confidence and does not require 
 a movement of the head. 
 
 To obtain further evidence experiments were made with the 
 approximately pure tones emitted from forks in association with 
 resonators; but in order to meet the objection that the first sound 
 of the fork, especially when struck, might give a clue, and so 
 vitiate the experiment, two similar forks and resonators, of pitch 
 256, were provided. These were held by two assistants, between 
 whom the observer stood midway. In each trial both forks were 
 struck, and afterwards one only was held to its resonator. The 
 results were perfectly clear. When the forks were to the right 
 and to the left, the observer could distinguish them instinctively 
 and without fail. But when he turned through a right angle, 
 so as to bring the forks to positions in front and behind him, no 
 discrimination was possible, and an attempt to pronounce was 
 felt to be only guessing. 
 
 That it should be impossible to distinguish whether a pure 
 tone comes from in front or from behind is intelligible enough. 
 On account of the symmetry the two ears would be affected alike 
 in both cases, and any difference of intensity due to the position 
 could not avail in the absence of information as to the original 
 intensity. The difficulty is rather to understand how the discrimi- 
 nation between front and rear is effected in other cases, e.g. of the 
 voice, where it is found to be easy. It can only be conjectured 
 
442 
 
 HEAD AS AN OBSTACLE. 
 
 [385. 
 
 that the quality of a compound sound is liable to modification by 
 the external ear, which is differently presented in the two cases. 
 
 The ready discrimination between right and left, even when 
 pure tones are concerned, is naturally attributed to the different 
 intensities with which the sound would be perceived by the two 
 ears. But this explanation is not so complete as might be sup- 
 posed. It is true that very high sounds, such as a hiss, are ill 
 heard with the averted ear ; but when the pitch is moderate, e.g. 
 256 per second, the difference of intensity on the two sides does 
 not seem very great. The experiment may easily be tried roughly 
 by stopping one ear with the finger and turning round backwards 
 and forwards while listening to a sound held steadily. Calcula- 
 tion ( 328) shews, moreover, that the human head, considered as 
 an obstacle to the waves of sound, is scarcely big enough in relation 
 to the wave length to throw a distinct shadow. As an illus- 
 tration I have calculated the intensity of sound due to a distant 
 source at various points on the surface of a fixed spherical obstacle. 
 The result depends upon the ratio (kc) between the circumference 
 of the sphere and the length of the wave. If we call the point 
 upon the spherical surface nearest to the source the anterior pole, 
 and the opposite point (where the shadow might be expected to be 
 most intense) the posterior pole, the results on three suppositions 
 as to the relative magnitudes of the sphere and wave length are 
 as follows : 
 
 
 kc = 2 
 
 kc = l 
 
 kc = $ 
 
 Anterior pole 
 
 69 
 
 50 
 
 29 
 
 Posterior pole 
 
 32 
 
 28 
 
 26 
 
 Equator 
 
 36 
 
 24 
 
 23 
 
 When for example the circumference of the sphere is but half 
 the wave length, the intensity at the posterior pole is only about 
 a tenth part less than at the anterior pole, while the intensity is 
 least of all in a lateral direction. When kc is less than J, the 
 difference of the intensities at the two poles is still less important, 
 amounting to about 1 per cent, when kc = J. 
 
 The case of the head and a pitch c' would correspond to kc '4 
 about, so that the differences of intensity indicated by theory are 
 decidedly small. The explanation of the power of discrimination 
 actually observed would be easier, if it were possible to suppose 
 
/ ^ 
 386.] EXCEPTIONS TO OHM'S LAW. 443 
 
 account taken of the different phases of the vibrations by which 
 the two ears are attacked 1 . 
 
 386. Passing on to another branch of our subject, we have 
 now to consider more closely the impression produced upon the 
 ear by an arbitrary sequence of aerial pressures fluctuating about 
 a certain mean value. According to the literal statement of 
 Ohm's law (27) the ear is capable of hearing as separate tones 
 all the simple vibrations into which the sequence of pressures may 
 be analysed by Fourier's theorem, provided that the pitch of these 
 components lies between certain limits. Components whose pitch 
 lies outside the limits would be ignored. Moreover, within the 
 limits of audibility the relative phases of the various components 
 would be a matter of indifference. 
 
 To the law stated in this extreme form there must obviously 
 be exceptions. It is impossible to suppose that the ear would 
 hear as separate tones simple components of extremely nearly the 
 same frequency. Such components, it is well known, give rise to 
 beats, and their relative phase is a material element in the question. 
 Again, it will be evident that the corresponding tone will not be 
 heard unless a vibration reaches a certain intensity. A finite 
 intensity would be demanded, even if the vibration stood by itself ; 
 and we should expect that the intensity necessary for audibility 
 would be greater in the presence of other vibrations, especially 
 perhaps when these correspond to harmonic undertones, It will 
 be advisable to consider these necessary exceptions to the univer- 
 sality of Ohm's law a little more in detail. 
 
 The course of events, when the interval between two simple 
 vibrations is gradually increased, has been specially studied by 
 Bosanquet 2 . As in 30, 65a, if the components be cos 
 cos 27rn 2 , we have for the resultant, 
 
 u = cos 27^ + cos 
 = 2 cos 7r(n<t n^t . cos 7r(n z + n^)t ........ (1) ; 
 
 shewing that the resultant of two simple vibrations of equal 
 amplitudes and of frequencies n l} 7i 2 can be represented mathe- 
 matically by a single vibration whose frequency is the mean, viz. 
 
 1 Nature, vol. xiv. p. 32, 1876. Phil. Mag. vol. in. p. 456, 1877; vol. vn. p. 149, 
 1879. 
 
 2 Phil. Mag. vol. xi. p. 420, 1881. 
 
444 BOSANQUET'S OBSERVATIONS. [386. 
 
 t an( J whose amplitude varies according to the cosine law, 
 involving a change of sign ( 65a), with a frequency (^ 2 - <%). This 
 single vibration is not simple. The question now arises under 
 which of the two forms in (1) will the ear perceive the sound. 
 According to the strict reading of Ohm's law the two tones ^ and 
 ?* 2 would be perceived separately. We know that when % and ?? 2 
 are nearly enough equal this does not and could not happen. 
 The second form then represents the phenomenon; and it indicates 
 beats, the tone (X + 7i 2 ) having an intensity which varies between 
 and 4 with a frequency (n z nj equal to the difference of fre- 
 quencies of the original tones. Mr Bosanquet found that " (a) the 
 critical interval at which two notes begin to be heard beside their 
 beats, or resultant displacements, is about two commas, throughout 
 that medium portion of the scale which is used in practical music ; 
 (/8) this critical interval appears to be not exactly the same for all 
 ears." But in both the cases examined the beats alone were heard 
 with an interval of one comma, and the two notes were quite clear 
 beside the beats with an interval of three commas. "As the 
 interval increases, the separate notes become more and more pro- 
 minent, and the beats diminish in loudness and distinctness, till, 
 by the time that a certain interval is reached, which is about a 
 minor third in the middle of the scale, the beats practically dis- 
 appear and the two notes alone survive." 
 
 On the second question as to the strength in which a com- 
 ponent simple vibration, of sufficiently distinct pitch, must be 
 present in order to assert itself as a separate tone there is but 
 little evidence, and that not very accordant. According to the 
 experiments of Brandt and Helmholtz ( 130) Young's law as to 
 the absence in certain cases of particular components from the 
 sound of a plucked string is verified. Observations of this kind 
 are easily made with resonators; but for the present purpose the 
 use of resonators is inadmissible, the question relating to the 
 behaviour of the unassisted ear. 
 
 On the other hand A. M. Mayer 1 found that sounds of consider- 
 able intensity when heard by themselves were liable to be completely 
 obliterated by graver sounds of sufficient force. In some experi- 
 ments the graver note was from an open organ-pipe which sounded 
 steadily, while the higher was that of a fork, excited vigorously 
 and then allowed to die down. The action of the fork could be 
 
 1 Phil. Mag. vol. n. p. 500, 1876. 
 
386.] MAYER'S OBSERVATIONS. 445 
 
 made intermittent by moving the hand to and fro over the mouth 
 of its resonance box. The results are thus described. " At first 
 every time that the mouth of the box is open the sound of the 
 fork is distinctly heard and changes the quality of the note of the 
 open pipe. But as the vibrations of the fork run down in ampli- 
 tude the sensations of its effect become less and less till they soon 
 entirely vanish, and not the slightest change can be observed in 
 the quality or intensity of the note of the organ-pipe, whether the 
 resonance box of the fork be open or closed. Indeed at this stage 
 of the experiment the vibrations of the fork may be suddenly and 
 totally stopped without the ear being able to detect the fact. But 
 if instead of stopping the fork when it becomes inaudible we stop 
 the sound of the organ-pipe, it is impossible not to feel surprised 
 at the strong sound of the fork which the open pipe had smothered 
 and had rendered powerless to affect the ear." 
 
 But "no sound, even when very intense, can diminish or 
 obliterate the sensation of a concurrent sound which is lower in 
 pitch. This was proved by experiments similar to the last, but 
 differing in having the more intense sound higher (instead of 
 lower) in pitch. In this case, when the ear decides that the 
 sound of the (lower and feebler) tuning-fork is just extinguished, 
 it is generally discovered on stopping the higher sound that the 
 fork, which should produce the lower sound, has ceased to vibrate. 
 This surprising experiment must be made in order to be appre- 
 ciated. I will only remark that very many similar experiments, 
 ranging through four octaves, have been made, with consonant 
 and dissonant intervals, and that scores of different hearers have 
 confirmed this discovery." 
 
 These results, which are not difficult to verify 1 , involve a 
 serious deduction from the universality of Ohm's law, and must 
 have an important bearing upon other unsettled questions relating 
 to audition. It is to be observed that in Mayer's experiments 
 the question is not merely whether a particular tone can be 
 heard as such. The higher sound of feebler intensity is not heard 
 at all 
 
 The audibility of a sound, even when isolated, is influenced by 
 the state of the ear as regards fatigue. The effect is especially 
 
 1 Instead of a box screwed to the fork, I found it better to use an independent 
 resonator, to the mouth of which the fork is made to approach and recede in a 
 definite manner. 
 
446 FATIGUE OF EAR. [386. 
 
 apparent with the very high notes of bird-calls ( 371). "A 
 bird-call was mounted in connection with a loaded gas-bag and a 
 water-manometer, by which means the pressure could be kept 
 constant for a considerable time. When the ear is placed at a 
 moderate distance from the instrument, a disagreeable sound is 
 heard at first, but after a short interval, usually not more than 
 three or four seconds, fades away and disappears altogether. A 
 very short intermission suffices for at any rate a partial recovery 
 of the power of hearing. A pretty rapid passage of the hand, 
 screening the ear for a fraction of a second, allows the sound to be 
 heard again 1 ." 
 
 But although Ohm's law is subject to important limitations, it 
 can hardly be disputed that the ear is capable of making a rough 
 analysis of a compound vibration into its simple parts. The 
 nature of the difficulty commonly met with has already been 
 referred to ( 25, 26), but a few further remarks may here be 
 made. 
 
 In resolving compound notes a certain control over the 
 attention is the principal requisite, and Helmholtz found that 
 the advantage does not always lie with musically trained ears. 
 Before a particular tone is listened for, it ought to be sounded 
 so as to become fixed in the memory, but not too loudly, lest 
 the sensitiveness of the ear be unduly impaired. As a rule the 
 uneven component tones, twelfth, higher third, &c., are more easily 
 recognised than the octaves. 
 
 On the pianoforte, for example, let g be first gently given, and 
 as soon as the key is released, let c be sounded strongly. The 
 tone g' on which the attention should be kept rivetted throughout, 
 may now be heard as part of the compound note c. A similar 
 experiment may be made with the higher third e", and an acute 
 ear may detect a slight fall in pitch. This is a consequence of the 
 equal temperament tuning ( 19), and shews clearly that the 
 apparent prolongation of the tone is not due to imagination. In 
 modern pianos the seventh and ninth component tones are often 
 weak or altogether absent, but on the harmonium these tones may 
 usually be heard. 
 
 It is still better when the tone to be listened for is first 
 obtained as a harmonic from the string c itself. In the case of 
 
 1 Phil. Mag. vol. xin. p. 344, 1882. 
 
387.] HEAKING OF OVERTONES. 447 
 
 the twelfth, for example, strike the key gently while the string is 
 lightly touched at one-third of its length, and then after removal 
 of the finger more strongly. The proper point may be con- 
 veniently found by sliding the finger slowly along the string, 
 while the key is continually struck. When a point of aliquot 
 division is reached, the corresponding harmonic rings out clearly ; 
 otherwise the sound is feeble and muffled. In this way Helmholtz 
 succeeded in hearing the overtones of thin strings as far as the 
 sixteenth. From this point they lie too close together to be 
 easily distinguished. 
 
 A further, slight modification of this method is especially 
 recommended by Helmholtz. Instead of using the finger, the 
 nodal point is touched with a small camel's hair brush. This 
 allows the degree of damping to be varied at pleasure, and a 
 gradual transition to be made from the pure harmonic, free from 
 all admixture of components which have not a node at the point 
 touched, to the natural note of the string. 
 
 But it is with the assistance of resonators that overtones are 
 most easily heard in the first instance. For this purpose a 
 resonator is chosen, tuned, say, to g', and the ear is placed in 
 communication with its cavity. When c is sounded, either on the 
 piano or harmonium, or with the human voice, the tone g' may 
 usually be heard very loud and distinct. Indeed on many 
 pianofortes a tone g' may be heard as loudly from its harmonic 
 undertones g or c. as from the string g' itself. When an overtone 
 has once been heard, the assistance of the resonator should be 
 gradually withdrawn, which may be done either by removing it 
 from the ear, or putting it out of tune by an obstacle (such as the 
 finger) held near its mouth. 
 
 387. If it be admitted that the ear is capable of analysing 
 a musical note into components, or partials, it follows almost of 
 necessity that these more elementary sensations correspond to 
 simple vibrations. So long as we keep within the range of the 
 principle of superposition, this is the kind of analysis effected by 
 mechanical appliances, such as resonators, and all the more patent 
 facts go to prove that the ear resolves according to the same laws. 
 Moreover, the d priori probabilities of the case seem to tend in 
 the same direction. It is difficult to suppose that physiological 
 effects electrical, chemical, or of some unknown character, are 
 
448 HELMHOLTZ'S THEORY. [387. 
 
 produced directly by the impact of sonorous waves involving 
 merely a variable fluid pressure. Helmholtz's theory of audition 
 is based upon the more natural supposition that the immediate 
 effect of the waves is to set into ordinary mechanical vibration 
 certain internal vibrators 1 , and that nervous excitation follows as a 
 secondary consequence. 
 
 The modus operandi is conceived to be as follows. When a 
 simple tone finds access to the ear, all the parts capable of motion 
 vibrate in synchronism with the source. If there be any part, 
 approximately isolated, whose natural period nearly agrees with 
 that of the sound, then the vibration of that part is far more 
 intense than it would otherwise be. Practically this part of the 
 system may be said to respond only to tones whose pitch lies 
 within somewhat narrow limits. Now it is supposed that the 
 auditory nerves are in communication with vibrating parts of the 
 kind described, whose natural pitch ranges at small intervals 
 between the limits of hearing in such a manner that when any 
 part vibrates the corresponding nerve is excited and conveys the 
 impression to the brain. In the case of a simple tone, one (or at 
 most a relatively small number) of the whole series of nerves is 
 excited, the excitation of the nerve being the proximate cause of 
 the hearing of the tone. 
 
 At this point the question presents itself whether more than 
 one simple vibration may not have the power of exciting the same 
 nerve ? A priori, this might well be the case ; for the vibrating 
 parts might be susceptible of more than one mode of vibration, 
 and therefore of more than one natural period. If we were to 
 suppose that the natural periods of any vibrating part formed a 
 harmonic scale, so that the same auditory nerve was excited by a 
 tone and its octave, the supposition would certainly give a very 
 ready explanation of the remarkable resemblance of octaves, and 
 would tend to mitigate some of the difficulties which at present 
 stand in the way of accepting Helmholtz's theory as a complete 
 account of the facts of audition 2 . As we shall see presently, 
 
 1 The drum-skin and its attachments are here regarded as external to the true 
 auditory mechanism. However important may be the part they play, it is analogous 
 rather to that of a hearing tube or of the disc of a mechanical telephone. 
 
 2 A curious question suggests itself as to what would happen in case the vibra- 
 tions capable of exciting the same nerve deviated sensibly or considerably from the 
 harmonic scale. In this way ears naturally confused in their appreciation of 
 musical relations may easily be imagined. 
 
388.] 
 
 DEGREE OF DAMPING. 
 
 449 
 
 Helmholtz would admit, or rather assert, that when the sounds- are 
 strong two originally simple vibrations, such as c and c, would 
 excite to some extent the same nerve, but he regards this as 
 depending upon a failure in the law of superposition, due to 
 excessive vibration. 
 
 388. It is evident that Helmholtz's theory gives a very 
 natural account of Ohm's law, as well as of the limitation to which 
 it is subject when two simple vibrations are in operation of nearly 
 the same pitch. Some of the internal vibrators are then within 
 the influence of both disturbing causes, and are accordingly 
 excited in an intermittent manner, giving rise to beats, when the 
 period is long, and to a sensation of roughness as the beats 
 become too quick to be easily perceived separately. But when 
 the interval between the two vibrations is increased, a point is 
 soon reached after which no internal vibrator is sensibly affected 
 by both disturbing causes, so that from this point onwards the 
 resulting sensation is free from beats or roughnesses, or at least 
 should be so according to the strict interpretation of the law. To 
 this point we shall return later. 
 
 The magnitude of the interval, over which a single internal 
 vibrator will respond sensibly, is an element of considerable 
 importance in the theory. It has already been shewn ( 49) that 
 there is a relation between this interval and the number of free 
 vibrations which can be executed by the vibrating body. Thus, if 
 the interval between the natural and the forced vibration required 
 to reduce the resonance to ^ of the maximum be a semitone, 
 this implies that after 9*5 free vibrations the intensity would be 
 reduced to y 1 ^ of its original value, and so on for other intervals. 
 From a consideration of the effect of trills in music, Helmholtz 
 concludes that the case of the ear corresponds somewhat to that 
 above specified, and he gives the accompanying table shewing the 
 
 Difference 
 
 Intensity of 
 
 Difference 
 
 Intensity of 
 
 of pitch 
 
 resonance 
 
 of pitch 
 
 resonance 
 
 o-o 
 
 100 
 
 0-6 
 
 7-2 
 
 0-1 
 
 74 
 
 0-7 
 
 5-4 
 
 0-2 
 
 41 
 
 0-8 
 
 4-2 
 
 0-3 
 
 24 
 
 0-9 
 
 3-3 
 
 0-4 
 
 15 
 
 Whole tone 
 
 2-7 
 
 Semitone 
 
 10 
 
 
 
 R. II. 
 
 29 
 
450 A. M. MAYER'S EXPERIMENTS. [388. 
 
 relation obtaining in this case between the difference of free and 
 forced pitch and the intensity of resonance, measured by the 
 square of the amplitude of vibration. 
 
 Although according to Helmholtz's theory the sensation of 
 dissonance is caused by intermittent excitation of those vibrating 
 parts which are within the range of two or more elements of the 
 sound, it is not to be inferred that the number of beats is a 
 sufficient measure of the dissonance. On the contrary it is found 
 that if the number of beats be retained constant (e.g. 33 per 
 second), the effect is more and more free from roughness as the 
 sounds are made deeper, the intervals being correspondingly 
 increased. 
 
 The experiments of A. M. Mayer 1 extend over a considerable 
 range of pitch, and have been made by two methods. In the first 
 method a sound, which would otherwise be a pure tone, is 
 rendered intermittent, and the rate of intermittence is gradually 
 raised to the point at which the effect upon the ear again becomes 
 smooth. The results are shewn in the accompanying table, in 
 which the first column gives the pitch of the sound and the 
 second the minimum number of intermittences per second 
 required to eliminate the roughness. 
 
 Pitch 
 
 Frequency of 
 Intermittence 
 
 (n) 
 
 (m) 
 
 
 64 
 
 23-1 
 
 
 128 
 
 36 
 
 
 256 
 
 62 
 
 
 320 
 
 73 
 
 
 384 
 
 88 
 
 
 512 
 
 108 
 
 
 640 
 
 126 
 
 
 768 
 
 143 
 
 
 1023 
 
 170 
 
 
 The theory of intermittent vibrations has already been given 
 65 a. It is to be remembered that by the nature of the case an 
 intermittent vibration cannot be simple. To a first approximation 
 it may be supposed to be equivalent to three simple vibrations of 
 frequencies, n m, n, n+m, and the roughness experienced by 
 
 1 Phil. Mag. vol. XLIX. p. 352, 1875; vol. xxxvu. p. 259, 1894. 
 
389.] 
 
 SMALLEST CONSONANT INTERVAL. 
 
 451 
 
 the ear may be looked upon as due to the beats of these three 
 tones. 
 
 Mayer has experimented also upon the "smallest consonant 
 intervals among simple tones," i.e. upon the intervals at which the 
 roughness due to beats just disappears, the plural being used 
 since it is found that the necessary interval varies at different 
 parts of the scale. 
 
 Pitch 
 
 K) 
 
 Additional 
 vibrations 
 required 
 ("2 - i) 
 
 Smallest 
 consonant 
 intervals in 
 semitones 
 
 48 
 
 
 none 
 
 64 
 
 
 none 
 
 96 
 
 i'i 
 
 6-15 
 
 128 
 
 38 
 
 4-50 
 
 192 
 
 48 
 
 3-86 
 
 256 
 
 58 
 
 3-53 
 
 316 
 
 68 
 
 3-34 
 
 432 
 
 85 
 
 3-12 
 
 575 
 
 107 
 
 2-95 
 
 766 
 
 130 
 
 2-70 
 
 1707 
 
 210 
 
 2-00 
 
 2304 
 
 245 
 
 1-76 
 
 2560 
 
 256 
 
 1-64 
 
 2806 
 
 266 
 
 1-54 
 
 Different observers agreed very closely as to the point at 
 which roughness disappeared. 
 
 According to the theory of intermittent sounds it is to be 
 expected that for a given pitch m in the first set of experiments 
 should be nearly the same as (n. 2 n^ in the second, and this is 
 pretty well verified by Mayer's numbers, at least over the middle 
 region of the scale. 
 
 389. From the degree of damping above determined it 
 follows that the natural pitch of the internal vibrators, which 
 respond sensibly to a given simple sound, ranges over about a 
 whole tone, and it may excite surprise that we are able to 
 compare with such accuracy the pitch of musical sounds heard in 
 succession. The explanation probably depends a good deal upon 
 the symmetry of the effects on the two sides of the maximum. A 
 comparison with the capabilities of the eye in a similar case may 
 
 292 
 
452 NUMBER OF IMPULSES NECESSARY [389. 
 
 be instructive. In setting the cross wires of a telescope upon the 
 centre of a symmetrical luminous band, e.g. an interference band, 
 it is found that the error need not exceed T J of the width. A 
 similarly accurate judgment as to the centre of the region excited 
 by a given musical note would lead to an estimation of pitch 
 accurate to about y^, agreeing well enough with the facts to be 
 explained. 
 
 In the light of the same principle we may consider how far 
 the perception of pitch would be prejudiced by a limitation of the 
 number of vibrations executed during the continuance of a sound. 
 According to the estimate of Helmholtz already employed ( 388) 
 the internal vibrations, excited and then left to themselves, would 
 remain sensible over about 10 periods. The number of impulses 
 required to produce nearly the full effect is of this order of 
 magnitude. If the number were increased beyond 20 or 30, 
 there would be little further concentration of effect in the 
 neighbourhood of the maximum, and therefore little foundation 
 for greater accuracy in the estimation of pitch. 
 
 Experiments upon this subject have been made by Seebeck 1 , 
 Pfaundler 2 , S. Exner 8 , Auerbach 4 , and W. Kohlrausch 5 , those of 
 the last being the most extensive. An arc of a circle carrying a 
 limited number of teeth was attached to a pendulum, which could 
 be let go under known conditions. In their passage the teeth 
 struck against a card suitably held ; and the sound thus generated 
 was compared with that of a monochord. By varying the length 
 in the usual manner the chord was tuned until the pitch was just 
 perceptibly higher, or just perceptibly lower, than that proceeding 
 from the card, and the interval between the two, called the 
 characteristic interval, determined the precision with which the 
 pitch could be estimated in the case of a given total number of 
 vibrations. The best results were obtained only after considerable 
 practice and in the entire absence of extraneous sounds. 
 
 Sixteen teeth appeared to define the pitch with all the 
 precision attainable, the characteristic interval (on the mean of a 
 number of experiments) being in this case '9922. Even with 9 
 
 1 Pogg. Ann. vol. LIU. p. 417, 1841. 
 
 2 Wien. Ber. vol. LXXVI. p. 561, 1877. 
 
 3 Pflilger's Archiv, vol. xm. p. 228, 1876. 
 
 4 Wied. Ann. vol. vi. p. 591, 1879. 
 6 Wien. Ann. vol. x. p. 1, 1880. 
 
390.] FOR ESTIMATION OF PITCH. 453 
 
 teeth the characteristic interval was '9903, shewing that this 
 small number of vibrations was capable of defining the pitch to 
 within one part in 200. But the most surprising results were 
 those obtained with a very low number of teeth. For 3 teeth 
 the characteristic interval was '9790, and for 2 teeth '9714. 
 
 The fact that pitch can be defined with considerable accuracy 
 by so small a sequence of vibrations has sometimes been regarded 
 as an objection to Helmholtz's theory of audition. I do not think 
 that there is any ground for this opinion. So far as there is a 
 difficulty, it is one that would tell equally against any other 
 theory that could be proposed. 
 
 It would seem that the delimitation of pitch in Kohlrausch's 
 experiments may have been greatly favoured by the approximate 
 discontinuity of the impulses. For it is to be remembered that 
 the internal vibrators concerned are not those only whose period 
 ranges round the interval between the taps, but also those whose 
 periods are approximately submultiples of this quantity. As 
 regards the vibrators in the octave, the number of impulses is 
 practically doubled, for the twelfth trebled, and so on, just as in 
 optics the resolving power of a grating with a limited number of 
 lines is increased in the spectra of the second and higher orders. 
 
 Vibrations limited to a moderate number of periods are some- 
 times generated by reflection of short sounds from railings or 
 steps. At Terling there is a flight of about 20 steps which returns 
 an echo of a clap of the hands as a note resembling the chirp of a 
 sparrow. In all such cases the action is exactly analogous to that 
 of a grating in optics. 
 
 390. When two sounds nearly in unison are compound, we 
 have to consider not only the beats of their first partials, or 
 primes, but also the beats of the overtones. The beats of the 
 octave components are twice, and those of the twelfth three times, 
 as quick as the simultaneous beats of the primes. In some cases, 
 especially where the pitch is very low, mistakes may easily be 
 made by overlooking the prime beats, which affect the ear but 
 feebly. If the octave beats be reckoned as though they were the 
 beats of the primes, the difference of pitch will be taken to be the 
 double of its true value. 
 
 But it is in the case of disturbed consonances other than the 
 unison that the importance of upper partials, or overtones, makes 
 
454 CONSONANCES DEFINED [390. 
 
 itself specially felt. For example, take the Fifth c g. The third 
 partial of c and the second partial of g coincide at g ' . If the 
 interval be true, there are no beats ; but if it be slightly disturbed 
 from the true ratio 3 : 2, the two previously coincident tones 
 separate from one another and give rise to beats. The frequency 
 of the beats follows at once from the manner of their genesis. 
 Thus if the lower note be disturbed from its original frequency by 
 one vibration per second, its third partial is changed by 3 
 vibrations per second, and 3 per second is accordingly the 
 frequency of the beats. But if the upper note undergoes a 
 disturbance of one vibration per second, while the lower remains 
 unaltered, the frequency of the beats is 2. This rule is evidently 
 general. If the consonance be such that the hth partial of the 
 lower note coincides with the kih partial of the upper note, then 
 when the lower note is altered by one vibration per second, the 
 frequency of the beats is h, and when the upper note is altered by 
 the same quantity, the frequency of the beats is k. 
 
 " We have stated that the beats heard are the beats of those 
 partial tones of both compounds which nearly coincide. Now it 
 is not always very easy on hearing a Fifth or an Octave which is 
 slightly out of tune, to recognise clearly with the unassisted ear 
 which part of the whole sound is beating. On listening we are 
 apt to feel that the whole sound is alternately reinforced and 
 weakened. Yet an ear accustomed to distinguish upper partial 
 tones, after directing its attention upon the common upper partials 
 concerned, will easily hear the strong beats of these particular 
 tones, and recognise the continued and undisturbed sound of the 
 primes. Strike the note (c), attend to its upper partial (g'\ and 
 then strike a tempered Fifth (g) ; the beats of (g r ) will be clearly 
 heard. To an unpractised ear the resonators already described 
 will be of great assistance. Apply the resonator for (cf\ and the 
 above beats will be heard with great distinctness. If, on the 
 other hand, a resonator, tuned to one of the primes (c) or (g}, be 
 employed, the beats are heard much less distinctly, because the 
 continuous part of the tone is then reinforced 1 ." 
 
 Experiments of this kind are conveniently made on the 
 harmonium. Small changes of pitch may be obtained by only 
 partially (instead of fully) depressing the key, the effect of which 
 is to flatten the note. The beats of the common overtone are 
 
 1 Sensations of Tone, 2nd ed. p. 181. 
 
390.] BY BEATS OF UPPER PARTIALS. 455 
 
 easily heard when . a (tempered) Fifth is sounded ; those of the 
 equal temperament Third are somewhat rapid. 
 
 The harmonium is also a suitable instrument for experiments- 
 illustrative of just intonation. A reed may be flattened by 
 loading the free end of the tongue with a fragment of wax, and 
 sharpened by a slight filing at the same place. It is easy, 
 especially with the aid of resonators, to tune truly the chords 
 c' e' g f , f a' c", whose consonance will then contrast favour- 
 ably with the unaltered tempered chord g' b' d". It is not 
 consistent with the plan of this work to enter at length into 
 questions of temperament and just intonation. Full particulars 
 will be found in the English edition of Helmholtz (with Ellis's 
 notes) and in Mr Bosanquet's treatise. 
 
 According to Helmholtz's theory it is mainly the beats of the 
 upper partials which determine the ordinary consonant intervals, 
 any departure from which is made evident by the beats of the 
 previously coincident overtones. But even when the notes are 
 truly tuned, the various consonances differ in degree, on account 
 of the disturbances which may arise from overtones which approach 
 one another too nearly. 
 
 The unison, octave, twelfth, double octave, etc., may be 
 regarded as absolute consonances, the second component intro- 
 ducing no new element but merely reinforcing a part of the other. 
 
 The remaining consonant intervals, such as the Fifth and the 
 Major Third, are in a manner disturbed by their neighbourhood to 
 other consonant intervals. In the case of the truly tuned Fifth, 
 for example, with frequencies represented by 3 and 2, there is 
 indeed coincidence between the second partial of the higher note 
 and the third partial of the graver note, but the partials which 
 define the Fourth, of pitch 3x3 = 9 and 4x2 = 8, are within a 
 whole Tone of one another and accordingly near enough to 
 produce disturbance. In like manner the Major Third may be 
 regarded as disturbed by its neighbourhood to the Fourth, and so 
 on in the case of other intervals. 
 
 The importance of these disturbances, and consequently the 
 order in which the various intervals stand in respect to their 
 degree of consonance, varies with the quality of the sounds. As 
 an example where overtones are present in considerable strength, 
 Helmholtz has estimated the degree of consonance of various 
 
456 
 
 DIFFERENCE TONES 
 
 [390. 
 
 intervals on the violin, and has exhibited the results in the form 
 of a curve 1 . 
 
 391. The principle of superposition ( 83), assumed in 
 ordinary acoustical discussions, depends for its validity upon the 
 assumption that the vibrations concerned are infinitely small, or 
 at any rate similar in their character to infinitely small vibrations, 
 and it is only upon this supposition that Ohm's law finds 
 immediate application. One apparent exception to the law has 
 long been known. This is the combination-tone discovered by 
 Sorge and Tartini in the last century. If two notes, at the 
 interval for example of a Major Third, be sounded together 
 strongly, there is heard a grave sound in addition to the two 
 others. In the case specified, where the primary sounds, or 
 generators, as they may conveniently be called, are represented by 
 the numbers 4 and 5, the combination- tone is represented by 1, 
 being thus two octaves below the graver generator. 
 
 In the above example the new tone has the period of the cycle 
 of the generating tones ; but Helm hoi tz found that this rule fails 
 in many cases. The following table 2 exhibits his results as 
 obtained by means of tuning-forks : 
 
 
 Kelative Frequency 
 
 Generators 
 
 Combination 
 tone 
 
 
 Generators 
 
 Combination 
 tone 
 
 j 
 
 
 
 b f 
 
 B 
 
 2 : 3 
 
 1 
 
 f b' 
 
 B 
 
 3 : 4 
 
 1 
 
 b d' 
 
 B 
 
 4 : 5 
 
 I 
 
 d' f 
 
 B 
 
 5 : 6 
 
 1 
 
 f as' 
 
 B l 
 
 6:7 1 
 
 b g' 
 
 es 
 
 3 : 5 
 
 2 
 
 d' as' 
 
 B 
 
 5 : 7 
 
 2 
 
 d' b' 
 
 f 
 
 5 : 8 
 
 3 
 
 In the last three cases the tones heard were not those in the 
 period of the complete cycle, but their frequencies are the differ- 
 ences of the frequencies of the generators. In virtue of this rule, 
 which was found to apply in all cases 3 , the combination-tones in 
 question are called difference-tones. 
 
 1 Sensations of Tone, p. 193. 
 
 2 Berlin Monatsber. , 1856. 
 
 :1 It is, however, disputed by other writers. 
 
391.] SOMETIMES EXTERNAL TO EAR. 457 
 
 According to Helmholtz it is necessary to the distinct audibility 
 of combination-tones that the generators be strong. We shall see 
 presently that this statement has been contested. "They are 
 most easily heard when the two generating tones are less than an 
 octave apart, because in that case the differential is deeper than 
 either of the two generating tones. To hear it at first, choose two 
 tones which can be held with great force for some time, and form 
 a justly intoned harmonic interval. First sound the low tone and 
 then the high one. On properly directing attention, a weaker low 
 tone will be heard at the moment that the higher note is struck ; 
 this is the required combinational tone. For particular instru- 
 ments, as the harmonium, the combinational tones can be made 
 more audible by properly tuned resonators. In this case the tones 
 are generated in the air contained within the instrument. But in 
 other cases where they are generated solely within the ear, the 
 resonators are of little or no use 1 ." 
 
 On the strength of some observations by Bosanquet and Preyer, 
 doubts have been expressed as to the correctness of Helmholtz's 
 statement that combination-tones may exist outside the ear, and 
 strangely enough they have been adopted by Ellis. The question 
 has an important bearing upon the theory of combination- tones ; 
 and it has recently been examined by Rticker and Edser 2 , who 
 used apparatus entirely independent of the ear. They conclude 
 that " Helmholtz was correct in stating that the siren produces 
 two objective notes the frequencies of which are respectively equal 
 to the sum and difference of the frequencies of the fundamentals." 
 My own observations have been made upon the harmonium, and 
 leave me at a loss to understand how two opinions are possible. The 
 resonator is held with its mouth as near as may be to the reeds 
 which sound the generating notes, and is put in and out of tune 
 to the difference-tone by slight movements of the finger. When 
 the tuning is good, the difference-tone swells out with considerable 
 strength, but a slight mistuning (probably of the order of a 
 semitone) reduces it almost to silence. In some cases, e.g. when 
 the interval between the generators is a (tempered) Fifth, the 
 difference-tone is heard to beat. 
 
 The last observation proves that in some cases there exist two 
 difference-tones of nearly the same pitch. Helmholtz finds the 
 
 1 Sensations of Tone, p. 153. 
 
 2 Phil. Mag. vol. xxxix. p. 357, 1895. 
 
458 BEATS OF DIFFERENCE TONE. [391. 
 
 explanation of this in the compound nature of the sounds. Thus 
 in the case of the Fifth, represented by the numbers 2 and 3, we 
 have not only the primes to consider, but the overtones 2x2, 
 3x2, etc., 2x3, 3x3, etc. Accordingly the difference-tone 1 
 may be derived from 2x2 = 4 and 3, as well as from 3 and 2, and 
 since the octave partial is usually strong, the one source may be 
 as important as the other. But if we substitute the Major Third 
 (5 : 4) for the Fifth, we do not get a second difference-tone 1 until we 
 come to the fourth partial (16) of the graver note and the third 
 (15) of the higher, and these would usually be too feeble to produce 
 much effect. 
 
 As regards the frequency of the beats, let us return to the case 
 of the Fifth, supposing it to be so disturbed that the frequencies 
 are 200 and 301. The difference tone due to the primes is 
 301 200 = 101, and that due to the octave partial is 
 
 2x200-301=99; 
 
 and these difference-tones sounding together will give beats with 
 frequency 2. This, it will be observed, is the same number of 
 beats as is due to the common overtone, viz. 2 x 301 - 3 x 200 ; 
 but while the latter beats are those of the tone 600, the beats of 
 the combination-tone are at pitch 100. 
 
 392. According to the views of the older theorists Chladni, 
 Lagrange, Young, etc., the explanation of the difference-tone 
 presented no particular difficulty. As the generators separate in 
 pitch, the beats quicken and at last become too rapid for apprecia- 
 tion as such, passing into a difference-tone, whose frequency is 
 continuous with the frequency of the beats. This view of the 
 matter, which has commended itself to many writers, was rejected 
 by Helmholtz, as inconsistent with Ohm's law ; and that physicist 
 has elaborated an alternative theory, according to which the 
 failure is not in Ohm's law, but in the principle of superposition. 
 
 Helmholtz's calculation of the effect of a want of symmetry in 
 the forces of restitution, when the vibrations of a system cannot be 
 regarded as infinitely small, has already been given ( 68). It 
 appears that in addition to the terms in pt, qt, corresponding to 
 the generating forces, there must be added other terms of the 
 second order in 2pt, 2qt, (p + q)t, (p q) t, the last of which repre- 
 sents the difference-tone. This explanation depends, as Hermann 1 
 
 1 Pfliiger'* Archiv, vol. XLIX. p. 507, 1891. 
 
392.J SUMMATION TONE. 459 
 
 has remarked, upon the assumed failure of symmetry. If, as in 
 67, we suppose a force of restitution proportional partly to the 
 first power and partly to the cube of the displacement, we do not __ 
 obtain a term in (p q) t, but in place of it terms of the third 
 order involving (2p q)t, (2q p)t, etc. This objection, however, 
 is of little practical importance, because the failure of symmetry 
 almost always occurs. It may suffice to instance the all important 
 case of aerial vibrations. Whether we are considering progressive 
 waves advancing from a source, or the stationary vibrations of a 
 resonator, there is an essential want of symmetry between conden- 
 sation and rarefaction, and the formation in some degree of octaves 
 and combination-tones is a mathematical necessity. 
 
 The production of external, or objective, combination-tones 
 demands the coexistence of the generators at a place where they 
 are strong 1 . This will usually occur only when the generating 
 sounds are closely associated, as in the polyphonic siren and in the 
 harmonium. In these cases the conditions are especially favourable, 
 because the limited mass of air included within the instrument is 
 necessarily strongly affected by both tones. When the generating 
 sources are two organ-pipes, even though they stand pretty near 
 together, the difference-tone is not appreciably strengthened by a 
 resonator, from which we may infer that but little of it exists 
 externally to the ear. 
 
 We have as yet said nothing about the summation- tone, corre- 
 sponding to the term in (p 4- q) t. The existence of this tone was 
 deduced by Helmholtz theoretically ; and he afterwards succeeded 
 in hearing it, not only from the siren and harmonium, where it 
 exists objectively and is reinforced by resonators, but also from 
 tuning-forks and organ-pipes. Helmholtz narrates also an experi- 
 ment in which he caused a membrane to vibrate in response to 
 the summation-tone, and similar experiments have recently been 
 carried out with success by Rticker and Edser (1. c.). 
 
 Nevertheless, it must be admitted that summation-tones are 
 extremely difficult to hear. Hermann (1. c.) asserts that he can 
 neither hear them himself nor find any one able to do so ; and he 
 regards this difficulty as a serious objection to Helmholtz's theory, 
 according to which the summation and the difference tone should 
 be about equally strong. 
 
 1 The estimates of condensation ( 384) for sounds just audible make it highly 
 improbable that the principle of superposition could fail to apply to sounds of that 
 order of magnitude. 
 
 ^ ; ^?>^ 
 
 ff ' 
 
 ( UNIVERSITY 
 
460 HELMHOLTZ'S VIEWS [392. 
 
 An objection of another kind has been raised by Konig 1 . He 
 remarks that even if a tone exist of the pitch of the summation- 
 tone, it may in reality be a difference-tone, derived from the upper 
 partials of the generators. As a matter of arithmetic this argu- 
 ment cannot be disputed ; for if p and q be commensurable, it will 
 always be possible to find integers h and k, such that 
 
 p + q = hp- kq. 
 
 But this explanation is plausible only when h and k are small 
 integers. 
 
 It seems to me that the comparative difficulty with which 
 summation-tones are heard is in great measure, if not altogether, 
 explained by the observations of Mayer ( 386). These tones are 
 of necessity higher in pitch than their generators, and are accord- 
 ingly liable to be overwhelmed and rendered inaudible. On the 
 other hand the difference-tone, being usually graver, and often 
 much graver, than either of its generators, is able to make itself 
 felt in spite of them. And even as regards difference-tones, it 
 had already been remarked by Helmholtz that they become more 
 difficult to hear when they cease to constitute the gravest element 
 of the sound by reason of the interval between the generators 
 exceeding an octave. 
 
 393. In the numerous cases where differential tones are 
 audible which are not reinforced by resonators, it is necessary in 
 order to carry out Helmholtz's theory to suppose that they have 
 their origin in the vibrating parts of the outer ear, such as the 
 drum-skin and its attachments. Helmholtz considers that the 
 structure of these parts is so unsymmetrical that there is nothing 
 forced in such a supposition. But it is evident that this explana- 
 tion is admissible only when the generating sounds are loud, i.e. 
 powerful as they reach the ear. Now, the opponents of Helmholtz's 
 views, represented by Hermann, maintain that this condition is 
 not at all necessary to the perception of difference-tones. Here 
 we have an issue as to facts, the satisfactory resolution of which 
 demands better experiments, preferably of a quantitative nature, 
 than any yet executed. My own experience tends rather to 
 support the view of Helmholtz that loud generators are necessary. 
 On several occasions stopped organ-pipes d'", e" f , were blown with 
 
 1 Fogy. Ann. vol. 157, p. 177, 1876. 
 
3.93.] AND CRITICISMS THEREON. 461 
 
 a steady wind, and were so tuned that the difference- tone gave 
 slow beats with an electrically maintained fork, of pitch 128, 
 mounted in association with a resonator of the same pitch. When 
 the ear was brought up close to the mouths of the pipes, the 
 difference-tone was so loud as to require all the force of the fork 
 in order to get the most distinct beats. These beats could be 
 made so slow as to allow the momentary disappearance of the 
 grave sound, when the intensities were rightly adjusted, to be 
 observed with some precision. In this state of things the two 
 tones of pitch 128, one the difference-tone and the other derived 
 from the fork, were of equal strength as they reached the observer; 
 but as the ear was withdrawn so as to enfeeble both sounds by 
 distance, it seemed that the combination-tone fell off more quickly 
 than the ordinary tone from the fork. It might be possible to 
 execute an experiment of this kind which should prove decisively 
 whether the combination-tone is really an effect of the second 
 order, or not. 
 
 In default of decisive experiments we must endeavour to 
 balance the a priori probabilities of the case. According to the 
 views of the older theorists, adopted by Konig, Hermann, and 
 other critics of Helmholtz, the beats of the generators, with their 
 alternations of swellings and pauses, pass into the differential tone 
 of like frequency, without any such failure of superposition as is 
 invoked by Helmholtz. The critics go further, and maintain that 
 the ear is capable of recognising as a tone any periodicity within 
 certain limits of frequency 1 . 
 
 Plausible as this doctrine is from certain points of view, a 
 closer examination will, I think, shew that it is encumbered with 
 difficulties. Among these is the ambiguity, referred to in 12, as 
 to what exactly is meant by period. A periodicity with frequency 
 128 is also periodicity with frequency 64. Is the latter tone to be 
 heard as well as the former ? So far as theory is concerned, such 
 questions are satisfactorily answered by Ohm's law. Experiment 
 may compel us to abandon this law, but it is well to remember 
 that there is nothing to take its place. Again, by consideration of 
 particular cases it is not difficult to prove that the general doctrine 
 above formulated cannot be true. Take the example above 
 mentioned in which two organ-pipes gave a difference-tone of 
 pitch 128. There is periodicity with frequency 128, and the 
 
 1 Hermann, loc. cit. p. 514. 
 
462 DIFFERENCE TONES. [393. 
 
 corresponding tone is heard 1 . So far, so good. But experiment 
 proves also that it is only necessary to superpose upon this another 
 tone of frequency 128, obtained from a fork, in order to neutralize 
 the combination- tone and reduce it to silence. The periodicity of 
 128 remains; if anything in a more marked manner than before, 
 but the corresponding tone is not heard. 
 
 I think it is often overlooked in discussions upon this subject 
 that a difference-tone is not a mere sensation, but involves a 
 vibration of definite amplitude and phase. The question at once 
 arises, how is the phase determined ? It would seem natural to 
 suppose that the maximum swell of the beats corresponds to one 
 or other extreme elongation in the difference-tone, but upon the 
 principles under discussion there seems to be no ground for a 
 selection between the alternatives. Again, how is the amplitude 
 determined ? The tone certainly vanishes with either of the 
 generators. From this it would seem to follow that its amplitude 
 must be proportional to the product of the amplitudes of the 
 generators, exactly as in Helmholtz's theory. If so, we come back 
 to difference-tones of the second order, and their asserted easy 
 audibility from feeble generators is no more an objection to one 
 theory than to another. 
 
 An observation, of great interest in itself, and with a possible 
 bearing upon our present subject, has been made by Kb'nig and 
 Mayer 2 . Experimenting both with forks and bird-calls, they have 
 found that audible difference-tones may arise from generators 
 whose pitch is so high that they are separately inaudible. Perhaps 
 an interpretation might be given in more than one way, but the 
 passage of an inaudible beat into an audible difference-tone seems 
 to be more easily explicable upon the basis of Helmholtz's theory. 
 
 Upon the whole this theory seems to afford the best ex- 
 planation of the facts thus far considered, but it presupposes a more 
 ready departure from superposition of vibrations within the ear 
 than would have been expected. 
 
 394. In 390 we saw that in the case of ordinary compound 
 sounds, containing upper partials fairly developed, the recognised 
 consonant intervals are distinguished from neighbouring intervals 
 
 1 In strictness, the periodicity is incomplete, unless p and q are multiples of 
 (f-). 
 
 2 Mayer, Rep. Brit. Ass. p. 573, 1894. 
 
394.] BEATS OF IMPERFECT CONSONANCES. 463 
 
 by well marked phenomena, of which there was no difficulty in 
 rendering a satisfactory account. We have now to consider the 
 more difficult subject of consonance among pure tones; and we 
 shall have to encounter considerable differences of opinion, not only 
 as to theoretical explanations, but as to matters of observation. 
 Here, as elsewhere, it will be convenient to begin with a statement 
 of Helmholtz's views 1 according to which, in a word, the beats 
 of such mis tuned consonances are due to combination-tones. 
 
 " If combinational tones were not taken into account, two 
 simple tones, as those of tuning-forks, or stopped organ-pipes, 
 could not produce beats unless they were very nearly of the 
 same pitch, and such beats are strong when their interval is 
 a minor or major second, but weak for a Third, and then only 
 recognisable in the lower parts of the scale, and they gradually 
 diminish in distinctness as the interval increases, without shewing 
 any special differences for the harmonic intervals themselves. For 
 any larger interval between two simple tones there would be 
 absolutely no beats at all, if there were no upper partial or 
 combinational tones, and hence the consonant intervals... would 
 be in no way distinguished from adjacent intervals ; there would 
 in fact be no distinction at all between wide consonant intervals 
 and absolutely dissonant intervals. 
 
 Now such wider intervals between simple tones are known 
 to produce beats, although very much weaker than those hitherto 
 considered, so that even for such tones there is a difference be- 
 tween consonances and dissonances, although it is very much 
 more imperfect than for compound tones 2 ." 
 
 Experiments upon this subject are difficult to execute satis- 
 factorily. In the first place it is not easy to secure simple tones. 
 As sources recourse is usually had to stopped organ-pipes or to 
 tuning-forks* but much precaution is required. From the free 
 ends of the vibrating prongs of a fork many overtones may usually 
 be heard 3 . Again, if a fork be employed after the manner of 
 musicians with its stalk pressed against a resonating board, the 
 octave is loud and often predominant 4 . The best way is to hold 
 
 1 Ascribed by him to Hallstrom and Scheibler. 
 
 2 Sensations of Tone, p. 199. 
 
 3 Konig's experiments shew that this is especially the case when the prongs are 
 thin. Wied. Ann. vol. xiv. p. 373, 1881. 
 
 4 The prime tone may even disappear altogether. If in their natural position 
 the prongs of a fork are closest below, an outward movement during the vibration 
 
464 HELMHOLTZ'S VIEWS. [394. 
 
 the free ends of the prongs over a suitably tuned resonator. But 
 even then we cannot be sufe that a loud sound thus obtained is 
 absolutely free from the octave partial. 
 
 In the case of the octave the differential tone already con- 
 sidered suffices. " If the lower note makes 100 vibrations per 
 second, while the imperfect octave makes 201, the first differential * 
 tone makes 201 100 = 101, and hence nearly coincides with the 
 lower note of 100 vibrations, producing one beat for each 100 
 vibrations. There is no difficulty in hearing these beats, and 
 hence it is easily possible to distinguish imperfect octaves from 
 perfect ones, even for simple tones, by the beats produced by 
 the former." 
 
 The frequency of the beats is the same as if it were due to 
 overtones; but there is one important difference between the 
 two cases noted by Ellis though scarcely, if at all, referred to by 
 Helmholtz. In the latter the beats would affect the octave tone, 
 whereas according to the above theory the beats will belong to 
 the lower tone. Bosanquet, Kb'nig and others are agreed that 
 in this respect the theory is verified. 
 
 Again, if the beats were due to combination-tones, they must 
 tend to disappear as the sounds die away. The experiment is 
 very easily tried with forks, and according to my experience the 
 facts are in harmony. When the sounds are much reduced, 
 the mistuning fails to make itself apparent. 
 
 " For the Fifth, the first order of differential tones no longer 
 suffices. Take an imperfect Fifth with the ratio 200 : 301 ; then 
 the differential tone of the first order is 101, which is too far 
 from either primary to generate beats. But it forms an imperfect 
 Octave with the tone 200, and, as just seen, in such a case beats 
 ensue. Here they are produced by the differential tone 99 
 arising from the tone 101 and the tone 200, and this tone 99 
 makes two beats in a second with the tone 101. These beats 
 then serve to distinguish the imperfect from the justly intoned 
 Fifth, even in the case of two simple tones. The number of these 
 beats is also exactly the same as if they were the beats due to 
 
 will depress the centre of inertia, the stalk being immovable, but if the prongs are 
 closest above, the contrary result may ensue. There must be some intermediate 
 construction for which the centre of inertia will remain at rest during the vibration. 
 In this case the sound from a resonance board is of the second order, and is 
 destitute of the prime tone. 
 
394.] DIFFERENTIAL TONES OF SECOND ORDER. 465 
 
 the upper partial tones. But to observe these beats the two 
 primary tones must be loud, and the ear must not be distracted 
 by any extraneous noise. Under favourable circumstances, how- 
 ever, they are not difficult to hear." 
 
 It is important to be clear as to the order of magnitude of 
 the various differential tones concerned. If the primary tones, 
 with frequencies represented by p and q, have amplitudes e and 
 f respectively, quantities of the first order, then ( G8) the first 
 difference and summation tones have frequencies corresponding 
 to 
 
 2p, 2q, p + q, p-q, 
 
 and are of the second order in e and /. A complete treatment 
 of the second differential tones requires the retention of another 
 term /3w 3 ( 67) in the expression of the force of restitution. From 
 this will arise terms of the third order in e and / with frequencies 
 corresponding to 
 
 3p, 2pq, p2q, 3?; 1 
 
 and there are in addition other terms of the same frequencies 
 and order of magnitude, independent of /9, arising from the full 
 development to the third order of au 2 . In the case of the disturbed 
 Fifth above taken, the beats are between the tone 2q p = 99, 
 which is of the third order of magnitude, and p q = 101 of the 
 second order. The exposition, quoted from Helmholtz, refers to 
 the terms last mentioned, which are independent of (3. 
 
 The beats of a disturbed Fourth or major Third depend upon 
 difference- tones of a still higher order of magnitude, and according 
 to Helmholtz's observations they are scarcely, if at all, audible, 
 even when the primary tones are strong. This is no more than 
 would have been expected ; the difficulty is rather to understand 
 how the beats of the disturbed Fifth are perceptible and those of 
 the disturbed Octave so easy to hear. 
 
 When more than two simple tones are sounded together, 
 fresh conditions arise. " We have seen that Octaves are precisely 
 limited even for simple tones by the beats of the first differential 
 tone with the lower primary. Now suppose that an Octave has 
 been tuned perfectly, and that then a third tone is interposed 
 to act as a Fifth. Then if the Fifth is not perfect, beats will ensue 
 from the first differential tone. 
 
 1 Bosanquet, Phil. Mag. vol. xi. p. 497, 1881. 
 R. II. 30 
 
466 CHORD OF THREE NOTES. [394. 
 
 Let the tones forming the perfect Octave have the pitch 
 numbers 200 and 400, and let that of the imperfect Fifth be 
 301. The differential tones are 
 
 400-301= 99 
 
 301-200 = 101 
 
 Number of beats 2. 
 
 These beats of the Fifth which lies between two Octaves are 
 much more audible than those of the Fifth alone without its 
 Octave. The latter depend on the weak differential tones of 
 the second order, the former on those of the first order. Hence 
 Scheibler some time ago laid down the rule for tuning tuning- 
 forks, first to tune two of them as a perfect Octave, and then to 
 sound them both at once with the Fifth, in order to tune the 
 latter. If Fifth and Octave are both perfect, they also give 
 together the perfect Fourth. 
 
 The case is similar, when two simple tones have been tuned 
 to a perfect Fifth, and we interpose a new tone between them to act 
 as a major Third. Let the perfect Fifth have the pitch numbers 
 400 and 600. On intercalating the impure major Third with the 
 pitch number 501 in lieu of 500, the differential tones are 
 
 600-501= 99 
 500 - 400 = 101 
 Number of beats 2." 
 
 395. In Helrnholtz's theory of imperfect consonances the 
 cycles heard are regarded as risings and fallings of intensity of 
 one or more of the constituents of the sound, whether these be 
 present from the first, or be generated by transformation, to use 
 Bosanquet's phrase, in the transmitting mechanism of the ear. 
 According to Ohm's law, such changes of intensity are the only 
 thing that could be heard, for the relative phases of the constitu- 
 ents (supposed to be sufficiently removed from one another in 
 pitch) are asserted to be matters of indifference. 
 
 This question of independence of phase-relation was examined 
 by Helmholtz in connection with his researches upon vowel sounds 
 ( 397). Various forks, electrically driven from one interrupter 
 ( 64), could be made to sound the prime tone, octave, twelfth 
 etc., of a compound note, and the intensities and phases of the 
 constituents could be controlled by slight modifications in the 
 
395.] QUESTION OF PHASE. 467 
 
 (natural) pitch of the forks and associated resonators. According 
 to Helmholtz's observations changes of phase were without 
 distinct effect upon the quality of the compound sound. 
 
 It is evident, however, that the question of the effect, if any, 
 upon the ear of a change in the phase relationship of the various 
 components of a sound can be more advantageously examined by 
 the method of slightly mistuned consonances. If, for example, an 
 Octave interval between two pure tones be a very little imperfect, 
 the effect upon the ear at any particular moment will be that of a 
 true interval with a certain relation of phases, but after a short 
 time, the phase relationship will change, and will pass in turn 
 through every possible value. The audibility of the cycle is 
 accordingly a criterion for the question whether or not the ear 
 appreciates phase relationship ; and the results recorded by 
 Helmholtz himself, and easily to be repeated, shew that in a 
 certain sense the answer must be in the affirmative. Otherwise 
 slow beats of an imperfect Octave would not be heard. The 
 explanation by means of combination-tones does not alter the 
 fact that the ear appreciates the phase relationship of two 
 originally simple tones, at any rate when they are moderately 
 loud 1 . 
 
 According to the observations of Lord Kelvin 2 the "beats of 
 imperfect harmonies," other than the Octave and Fifth, are not so 
 difficult to hear as Helmholtz supposed. The tuning-forks employed 
 were mounted upon box resonators, and it might indeed be argued 
 that the sounds conveyed down the stalks were not thoroughly 
 purged from Octave partials. But this consideration would hardly 
 affect the result in some of the cases mentioned. It appeared that 
 the beats on approximations to each of the harmonies 2 : 3, 3 : 4, 
 4 : 5, 5 : 6, 6 : 7, 7 : 8, 1 : 3, 3 : 5 could be distinctly heard, and 
 that they all " fulfil the condition of having the whole period of the 
 imperfection, and not any sub-multiple of it, for their period," the 
 same rule as would apply were the beats due to nearly coincident 
 overtones. As regards the necessity for loud notes, Kelvin found 
 that the beats of an imperfectly tuned chord 3:4:5 were some- 
 times the very last sound heard, as the vibrations of the forks 
 died down, when the intensities of the three notes chanced at the 
 end to be suitably proportioned. 
 
 1 Konig, Wied. Ann. vol. xiv. p. 375, 1881. 
 
 2 Proc. Roy. Soc. Edin. vol. ix. p. 602, 1878. 
 
 302 
 
468 KONIG'S OBSERVATIONS. [395. 
 
 The last observation is certainly difficult to reconcile with a 
 theory which ascribes the beats to combination-tones. But on the 
 other side it may be remarked that the relatively easy audibility 
 of the beats from a disturbed Octave and from a disturbed chord 
 of three notes (3:4: 5), which would depend upon the first differ- 
 ential tone, is in good accord with that theory, and (so far as 
 appears) is not explained by any other. 
 
 396. But the observations most difficult of reconciliation 
 with the theory of Helmholtz are those recorded by Konig 1 , who 
 finds tones, described as beat-tones, not included among the 
 combination-tones; and these observations, coming from so 
 skilful and so well equipped an investigator, must carry great 
 weight. The principal conclusions are thus summarised by 
 Ellis 2 . "If two simple tones of either very slightly or greatly 
 different pitches, called generators, be sounded together, then 
 the upper pitch number necessarily lies between two multiples 
 of the lower pitch number, one smaller and the other greater, and 
 the differences between these multiples of the pitch number of the 
 lower generator and the pitch number of the upper generator give 
 two numbers which either determine the frequency of the two sets 
 of beats which may be heard or the pitch of the beat-notes which 
 may be heard in their place. 
 
 The frequency arising from the lower multiple of the lower 
 generator is called the frequency of the lower beat or lower beat- 
 note, that arising from the higher multiple is called the frequency 
 of the higher beat or beat-note, without at all implying that one 
 set of beats should be greater or less than the other, or that one 
 beat-note should be sharper or flatter than the other. They are in 
 reality sometimes one way and sometimes the other. 
 
 Both sets of beats, or both beat-notes, are not usually heard 
 at the same time. If we divide the intervals examined into groups 
 (1) from 1 : 1 to 1 : 2, (2) from 1 : 2 to 1 : 3, (3) from 1 : 3 to 1 : 4, 
 (4) from 1 : 4 to 1 : 5, and so on, the lower beats and beat- tones 
 extend over little more than the lower half of each group, and the 
 upper beats and beat-tones over little more than the upper half. 
 For a short distance in the middle of each period both sets of beats, 
 or both beat-notes are audible, and these beat-notes beat with each 
 
 1 Pogg. Ann. vol. CLVII. p. 177, 1876. 
 
 2 Sensations of Tone, p. 529. 
 
396.] KONIG'S OBSERVATIONS. 469 
 
 other, forming secondary beats, or are replaced by new or secondary 
 beat-notes." 
 
 In certain cases the beat-notes coincide with the differential 
 tone, but Konig considers that the existence of combinational 
 tones has not been proved with certainty. It is to be observed that 
 in these experiments the generating tones were as simple as Konig 
 could make them ; but the possibility remains that overtones, not 
 audible except through their beats, may have arisen within the 
 ear by transformation. This is the view favoured by Bosanquet, 
 who has also made independent observations with results less diffi- 
 cult of accommodation to Helmholtz's views. 
 
 It will be seen that Konig adopts in its entirety the opinion 
 that beats, when quick enough, pass into tones. Some objections 
 to this idea have already been pointed out ; and the question must 
 be regarded as still an open one. Experiments upon these subjects 
 have hitherto been of a merely qualitative character. The diffi- 
 culties of going further are doubtless considerable ; but I am 
 disposed to think that what is most wanted at the present time 
 is a better reckoning of the intensities of the various tones dealt 
 with and observed. If, for example, it could be shewn that the 
 intensity of a beat-tone is proportional to that of the generators, 
 it would become clear that something more than combination-tones 
 is necessary to explain the effects. 
 
 Konig has also examined the question of the dependence of 
 quality upon phase relation, using a special siren of his own con- 
 struction 1 . His conclusion is that while quality is mainly deter- 
 mined by the number and relative intensity of the harmonic tones, 
 still the influence of phase is not to be neglected. A variation of 
 phase produces such differences as are met with in different 
 instruments of the same class, or in various voices singing the 
 same vowel. A ready appreciation of such minor differences re- 
 quires a series of notes, upon which a melody can be executed, and 
 they may escape observation when only a single note is available. 
 To me it appears that these results are in harmony with the view 
 that would ascribe the departure from Ohm's law, involved in any 
 recognition of phase relations, to secondary causes. 
 
 397. The dependence of the quality of musical sounds of given 
 pitch upon the proportions in which the various partial tones are 
 
 1 Wied. Ann. vol. xiv. p. 392, 1881. 
 
470 WILLIS' EXPERIMENTS [397. 
 
 present has been investigated by Helmholtz in the case of several 
 musical instruments. Further observations upon wind instru- 
 ments will be found in a paper by Blaikley \ But the most 
 interesting, and the most disputed, application of the theory is to 
 the vowel sounds of human speech. 
 
 The acoustical treatment of this subject may be considered to 
 date from a remarkable memoir by Willis 2 . His experiments 
 were conducted by means of the free reed, invented by Kratzen- 
 stein (1780) and subsequently by Grenie, which imitates with fair 
 accuracy the operation of the larynx. Having first repeated success- 
 fully Kempelen's experiment of the production of vowel sounds by 
 shading in various degrees the mouth of a funnel-shaped cavity in 
 association with the reed, he passed on to examine the effect of 
 various lengths of cylindrical tube, the mounting being similar to 
 that adopted in organ-pipes. The results shewed that the vowel 
 quality depended upon the length of the tube. From these and other 
 experiments he concluded that cavities yielding (when sounded in- 
 dependently) an identical note " will impart the same vowel quality 
 to a given reed, or indeed to any reed, provided the note of the 
 reed be flatter than that of the cavity." Willis proceeds (p. 243) : 
 " A few theoretical considerations will shew that some such effects 
 as we have seen, might perhaps have been expected. According 
 to Euler, if a single pulsation be excited at the bottom of a tube 
 closed at one end, it will travel to the mouth of this tube with the 
 velocity of sound. Here an echo of the pulsation will be formed 
 which will run back again, be reflected from the bottom of the 
 tube, and again present itself at the mouth where a new echo will 
 be produced, and so on in succession till the motion is destroyed 
 by friction and imperfect reflection..,. The effect therefore will be 
 a propagation from the mouth of the tube of a succession of 
 equidistant pulsations alternately condensed and rarefied, at 
 intervals corresponding to the time required for the pulse to 
 travel down the tube and back again ; that is to say, a short burst 
 of the musical note corresponding to a stopped pipe of the length 
 in question, will be produced. 
 
 Let us now endeavour to apply this result of Euler's to the 
 case before us, of a vibrating reed, applied to a pipe of any length, 
 
 1 Phil Mag. vol. vi. p. 119, 1878. 
 
 2 On the Vowel Sounds, and on Keed Organ-pipes. Camb. Phil. Trans, vol. in. 
 p. 231, 1829. 
 
397. J UPON VOWEL SOUNDS. 471 
 
 and examine the nature of the series of pulsations that ought 
 to be produced by such a system upon this theory. 
 
 The vibrating tongue of the reed will generate a series of 
 pulsations of equal force, at equal intervals of time, but alternately 
 condensed and rarefied, which we may call the primary pulsations ; 
 on the other hand each of these will be followed by a series 
 of secondary pulsations of decreasing strength, but also at equal 
 intervals from their respective primaries, the interval between 
 them being, as we have seen, regulated by the length of the 
 attached pipe." 
 
 And further on (p. 247) : " Experiment shews us that the series 
 of effects produced are characterized and distinguished from each 
 other by that quality we call the vowel, and it shews us more, it 
 shews us not only that the pitch of the sound produced is always 
 that of the reed or primary pulse, but that the vowel produced is 
 always identical for the same value of s [the period of the secondary 
 pulses]. Thus, in the example just adduced, g" is peculiar to the 
 vowel A [as in Paw, Nought] ; when this is repeated 512 times in 
 a second, the pitch of the sound is c, and the vowel is A : if by 
 means of another reed applied to the same pipe it were repeated 
 340 times in a second, the pitch would be /, but the vowel still A. 
 Hence it would appear that the ear in losing the consciousness of 
 the pitch of s, is yet able to identify it by this vowel quality." 
 
 From the importance of his results and from the fact that the 
 early volumes of the Cambridge Transactions are not everywhere 
 accessible, I have thought it desirable to let Willis speak for 
 himself. It will be seen that so far as general principles are 
 concerned, he left little to be effected by his successors. Some- 
 what later in the same memoir (p. 249) he gives an account of a 
 special experiment undertaken as a test of his theory. " Having 
 shewn the probability that a given vowel is merely the rapid 
 repetition of its peculiar note, it should follow that if we can 
 produce this rapid repetition in any other way, we may expect to 
 hear vowels. Robison and others had shewn that a quill held 
 against a revolving toothed wheel, would produce a musical note 
 by the rapid equidistant repetition of the snaps of the quill upon 
 the teeth. For the quill I substituted a piece of watch-spring 
 pressed lightly against the teeth of the wheel, so that each snap 
 became the musical note of the spring. The spring being at the 
 same time grasped in a pair of pincers, so as to admit of any 
 
472 HELMHOLTZ'S VIEWS. [397. 
 
 alteration in length of the vibrating portion. This system 
 evidently produces a compound sound similar to that of the pipe 
 and reed, and an alteration in the length of the spring ought 
 therefore to produce the same effect as that of the pipe. In effect 
 the sound produced retains the same pitch as long as the wheel 
 revolves uniformly, but puts on in succession all the vowel 
 qualities, as the effective length of the spring is altered, and that 
 with considerable distinctness, when due allowance is made for 
 the harsh and disagreeable quality of the sound itself." 
 
 In his presentation of vowel theory Helmholtz, following 
 Wheatstone 1 , puts the matter a little differently. The aerial 
 vibrations constituting natural or artificial vowels are, when a 
 uniform regime has been attained ( 48, 66, 322 k), truly periodic, 
 and the period is that of the reed. According to Fourier's 
 theorem they are susceptible of analysis into simple vibrations, 
 whose periods are accurately submultiples of the reed period. 
 The effect of an associated resonator can only be to modify the 
 intensity and phase of the several components, whose periods are 
 already prescribed. If the note of the resonating cavity the 
 mouth-tone coincide with one of the partial tones of the voice- 
 or larynx-note, the effect must be to exalt in a special degree the 
 intensity of that tone ; and whether there be coincidence or not, 
 those partial tones whose pitch approximates to that of the 
 mouth-tone will be favoured. 
 
 This view of the action of a resonator is of course perfectly 
 correct ; but at first sight it may appear essentially different from, 
 or even inconsistent with, the account of the matter given by 
 Willis. For example, according to the latter the mouth-tone may 
 be, and generally will be, inharmonic as regards the larynx-tone. 
 In order to understand this matter we must bear in mind two 
 things which are often imperfectly appreciated. The first is the 
 distinction between forced and free vibrations. Although the 
 natural vibrations of the oral cavity may be inharmonic, the forced 
 vibrations can include only harmonic partials of the larynx 
 note. And again, it is important to remember the definition 
 of simple vibrations, according to which no vibrations can be 
 simple that are not permanently maintained without variation of 
 amplitude or phase. The secondary vibrations of Willis, which 
 
 1 London and Westminster Review, Oct. 1837 ; Wheatstone'* Scientific Papers, 
 London, 1879, p. 348. 
 
397.] FIXED VERSUS RELATIVE PITCH. 473 
 
 die down after a few periods, are not simple. When the complete 
 succession of them is resolved by Fourier's theorem, it is repre- 
 sented, not by one simple vibration, but by a large or infinite 
 number of such. 
 
 From these considerations it will be seen that both ways 
 of regarding the subject are legitimate and not inconsistent with 
 one another. When the relative pitch of the mouth-tone is low, 
 so that, for example, the partial of the larynx note most reinforced 
 is the second or the third, the analysis by Fourier's series is the 
 proper treatment. But when the pitch of the mouth-tone is high, 
 and each succession of vibrations occupies only a small fraction of 
 the complete period, we may agree with Hermann that the 
 resolution by Fourier's series is unnatural, and that we may 
 do better to concentrate our attention upon the actual form 
 of the curve by which the complete vibration is expressed. More 
 especially shall we be inclined to take this course if we entertain 
 doubts as to the applicability of Ohm's law to partials of high 
 order. 
 
 Since the publication of Helmholtz's treatise the question has 
 been much discussed whether a given vowel is characterized by 
 the prominence of partials of given order (the relative pitch 
 theory), or by the prominence of partials of given pitch (the fixed 
 pitch theory), and every possible conclusion has been advocated. 
 We have seen that Willis decided the question, without even 
 expressly formulating it, in favour of the fixed pitch theory. 
 Helmholtz himself, if not very explicitly, appeared to hold the 
 same opinion, perhaps more on a priori grounds than as the result 
 of experiment. If indeed, as has usually been assumed by 
 writers on phonetics, a particular vowel quality is associated with 
 a given oral configuration, the question is scarcely an open one. 
 Subsequently under Helmholtz's superintendence the matter was 
 further examined by Auerbach 1 , who along with other methods 
 employed a direct analysis of the various vowels by means 
 of resonators associated with the ear. His conclusion on the 
 question under discussion was the intermediate one that loth 
 characteristics were concerned. The analysis shewed also that in 
 all cases the first, or fundamental tone, was the strongest element 
 in the sound. 
 
 A few years later Edison's beautiful invention of the phono- 
 1 Pogg. Ann. Erganzung-band vin. p. 177, 1876. 
 
474 EXPERIMENTS WITH PHONOGRAPH. [397. 
 
 graph stimulated anew inquiry upon this subject by apparently 
 affording easy means of making an experimentum crucis. If 
 vowels were characterized by fixed pitch, they should undergo 
 alteration with the speed of the machine ; but if on the other 
 hand the relative pitch theory were the true one, the vowel 
 quality should be preserved and only the pitch of the note 
 be altered. But, owing probably to the imperfection of the earlier 
 instruments, the results arrived at by various observers were still 
 discrepant. The balance of evidence inclined perhaps in favour of 
 the fixed pitch theory 1 . Jenkin and Ewing 2 analysed the 
 impressions actually made upon the recording cylinder, and their 
 results led them to take an intermediate view, similar to that of 
 Auerbach. It is clear, they say, "that the quality of a vowel 
 sound does not depend either on the absolute pitch of reinforce- 
 ment of the constituent tones alone, or on the simple grouping of 
 relative partials independently of pitch. Before the constituents 
 of a vowel can be assigned, the pitch of the prime must be given ; 
 and, on the other hand, the pitch of the most strongly reinforced 
 partial is not alone sufficient to allow us to name the vowel." 
 
 With the improved phonographs of recent years the question 
 can be attacked with greater advantage, and observations have been 
 made by McKendrick and others, but still with variable results. 
 Especially to be noted are the extensive researches of Hermann 
 published in P finger s Archiv. Hermann pronounces unequivocally 
 in favour of the fixed pitch characteristic as at any rate by far the 
 more important, and his experiments apparently justify this 
 /conclusion. He finds that the vowels sounded by the phonograph 
 are markedly altered when the speed is varied. 
 
 Hermann's general view, to which he was led independently, 
 is identical with that of Willis. " The vowel character consists in 
 a mouth-tone of amplitude variable in the period of the larynx 
 tone 3 ." The propriety of this point of view may perhaps be 
 considered to be established, but Hermann somewhat exaggerates 
 the difference between it and that of Helmholtz. 
 
 His examination of the automatically recorded curves was 
 effected in more than one way. In the case of the vowel A 4 the 
 
 1 Graham Bell, Ann. Journ. of Otology, vol. i. July, 1879. 
 
 2 Edin. Trans, vol. xxvni. p. 745, 1878. 
 :{ Pfliig. Arch. vol. XLVII. p. 351, 1890. 
 
 4 The vowel signs refer of course to the continental pronunciation. 
 
397.] HERMANN'S EXPERIMENTS. 475 
 
 amplitudes of the various partials, as given by the Fourier 
 analysis, are set forth in the annexed table, from which it appears 
 that the favoured partial lies throughout between e* and g\ 
 
 VOWEL A. 
 
 Note 
 
 123 
 
 Ordinal 
 4 
 
 number of 
 5 6 
 
 partial. 
 
 7 8 
 
 9 
 
 10 
 
 G 
 
 
 
 
 12 
 d 2 
 
 37 '42 
 
 <f2 g 2 
 
 11 
 
 a 2 
 
 12 
 
 h 2 
 
 A 
 
 
 
 13 
 
 cis 2 
 
 30 
 
 e 2 
 
 33 -10 
 
 <g 2 a 2 
 
 09 
 
 h 2 
 
 08 
 cis 3 
 
 H 
 
 05 -09 
 
 H fis' 
 
 22 
 t' 
 
 37 
 
 dis 2 
 
 45 
 
 fis 2 
 
 10 '15 
 
 <a 2 h 2 
 
 
 
 c 
 
 11 
 
 c 
 
 19 
 
 c 2 
 
 54 
 
 e 2 
 
 38 
 g 2 
 
 16 '09 
 
 <ais 2 c 3 
 
 10 
 
 d 3 
 
 
 d 
 
 
 29 
 
 d 2 
 
 52 
 
 fis 2 
 
 08 
 a 2 
 
 18 
 <c 3 
 
 06 
 e 3 
 
 
 e 
 
 13 
 
 h' 
 
 55 
 
 e 2 
 
 28 
 gis 2 
 
 24 
 h 2 
 
 07 
 <d 2 
 
 
 
 fis 
 
 30 
 
 cis 2 
 
 61 
 
 fis 2 
 
 07 
 ais 2 
 
 11 
 
 cis 2 
 
 11 
 <e 2 
 
 
 
 g 
 
 11 -39 
 
 g d 2 
 
 55 
 
 g 2 
 
 21 
 
 h 2 
 
 11 
 
 d 3 
 
 08 
 <f 3 
 
 
 
 a 
 
 71 
 
 e 2 
 
 18 
 
 a 2 
 
 18 
 
 cis 3 
 
 09 
 
 e 3 
 
 
 
 
 h 
 
 74 
 
 fis 2 
 
 17 
 
 h 2 
 
 13 
 
 dis 3 
 
 
 
 
 
 c' 
 
 41 -54 
 
 C 3 g 2 
 
 40 
 
 C 3 
 
 11 
 
 e 3 
 
 
 
 
 
 d' 
 
 71 -31 
 
 d 2 a 2 
 
 26 
 
 d 3 
 
 
 
 
 
 
 The analysis of the curves into their Fourier components 
 involves a great deal of computation, and Hermann is of opinion 
 that the principal result, the pitch of the vowel characteristic, can 
 be obtained as accurately and far more simply by direct measure- 
 ment on the diagram of the wave-lengths of the intermittent 
 vibrations. The application of this method to the curves for A 
 before used gave 
 
476 
 
 VOWEL CHARACTERISTIC. 
 
 [397, 
 
 VOWEL A. 
 
 Note 
 
 L 
 mm. 
 
 I 
 mm. 
 
 Chai 
 Frequency 
 
 -acteristic tone 
 Note 
 
 G 98 
 
 18-5 
 
 2-4 
 
 756 
 
 >fis 2 (740) 
 
 A 110 
 
 16-3 
 
 2-5 
 
 717 
 
 > f 2 (698-5) 
 
 H 123-5 
 
 14-9 
 
 2-6 
 
 708 
 
 >f 2 (698-5) 
 
 c 130-8 
 
 13-6 
 
 2-55 
 
 698 
 
 f 2 
 
 d 146-8 
 
 11-6 
 
 2-4 
 
 710 
 
 >f 2 (698-5) 
 
 e 164-8 
 
 10-9 
 
 2-3 
 
 781 
 
 <g 2 (784) 
 
 fis 185 
 
 9-8 
 
 2-5 
 
 725 
 
 <fis 2 (744) 
 
 g 196 
 
 9-1 
 
 2-5 
 
 714 
 
 >f 2 (698-5) 
 
 a 220 
 
 8-2 
 
 2-5 
 
 714 
 
 > f 2 (698-5) 
 
 h 246-9 
 
 7-3 
 
 2-6 
 
 693 
 
 <f 2 (698-5) 
 
 c' 261-7 
 
 6-8 
 
 1 
 
 1 
 
 
 d' 293-7 
 
 6-2 
 
 I 
 
 ? 
 
 
 Here L is the double period of the complete vibration and I the 
 double period of the vowel characteristic. It appears plainly 
 that I preserves a nearly constant value when L varies over a 
 considerable range. 
 
 A general comparison of his results with those obtained by 
 other methods has been given by Hermann, from which it will be 
 seen that much remains to be done before the perplexities 
 involving this subject can be removed. Some of the discrepancies 
 
 
 Characteristic 
 
 Mouth-tones according to 
 
 
 tone from 
 
 
 Vowel 
 
 graphical records 
 Hermann 
 
 Bonders 
 
 Helmholtz 
 
 Konig 
 
 Auerbach 
 
 A 
 
 e 2 - g is 9 
 
 b' 
 
 b 2 
 
 b 2 
 
 f 8 
 
 E 
 
 h 3 -c 4 
 
 cis 3 
 
 i , b 
 
 b 3 
 
 a' ( g') 
 
 I 
 
 d 4 g 4 
 
 f 3 
 
 f d 
 
 b 4 
 
 f 
 
 O 
 
 d 2 e 2 
 
 d' 
 
 b' 
 
 b' 
 
 a' 
 
 U 
 
 c 2 d 2 
 
 f 
 
 f 
 
 b 
 
 r 
 
 that have been encountered may probably have their origin in 
 real differences of pronunciation to which only experts in phonetics 
 are sufficiently alive 1 . Again, the question of double resonance 
 has to be considered, for the known shape of the cavities concerned 
 1 Lloyd, Phonetische Studien, vol. in. part J. 
 
397.] IS THE PRIME TONE SENSIBLE? 477 
 
 renders it not unlikely that the complete characterization of a 
 vowel is of a multiple nature ( 310). It should be mentioned 
 that in Lloyd's view the double characteristic is essential, and 
 that the identity of a vowel depends not upon the absolute pitch 
 of one or more resonances, but upon the relative pitch of two or 
 more. In this way he explains the difficulty arising from the fact 
 that the articulation for a given vowel appears to be the same for 
 an infant and for a grown man, although on account of the great 
 difference in the size of the resonating cavities the absolute pitch 
 must vary widely. 
 
 It would not be consistent with the plan of this work to 
 go further into details with regard to particular vowels ; but 
 one remarkable discrepancy between the results of Hermann 
 and Auerbach must be alluded to. The measurements by the 
 former of graphical records shew in all cases a nearly complete 
 absence of the first, or fundamental, tone from the general sound, 
 which Auerbach on the contrary, using resonators, found this tone 
 the most prominent of all. Hermann, while admitting that the 
 tone is heard, regards it as developed within the ear after 
 the manner of combination-tones ( 393). I have endeavoured 
 to repeat some of Auerbach's observations, and I find that for all 
 the principal vowels (except perhaps A) the fundamental tone is 
 loudly reinforced, the contrast being very marked as the resonator 
 is put in and out of tune by a movement of the finger over 
 its mouth. This must be taken to prove that the tone in 
 question does exist externally to the ear, as indeed from the 
 manner in which the sound is produced could hardly fail to be the 
 case; and the contrary evidence from the records must be 
 explained in some other way. 
 
 An important branch of the subject is the artificial imitation 
 of vowel sounds. The actual synthesis by putting together in 
 suitable strengths the various partials was effected by Helm- 
 holtz \ For this purpose he used tuning-forks and resonators, the 
 forks being all driven from a single interrupter ( 63, 64). These 
 experiments are difficult, and do not appear to have been repeated. 
 Helmholtz was satisfied with the reproduction in some cases, 
 although in others the imitation was incomplete. Less satisfactory 
 results were attained when organ-pipes were substituted for 
 the forks. 
 
 1 Sensations of Sound, ch. vi. 
 
478 ARTIFICIAL VOWELS. [397. 
 
 Vowel sounds have been successfully imitated by Preece and 
 Stroh 1 , who employed an apparatus upon the principle of the 
 phonograph, in which the motion of the membrane was controlled 
 by specially shaped teeth, cut upon the circumference of a re- 
 volving wheel. They found that the vowel quality underwent 
 important changes as the speed of rotation was altered. 
 
 For artificial vowels, illustrative of his special views, Hermann 
 recommends the polyphonic siren ( 11). If when the series of 
 12 holes is in operation and a suitable velocity has been attained, 
 the series of 18 holes be put alternately into and out of action, 
 the difference-tone (6) is heard with great loudness and it 
 assumes distinctly the character of an 0. At a greater speed the 
 vowel is Ao, and at a still higher speed an unmistakable A. 
 
 With the use of double resonators, suitably proportioned, 
 Lloyd has successfully imitated some of the whispered vowels. 
 
 In the account here given of the vowel question it has only 
 been possible to touch upon a few of the more general aspects of 
 it. The reader who wishes to form a judgment upon controverted 
 points and to pursue the subject into detail must consult the 
 original writings of recent workers, among whom may be specially 
 mentioned Hermann, Pipping, and Lloyd. The field is an 
 attractive one ; but those who would work in it need to be well 
 equipped, both on the physical and on the phonetic side. 
 
 1 Proc. Roy. Soc. vol. xxvm. p. 358, 1879. 
 
479 
 
 NOTE TO 86 1 . 
 
 IT may be observed that the motion of any point belonging to 
 a system of n degrees of freedom, which executes a harmonic motion, is. 
 in general linear. For, if x, y, z be the space coordinates of the point,, 
 we have 
 
 x = X cos nt, y = T cos nt, z Z cos ntj 
 
 where X, Y, Z are certain constants ; so that at all times 
 x:y.z = X: Y : Z. 
 
 If there be more than one mode of the frequency in question, 
 the coordinates are not necessarily in the same phase. The most 
 general values of x, y, z, subject to the given periodicity, are then 
 
 x = X l cos nt + X% sin nt, 
 y- Y l cos nt + Y 2 sin nt, 
 z = Z^ cos nt + Z^ sin nt, 
 equations which indicate elliptic motion in the plane 
 
 x (Y,Z, - Z, T,) +y(Z l X* - X,Z 9 ) + z (X, Y,- Y,X 2 ) = 0. 
 
 1 This note appears now for the first time. 
 
APPENDIX TO CHAPTER V 1 . 
 
 ON THE VIBRATIONS OF COMPOUND SYSTEMS WHEN THE AMPLITUDES 
 ARE NOT INFINITELY SMALL. 
 
 IN 67, 68 we have found second approximations for the vibrations 
 of systems of one degree of freedom, both in the case where the 
 vibrations are free and where they are due to the imposition of given 
 forces acting from without. It is now proposed to extend the investi- 
 gation to cases where there is more than one degree of freedom. 
 
 In the absence of dissipative and of impressed forces, everything 
 may be expressed ( 80) by means of the functions T and V. In 
 the case of infinitely small motion in the neighbourhood of the 
 configuration of equilibrium, T and V reduce themselves to quadratic 
 functions of the velocities and displacements with constant coefficients, 
 and by a suitable choice of coordinates the terms involving products 
 of the several coordinates may be made to disappear ( 87). Even 
 though we intend to include terms of higher order, we may still avail 
 ourselves of this simplification, choosing as coordinates those which 
 have the property of reducing the terms of the second order to sums of 
 squares. Thus we may write 
 
 2Hr4n& > +i^& a +'>^4-& + 4t4i& + (1), 
 
 in which A n , A^,... are functions of <f> lt <j> 2) ... including constant terms 
 !, 6t 2 , ..., while AW, A wt ... are functions of the same variables without 
 constant terms : 
 
 v=fa<t>* + fa<i>*+...+-v s + r 4 + (2), 
 
 where F 8 , F 4 , ... denote the parts of V which are of degree 3, 4, ... 
 in < < 2 , ... 
 
 For the first approximation, applicable to infinitely small vibrations, 
 we have 
 
 A ll = a lt A y2 =a 2 , ... J 13 = 0, 4 U = 0..., F, = 0, F 4 =0, ...; 
 1 This appendix appears now for the first time. 
 
VIBRATIONS OF THE SECOND ORDER. 481 
 
 so that (87) Lagrange's equations are 
 
 Q, &c ............. (3), 
 
 in which the coordinates are separated. The solution relative to fa 
 may be taken to be 
 
 fa = ff lG osnt, fa = 0, < 3 = 0, ...&c ............. (4), 
 
 where c l n i a l =Q .............................. (5). 
 
 Similar solutions exist relative to the other coordinates. 
 
 The second approximation, to which we now proceed, is to be 
 founded on (4), (5); and thus < 2 , fa, ... are to be regarded as small 
 quantities relatively to fa. 
 
 For the coefficients in (1) we write 
 
 A n = a l + a n fa + a l2 fa+ ..., A l2 = a z fa + ... t A 13 = asfa+... 
 
 ......... (6), 
 
 and in (2) V, = y,fa* + y,fa*fa + ..................... (7); 
 
 so that for a further approximation 
 
 dT/dfa = (! + a n fa) fa + a z fafa + a s fafa + .. . , 
 
 - (- ) = (^ + a u fa) fa 
 
 a u fa z 
 dt \dfa/ 
 
 + a 2 fafa 
 
 dT/dfa = %a u <l>i +a 2 fafa + a s fafa+ ... 
 
 Thus as the equation ( 80) for fa, terms of the order fa 2 being retained, 
 we get 
 
 (a l + a n fa)fa + a ll fa* + Cl fa+3 yi fa 2 = ............ (8). 
 
 To this order of approximation the coordinate fa is separated from the 
 others, and the solution proceeds as in the case of but one degree 
 of freedom ( 67). We have from (4) 
 
 fafa = - n 2 H* cos 2 nt = - %ri*H* (1 + cos 2w<), 
 fa* = tfH* sin 2 nt = %ri*H? (1 - cos 2nt), 
 fa 2 - H* cos 2 nt = JflV 2 (1 + cos 2nt) ; 
 so that (8) becomes 
 
 a, fa + c^fa + (- in 2 a n + 3 7l ) H* + (- f wXi + |yj) H? cos 2nt = 
 
 ......... (9). 
 
 The solution of (9) may be expressed in the form 
 
 fa = H + ZT; cos n + IT, cos 2n + ............... (10), 
 
 and a comparison gives 
 
 R. II. 31 
 
482 VIBRATIONS OF THE SECOND ORDER. 
 
 Thus to a second approximation 
 
 ^^-^)H^ n ^(^^ y} H f(M 
 
 c x Cj - 4w 2 a x 
 
 and the value of n is the same, i.e. ^(c^a^, as in the first approxi- 
 mation. 
 
 We have now to express the corresponding values of < 2 , </> 3 ____ 
 From (6) 
 
 dT/d<j> 2 = az<t> l <j> l + a 2 < 2 + . . . , 
 
 and Lagrange's equation becomes, terms of order ^ being retained, 
 
 a^ + C 2 <f> 2 + a^'4i + (a 2 - Ja 12 ) <^ 2 + y^ = 0, 
 or on substitution from (4) in the small terms 
 
 * COS Znt = 
 
 ......... (12). 
 
 Accordingly, if 
 
 <f) 2 = K Q + KI cos nt + K% cos 2^ + 
 
 we find on comparison with (12) 
 
 (15), 
 ......... (16). 
 
 Thus j5Tj = 0, and the introduction of the values of K Q and K 2 from (14), 
 (16) into (13) gives the complete value of </> 2 to the second approxima- 
 tion. 
 
 The values of < 3 , < 4 , &c. are obtained in a similar manner, and 
 thus we find to a second approximation the complete expression for 
 those vibrations of a system of any number of degrees of freedom 
 which to a first approximation are expressed by (4). 
 
 The principal results of the second approximation are (i) that the 
 motion remains periodic with frequency unaltered, (ii) that terms, 
 constant and proportional to cos 2 ft , are added to the value of that 
 coordinate which is finite in the first approximation, as well as to those 
 which in the first approximation are zero. 
 
 We now proceed to a third approximation ; but for brevity we will 
 confine ourselves to the case (a) where there are but two degrees 
 of freedom, and (/?) where the kinetic energy is completely expressed as 
 a sum of squares of the velocities with constant coefficients. This will 
 include the vibrations of a particle moving in two dimensions in 
 the neighbourhood of a place of equilibrium. 
 
VIBRATIONS OF THE SECOND ORDER. 483 
 
 We have 
 
 T = ia 1 <j>* + a 2 <j> 2 2 , 7=1^ + Jc 2 2 2 + V,+ F 4 , 
 
 where F 3 - 7l ^ 3 + y,<tfk + y'^tf + .................. (17), 
 
 7, = ^ +8,^ + ........................... (18); 
 
 so that Lagrange's equations are 
 
 ........ (19), 
 
 ......... (20). 
 
 As before, we are to take for the first approximation 
 
 <f> l = ff l cosnt, < 2 = .................. (21). 
 
 For the solution of (19), (20) we may write 
 
 cos 2ra + H 3 cos3nt + ......... (22), 
 
 cos2nt + K 3 cos 3^ + ......... (23). 
 
 In (22), (23) ZT , # 2 , K , A" 2 are quantities of the second order in H lf 
 whose values have already been given, while K lt H 3 , K% are of the 
 third order. Retaining terms of the third order, we have 
 
 X 2 = i/^ 2 + (ZH^H^ + H^HJ) cos nt + ^H-f cos 2nt + H^H^ cos 3nt, 
 0J02 = (#! JT + J^Jfa) cos w + J^/Ta cos 3w, 
 0! 3 = IH* cos 7i + \H* cos 3n^. 
 
 Substituting these values in the small terms of (19), (20), and from 
 (22), (23) in the two first terms, we get the following 8 equations, 
 correct to the third order, 
 
 c^ + fy^^O ..................... (24), 
 
 Cl - wX + 3 7l (2# + # 2 ) + 2y 2 (K + J^ 2 ) + 3^^ = 0...(25), 
 ( Cl -4^X)^ 2 + |yi^i 2 = ..................... (26), 
 
 (c, - Ma,) H, + Zy&H, + y.H.K, + ^H* = 0...(27) ; 
 c^ + Jy^^O ..................... (28), 
 
 y'H, (2^T + ^T 2 ) + f 8,^= 0...(29), 
 
 2 + iy 2 ^i 2 - ..................... (30), 
 
 (c 2 - 9n 2 2 ) JT 8 + y&fft + y'H^ + faH* = O...(31). 
 
 Of these (24), (26), (28), (30) give immediately the values of H , H 9 , 
 K Q , K^ which are the same as to the second order of approximation, 
 and the substitution of these values in (27), (29), (31) determines 
 H 31 K lt K z as quantities of the third order. The remaining equation 
 (25) serves to determine n. We find as correct to this order 
 
 _ _ _ 
 
 - c,-4o 
 
 312 
 
484 VIBRATIONS OF THE SECOND ORDER. 
 
 If y 2 = 0, this result will be found to harmonize with (9) 67, when the 
 (jifferences of notation are allowed for, and the first approximation to n 
 is substituted in the small terms. 
 
 The vibration above determined is that founded upon (21) as first 
 approximation. The other mode, in which approximately fa = 0, can 
 be investigated in like manner. 
 
 If V be an even function both of fa and fa, y 15 y 2 , / &j vanish, and 
 the third approximation is expressed by 
 
 Indeed under this condition fa vanishes to any order of approxi- 
 mation. 
 
 These examples may suffice to elucidate the process of approximation. 
 An examination of its nature in the general case shews that the 
 following conclusions hold good however far the approximation may be 
 carried. 
 
 (a) The solution obtained by this process is periodic, and the 
 frequency is an even function of the amplitude of the principal 
 term (H-^. 
 
 (b) The Fourier series expressive of each coordinate contains 
 cosines only, without sines, of the multiples of nt. Thus the whole 
 system comes to rest at the same moment of time, e.g. t = 0, and then 
 retraces its course. 
 
 (c) The coefficient of cos rnt in the series for any coordinate is of 
 the rth order (at least) in the amplitude (H^) of the principal term. 
 For example, the series of the third approximation, in which higher 
 powers of H-^ than H? are neglected, stop at cos 3nt. 
 
 (d) There are as many types of solution as degrees of freedom ; 
 but, it need hardly be said, the various solutions are not superposable. 
 
 One important reservation has yet to be made. It has been 
 assumed that all the factors, such as (c 2 4n 2 a 2 ) in (30), are finite, that 
 is, that no coincidence occurs between a harmonic of the actual 
 frequency and the natural frequency of some other mode of infinitesimal 
 vibration. Otherwise, some of the coefficients, originally assumed to 
 be subordinate, e.g. 7T 2 in (30), become infinite, and the approximation 
 breaks down. We are thus precluded from obtaining a solution in 
 some of the cases where we should most desire to do so. 
 
 As an example of this failure we may briefly notice the gravest 
 vibrations in one dimension of a gas, obeying Boyle's law, and 
 
VIBRATIONS OF THE SECOND ORDER. 485 
 
 contained in a cylindrical tube with stopped ends. The equation to be 
 satisfied throughout, (4) 249, is of the form 
 
 JxJ dt* 
 and the procedure suggested by the general theory is to assume 
 
 y = x + 2/ + 2/j cos nt + y z cos 2nt + . . . , 
 
 where 2/ = # 01 sin x + # 02 sin 2x + # 03 sin 3x + . . . , 
 
 2/j = # n sin cc + # 12 sin 2x + H 13 sin 3x + . . . , 
 2/2 = # 2 i sin x + #22 sin 2x + H<% sin 3x + . . . , 
 and so on. In the first approximation 
 
 y = x + # n sin x cos nt, 
 with n l. But when we proceed to a second approximation, we find 
 
 still with n equal to 1, so that the method breaks down. The term 
 # 22 sin 2x cos 2nt in the value of y, originally supposed to be subordinate, 
 enters with an infinite coefficient. 
 
 It is possible that we have here an explanation of the difficulty of 
 causing long narrow pipes to speak in their gravest mode. 
 
 The behaviour of a system vibrating under the action of an 
 impressed force may be treated in a very similar manner. Taking, for 
 example, the case of two degrees of freedom already considered in 
 respect of its free vibrations, let us suppose that the impressed 
 forces are 
 
 ^ 1 = E 1 cospt, $2 = (33)) 
 
 so that the solution to a first approximation is 
 
 *. = <> (34). 
 
 With substitution of p for n equations (22), (23) are still applicable, 
 and also the resulting equations (24) to (31), except that in (25) the 
 left-hand member is to be multiplied by H and that on the right E^ is 
 to be substituted for zero. This equation now serves to determine H lt 
 instead of, as before, to determine n. 
 
 It is evident that in this way a truly periodic solution can always 
 be built up. The period is that of the force, and the phases are such 
 that the entire system comes to rest at the moment when the force is 
 at a maximum (positive or negative). After this the previous course 
 is retraced, as in the case of free vibrations, each series of cosines 
 remaining unchanged when the sign of t is reversed. 
 
NOTE TO 273 1 . 
 
 A METHOD of obtaining Poisson's solution (8) given by Liouville 2 is 
 worthy of notice. 
 
 If r be the polar radius vector measured from any point 0, and the 
 general differential equation be integrated over the volume included 
 between spherical surfaces of radii r and r + dr, we find on transforma- 
 tion of the second integral by Green's theorem 
 
 d?(r\)_ d*(rX) 
 
 ~dT- a df- ........................... w> 
 
 in which \ = ff<f>d<r, that is to say is proportional to the mean value 
 of <f> reckoned over the spherical surface of radius r. Equation (a) may 
 be regarded as an extension of (1) 279; it may also be proved 
 from the expression (5) 241 for V 2 < in terms of the ordinary polar 
 co-ordinates r, 0, o>. 
 
 The general solution of (a) is 
 
 r\ = x (at+r) + 6(at-r) ........................ (ft), 
 
 where x and are arbitrary functions ; but, as in 279, if the pole be 
 not a source, x ( at ) + # ( at ) = > so tnat 
 
 r\ = x(at + r)- x (at-r) ........................ (7). 
 
 It appears from (y) that at 0, when r 0, X = 2^' (at), which is 
 therefore also the value of 47r< at at time t. Again from (y) 
 
 _ , , . d (r\) d (r\) 
 
 so that 
 
 _ at 
 
 or in the notation of 273 
 
 By writing at in place of r in (8) we obtain the value of 2^ (at), or 
 4ir</>, which agrees with (8) 273. 
 
 1 This note appeared in the first edition. 
 
 2 Liouville, torn. i. p. 1, 1856. 
 
APPENDIX A. ( 307 \) 
 
 CORRECTION FOR OPEN END. 
 
 THE problem of determining the correction for the open end of a 
 tube is one of considerable difficulty, even when there is an infinite 
 flange. It is proved in the text ( 307) that the correction a is greater 
 than JTT R, and less than (8/877) R. The latter value is obtained by 
 calculating the energy of the motion on the supposition that the velocity 
 parallel to the axis is constant over the plane of the mouth, and 
 comparing this energy with the square of the total current. The actual 
 velocity, no doubt, increases from the centre outwards, becoming infinite 
 at the sharp edge ; and the assumption of a constant value is a some- 
 what violent one. Nevertheless the value of a so calculated turns out 
 to be not greatly in excess of the truth. It is evident that we 
 should be justified in expecting a very good result, if we assume an 
 axial velocity of the form 
 
 r denoting the distance of the point considered from the centre of the 
 mouth, and then determine /x and fj! so as to make the whole energy a 
 minimum. The energy so calculated, though necessarily in excess, must 
 be a very good approximation to the truth. 
 
 In carrying out this plan we have two distinct problems to deal with, 
 the determination of the motion (1) outside, and (2) inside the cylinder. 
 The former, being the easier, we will take first. 
 
 The conditions are that <$> vanish at infinity, and that when x = 0, 
 d<j>/dx vanish, except over the area of the circle r = R, where 
 
 d<j>ldx = l+^IR* + p!r*IR* .................. (1). 
 
 Under these circumstances we know ( 278) that 
 
 27T 
 
 where p denotes the distance of the point where <f> is to be estimated 
 from the element of area da: Now 
 
 1 This appendix appeared in the first edition. 
 
 2 The density of the fluid is supposed to be unity. 
 
488 CORRECTION FOR OPEN END. 
 
 if P represent the potential on itself of a disc of radius R, whose 
 density = 1 + /xr 2 /^ 2 + p'r^R*. 
 
 The value of P is to be calculated by the method employed in the text 
 ( 307) for a uniform density. At the edge of the disc, when cut down 
 to radius #, we have the potential 
 
 and thus 
 
 14 5 314 , 214 , 89 
 
 on effecting the integration. This quantity divided by TT gives twice 
 the kinetic energy of the motion defined by (1). 
 
 The total current 
 
 (5). 
 
 We have next to consider the problem of determining the motion of 
 an incompressible fluid within a rigid cylinder under the conditions 
 that the axial velocity shall be uniform when x = co , and when x = 
 shall be of the form 
 
 d<j>/dx = 1 + /xr 2 /^ 2 + /xV 4 /^ 4 . 
 
 It will conduce to clearness if we separate from <f>, that part of it which 
 corresponds to a uniform flow. Thus, if we take 
 
 d<f>/dx =l+^/x + J/x' + d\f//dx, 
 
 \l/ will correspond to a motion which vanishes when x is numerically 
 great. When x = Q, 
 
 ^/^^x,(r 2 -l) + / x'(r 4 -l) .................. (6), 
 
 if for the sake of brevity we put R = 1. 
 Now \j/ may be expanded in the series 
 
 t = *a p e>"J (pr) ........................... (7), 
 
 where p denotes a root of the equation 
 
 Each term of this series satisfies the condition of giving no radial 
 
 1 The numerical values of the roots are approximately 
 
 p 1= 3-831705, p 2 = 7-015, p 3 = 10174, 
 
 #4= 13-324, ^5 = 16-471, p 6 =19'616. 
 
CORRECTION FOR OPEN END. 489 
 
 velocity, when r = 1 ; and no motion of any kind, when x - oo . It 
 remains to determine the coefficients a p so as to satisfy (6), when x = Q. 
 From r = to r = 1, we must have 
 
 ;<pr)-/*(f*-j)+/<f-& 
 
 whence multiplying by J (pr) rdr and integrating from to 1, 
 
 every term on the left, except one, vanishing by the property of the 
 functions. For the right-hand side we have 
 
 / rdrJ (pr) = Q, 
 Jo 
 
 so that 
 
 The velocity-potential <f> of the whole motion is thus 
 
 the summation extending to all the admissible values of ;?. We have 
 now to find the energy of motion of so much of the fluid as is included 
 between x = 0, and x = l, where I is so great that the velocity is there 
 sensibly constant. 
 
 By Green's theorem 
 2 (kinetic energy) = ( < -f- ^r dr (x = 0) - ( <jf> -^ 27rr dr (x = -l). 
 
 JQ CiOC JQ d3C 
 
 Now, when x = l, 
 
 <t> = - (1 + 1/x + I/) I, 
 d<j>/dx=l + J/x + J/; 
 so that the second term is 7rl(l + i/x + J/w,') 2 . 
 
 In calculating the first term, we must remember that if p l and p 2 
 be two different values of p, 
 
490 CORRECTION FOR OPEN END. 
 
 Thus 
 
 Accordingly, on restoring R, 
 
 2 (kinetic energy) = irlPl (1 + //, + 
 
 To this must be added the energy of the motion on the positive side 
 of x - 0. On the whole 
 2 kinetic energy _ I _ 16 _ ( Q , / _ _8 \| 2 _ 5 
 
 current 2 = ^~R* + irR 1 + i/i + J/x') 2 ^ \^ H ^ V pVJ ^ 
 
 (current) 2 R* irR (1 + i/i + 
 
 1 + H/* + A/* 2 + I M/ + III W*' + A 9 5 
 
 
 Hence, if a be the correction to the length, 
 
 + (67T 
 
 + {247T (Sp- B 
 By numerical calculation from the values of p 
 
 2^-5 = -00128266; S/r 5 - 8Sjr 7 = -00061255., 
 5p- 5 - U2p- 7 + 642jtr 9 = -00030351, 
 and thus 37ra/872 = [1 + -9333333^ + -5980951 pf 
 
 + -2622728 ^ + -363223 /x/ + -1307634 /x' 2 ]-i-(l + }/t + 
 0666667^ + -0685716/- -0122728 /x 2 - -029890/x^ - -Q196523/ 2 
 
 ..................... (11). 
 
 The fraction on the right is the ratio of two quadratic functions of 
 /*, //, and our object is to determine its maximum value. In general if 
 S and S 1 be two quadratic functions, the maximum and minimum values 
 of * = S+S' are given by the cubic equation 
 
 - As- 3 + z- 2 - V 1 + A' = 0, 
 where S = ap? + 6// 2 + c + 2/// + 2#/x + 
 
 AS" = a> 8 + 6> /2 + c' + 2/y 
 
 A = aftc i 
 
 _/2) a ' + ( ca _ ^ 6 / + ^ _ ,^ c / 
 
 + 2 (srA - a/)/' + 2 (A/- %) ^ + 2 (/<? - cA) A', 
 

 CORRECTION FOR OPEN END. 491 
 
 and ', A', are derived from and A by interchanging the accented and 
 unaccented letters. 
 
 In the present case, since S' is a product of linear factors, A' = ; 
 and since the two factors are the same, ' = 0, so that z = A -=- simply. 
 Substituting the numerical values, and effecting the calculations, we 
 find z = '0289864, which is the maximum value of the fraction consistent 
 with real values of p, and ju,'. 
 
 The corresponding value of a is '82422 R, than which the true 
 correction cannot be greater. 
 
 If we assume /x.' = 0, the greatest value of z then possible is '024363, 
 which gives 
 
 a = -828146 R\ 
 
 On the other hand if we put /A = 0, the maximum value of z comes 
 out -027653, whence 
 
 a = -825353 R. 
 
 It would appear from this result that the variable part of the 
 normal velocity at the mouth is better represented by a term varying 
 as r 4 , than by one varying as r 2 . 
 
 The value a = -8242 R is probably pretty close to the truth. If the 
 normal velocity be assumed constant, a = '848826 R if of the form 
 1 + /xr 2 , a = -82815 R, when /x is suitably determined ; and when the 
 form 1 + /xr 2 + /xV 4 , containing another arbitrary constant, is made 
 the foundation of the calculation, we get a = '8242 R. 
 
 The true value of a is probably about '82 R. 
 
 In the case of /x = 0, the minimum energy corresponds to /x' = 1-103, 
 so that 
 
 On this supposition the normal velocity of the edge (r = R) would 
 be about double of that near the centre. 
 
 1 Notes on Bessel's functions. Phil. Mag. Nov. 1872. 
 
INDEX OF AUTHOES. 
 
 Airy, i. 49 ; n. 127 
 
 Arago, i. 2 
 
 Aristotle, i. 182 
 
 Auerbach, n. 452, 473, 476, 477 
 
 Barrett, n. 401, 402, 404, 407 
 
 Barry, n. 402 
 
 Basset, i. 432; n. 13, 341, 375, 376, 428 
 
 Beetz, n. 370 
 
 Bell, C., n. 368, 409 
 
 Bell, Graham, i. 470, 471; n. 474 
 
 Bernard, i. 347 
 
 Bernoulli, i. 133, 181, 255 ; n. 114, 201 
 
 Bertrand, i. 100 
 
 Betti, i. 157 
 
 Bidone, n. 355, 357 
 
 Billet, n. 119 
 
 Blaikley, n. 115, 203, 219, 235 
 
 Boltzmann, n. 223, 433 
 
 Boole, n. 37 
 
 Bosanquet, n. 17, 47, 201, 202, 216, 
 
 443, 444, 455, 457, 464, 466 
 Bosscha, ii. 48, 233 
 Bourget, i. 329, 330, 346, 347 
 Boutey, n. 115 
 Boys, n. 45 
 Brandt, i. 192; n. 444 
 Bravais, i. 2 ; n. 48 
 Bryan, i. 387 
 Buff, n. 356 
 Buijs Ballot, n. 155 
 Burton, i. 78; n. 38, 207, 276 
 
 Cagniard de la Tour, i. 5 
 Cauchy, i. 255,417; n. 6 
 
 Cavaille'-Coll, n. 219 
 
 Chladni, i. 254, 361, 362, 367, 371, 377, 
 
 379, 381, 387; n. 61, 227, 346, 458 
 Chree, i. 252; n. 272, 428 
 Christiansen, i. 168 
 Chrystal, i. 445, 449 
 Clarke, i. 60, 87 
 
 Clebsch, i. 297, 345, 353 ; n. 429 
 Clement, n. 21, 22 
 Colladon, i. 3; n. 317 
 Crum Brown, i. 72 
 
 D'Alembert, i. 170, 225 
 
 Davis, n. 225 
 
 De la Eive, n. 230 
 
 De Morgan, n. 23 
 
 Derham, n. 137 
 
 Deschanel, i. 7 
 
 Desor, n. 137 
 
 Desormes, n. 21, 22 
 
 Donders, n. 476 
 
 Donkin, i. 196, 209, 212, 253, 285, 286, 
 
 297, 298, 299, 353 
 Doppler, n. 155, 156 
 Dove, i. 446; n. 440 
 Duhamel, n. 70, 114, 297 
 Duhem, i. 150 
 Dulong, n. 61 
 Dvorak, n. 41, 43, 60, 333, 334 
 
 Earnshaw, n. 35, 37, 38 
 Edison, i. 474; n. 473 
 Edser, n. 457, 459 
 Ellis, n. 457, 464, 468 
 Elsas, i. 349, 350, 351 
 
INDEX OF AUTHORS. 
 
 493 
 
 Euler, i. 170, 181, 255; n. 51, 201 
 Everett, i. 7; n. 130 
 Ewing, n. 474 
 Exner, n. 452 
 
 Faraday, i. 82, 368 ; n. 225, 227, 346 
 
 Fenkner, i. 387 
 
 Fermat, n. 126, 132 
 
 Ferraris, i. 473 
 
 Ferrers, n. 236, 248 
 
 Forbes, n. 225 
 
 Fourier, i. 24, 25, 118, 203, 302 
 
 Fresnel, n. 82, 119, 120, 121 
 
 Froude, i. 212, 475, 477 ; n. 430 
 
 Gauss, i. 410, 416 
 
 Germain, S., i. 369 
 
 Goldingham, n. 47 
 
 Govi, n. 402 
 
 Gray, n. 399 
 
 Green, i. 352; n. 10, 85, 144, 248 
 
 Greenhill, i. 278 
 
 Gripon, n. 202, 223 
 
 Guthrie, n. 43, 300 
 
 Hallstrom, n. 463 
 
 Handel, i. 9 
 
 Hansen, i. 321, 329 
 
 Heaviside, i. 433, 450, 453, 457, 462, 
 463, 464, 467, 469, 470 
 
 Heine, n. 290, 297 
 
 Helmholtz, i. 12, 13, 14, 16, 52, 59, 68, 
 154, 157, 190, 192, 197, 198, 209, 212, 
 231, 434, 435; 11. 5, 11, 35, 94, 106, 
 143, 145, 146, 154, 175, 177, 187, 195, 
 201, 208, 215, 220, 221, 234, 235, 252, 
 319, 376, 408, ch. xxm passim 
 
 Henry, i. 434, 438, 440 ; n. 129, 135, 138 
 
 Hermann, n. 458, 459, 460, 461, 474, 
 475, 476, 477, 478 
 
 Herschel, J., i. 148; n. 65, 210, 246 
 
 Hertz, i. 433,474; n. 427 
 
 Hill, i. 85 
 
 Hopkins, n. 201 
 
 Hoppe, i. 304, 386 
 
 Hughes, i. 446, 447, 453, 455, 456, 457, 
 474 
 
 Huygens, n. 119, 122, 123, 124 
 
 Jaeriscb, n. 429 
 Jellet, i. 423 
 
 Jenkin, n. 474 
 Joule, n. 22 
 
 Kelvin, i. 24, 25, 98, 99, 100, 104, 108, 
 109, 139, 253, 352, 353, 402, 433, 434, 
 436, 442, 466, 476, 478; n. 7, 10, 11, 
 12, 16, 22, 43, 44, 106, 129, 137, 289, 
 297, 313, 344, 345, 350, 370, 376, 400, 
 467 
 
 Kircbboff, i. 296, 345, 353, 358, 362, 363, 
 369, 370, 371; n. 99, 148, 216, 319, 
 320, 321, 323, 324, 326, 327, 333 
 
 Koenig, B., i. 85, 383 ; n. 48, 62, 155, 
 
 460, 461, 462, 463, 464, 467, 468, 476 
 Koenig, W., n. 44, 46, 427 
 Kohlrausch, W., n. 412, 452, 453 
 Kolacek, n. 193 
 
 Kundt, i. 168; n. 30, 46, 47, 57, 58, 59, 
 60, 74, 75, 326 
 
 La Cour, i. 67 
 
 Lagrange, 1. 101, 109, 157, 170, 172, 181, 
 
 429; n. 4, 5, 6, 51, 201,458 
 Lamb, i. 157, 305, 417, 418, 419, 432, 
 
 462, 463 ; n. 13, 266, 376, 391, 429 
 Laplace, i. 109, 148; n. 19, 20, 86, 236 
 Lame, i. 315, 318, 353 
 Leconte, n. 401, 402 
 Lenard, n. 374 
 Le Eoux, n. 48 
 Leslie, n. 225, 239, 244, 246 
 Liouville, i. 217, 219, 221, 222, 294; 
 
 n. 486 
 
 Lipscbitz, n. 167 
 Liscovius, ii. 187 
 Lissajous, i. 28, 31, 32, 34, 62, 255, 285; 
 
 n. 350 
 
 Lloyd, n. 476, 477, 478 
 Lodge, A., i. 366 
 Lodge, 0., i. 433, 449, 456 
 Lommel, i. 320; n. 143, 167, 399 
 Love, i. 305, 402, 407, 409, 415, 432; 
 
 n. 428 
 
 Mach, n. 110, 155 
 
 Magnus, n. 356, 357, 364, 371 
 
 Mathieu, i. 343, 370 
 
 Mathews, n. 399 
 
 Matthiessen, n. 345 
 
 Maxwell, i. 160, 374, 433, 435, 437, 460, 
 
 461, 464 ; n. 163, 182, 249, 313, 320 
 
494 
 
 INDEX OF AUTHORS. 
 
 Mayer, n. 22 
 
 Mayer, A. M., i. 71, 88; n. 30, 41, 56, 
 
 118, 156, 411, 444, 450, 451, 460, 
 
 462 
 
 McKendrick, n. 474 
 McLeod, i. 60, 87 
 McMahon, i. 329 ; n. 267, 298 
 Melde, i. 81, 197, 348, 387; n. 220 
 Mersenne, i. 8, 181, 182, 183 
 Michell, i. 214, 305, 344 
 Moll, i. 2; n. 47 
 
 Newton, n. 18, 19 
 
 Oberbeck, i. 462 
 Ohm, i. 12, 16, 18 
 Oppel, n. 72 
 
 Page, n. 225 
 
 Pearson, i. 296 
 
 Perry, n. 250 
 
 Petzval, n. 155 
 
 Pfaundler, n. 452 
 
 Pipping, n. 478 
 
 Plateau, i. 34; n. 360, 363, 364, 368, 
 
 371, 375, 402, 406 
 Pockels, i. 343, 346 
 Pockhammer, i. 252, 257 ; n. 428 
 Poisson, i. 255, 264, 345, 353, 362, 369 ; 
 
 n. 21, 34, 35, 38, 41, 57, 64, 85, 97, 
 
 107, 249,315, 417, 429, 486 
 Poncelet, n. 350 
 Preece, i. 473, 474 ; n. 478 
 Preston, n. 30 
 Preyer, n. 432, 433, 457 
 
 Quincke, n. 65, 210 
 
 Eankine, i. 477 ; n. 23 
 
 Kegnault, i. 2; n. 23, 47, 48 
 
 Reynolds, i. 476, 477 ; n. 129, 135, 138 
 
 Riccati, i. 255 
 
 Ridout, n. 402 
 
 Riemann, i. 181, 311, 328, 342 ; n. 38, 
 
 39, 40 
 
 Riess, n. 233 
 Rijke, n. 232, 233 
 Rowland, n. 96 
 Routh, i. 109, 120, 122, 123, 125, 139, 
 
 141, 142 
 Riicker, n. 457, 459 
 
 Salmon, i. 125 
 
 Savart, i. 253, 346, 347, 349, 362, 364, 
 368, 369; n. 61, 201, 215, 360, 362, 
 363, 364, 368, 370, 371, 402, 406, 430 
 
 Savart, N., n. 77 
 
 Schaffgotsch, n. 228 
 
 Sclieibler, i. 60, 61, 62, 88, 183 ; n. 463 
 
 Sclmeebeli, n. 220, 326 
 
 Schaik, van, n. 221, 432 
 
 Schuster, n. 119 
 
 Schwartz, i. 150 
 
 Scott Russell, n. 155, 350 
 
 Seebeck, i. 184, 255, 284, 285, 286, 299, 
 300, 301; n. 225, 452 
 
 Seebeck, A., n. 326 
 
 Sellmeier, i. 168 
 
 Sidgwick, Mrs, n. 407 
 
 Simpson, i. 393 
 
 Smith, n. 220 
 
 Somof, i. 109 
 
 Sondhauss, n. 83, 175, 187, 188, 203, 
 220, 223, 227, 228, 230, 231, 410 
 
 Soret, n. 142 
 
 Stampfer, n. 48 
 
 Stokes, i. 128, 129, 185, 303, 329, 475; 
 n. 6, 12, 24, 25, 26, 28, 35, 36, 99, 
 101, 103, 110, 132, 237, 239, 240, 243, 
 246, 284, 304, 306, 315, 320, 341, 408, 
 417, 427 
 
 Stone, E. J., u. 48 
 
 Stone, W.H., n. 235 
 
 Strehlke, i. 255, 277, 285, 362, 363, 375 
 
 Stroh, n. 478 
 
 Strouhal, n. 412, 413, 414 
 
 Sturm, i. 217, 219, 221, 294 
 
 Tait, i. 24, 25, 72, 98, 104, 108, 109, 
 
 139, 253, 352, 402, 473, 475 ; n. 10, 
 
 44, 106, 289, 313 
 Taylor Brook, i. 181 
 Terquem, i. 253 
 Thompson, S. P., n. 440 
 Thomson, James, n. 130 
 Thomson, J. J., i. 433, 442, 446, 464 
 Thomson, W., see Kelvin 
 Todhunter, i. 296; n. 236, 289, 296 
 Topler, i. 35 ; n. 110, 223, 228, 433 
 Trevelyan, n. 224, 225 
 Tyndall, i. 13 ; ir. 60, 83, 135, 136, 137, 
 
 147, 225, 228, 311, 328, 400, 401, 404, 
 
 405, 406 
 
INDEX OF AUTHORS. 495 
 
 Verdet, n. 138 Wheatstone, i. 3, 32, 377, 383, 451, 453; 
 
 Violle, ii. 48 n. 228, 472 
 
 Voigt, i. 372 Wien, n. 440 
 
 Volkmann, n. 19 Willis, n. 470, 471, 473 
 
 Waterston, n. 30 Young, i. 7, 88, 181, 182, 187, 191, 231, 
 
 Webb, n. 373 235, 239, 241, 377 ; n. 72, 86, 458 
 
 Weber, i. 377 
 
 Weber, H., i. 457 Zamminer, i. 11 
 
 Wertheim, i. 353; n. 48, 61, 187, 202 
 
INDEX OF SUBJECTS. 
 
 Abnormal dispersion, i. 168 
 Absolute pitch, i. 85, 183 
 Aeolian harp, i. 212 ; n. 413 
 
 ,, tones, n. 412 
 Aerial vibrations, general equations of, 
 
 n. 97 
 ,, ,, in rectangular chamber, 
 
 n. 70 
 
 , , , , maintained by Heat, n. 226 
 Air, Velocity of Sound in, i. 2 
 
 ,, viscosity of, n. 313 
 Amplitude, i. 19 
 
 ,, of but just audible sounds, n. 433 
 Ampton bell, i. 392 
 Analogy of fluid motions with heat and 
 
 electricity, n. 13 
 Analysis of sounds by ear, i. 14, 191 ; 
 
 n. ch. xxin 
 
 Aperture, conductivity of elliptic, 11. 177 
 Approximately simple systems, i. 113 
 Approximation, second, i. 76, 78 ; n. 480 
 Arbitrary initial disturbance, n. 98, 417 
 Atmospheric refraction, n. 130 
 Attenuation by distance, i. 3 
 Attractions due to fluid motion, n. 43 
 Audibility, amplitude necessary for, n. 
 
 433 
 of one sound in presence of 
 
 another, n. 444 
 Audition, binormal. 
 
 ,, Facts and Theories of, ch. xxui 
 
 Balance, induction, of Hughes, i. 446 
 
 Barrett's observations upon sensitive 
 flames, n. 401 
 
 Bar, loaded, i. 249 
 
 Bars, circular, i. 304 
 
 expression for V, i. 267 
 Fourier's solution, i. 302 
 
 Bars, groups of waves, i. 301 
 ,, initial conditions, i. 269 
 ,, lateral vibrations of, i. 255 
 loaded, i. 288 
 
 longitudinal vibrations of, i. 242 
 ,, normal functions in various cases, 
 
 i. 272 
 
 ,, permanent tension, i. 296 
 ,, positions of nodes, i. 287 
 ,, variable density, i. 294 
 Beat-notes, n. 468 
 Beats, i. 22; n. 444 
 
 due to overtones, i. 26 ; n. 454 
 of bells, i. 389 
 ,, of chords, n. 465 
 of fifth, n. 464 
 of fourth, n. 465 
 ,, of octave, n. 464 
 of third, ii. 465 
 ,, of upper partials, 11. 453 
 ,, slow versus quick, i. 61 
 Bell emits no sound along axis, 11. 117 
 
 ,, sounded in hydrogen, n. 239 
 Bell's experiments, n. 409 
 
 on liquid jets, n. 368 
 
 Bells, beats of, i. 389 
 Belgian, i. 393 
 church, i. 391 
 ,, false octaves, i. 394 
 ,, nodal meridians, i. 389 
 ,, observations upon glass, i. 390 
 Bending, potential energy of, i. 257, 352, 
 
 353, 411, 425 
 Bertrand's theorem, i. 100 
 Bessel's functions, n. 91, 163, 293, 297, 
 
 354, 397 
 
 ,, ,, imaginary argument, i. 366 
 ,, roots of, i. 329 
 table of, i. 321 
 
INDEX OF SUBJECTS. 
 
 497 
 
 Bidone's observations, u. 356 
 Bird-call, n. 141, 410 
 Blackburn's pendulum, i. 31 
 Bow, action of, i. 212 
 Boyle's law, n. 19 
 Branched tube, n. 63 
 Bridge, Wheatstone's, i. 449 
 
 Cable formula of Lord Kelvin, i. 466 
 
 Capillarity, n. 343 
 
 Capillary tension determined by ripples, 
 
 n. 346 
 
 Chamber, rectangular, n. 156 
 Character of Sounds, i. 13 ; n. 470 
 Characteristic of vowels, n. 473 
 Chladni's figures, i. 368 
 Chords, beats of, n. 465 
 Circulation, n. 7 
 Clement and Desormes' experiments, n. 
 
 21 
 
 Clouds, acoustic, n. 136 
 Coexistence of small motions, i. 105 
 Collisions of drops, n. 369 
 Comma, i. 10 
 Composition of isoperiodic vibrations, i. 
 
 19 
 
 ,, ,, random vibrations, i. 36 
 
 Conditions, initial, i. 127 
 Conducting screen, i. 460 
 Conductivity of apertures, n. 173, 175 
 ,, necks, n. 181 
 
 ,, approximately cylindri- 
 
 cal tubes, ii. 184 
 Conductors in parallel, i. 441 
 Conjugate property, i. 127, 263, 358 
 Conical shell, i. 399 
 
 tube, n. 113, 114 
 Consonances, Beats of imperfect, n. 
 
 463 
 ,, defined by beats of upper 
 
 partials, n. 454 
 
 Constraint, one imposed, i. 119 
 Constraints, several, i. 123 
 Continuity, equation of, n. 3 
 Conveyance of Sound by Wires, i. 3 
 Coordinates generalised, i. 91 
 
 ,, normal, i. 107 
 
 Core, conducting, i. 463 
 Correction for open end, n. 487 
 Corrugated surface, bounding an impene- 
 trable medium, n. 92 
 ,, reflection from, n. 89 
 
 R. II. 
 
 Crispations, investigated by Faraday, 
 
 n. 346 
 
 Currents, free, in cylinder, j. 461 
 ,, induced electrical, i. 436 
 ,, initial, i. 439 
 Cylinder, vibrations within a closed, n. 
 
 300 
 Cylinders, liquid, under capillary force, 
 
 n. 352 
 Cylindrical obstacle, n. 309 
 
 ,, shell, conditions of in exten- 
 sion, i. 399 
 
 ,, shell, effect of friction, i. 388 
 , , shell, observations by Fenkner, 
 
 i. 387 
 ,, shell, potential and kinetic 
 
 energies, i. 385 
 
 ,, shell vibrating in two dimen- 
 sions, i. 384 
 
 ,, shell, tangential vibrations, i. 
 388 
 
 D'Alembert's solution of differential 
 
 equation, i. 225 
 
 Damping of vibrations within the ear 
 according to Helmholtz's 
 theory, n. 449 
 various degrees of, i. 51 
 Density, string of variable, i. 215 
 Difference-tones, 11. 456 
 
 ,, Helmholtz's theory of, 
 
 n. 458, 460 
 
 ,, of second order, n. 464 
 
 ,, order of magnitude, ii. 
 
 465 
 ,, sometimes external to 
 
 ear, n. 457 
 Diffraction, n. 139 
 
 ,, spectra, n. 89 
 
 Dilatational waves in an elastic solid, 
 
 n. 416, 418 
 
 Dimensions, method of, i. 54 ; n. 429 
 Disc, shadow of circular, n. 143 
 ,, stroboscopic, i. 35; ii. 407 
 ,, suspended, forces upon, n. 44 
 Dissipation, case of small, i. 136 
 ,, function, i. 102 
 
 ,, for viscous fluid, 
 
 n. 315 
 
 Dissipative forces, i. 45, 102, 130 
 Distortional waves, n. 419 
 Divergent wave in two dimensions, ii. 304 
 
 32 
 
498 
 
 INDEX OF SUBJECTS. 
 
 Diverging waves, n. 123, 239 
 
 Dominant, i. 8 
 
 Drums, i. 348 
 
 Doppler's principle, n. 154 
 
 Driving point, i. 158 
 
 Drops, collisions of, n. 369 
 
 electrified, n. 374 
 
 vibrations of, n. 371 
 Dynamical similarity, n. 429 
 
 Echoes, harmonic, n. 152 
 Elastic solid, potential of strain, n. 313 
 Elasticity, comparison of notations, i. 353 
 Electrical system, i. 126 
 
 vibrations, i. 433 
 Electric fork, i. 65 
 
 Electricity, conducting screen, i. 460 
 ,, conductors in parallel, i. 441 
 
 ,, Edison's transmitter, i. 474 
 
 ,, effect upon a small fountain, 
 
 n. 369 
 
 ,, free currents in cylinder, i. 461 
 
 ,, Heaviside's theory of wires, 
 
 1.467 
 
 ,, Hughes' apparatus, i. 453 
 
 induced currents, i. 436 
 
 ,, induction balance, Wheat- 
 
 stone's bridge, i. 449 
 inductometer, i. 457 
 
 ,, initial currents, i. 439 
 
 resistance of wires to alter- 
 
 nating currents, i. 464 
 ,, telephone, i. 471 
 
 transmitter, i. 470 
 
 Electrified drops, n. 374 
 Electromagnet and leyden, i. 434 
 
 ,, forced vibrations, i. 435 
 
 Elliptic aperture, conductivity of, n. 177 
 ,, comparison with circular, n. 179 
 Enclosure, vibrations in two dimensions 
 
 within a circular, n. 297 
 Enclosure, vibrations of a gas contained 
 
 within a spherical, n. 264 
 End, correction for open, u. 487 
 Energy emitted from vibrating spherical 
 
 surface, n. 252 
 ,, kinetic, i. 96 
 
 law of, verified in reflection, n. 85 
 of spherical waves, n. 112 
 .. ,, when confined in a 
 conical tube, n. 
 113 
 
 Energy, potential, i. 92 
 
 ,, ,, of condensation, n. 17 
 
 ,, ,, ,, bending, i. 256 
 
 ,, transmission of energy in plane 
 aerial waves, 11. 16 
 
 Equal roots of determinantal equation, 
 
 i. 109 
 ,, Temperament, i. 11 
 
 Equations, Lagrange's, i. 100 
 
 Equilibrium theory, i. 133 
 
 Fabrics, interference of partial reflections 
 
 from, n. 311 
 
 passage of sound through, n. 311 
 Faraday's investigations on crispations, 
 
 n. 346 
 
 Fatigue of ear, n. 446 
 Fermat's principle of least time, n. 126 
 Fifth, i. 8 
 
 beats of, u. 464 
 Flame, reflection of sound from, u. 83 
 sensitive for diffraction experi- 
 ments, ii. 141 
 Flames, sensitive, n. 400 
 ,, singing, n. 227 
 ,, ,, Sondhauss' experiments 
 
 upon, ii. 227 
 Fluid, perfect, n. 1 
 Fog signals, n. 137 
 Force applied at a single point, i. 134 
 at one point of elastic solid, n. 
 
 425 
 
 Forced electrical vibrations, i. 435 
 vibration, i. 46, 63, 145 
 , >, of string, i. 192 
 
 Fork for intermittent illumination, i. 34 
 ,, electric, i. 65 
 ,, ideal, i. 58 
 ,, opposing action of two prongs, n. 
 
 306 
 Forks for experiments on interference, 
 
 n. 117 
 
 tuning-, i. 59 
 
 Fountain, disturbed by electricity, n. 369 
 Fourier's solution for transverse vibra- 
 tions of bars, i. 302 
 ,, Theorem, i. 25 
 Fourth, i. 8 
 
 beats of, n. 465 
 
 Free vibration, i. 46, 74, 105, 109 
 Frequency, i. 7 
 
 Table of, i. 11 
 
INDEX OF SUBJECTS. 
 
 499 
 
 Fresnel's expressions for reflected and 
 
 refracted waves, n. 82 
 ,, zones, ii. 119 
 Friction fluid, n. 312 
 Functions, normal, i. 118 
 
 Galleries, whispering, n. 127 
 
 General equations of aerial vibration, n. 
 
 97 
 
 ,, ,, ,, free vibration, 1. 138 
 
 Generalized Coordinates, i. 91 
 Grating circular, n. 142 
 Green's investigation of reflection and 
 refraction, 11. 78 
 
 I,, theorem, Helmholtz's extension 
 of, n. 144 
 
 Groups of waves, i. 301 
 Gyrostatic terms, i. 104 
 
 Harmonic curve, i. 21 
 ,, echoes, n. 152 
 ,, scale, i. 8 
 ,, vibrations, i. 19, 44 
 Harmonics, i. 8, 12 
 Harmonies, beats of imperfect, n. 467 
 Harmonium, absolute pitch by, i. 88 
 Harp, ^olian, i. 212 ; n. 413 
 Head as an obstacle, n. 442 
 Heat, analogy with fluid motion, n. 13 
 ,, conduction, effects of, n. 321 
 ,, maintenance of vibrations by 
 
 means of, n. 224 
 Heats, specific, n. 20 
 Heaviside's theory of electrical propaga- 
 tion in wires, i. 467 
 
 Helmholtz's extension of Green's theo- 
 rem, ii. 144 
 
 ,, reciprocal theorem, ii. 145 
 
 Hooke's law, i. 171 
 Huygens' principle, n. 119 
 Hughes' apparatus, i. 453 
 Hydrogen, bell sounded in, n. 239 
 ,, flames, n. 227 
 
 Impulses, i. 96 
 
 ,, number necessary to define 
 
 pitch, n. 452 
 Incompressible fluid, n. 9 
 Induction balance, i. 446 
 Inductometer, i. 457 
 Inertia, lateral, of bars, i. 251 
 Inexorable motions, i. 149 
 
 Infinities occurring when n + KU=Q, ii, 
 
 398 
 
 Initial conditions, i. 127 
 Instability, i. 75, 143 
 
 ,, of electrified drops, ii. 374 
 
 jets, n. 360 
 
 ,, ,, vortex motion, n. 378 
 
 Intensity, mean, i. 39 
 Interference, i. 20 
 Intermittent Illumination, i. 34 
 
 ,, vibrations, i. 71, 165 ; ii. 440 
 
 Interrupter, fork, i. 68, 455 
 Interval, smallest consonant, ii. 451 
 Intervals, i. 7, 8 
 Inversion of Intervals, i. 8 
 Irrotational motion, n. 10 
 
 Jet interrupter, i. 456; n. 368 
 Jets, Bell's experiments, n. 368 
 ,, Bidone's observations, n. 356 
 ,, instability of, ii. 361 
 , , instability of, due to vorticity , n. 380 
 ,, Savart's observations upon, n. 363 
 ,, wave length of maximum instabi- 
 lity, n. 361 
 
 ,, under electrical influence, n. 369 
 ,, used to find the tension of recently 
 
 formed surfaces, ii. 359 
 ,, varicose or sinuous?, n. 402 
 , , vibrations about a circular figure, ii. 
 357 
 
 Kaleidophone, i. 32 
 
 Kelvin's Theorem, i. 99 
 
 Kettle-drums, i. 348 
 
 Key-note, i. 8 
 
 Kinetic energy, i. 96 
 
 Kirchhoff's investigation of propagation 
 
 of sound in narrow tubes, ii. 319 
 Kcenig's apparatus for absolute pitch, 
 
 i. 85 
 Kundt's tube, n. 47, 57, 333 
 
 Lagrange's equations, i. 100 
 
 ,, theorem in fluid motion, ii. 6 
 Laplace's correction to velocity of Sound, 
 
 n. 19, 20 
 
 ,, functions, applications of, n. 236 
 Lateral inertia of bars, i. 251 
 
 ,, vibrations of bars, i. 255 
 Leconte's observation of sensitive 
 flames, n. 401 
 
500 
 
 INDEX OF SUBJECTS. 
 
 Leslie's experiment of bell struck in 
 
 hydrogen, n. 239 
 Ley den and electromagnet, i. 434 
 Liouville's theorem, i. 222 
 Liquid cylinder and capillary force, n. 352 
 Lissajous' Figures, i. 28 
 
 ,, phenomenon, n. 349 
 Load carried by string, i. 53 
 Loaded spring, i. 57 
 Longitudinal Vibrations, i. 242 
 Loudness of Sounds, i. 13 
 Low notes from flames, u. 228 
 
 Maintenance of aerial vibrations by heat, 
 
 n. 226 
 
 ,, vibrations, i. 79, 81 
 
 Mass, effect of increase in, i. Ill 
 Melde's experiment, i. 81 
 Membranes, boundary an approximate 
 
 circle, i. 337 
 Bourget's observations on, 
 
 i. 347 
 
 ,, circular, i. 318 
 
 ,, elliptical boundary, i. 343 
 
 ,, forced vibrations, i. 349 
 
 ,, form of maximum period, 
 
 i. 341 
 
 loaded, i. 334 
 
 ,, nodal figures of, i. 331 
 
 potential energy, i. 307 
 
 ,, rectangular, i. 307 
 
 ,, triangular, i. 317 
 
 Mersenne's laws for vibration of strings, 
 
 i. 182 
 
 Microscope, vibration, i. 34 
 Modulation, i. 10 
 Moisture, effect of, on velocity of Sound, 
 
 n. 30 
 
 Motional forces, i. 104 
 Motions, coexistence of small, i. 105 
 Multiple sources, n. 249 
 Multiply-connected spaces, n. 11 
 Musical sounds, i. 4 
 
 Narrow tubes, propagations of sound in, 
 
 n. 319 
 Nodal lines for circular membrane, i. 331 
 
 ,, rectangular membrane, 
 i. 314 
 
 of square plates, i. 374 
 
 meridians of bells, i. 389, 391 
 Nodes and Loops, n. 51, 77, 403 
 
 Nodes of vibrating strings, i. 223 
 Normal coordinates, i. 107 
 ,, functions, T. 118 
 ,, ,, for lateral vibrations 
 
 of bars, i. 262 
 Notations, comparison of (elasticity), 
 
 i. 353 
 
 Notes and Noises, i. 4 
 ,, Tones, i. 13 
 
 Obstacle, cylindrical, n. 309 
 ,, in elastic solid, n. 420 
 ,, linear, n. 423 
 ,, spherical, i. 272 
 Octave, Beats of, n. 464 
 
 corresponds to 2 : 1, i. 7, 8 
 Ohm's law, exceptions to, 11. 443 
 One degree of freedom, i. 43 
 Open end, condition for, n. 52, 196 
 ,, ,, correction for, n. 487 
 ,, ,, experiments upon correction 
 
 for, ii. 201 
 
 Order, vibrations of the second, n. 480 
 Organ-pipes, n. 218 
 
 ,, influence of wind in dis- 
 
 turbing pitch, ii. 219 
 ,, maintenance of vibration, 
 
 n. 220 
 
 mutual influence of, n. 222 
 
 ,, overtones of, n. 221 
 
 Overtones, i. 13 
 
 ,, absolute pitch by, i. 88 
 ,, best way of hearing, n. 446 
 
 Pendulous vibration, i. 19 
 
 Period, i. 19 
 
 calculation of, i. 44 
 
 Periodic vibration, i. 5 
 
 Periods of free vibrations, i. 109 
 
 ,, ,, lateral vibration of bars, i. 277 
 ,, for rectangular membrane, i. 311 
 ,, stationary in value, i. 109 
 
 Permanent type, waves of, ii. 32 
 
 Persistances, theorem respecting, i. 126 
 
 Phase, i. 19 
 
 ,, does it influence quality? n. 467, 
 469 
 
 Phases at random, i. 36 
 
 Phonograph, n. 473 
 
 Phonic wheel, i. 67 
 
 Pianoforte string, i. 191 
 
 Pisa Baptistery, resonance in, ii. 128 
 
INDEX OF SUBJECTS. 
 
 501 
 
 Pitch, i. 4, 13 
 
 ,, absolute, i. 85 
 
 ,, estimation of, u. 433 
 
 ,, high, bird-calls of, n. 411 
 
 number of impulses necessary for 
 
 definition of, n. 452 
 range of audibility, n. 432 
 ,, related to Frequency, i. 6 
 ,, standard, i. 9 
 Plane waves of aerial vibration, n. 15 
 
 ,, ,, reflection of, n. 427 
 Plateau's theory of jets, n. 364 
 Plate plane, i. 404 
 ,, vibrating circular, reaction of air 
 
 upon, ii. 162 
 Plates, circular, i. 359 
 ,, clamped edge, i. 367 
 ,, comparison with observation, i. 
 
 362 
 
 ,, conditions for free edge, i. 357 
 ,, curved, i. 395 
 gravest mode of square, i. 379 
 Kirchhoff ' s theory, i. 363, 370 
 ,, nodal lines by symmetry, i. 381 
 ,, oscillation of nodes, i. 365 
 , , potential energy of bending, i. 353 
 ,, rectangular, i. 371 
 ,, theory of a special case, i. 372 
 ,, vibrations of, i. 352 
 Point, most general motion of a, of a 
 system executing simple vibrations, 
 ii. 479 
 Poisson's integral, n. 38, 41 
 
 ,, solution for arbitrary initial 
 
 disturbance, n. 99 
 Porous walls, n. 328 
 Potential energy, i. 92, 353 
 
 ,, ,, of bending, i. 256 
 
 Pressure, equations of, n. 2, 14 
 Probability of various resultants, i. 41 
 Progressive waves, i. 475 
 
 ,, ,, subject to damping, 
 
 i. 232 
 Propagation of sound in water, i. 3 
 
 Quality of sounds, i. 13; n. 467, 469 
 Quincke's tubes, ii. 210 
 
 Eadiation, effect of, on propagation of 
 
 Sound, n. 24 
 Eankine's calculation of specific heats, 
 
 ii. 23 
 
 Eeaction at driving point, i. 158 
 
 ,, of a dependent system, i. 167 
 Keciprocal relation, i. 93, 95, 98, 150 
 
 ,, theorem, n. 145 
 Rectangular chamber, n. 70, 156 
 ,, membrane, i. 307 
 
 ,, Plate of air, ii. 74 
 
 Eeed instruments, n. 234 
 
 ,, interrupter, i. 457 
 Reflection and refraction of plane waves, 
 
 n. 78 
 from a corrugated surface, n. 
 
 89 
 ,, plate of air of finite 
 
 thickness, ii. 87 
 
 ,, ,, porous wall, n. 330 
 
 ,, ,, curved surfaces, n. 125 
 
 ,, ,, strata of varying tempe- 
 
 rature, n. 83 
 ,, wall, n. 77 
 
 ,, of waves at a junction of two 
 
 strings, i. 234 
 ,, ,, waves in elastic solid, ii. 427 
 
 total, n. 84 
 
 Eefraction, atmospheric, n. 130 
 ,, by wind, n. 133, 135 
 
 Eegnault's experiments on specftc 
 
 heats, n. 23 
 Eesistance, i. 160, 437 
 
 ,, forces of, i. 137 
 generalised, i. 449 
 ,, of wires to alternating cur- 
 
 rents, i. 464 
 Eesonance, i. 70 
 
 cases, i. 59 
 
 in buildings, n. 128, 333 
 
 ,, multiple, n. 189 
 
 Eesonator, n. 447 
 
 ,, absorption of Sound by, n. 209 
 
 ,, and double source, n. 214 
 
 ,, close to source, n. 211 
 
 ,, excitation by flames, n. 227 
 
 ,, ,, of, n. 218 
 
 ,, experiments upon pitch of, 
 
 n. 203 
 
 ,, forced vibration of, ii. 195 
 , , loss of energy from, ii. 193 
 two or more, n. 215 
 Eesonators and forks, i. 85 
 
 , , comparison with experiment , 
 
 n. 187 
 ,, natural pitch of, n. 174 
 
502 
 
 INDEX OF SUBJECTS. 
 
 Eesonators, repulsion of, n. 42 
 
 theory of, n. 170 
 Kiemann's equations, n. 39 
 Rijke's Sound, n. 232 
 King, vibrations of, i. 383 
 Rings, circular, vibrations of, i. 304 
 Eipples, used for determination of capil- 
 lary tension, n. 346 
 
 Boots of determinantal equation, i. 139 
 Routh's theorems, i. 140 
 
 Sand, movements of, i. 368 
 
 Savart's observations upon jets, n. 363, 
 
 371 
 Second approximation, i. 76, 78 ; n. 480 
 
 ,, order, phenomena of, n. 41 
 Secondary circuit, influence of, i. 160, 
 
 437 
 ,, waves, due to variation of 
 
 medium, n. 150 
 Self-induction, i. 160, 437, 434 
 Sensitive flames, n. 400 
 
 jets of liquid in liquid, n. 406 
 Shadow caused by sphere, n. 255 
 
 ,, of circular disc, n. 143 
 Shadows, n. 119 
 Shell, cylindrical, i. 384 
 ,, effect of rotation, i. 387 
 observations by Fenker, i. 387 
 ,, potential and kinetic energies, i. 
 
 385 
 
 ,, tangential vibrations, i. 388 
 Shells, i. 395 
 
 ,, conditions of inextension, i. 398 
 
 conical, i. 399 
 
 ,, cylindrical, potential energy, i. 
 
 403 
 
 M ,, extensional vibra- 
 
 tions, i. 407 
 ,, potential energy of bending, i. 
 
 411 
 
 ,, flexural and extensional vibra- 
 tions, i. 396 
 ,, normal inextensional modes, i. 
 
 401 
 
 spherical, i. 401, 417, 420 
 Signals, fog, n. 135 
 Silence, points of, due to interference, 
 
 n. 116 
 
 Similarity, dynamical, n. 410, 413, 429 
 Singing flames, n. 227 
 Smoke jets, sensitive, n. 401 
 
 Smoke jets, periodic view of, n. 405 
 Solid bodies, vibrations of, n. 415 
 ,, elastic plane waves, n. 416 
 ,, limited initial disturbance, n. 417 
 ,, small obstacle in, n. 420 
 Sondhauss' observations upon bird-calls, 
 
 n. 410 
 
 Sonometer, i. 183 
 Sound, movements of, i. 368 
 Source, linear, n. 421 
 
 of harmonic type, n. 105 
 ,, of sound, direction of, n. 441 
 Sources, multiple, n. 249 
 
 simple and double, n. 146 
 Sparks for intermittent illumination, i. 34 
 Speaking trumpet, n. 113, 138 
 
 tubes, i. 3 
 Specific heats, n. 20 
 Sphere, communication of motion to air 
 
 from vibrating, n. 328 
 ,, obstructing, on which plane 
 waves impinge, n. 272 
 ,, ,, pressure upon, n. 279 
 
 Spherical enclosure, gas contained with- 
 in a, ii. 264 
 ,, waves, energy propagated, n. 
 
 112 
 ,, harmonics, table of zonal, 11. 
 
 251 
 
 ,, sheet of gas, n. 285 
 ,, ,, transition to two dimen- 
 
 sions, ii. 296 
 ,, waves, ii. 109 
 Spring, i. 57 
 
 Standard of pitch, i. 9, 60 
 Standing waves on running water, n. 350 
 
 ,, jets of liquid in liquid, n. 406 
 Statical theorems, i. 92, 95 
 Steel, velocity of sound in a wire of, i. 
 
 245 
 
 Steps, reflection from, n. 453 
 Stokes, investigation of communication 
 of vibration from sounding 
 body to a gas, n. 239 
 on effect of radiation on propa- 
 gation of Sound, ii. 24 
 ,, theorem, i. 128 
 
 Stop-cock, effect of, in disturbing sensi- 
 tive flames, n. 404 
 Stream-function, n. 4 
 Striations in Kundt's tubes, n. 47 
 String, employed in experiments on the 
 
INDEX OF SUBJECTS. 
 
 503 
 
 analysis of sounds by the ear, 
 i. 191 ; n. 444 
 String extremities not absolutely fixed, 
 
 i. 200 
 
 finite load, i. 204 
 ,, forced vibrations of. i. 192 
 ,, imperfect flexibility, i. 239 
 ,, mass concentrated in equidistant 
 
 points, i. 172 
 
 ,, nodes under applied force, i. 223 
 ,, normal Modes, i. 185 
 ,, of pianoforte, i. 191 
 ,, ,, variable density, i. 115, 215 
 partial Differential Equation, i. 
 
 177 
 
 ,, propagation of waves along, i. 224 
 ,, reflection at a junction, i. 235 
 Seebeck's observations, i. 184 
 ,, stretched on spherical surface, 
 
 1.213 
 
 ,, tones form a musical note, 1. 181 
 ,, transverse vibrations of, i. 170 
 
 values of T and V, i. 178 
 ,, vibrations started by plucking, i. 
 
 188 
 
 ,, ,,ablow,i.!88 
 
 violin, i. 209 
 ,, with load, i. 53 
 ,, two attached masses, 1. 165 
 Stroboscopic disc, i. 35 ; n. 407 
 Strouhal's observations upon asolian 
 
 tones, ii. 413 
 Sturm's theorems, i. 217 
 Subsidence, rates of, i. 138 
 Summation-tone, n. 459 
 Superposition, principle of, i. 49 
 Supply tube, influence of, in sensitive 
 
 flames, n. 229 
 Syren, i. 5 ; n. 469 
 
 ,, for determining pitch, i. 9 
 
 Telephone experiment on conducting 
 
 screen, i. 460 
 ,, minimum current audible, 
 
 i. 473 
 
 plate, i. 367 
 ,, (see Electricity), theory of, 
 
 i. 471 
 Temperament, i. 10 ; 11. 445 
 
 equal, i. 10 
 
 Temperature, effect of, in altering vis- 
 cosity, ii. 408 
 
 Temperature, effect of, on forks, i. 60, 
 
 86 
 ,, influence on velocity of 
 
 sound, n. 29 
 
 Tension, capillary, determined by me- 
 thod of ripples, n. 346 
 Terling bells, i. 393 
 Theory, Helmholtz's, of audition, ii. 
 
 448 
 Third, i. 8 
 
 ,, major, beats of, ii. 465 
 Time, principle of least, n, 126 
 Tone corresponds to simple vibration, 
 
 i. 17 ; n. 447 
 Tones and Notes, i. 13 
 
 ,, pure from forks, i. 59; ii. 463 
 Tonic, i. 8 
 
 Tonometer, Scheibler's, i. 62 
 Torsional vibrations of bars, i. 253 
 Transformation to sums of squares, i. 
 
 108 
 
 Transition, gradual, of density, i. 235 
 Transverse vibrations in elastic solids, 
 
 n. 416 
 
 Trevelyan's rocker, n. 224 
 Triangular membrane, i. 317 
 Trumpet, speaking, n. 113, 138 
 Tube, unlimited, containing simple 
 
 source, ii. 158 
 Tubes, branched, ii. 65 
 
 Kundt's, ii. 47; n. 334 
 
 ,, rectangular, ii. 73 
 
 ,, variable section, n. 67 
 
 ,, vibrations in, n. 49 
 Tuning by beats, i. 23 
 Twelfth (3 : 1), i. 7 
 Two degrees of freedom, i. 160 
 Tyndall's high pressure sensitive flame, 
 
 n. 401 
 Type, change of, n. 34 
 
 Variable section, tubes of, n. 67 
 Vehicle necessary, i. 1 
 Velocity and condensation, relation be- 
 tween, n. 15, 35 
 ,, in Air, i. 2 
 ,, independent of Intensity and 
 
 Pitch, i. 2 
 ,, minimum, of waves on water, 
 
 ii. 345 
 
 ,, of sound, dependent on tempe- 
 rature, n. 29 
 
504 
 
 INDEX OF SUBJECTS. 
 
 Velocity of sound in air, observations 
 
 upon, ii. 47 
 
 ,, ,, in water, n. 30 
 
 ,, Laplace's correction, 
 
 n. 19, 20 
 ,, ,, Newton's calculation, 
 
 n. 18 
 
 ,, -potential, n. 4, 8, 15 
 Velocities, system started with given, i. 99 
 Vibration, forced, i. 63 
 Vibrations, forced and free, i. 49 
 
 ,, of the second order, n. 480 
 Violin string, i. 209 
 Viscosity, analogy with elastic strain, 
 
 n. 313 
 
 denned, n. 312 
 
 ,, narrow tubes with small, n. 325 
 of air, n. 313 
 varied by temperature, n. 408 
 Viscous fluid, propagation of plane waves 
 
 in, n. 315, 322 
 ,, threads of, n. 375 
 ,, ,, transverse vibrations in, 
 
 n. 317 
 
 Vortex motion and sensitive jets, n. 376 
 Vortices in Kundt's tubes, n. 340 
 Vorticity, case of stability, n. 384 
 
 ,, general equation for stratified, 
 
 ii. 383 
 
 layers of uniform, ii. 385 
 Vowel A, Hermann's results, n. 475, 476 
 Vowels, artificial, ii. 471, 477, 478 
 
 investigated by phonograph, n. 
 
 474 
 
 ,, pitch of characteristic, two theo- 
 ries, ii. 473 
 
 presence of prime tone, n. 477 
 ,, question of double resonance, 
 
 n. 477 
 Wheatstone and Helmholtz's, 
 
 n. 472 
 
 ,, Willis's experiments and theo- 
 ries, ii. 470 
 
 Wall, porous, n. 328 
 
 reflexion from fixed, n. 77, 108 
 Water, propagation of sound in, i. 3 ; 
 
 ii. 30 
 
 Water, surface waves on, n. 344 
 
 ,, waves on running, n. 350 
 Waves, aerial, diverging in two dimen- 
 sions, ii. 304 
 ,, dilatational, in an elastic solid, 
 
 n. 416 
 
 diverging, n. 123 
 ,, of permanent type, n. 32 
 ,, on water, n. 344 
 ,, plane, energy half potential and 
 
 half kinetic, n. 17 
 ,, exact investigation of, ii. 
 
 31 
 
 ,, ,, of aerial vibration, n. 15 
 
 ,, ,, of transverse vibration, 
 
 ii. 416 
 
 ,, positive and negative, i. 227 
 progressive, i. 475 
 subject to damping, 
 
 i. 232 
 ,, secondary, due to variation of 
 
 medium, ii. 150 
 ,, spherical, ii. 109 
 ,, standing, on running water, n. 
 
 350 
 
 ,, stationary, i. 227 
 two trains crossing obliquely, 
 
 n. 76 
 Wheatstone's bridge, i. 449 
 
 ,, kaleido phone, i. 32 
 
 Wheel, phonic, i. 67 
 Whispering galleries, n. 127 
 Whistle, steam, n. 223 
 Whistling by the mouth, n. 224 
 Wind, refraction by, n. 132, 135 
 Windows, how affected by explosions, 
 
 n. Ill 
 
 Wires, conveyance of sound by, i. 3, 251 
 ,, electrical currents in, i. 464 
 
 Young's modulus, i. 243 
 
 ,, theorem regarding vibrations of 
 strings, i. 187 
 
 Zonal spherical harmonics, n. 251 
 Zones of Huygens or Fresnel, n. 119, 
 141 
 
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