'-3 THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES AN ELEMENTARY TREATISE ON HYDRODYNAMICS AND SOUND AN ELEMENTARY TREATISE ON HYDKODYNAMICS AND SOUND BY A. B. BASSET, M.A., F.RS. TRINITY COLLEGE, CAMBRIDGE. SECOND EDITION, REVISED AND ENLARGED. CAMBRIDGE DEIGHTON BELL AND CO. LONDON GEORGE BELL AND SONS 1900 [All Rights reserved.] Camfarfoge : FEINTED BY J. AND C. F. CLAY AT THE UNIVERSITY PBESS. f// e^ PREFACE. HHHE treatise on Hydrodynamics, which I published in 1888, was intended for the use of those who are acquainted with the higher branches of mathematics, and its aim was to present to the reader as comprehensive an account of the whole subject as was possible. But although a somewhat formidable battery of mathematical artillery is indispensable to those who desire to . possess an exhaustive knowledge of any branch of mathematical " physics, yet there are a variety of interesting and important ^investigations, not only in Hydrodynamics but also in Electricity ^and other physical subjects, which are well within the reach of Sievery one who possesses a knowledge of the elements of the Differential and Integral Calculus and the fundamental principles ^of Dynamics. I have accordingly, in the present work, abstained Kfrom introducing any of the more advanced methods of analysis, \A IN such as Spherical Harmonics, Elliptic Functions and the like ; and, as regards the dynamical portion of the subject, I have endeavoured to solve the various problems which present them- selves, by the aid of the Principles of Energy and Momentum, and have avoided the use of Lagrange's equations. There are a few problems, such as the helicoidal steady motion and stability of a solid of revolution moving in an infinite liquid, which cannot be conveniently treated without having recourse to moving axes ; but as the theory of moving axes is not an altogether easy branch of Dynamics, I have as far as possible abstained from introducing VI PREFACE. them, and the reader who is unacquainted with the use of moving axes is recommended to omit those sections in which they are employed. The present work is principally designed for those who are reading for the Mathematical Tripos and for other examinations in which an elementary knowledge of Hydrodynamics and Sound is required; but I also trust that it will not only be of service to those who have neither the time nor the inclination to become conversant with the intricacies of the higher mathematics, but that it will also prepare the way for the acquisition of more elaborate knowledge, on the part of those who have an opportunity of devoting their attention to the more recondite portions of these subjects. The first part, which relates to Hydrodynamics, has been taken with certain alterations and additions from my larger treatise, and the analytical treatment has been simplified as much as possible. I have thought it advisable to devote a chapter to the discussion of the motion of circular cylinders and spheres, in which the equations of motion are obtained by the direct method of calcu- lating the resultant pressure exerted by the liquid upon the solid ; inasmuch as this method is far more elementary, and does not necessitate the use of Green's Theorem, nor involve any further knowledge of Dynamics on the part of the reader than the ordinary equations of motion of a rigid body. The methods of this chapter can also be employed to solve the analogous problem of deter- mining the electrostatic potential of cylindrical and spherical conductors and accumulators, and the distribution of electricity upon such surfaces. The theory of the motion of a solid body and the surrounding liquid, regarded as a single dynamical system, is explained in Chapter III., and the motion of an elliptic cylinder in an infinite liquid, and the motion of a circular cylinder in a liquid bounded by a rigid plane are discussed at length. The Chapters on Waves and on Rectilinear Vortex Motion comprise the principal problems which admit of treatment by elementary methods, and I have also included an investigation PREFACE. Vll due to Lord Rayleigh, respecting one of the simpler cases of the instability of fluid motion. In the second part, which deals with the Theory of Sound, I have to acknowledge the great assistance which I have received from Lord Rayleigh's classical treatise. This part contains the solution of the simpler problems respecting the vibrations of strings, membranes, wires and gases. A few sections are also devoted to the Thermodynamics of perfect gases, principally for the sake of supplementing Maxwell's treatise on Heat, by giving a proof of some results which require the use of the Differential Calculus. The present edition has been carefully revised throughout, and a certain amount of new matter has been added. I have devoted Chapter IX. to the flexion and vibrations of naturally straight wires and rods ; whilst an entirely new chapter has been added on the finite deformation of naturally straight and curved wires, in which I have discussed a variety of questions which admit of fairly simple mathematical treatment. FLEDBOROUGH HALL, HOLYPORT, BERKS. CONTENTS. PART I. HYDRODYNAMICS. CHAPTER I. ON THE EQUATIONS OF MOTION OF A PERFECT FLUID. AKT. PAGE 1. Introduction 1 2. Definition of a fluid 1 3. Kinernatical theorems. Lagrangian and flux methods . . . 2 4. Velocity and acceleration. The Lagrangian method . . . . 2 5. do. The flux method 3 6. The equation of continuity 4 7. The velocity potential 5 8. Molecular rotation . 6 9-10. Lines of flow and stream lines 6 11. Earnshaw's and Stokes' current function 7 12. The bounding surface .......... 8 13. Dynamical Theorem's .......... 9 14. Proof of the Principle of Linear Momentum 10 15. Pressure at every point of a fluid is equal in all directions . . 10 16. The equations of motion -. 11 17-18. Another proof of the equations of motion 12 19. Pressure is a function of the density . 15 20. Equations satisfied by the components of molecular rotation . . 15 21. Stokes' proof that a velocity potential always exists, if it exists at any particular instant . . . 16 22. Physical distinction between rotational and irrotational motion . . 17 23. Integration of the equations of motion when a velocity potential exists . 18 24. Steady motion. Bernoulli's theorem . 18 25. Impulsive motion > 20 26. Flow and circulation 21 X CONTENTS. ART. PAGE 27. Cyclic and acyclic irrotational motion. Circulation is independent of the time 22 28. Velocity potential due to a source 23 29. do. due to a doublet 24 30. do. due to a source in two dimensions .... 24 31. do. due to a doublet in two dimensions . . .25 32. Theory of images . .25 33. Image of a source in a plane 25 34. Image of a doublet in a sphere, whose axis passes through the centre of the sphere 26 35. Motion of a liquid surrounding a sphere, which is suddenly annihilated 27 36. Torricelli's theorem 29 37. The vena contracta 30 38. Giffard's injector . . . . , f , . . . .31 Examples 32 CHAPTER II. MOTION OF CYLINDERS AND SPHERES IN AN INFINITE LIQUID. 39. Statement of problems to be solved 37 40. Boundary conditions for a cylinder moving in a liquid ... 38 41. Velocity potential and current function due to the motion of a circular cylinder in an infinite liquid 40 42. Motion of a circular cylinder under the action of gravity ... 40 43. Motion of a cylinder in a liquid, which is bounded by a concentric cylindrical envelop 42 44. Current function due to the motion of a cylinder, whose cross section is a lemniscate of Bernoulli 43 45. Motion of a liquid contained within an equilateral prism ... 44 46. do. do. an elliptic cylinder ... 45 47. Conjugate functions 45 48. Current function due to the motion of an elliptic cylinder ... 46 49. Failure of solution when the elliptic cylinder degenerates into a lamina. Discontinuous motion 47 50. Motion of a sphere under the action of gravity 49 51. Motion may become unstable owing to the existence of a hollow . 51 52. Definition of viscosity ; and its effect upon the motion of sphere . 52 53. Resistance experienced by a ship in moving through water . . 54 54. Motion of a spherical pendulum, which is surrounded by liquid . 54 55. Motion of a spherical pendulum, when the liquid is contained within a rigid spherical envelop . 55 Examples 57 CONTENTS. XI CHAPTER III. MOTION OF A SINGLE SOLID IN AN INFINITE LIQUID. ART. PAGE 56. Different methods of solving the problem 61 57. Bertrand's theorem 62 58. Green's theorem 63 59-63. Applications of Green's theorem 64 64. Conditions which the velocity potential must satisfy .... 66 65. Kinetic energy of liquid is a homogeneous quadratic function of the velocities of the moving solid 67 66. Values of the components of momentum 68 67. Short proof of the expressions for the kinetic energy and momentum 70 68. Motion of a sphere 71 69-71. Motion of an elliptic cylinder under the action of no forces . 72 72. Motion of an elliptic cylinder under the action of gravity . . 77 73. Helicoidal steady motion of a solid of revolution .... 78 74. Conditions of stability 80 75. Application to gunnery 81 76-78. Motion of a circular cylinder parallel to a plane .... 81 Examples ............ 85 CHAPTER IV. WAVES. 79. Kinematics of wave-motion 87 80. Progressive waves and stationary waves 89 81. Conditions of the problem of wave-motion 90 82-84. Waves in a liquid under the action of gravity .... 91 85-87. Waves at the surface of separation of two liquids ... 93 88-91. Stable and unstable motion 95 92. Long waves in shallow water 99 93. Analytical theory of long waves 100 94. Stationary waves in flowing water ....... 101 95. Theory of group velocity 103 96. Capillarity 103 97. Capillary waves conditions at the free surface 104 98. Capillary waves under the action of gravity 105 99. Discussion of results 105 100. Capillary waves produced by wind . . .... . . 106 Examples 108 Xll CONTENTS. CHAPTER V. RECTILINEAR VORTEX MOTION. AST. PAGE 101. Vortex motion in two dimensions Ill 102. Definition of vorticity 112 103. Velocity due to a single vortex 113 104. Velocity potential due to a vortex 114 105. Conditions which the pressure must satisfy 114 106. Kirchhoff's elliptic vortex 115 107. Discussion on the stability of a vortex 117 108. Motion of two vortices of equal vorticities 118 109. Motion of two vortices of equal and opposite vorticities . . .118 110. Motion of a vortex in a square corner 119 111. Motion of a vortex inside a circular cylinder 120 112. Rankine's free spiral vortex 121 113. Fundamental properties of vortex motion 122 114. Proof that the vorticity is an absolute constant .... 124 Examples 125 PAET II. THEORY OF SOUND. CHAPTER VI. INTRODUCTION. 115. Noises and musical notes . . 131 116. Connection between the characteristics of a note and the geometrical constants of a wave 132 117. Velocity of propagation of sound in gases and liquids .... 132 118. Intensity 132 119. Pitch 132 120. Compound notes and pure tones . 133 121. Timbre 134 122. Beats 134 CHAPTER VII. VIBRATIONS OF STRINGS AND MEMBRANES. 123. Transverse and longitudinal vibrations 136 124. Equation of motion for transverse vibrations 137 125. Solution for a string whose ends are fixed 138 126. Initial conditions . 139 CONTENTS. Xlll ART. PAGE 127. Motion produced by a given displacement 140 128. Motion produced by an impulse applied at a point .... 141 129. Motion produced by a periodic force ....... 142 130. Free vibrations gradually die away on account of friction . . . 143 131. Forced vibrations 144 132. Normal coordinates. Kinetic and potential energy .... 144 133. Longitudinal vibrations 145 134. Transverse vibrations of membranes 146 135. Nodal lines of a square membrane 147 136. Circular membrane 148 Examples 149 CHAPTER VIII. FLEXION OF WIRES. 137. Equations of equilibrium of a wire 151 138. Value of the flexural couple 152 139. Conditions to be satisfied at the ends 153 140. The elastica . 154 141. Kirchhoff's kinetic analogue 156 142143. Stability under thrust 156 144. Greatest height consistent with stability 159 145. Equations of motion of a wire 161 146. Equation of motion for the flexural vibrations of a naturally straight wire 161 147. Conditions at a free end 162 148. Equation of motion and conditions at a free end, when the rotatory inertia is neglected 162 149. Period of an infinite wire 163 150. Flexural vibrations of a wire of given length 163 151. Period equations 163 152. Extensional vibrations 165 Examples 166 CHAPTER IX. THEORY OF CURVED WIRES. 153. General equations of motion 170 154. Value of the flexural couple . 171 155. Its components about two arbitrary axes are proportional to the changes of curvature 173 156. Value of the torsional couple 175 157. Potential energy of a deformed wire 176 158. Torsional couple is constant when the wire is naturally straight . 176 159. Integration of the equations of equilibrium of a naturally straight wire . 177 XIV CONTENTS. ART. PAGE 160. Discussion of the three first integrals . . . . . 178 161. A helix is a possible figure of equilibrium 178 162. Discussion of the terminal stresses 179 163. Stability of a naturally straight wire under thrust and twist . . 180 164 16?.' Equilibrium and stability of a naturally straight wire which is deformed into a circle 181 166. Vibrations of a circular ring 183 CHAPTER X. EQUATIONS OF MOTION OF A PERFECT GAS. 167. Fundamental equations of the small vibrations of a gas . . . 186 168. Displacement in a plane wave is perpendicular to the wave front . 187 169. Newton's value of the velocity of sound 188 170. Thermodynamics of gases 189 171. The second law of Thermodynamics . . . ' . . . . . 191 172. Specific heats of a gas 192 173. Specific heats of air . . 192 174. Equations of the adiabatic and isothermal lines of a perfect gas . 193 175. Elasticity of a perfect gas 194 176. Velocity of sound in air 195 177. Intensity of sound 195 CHAPTER XI. PLANE AND SPHERICAL WAVES. 178. Motion in a closed vessel 197 179 181. Motion in a cylindrical pipe 197 182. Reflection and refraction 199 183. Change of phase, when reflection is total . . . . . . 202 184. Spherical waves 203 185. Symmetrical waves in a spherical envelop 204 186. Waves in a conical pipe 205 187. Sources of sound 205 188. Diametral vibrations 205 189. Motion of a spherical pendulum surrounded by air . . ... 206 190. Scattering of a sound wave by a small rigid sphere .... 208 Examples 210 PAKT I. HYDRODYNAMICS. CHAPTER I. ON THE EQUATIONS OF MOTION OF A PERFECT FLUID. 1. THE object of the science of Hydrodynamics is to in- vestigate the motion of fluids. All fluids with which we are acquainted may be divided into two classes, viz. incompressible fluids or liquids, and compressible fluids or gases. It must however be recollected that all liquids experience a slight com- pression, when submitted to a sufficiently large pressure, and therefore in strictness a liquid cannot be regarded as an incom- pressible fluid ; but inasmuch as the compression produced by such pressures as ordinarily occur is very small, liquids may be usually treated as incompressible fluids without sensible error. The physical interest arising from the study of the motion of gases, is due to the fact that air is the vehicle by means of which sound is transmitted. We shall therefore devote the first part of this volume to the discussion of incompressible fluids or liquids, reserving the discussion of gases for the second part, which deals with the Theory of Sound. We must now define a fluid. 2. A fluid may be defined to be an aggregation of molecules, which yield to the slightest effort made to separate them from each other, if it be continued long enough. A perfect fluid, is one which is incapable of sustaining any tangential stress or action in the nature of a shear; and it will be shown in 15 that the consequence of this property is, that the pressure at every point of a perfect fluid is equal in all directions, whether the fluid be at rest or in motion. A perfect fluid is however an entirely ideal substance, since all fluids B. H. 1 2 EQUATIONS OF MOTION OF A PERFECT FLUID. with which we are acquainted are capable of offering resistance to tangential stress. This property, which is known as viscosity, gives rise to an action in the nature of friction, by which kinetic energy is gradually converted into heat. In the case of gases, water and many other liquids, the effect of viscosity is so small that such fluids may be approximately regarded as perfect fluids. The neglect of viscosity very much simplifies the mathematical treatment of the subject, and in the present treatise, we shall confine our attention to perfect fluids. Before entering upon the dynamical portion of the subject, it will be convenient to investigate certain kinematical proposi- tions, which are true for all fluids. Kinematical Theorems. 3. The motion of a. fluid may be investigated by two different methods, the first of which is called the Lagrangian method, and the second the Eulerian or flux method, although both are due to Euler. In the Lagrangian method, we fix our attention upon an element of fluid, and follow its motion throughout its history. The variables in this case are the initial coordinates a, b, c of the particular element upon which we fix our attention, and the time. This method has been successfully employed in the solution of very few problems. In the Eulerian or flux method, we fix our attention upon a particular point of the space occupied by the fluid, and observe what is going on there. The variables in this case are the coordinates x, y, z of the particular point of space upon which we fix our attention, and the time. Velocity and Acceleration. 4. In forming expressions for the velocity and acceleration of a fluid, it is necessary to carefully distinguish between the Lagrangian and the flux method. I. The Lagrangian Method. Let u, v, w be the component velocities parallel to fixed axes, of an element of fluid whose coordinates are x, y, z and x + 8x, y 4- &/, z + 8z at times t and t + 8t respectively, then u = dxjdt = x, v=y, w = z ( 1 ), VELOCITY AND ACCELERATION. 3 where in forming x, y, z we must suppose ac, y, z to be expressed in terms of the initial coordinates a, b, c and the time. The expressions for the component accelerations are f*=u=x, f v = y> f z = z .................. (2), where u, v, w are supposed to be expressed in terms of a, b, c and t, II. The Flux Method. 5. Let 8Q be the quantity of fluid which in time St flows across any small area A, which passes through a fixed point P in the fluid ; let p be the density of the fluid, q its resultant velocity, and e the angle which the direction of q makes with the normal to A drawn towards the direction in which the fluid flows. Then BQ = pqA&t cose, , 1 dQ therefore q = . -7- . pA cos e at Now A cos e is the projection of A upon a plane passing through P perpendicular to the direction of motion of the fluid ; hence SQ is independent of the direction of the area, and is the same for all areas whose projections upon the above-mentioned plane are equal. Hence the velocity is equal to the rate per unit of area divided by the density, at which fluid flows across a plane perpendicular to its direction of motion. The velocity is therefore a function of the position of P and the time. In the present treatise the flux method will almost exclu- sively be employed. We may therefore put u = F (ac, y, z, t) ; whence if u + Su be the velocity parallel to x at time t + St of the element of fluid which at time t was situated at the point t, y + vSt, z + wSt, t + St)-F(x, y, z, t). Therefore the acceleration, Bu du du du du f x = hm^- = J.+U-T- + V-J- + W-7-. J ct dt dx dy dz Hence if dfdt denotes the operator d/dt + udjdx + vdjdy + wdjdz, the component accelerations will be given by the equations f _ou , _rto f _^w /o\ J*=fo> Jy-^t' Jz ~ dt" 12 4 EQUATIONS OF MOTION OF A PERFECT FLUID. The Equation of Continuity. 6. If an imaginary fixed closed surface be described in a fluid, the difference between the amounts of fluid which flow in and flow out during a small interval of time 8t, must be equal to the increase in the amount of fluid during the same interval, which the surface contains. The analytical expression for this fact is called the equation of continuity. Let Q be any point (x, y, z), and consider an elementary parallelepiped SxSy&z. The amount of fluid which flows in across the face GB in time St is puSySzSt. The amount which flows out across the opposite face AD is puSySzSt + -j- (pu) SxSySzSt, Qw/ whence the gain of fluid due to the fluxes across the faces GB, AD is -T- (pu) 8xSySz8t. Treating the other faces in a precisely similar manner, it follows that the total gain is EQUATION OF CONTINUITY. 5 The amount of fluid within the element at time t is pSxSySz, and therefore the amount at time t + St is p + -n &t) SxSySz. at / The gain is therefore dp Equating this to (4) we obtain the equation dp d (pu) d (pv) d (pw) _ ~77 "I 7 I 7 ~\ 7 \J ( ). at ax ay az This equation is called the equation of continuity. In the case of a liquid, p is constant, and (5) takes the simple form du dv dw y + -y 'f -5- U (O> dx ay az We shall hereafter require the equation of continuity of a liquid referred to polar coordinates. This may be obtained in a similar manner by considering a polar element of volume r 2 sin O&r&d&Q), and it can be shown that if u, v, w be the velocities in the directions in which r, 0, &> increase, the required equation is d(vsin0) dw _ /( _, + r \j/ + r ^~ = Q ( 7 > - - ^- dr dd da If or, 6, z be cylindrical coordinates, the equation is d(tsm) dv dw_ j h -Tn + & -y- = diff dd dz The Velocity Potential. 7. In a large and important class of problems, the quantity ,.t.n udx + vdy + wdz is a perfect differential of a function of x, y, z ( which we shall call

, dd> dd> dd> whence u = -r -, v = ^r~, w= , (9). dx ay dz Substituting these values of u, v, w in (6) we obtain 5"^ "I" T ~^ T """" ^ (-l-v)j or as it is usually written 6 EQUATIONS OF MOTION OF A PERFECT FLUID. This equation is called Laplace's equation, from the name of its discoverer ; it is a very important equation, which continually occurs in a variety of branches of physics. The operator V 2 is called Laplace's operator. We can now obtain the transformation of Laplace's equation when polar coordinates are employed. For in this case udr + vrdQ + wr sin Odw = d, d(f) 1 d(f> 1 d whence u = -~ , t = --j2, w = = a -r- ............ (11). dr r ad r sin dw Substituting in (7) we obtain 1 d ( . d$ d ( 2 d\ dr\ T dr) __ -"- (l The equation of continuity and the theory of the velocity potential may therefore be employed to effect transformations, which it would be very laborious to work out by the usual methods for the change of the independent variables. 8. The existence of a velocity potential involves the conditions that each of the three quantities dw dv du dw dv du dy dz ' dz dx' dx dy should be zero ; when such is not the case we shall denote these quantities by 2, 2??, 2 The quantities , rj, for reasons which will be explained hereafter, are called components of molecular rotation, they evidently satisfy the equation d% .dy d ~J -- r ~l -- r 1~ " dx dy dz When a velocity potential exists, the motion is called irrota- tional ; and when a velocity potential does not exist, the motion is called rotational or vortex motion. Lines of Flow and Stream Lines. 9. DEF. A line of flow is a line whose direction coincides with the direction of the resultant velocity of the fluid. The differential equations of a line of flow are dx _ dy _ dz u v w' LINES OF FLOW AND STREAM LINES. 7 Hence if Xi( x > y> z > ^) =ct i' X?( x > U> z > = a 2 be any two independent integrals, the equations % = const., ^ = const., are the equations of two families of surfaces whose intersections determine the lines of flow. DEF. A stream line or a line of motion, is a line whose direction coincides with the direction of the actual paths of the elements of fluid. The equations of a stream line are determined by the simul- taneous differential equations, x = u, y = v, z = w, where ae, y, z must be regarded as unknown functions of t. The integration of these equations will determine x, y, z in terms of the initial coordinates and the time. 10. When a velocity potential exists, the equation udx + vdy + wdz = is the equation of a family of surfaces, at every point of which the velocity potential has a definite constant value, and which may be called surfaces of equi-velocity potential. If P be any point on the surface, = const., and dn be an element of the normal at P which meets the neighbouring surface (/> + S at Q, the velocity at P along PQ, will be equal to d/dn ; hence d must be positive, and therefore a fluid always flows from places of lower to places of higher velocity potential. The lines of flow evidently cut the surfaces of equi-velocity potential at right angles. 11. The solution of hydrodynamical problems is much sim- plified by the use of the velocity potential (whenever one exists), since it enables us to express the velocities in terms of a single function . But when a velocity potential does not exist, this cannot in general be done, unless the motion either takes place in two dimensions, or is symmetrical with respect to an axis. In the case of a liquid, if the motion takes place in planes parallel to the plane of xy, the equation of the lines of flow is udy vdx = (13). The equation of continuity is du dv _ dx dy 8 EQUATIONS OF MOTION OF A PERFECT FLUID. which shows that the left-hand side of (13) is a perfect differ- ential cfyr, whence The function -|r is called Earnshaw's current function. When the motion takes place in planes passing through the axis of 2, the equation of the lines of flow may be written TO- (wd-sr udz) = ..................... (15), where ty, z are cylindrical coordinates. By (8) the equation of continuity is d(vu) dw _ ~~~j -- > OT ~T~ = "j ttCT (LZ which shows that the left-hand side of (15) is a perfect differential dty, whence 1 . (27 ). dt \pj pdy pdy p dy < d / \ _ f du 77 dv dw dt \pJ p dz p dz p dz These equations may also be written in the form 9 /\ du 77 du t du , r7-=--j- + -j-+-- r -,&c. &c. dt \p/ p dx p dy p dz 21. It was stated in 7, that in many important problems, the motion is such that a velocity potential exists. The con- dition that such should be the case is, that , 77, should each vanish. We shall now prove that, when the fluid is under the action of a conservative system of forces, a velocity potential will always exist whenever it exists at any .particular instant. Let us choose the particular instant at which a velocity potential exists, as the origin of the time ; then by hypothesis |, 77, vanish when t = ; also the coefficients of these quantities in (27), will not become infinite at any point of the interior of the fluid; it will therefore be possible to determine a quantity L, which shall be a superior limit to the numerical values of these coefficients. Hence f, 77, cannot increase faster than if they satisfied the equation But if + 77 + = lp, we obtain by adding the above equations dt whence fl = Ae? Lt . Now H = when t = 0, therefore A = ; and since O is the sum of three quantities each of which is essentially positive, it follows that , 77, must always remain zero, if they are so at any particular instant. The above proof is due to Sir G. Stokes 1 . 1 "On the friction of fluids in motion," Section II, Trans. (Jamb, Phil. Soc. vol, VHI. MOLECULAR ROTATION. 17 22. There is, as was first shown by Sir G. Stokes, an important physical distinction in the character of the motion which takes place, according as a velocity potential does or does not exist. Conceive an indefinitely small spherical element of a fluid in motion to become suddenly solidified, and the fluid about it to be suddenly destroyed. By the instantaneous solidification, velocities will be suddenly generated or destroyed in the different portions of the element, and a set of mutual impulsive forces will be called into action. Let x, y, z be the coordinates of the centre of inertia G of the element at the instant of solidification, x + x', y + y', z + z those of any other point P in it ; let u, v, w be the velocities of G along the three axes just before solidification, u', v', w' the velocities of P relative to G] also let u, v, w be the velocities of G, u 1} v ly w^ the relative velocities of P, and f, 77, the angular velocities just after solidification. Since all the impulsive forces are internal, we have u u, v = v, w = w. We have also by the Principle of Conservation of Angular Momentum, 2m \y' (w l - w') - z' (v^ - v')} = 0, &c. m denoting an element of the mass of the element considered. But MJ = rjz' y', and u' is ultimately equal to du , du t du , and similar expressions hold good for the other quantities. Sub- stituting in the above equation, and observing that r/ = 0, and 2m#' 2 = 2m?/' 2 = 2m/ 2 , fdw ( dz; , <. T we have c= * * 2 \ We see then that an indefinitely small spherical element of the fluid, if suddenly solidified and detached from the rest of the fluid, will begin to move with a motion of translation alone, or a motion of translation combined with one of rotation, according as udx + vdy + wdz is, or is not, an exact differential, and in the latter case the angular velocities will be determined by the equations ,. _ dw _ cfo _ du _ dw 9 .. _ dp _ du dy dz' dz dx' dx dy' B. H. 18 EQUATIONS OF MOTION OF A PERFECT FLUID. On account of the physical meaning of the quantities f, 77, , they are called the components of molecular rotation, and motion which is such that they do not vanish is called rotational or vortex motion; when they vanish, the motion is called irrotational. In the foregoing investigation, it has been assumed that the pressure is a function of the density, and also that the fluid is under the action of a conservative system of forces ; it therefore follows that vortex motion cannot be produced and, if once set up, cannot be destroyed by such a system of forces. It can however be shown that the theorem is not true if the pressure is not a function of the density. If therefore by reason of any chemical action, the pressure should cease to be a function of the density during any interval of time however short, vortex motion might be produced, or if in existence might be destroyed. 23. The equations of motion can be integrated whenever a force and a velocity potential exist; for putting = _p:- F, J P and multiplying (22) by dx, dy, dz respectively and adding, we obtain ,~ du , . dv , dw , Now in the present case du du du dv dw ^i = -n+u-j- + v- r +w- r - ot at dx ax dx where q* = F(t) (28), where F is an arbitrary function. 24. When the motion of a liquid is steady, du/dt, dv/dt and dw/dt are each zero, and in this case the general equations of motion can always be integrated. It will however be necessary to distinguish between irrotational and rotational motion. STEADY MOTION. 19 In the former case d/dt = and F (t) is constant ; whence a first integral is p/p+V+1>q*=C ..................... (29). This result, from the name of its discoverer, is called Bernoulli's Theorem. When the motion is rotational, let and the general equations of motion may be written du _ dR} dt das dv dw ,. dR .(30). (i) Let the motion be steady and in two dimensions; then w = % = r) = 0, and none of the quantities are functions of z or t ; whence substituting the values of u and v from (14) in terms of i/r we obtain ^? = _ 2 ^ r ^? = _2^ da! dx ' dy dy ' which shows that 2=F' (-\/r), where jPis an arbitrary function; and therefore .................. (31). P From these results it follows that the sum of the first three terms is constant along a stream line, but varies as we pass from one stream line to another; also that the molecular rotation is constant along a stream line. (ii) In the same way it can be shown that when the motion is symmetrical with respect to an axis, .................. (32), where CD is the molecular rotation, and i/r is Stokes' current function. (iii) In the general case of the steady motion of a liquid, multiply (30) by u, v, w and add and we obtain dR dR dR _ /00 . .................. (33) - 22 20 EQUATIONS OF MOTION OF A PERFECT FLUID. Multiply by , rj, % and add, and we obtain These equations show that R is constant along a stream line and a vortex line ; whence it is possible to draw a family of surfaces each of which is covered with a network of stream lines and vortex lines. Let \ = const, be such a surface ; then since X contains stream lines and vortex lines, it follows that u\ x + v\ y + w\ z = 0, where \ x = d\/dx, &c. These equations show that if we write R= F(\) equations (33) and (34) will be satisfied; whence a first integral of the equations of motion is f where \ is a surface which contains both stream lines and vortex lines. Impulsive Motion. 25. The equations which determine the change of motion when a fluid is acted upon by impulsive forces, may be deduced in manner similar to that employed in 16. Let u, v, w and u', v', w' be the velocities of the fluid just before and just after the impulse ; p the impulsive pressure. Since impulsive forces are equal to the change of momentum which they produce, it follows by considering the motion of a small parallelepiped 8x8y8z, that p (u' u) 8x8y8z = p8y8z ( p + --f- 8x \ 8y8z, whence the equations of impulsive motion are f>(u '- u} = -f x P( ' ~dy .(36). FLOW AND CIRCULATION. 21 Multiplying by das, dy, dz and adding we obtain - dp/p = (u r - u) dx + (v f - v) dy + (w 1 - w) dz (37). In the case of a liquid p is constant, whence differentiating with respect to x, y, z, adding and taking account of the equation of continuity, we obtain V 2 p = (38). If the liquid were originally at rest, it is clear that the motion produced by the impulse must be irrotational, whence if $ be its velocity potential P = -P* (39), which is a very important result. Flow and Circulation. 26. The line integral f(udx + vdy+wdz), taken along any curve joining a fixed point A with a variable point P, is called the flow from A to P. If the points A and P coincide, so that the curve along which the integration takes place is a closed curve, this line integral is called the circulation round the closed curve. If the motion of a liquid is irrotational, and (f> A , P A , and is independent of the path from A to P; also the circulation round any closed curve is zero, provided (f) be a single-valued function. Cases however occur in which is a many-valued function ; and when this is the case, the value of the circulation will depend upon the position of the closed curve round which the integration is taken, being zero for some curves, whilst for others it has a finite value. For example, when the motion is in two dimensions, < satisfies the equation ttljS-o da?^ dy* and it can be verified by trial, that a particular solution of this equation is = m tan" 1 y/as. This value of $ therefore gives a possible kind of irrotational motion. Let 6 be the least value of the angle tan" 1 y/x; then since 22 EQUATIONS OF MOTION OF A PERFECT FLUID. the equation 6 = tan" 1 y\x is satisfied by 9 4- 2w7r, where n is any positive or negative integer, it follows that the most general value of is at any other point P is a function of the distance alone of P from the origin ; and Laplace's equation becomes ^ + ?^ =0 . dr 2 r dr Therefore m and j =-5' dr r 2 The origin is therefore a singular point, from or to which the stream lines diverge or converge, according as m is positive or negative. In the former case the singular point is called a source, in the latter case a sink. The flux across any closed surface surrounding the origin is, where dfl is the solid angle subtended by dS at the origin, and e is the angle which the direction of motion makes with the normal to 8 drawn outwards. The constant m is called the strength of the source. 24 EQUATIONS OF MOTION OF A PERFECT FLUID. 29. A doublet is formed by the coalescence of an equal source and sink. To find its velocity potential, let there P be a source and sink at S and H respectively, and let be the middle point of SH, then H S m m + HP _mSHcosSOP OP 2 Now let SH diminish and m increase inde- finitely, but so that the product m . SH remains finite and equal to /*, then /* cos SOP flZ if the axis of z coincides with OS. Hence the velocity potential due to a doublet, is equal to the magnetic potential of a small magnet whose axis coincides with the axis of the doublet, and whose negative pole corresponds to the source end of the doublet. 30. When the motion is in two dimensions, and is sym- metrical with respect to the axis of z, Laplace's equation becomes Therefore ^ T* I _ ""r r\ dr* r dr = m log r, d __ m dr~r' where r is the distance of any point from the axis. This value of = m log SP - m log HP SH u, cos SOP = m 75 cos SOP = - OP r Theory of Images. 32. Let H lt H 2 be any two hydrodynamical systems situated in an infinite liquid. Since the lines of flow either form closed curves or have their extremities in the singular points or bound- aries of the liquid, it will be possible to draw a surface S, which is not cut by any of the lines of flow, and over which there is therefore no flux, such that the two systems H l} H 2 are completely shut off from one another. The surface S may be either a closed surface such as an ellipsoid, or an infinite surface such as a paraboloid. If therefore we remove one of the systems (say H 2 ) and substitute for it such a surface as S, everything will remain unaltered on the side of S on which H 1 is situated ; hence the velocity of the liquid due to the combined effect of HI and H 2 will be the same as the velocity due to the system H l in a liquid which is bounded by the surface S. The system H 2 is called the image of H^ with respect to the surface S, and is such that if _fiT 2 were introduced and S removed, there would be no flux across S. The method of images was invented by Lord Kelvin, and has been developed by Helmholtz, Maxwell and other writers; it affords a powerful method of solving many important physical problems. 33. We shall now give some examples. Let S, S' be two sources whose strengths are m. Through A the middle point of SS' draw a plane at right angles to SS'. The normal component of the velocity of the liquid at any point P on this plane is m SI* m cos PSA + cos P8'A = 0. o r* 26 EQUATIONS OF MOTION OF A PERFECT FLUID. Hence there is no flux across AP. If therefore Q be any point on the right-hand side of AP, the velocity potential due to a source at S, in a liquid which is bounded by the fixed plane AP, is mm (j)= ~sQ"srQ' Hence the image of a source 8 with respect to a plane is an equal source, situated at a point S' on the other side of the plane, whose distance from it is equal to that of S. 34. The image in a sphere, of a doublet whose axis passes through the centre of the sphere, can also be found by elementary methods. Let 8 be the doublet, the centre of the sphere, a its radius, and let OS =/. The velocity potential of a doublet situated at the origin and whose axis coincides with OS, has already been shown to be racos# 9= ^ 5 whence if R, be the radial and transversal velocities -n _d _2mcos# "> == ~T n > dr r 3 _ 1 d(f> _ m sin 6 r dd r 3 Hence if we have a doublet at S, the component velocity along OP is cos OSP cos OPS - ^ sin OSP sin OPS cos OSP cos OPS + cos (OPS - OSP)} (41). Let us take a point H inside the sphere such that OH = a 2 //; IMAGE OF A DOUBLET IN A SPHERE. 27 then it is known from geometry that the triangles OPH and OSP are similar, and therefore the preceding expression may be written - - (cos OPH cos OHP + cos SPH}. But the normal velocity due to a doublet of strength m' placed at H is by (41) {cos OPH cos OHP + cos SPH}, and therefore the normal velocity will be zero if m/ for all positions of P. But by a well-known theorem, _ SP~ HP' and therefore the condition that the normal velocity should vanish, is that m' = ma 3 // 3 . Whence the image of a doublet of strength m in a liquid bounded by a sphere is another doublet placed at the inverse point H, whose strength is ma 3 // 3 . The theory of sources, sinks and doublets furnishes a powerful method of solving certain problems relating to the motion of solid bodies in a liquid 1 . We shall conclude this chapter by working out some examples. 35. A mass of liquid whose external surface is a sphere of radius a, and which is subject to a constant pressure II, surrounds a solid sphere of radius b. The solid sphere is annihilated, it is required to determine the motion of the liquid. It is evident that the only possible motion which can take 1 If a magnetic system be suddenly introduced into the neighbourhood of a conducting spherical shell, it can be shown that the effect of the induced currents at points outside the shell, is initially equivalent to a magnetic system inside the shell, which is the hydrodynamical image of the external system ; and that the law of decay of the currents, is obtained by supposing the radius of the shell to diminish according to the law ae~ a , where a- is the specific resistance of the shell. Analogous results hold good in the case of a plane current sheet ; hence all results concerning hydrodynamical images in spheres and planes are capable of an electro- magnetic interpretation. See C. Niven, Phil. Trans. 1881. 28 EQUATIONS OF MOTION OF A PERFECT FLUID. place, is one in which each element of liquid moves towards the centre, whence the free surfaces will remain spherical. Let R', R be their external and internal radii at any subsequent time, r the distance of any point of the liquid from the centre. The equation of continuity is whence r 2 v = F(t). The equation for the pressure is 1 dp _ dv dv p dr dt dr F'(t) dv* r 2 2 dr . T7II /\ whence = A + - } - Aw 2 . p r When r = R', p = II ; and when r = R, p = ; whence if V, V be the velocities of the internal and external surfaces Since the volume of the liquid is constant, also whence (R 3 + c 3 ) Putting z = R*V 2 , multiplying by 2R 2 and integrating, we obtain )* ^1 ' which determines the velocity of the inner surface. If the liquid had extended to infinity, we must put c = oo , and we obtain TORRICELLl's THEOREM. 29 whence if t be the time of filling up the cavity = V2nJo VF^Tl 3 ' Putting b 3 x = R 3 , this becomes The preceding example may be solved at once by the Principle of Energy. The kinetic energy of the liquid is S ' ' Ur -- - il ( R (R 3 + c 3 )^ The work done by the external pressure is T whence |H (b* - R 3 ) = V*R*p 4-n-n r*dr = fH-Tr (a 3 - (R 3 36. The determination of the motion of a liquid in a vessel of any given shape is one of great difficulty, and the solution has been effected in only a comparatively few number of cases. If, however, liquid is allowed to flow out of a vessel, the inclinations of whose sides to the vertical are small, an approximate solution may be obtained by neglecting the horizontal velocity of the liquid. This method of dealing with the problem is called the hypothesis of parallel sections. Let us suppose that the vessel is kept full, and the liquid is allowed to escape by a small orifice at P. Let h be the distance of P below the free surface, and z that of any element of liquid. Since the motion is steady, the equation for the pressure will be 30 EQUATIONS OF MOTION OF A PERFECT FLUID. Now if the orifice be small in comparison with the area of the top of the vessel, the velocity at the free surface will be so small that it may be neglected ; hence if II be the atmospheric pressure, when z = 0, p = Tl,v = and therefore C = II/p. At the orifice p = II, z = h, whence the velocity of efflux is and is therefore the same as that acquired by a body falling from rest, through a height equal to the depth of the orifice below the upper surface of the liquid. This result is called Torricelli's Theorem. The Vena Contracta. 37. When a jet of liquid escapes from a small hole in the bottom of a cistern, it is found that the area of the jet is less than the area of the hole ; so that if a- be the area of the hole and a' that of the jet, the ratio }. If A be the sectional area of the jet at the vena contracta, the quantity of fluid which escapes per unit of time is The momentum per unit of time is Apq* = 2A(p- H). The energy per unit of time is In Giffard's Injector, a jet of steam issuing by a pipe from the upper part of the boiler, is directed at an equal pipe leading back into the lower part of the boiler, the jet being kept constantly just surrounded with water. Now if we assume that the velocity of the steam jet is equal to the velocity at which the water flows into the pipe leading to the lower part of the boiler, which must be very nearly true ; it follows from the preceding equations that velocity of steam jet _ /p velocity of water jet V " ' . quantity of steam jet _ /, w! the angular velocities of the element in latitude and longitude respectively. 4. Fluid is moving in a fine tube of variable section K, prove that the equation of continuity is where v is the velocity at the point s. 5. If F (x, y, z, f) is the equation of a moving surface, the velocity of the surface normal to itself is - ~ d f where R 2 = (dFfdx) 2 + (dF/dy) 2 + (dF/dz)*. -ft Ctt 6. If x, y and z are given functions of a, b, c and t, where a, b and c are constants for any particular element of fluid, and if u, v and w are the values of x, y, z when a, 6, c are eliminated, prove analytically that d 2 x _ du du du du dt 2 dt dx dy dz' 7. If the lines of flow of a fluid lie on the surfaces of coaxial cones having the same vertex, prove that the equation of con- tinuity is r dt + T dr ( Up ) + 2pU + COS6C e dd> ^ = ' 8. Show that x 2 /(akt 2 ) 2 + Jet 2 {(y/b) 2 + 0/c) 2 } = 1 is a possible form of the bounding surface at time t of a liquid. 9. A fine tube whose section k is a function of its length s, in the form of a closed plane curve of area A filled with ice, is moved in any manner. When the component angular velocity of the tube about a normal to its plane is O, the ice melts without change of volume. Prove that the velocity of the liquid relatively B. H. 3 34 EQUATIONS OF MOTION OF A PERFECT FLUID. to the tube at a point where the section is K, at any subsequent time when co is the angular velocity is where 1/c = jk~^ds t the integral being taken once round the tube. 10. A centre of force attracting inversely as the square of the distance, is at the centre of a spherical cavity within an infinite mass of liquid, the pressure on which at an infinite distance is -BT, and is such that the work done by this pressure on a unit of area through a unit of length, is one half the work done by the attrac- tive force on a unit of volume of the liquid from infinity to the initial boundary of the cavity ; prove that the time of filling up the cavity will be '3\*1 a being the initial radius of the cavity, and p the density of the liquid. 11. A solid sphere of radius a is surrounded by a mass of liquid whose volume is 4?rc 3 /3, and its centre is a centre of attrac- tive force varying directly as the square of the distance. If the solid sphere be suddenly annihilated, show that the velocity of the inner surface when its radius is x, is given by &a? {(a* + c 3 )' - x\ = ( ~ + | ytic 3 ) (a 3 - a?) (c 3 + orf, V op u / where p is the density, II the external pressure and //, the absolute, force. 12. Prove that if or be the impulsive pressure, , ' the velocity potentials immediately before and after an impulse acts, V the potential of the impulses, w + pV+ p ($' ) = const. 13. If the motion of a homogeneous liquid be given by a single valued velocity potential, prove that the angular momentum of any spherical portion of the liquid about its centre is always zero. 14. Homogeneous liquid is moving so that , w= 0, EXAMPLES. 35 and a long cylindrical portion whose section is small, and whose axis is parallel to the axis of z, is solidified and the rest of the liquid destroyed. Prove that the initial angular velocity of the cylinder is Bft-Aa-ZFy A+B where A, B, F are the moments and products of inertia of the section of the cylinder about the axes. 15. Fluid is contained within a sphere of small radius ; prove that the momentum of the mass in the direction of the axis of x is greater than it would be if the whole were moving with the velocity at the centre by Ma? -R- Op where p x = dp/da; &c. 16. The motion of a liquid is in two dimensions, and there is a constant source at one point A in the liquid and an equal sink at another point B ; find the form of the stream lines, and prove that the velocity at a point P varies as (AP . -BP)" 1 , the plane of the motion being unlimited. If the liquid is bounded by the planes a; = 0, a; = a, y = 0, y = a, and if the source is at the point (0, a) and the sink at (a, 0), find an expression for the velocity potential. 17. The boundary of a liquid consists of an infinite plane having a hemispherical boss, whose radius is a and centre 0. A doublet of unit strength is situated at a point 8, whose axis coincides with OS, where OS is perpendicular to the plane. P is any point on the plane, OP = y, OS=f. Prove that the velocity of the liquid at P is 18. Prove that =f(t) {(r 2 + a 2 - 2as)- + (r 2 + a 2 + 2as)~* - r- 1 } + ^ (t) is the velocity potential of a liquid, and interpret it. Find the surfaces of equal pressure if gravity in the negative direction of the axis of z be the only force acting. ^ 9 O - 36 EQUATIONS OF MOTION OF A PERFECT FLUID. 19. Liquid enters a right circular cylindrical vessel by a supply pipe at the centre 0, and escapes by a pipe at a point A in the circumference ; show that the velocity at any point P is proportional to PBfPA . PO, where B is the other end of the diameter AO. The vessel is supposed so shallow that the motion is in two dimensions. CHAPTER II. MOTION OF CYLINDERS AND SPHERES IN AN INFINITE LIQUID. 39. THE present chapter will be devoted to the consideration of certain problems of two-dimensional motion, and we shall also discuss the motion of a sphere in an infinite liquid. If a right circular cylinder is moving in a liquid, the pressure of the liquid at any point of the cylinder passes through its axis, and therefore the resultant pressure of the liquid on the cylinder reduces to a single force, which can be calculated as soon as the pressure has been determined. Now the pressure at any point of the liquid is found by means of the equation and therefore p can be determined as soon as the velocity potential is known. Hence the first step towards the solution of problems of this character is to find the velocity potential. If the cylinder is not circular, the resultant pressure of the liquid upon its surface will usually be reducible to a single force and a couple, and the problem becomes more complicated. The motion of cylinders, which are not circular, can be most con- veniently treated by means of the dynamical methods explained in the next chapter. In the present chapter we shall show how to find the motion of an infinite liquid, in which cylinders of certain given forms are moving, and we shall also work out the solution of certain special problems relating to the motion of circular cylinders and spheres. 38 MOTION OF CYLINDERS AND SPHERES. 40. If the liquid be at rest, and a cylinder of any given form be set in motion in any manner, the subsequent motion of the liquid will be irrotational and acyclic, and is therefore completely determined by means of a velocity potential. It is however more convenient to employ Earnshaw's current function i/r. This function, when the motion is irrotational, satisfies the equation _ da? df at all points of the liquid. The integral 1 of this equation is TJr = f(a>+iy) + F(a:-iy) .................. (2), also u = ~, v = -- ....................... (3). dy dx We must now consider the boundary conditions to be satisfied by ty. If the liquid is at rest at infinity (which will usually be the case), d-^rjdx and d-fr/dy must vanish at infinity. If any portions of the boundary consist of fixed surfaces, the normal component of the velocity must vanish at such fixed boundaries, and there- fore the fixed boundaries must coincide with a stream line. This requires that ty = const, at all points of fixed boundaries. When the cylindrical boundary is in motion, the component velocity of the liquid along the normal, must be equal to the component velocity of the cylinder in the same direction. (i) Let the cylinder be moving with velocity U parallel to the axis of a, and let 6 be the angle which the normal to the cylinder makes with this axis ; then at the surface u cos + v sin 6 = Ucos 9. Now cos 6 = dyjds ; sin 6 = dxjds ; therefore by (3) d =v -dy ds ds' Integrating along the boundary, we obtain where A is a constant. 1 The easiest way of showing that (2) is a solution of (1), is to differentiate the right-hand side of (2) twice with respect to x, and twice with respect to y and add. Since the result is zero, this shows that (2) satisfies (1) ; also since (2) contains two arbitrary functions, it is the most general solution that can be obtained. BOUNDARY CONDITIONS. 39 (ii) If the cylinder be moving with velocity V parallel to the axis of y, it can be shown in the same manner that the surface condition is ^ = -Vx + B ......................... ,.(5). (iii) Let the cylinder be rotating with angular velocity w ; then at the surface u cos + v sin 9 = wy cos 6 + wx sin 6, d-dr dr or -~ = u>r -7- , as as Therefore ^r = -^wr z + C ........................ (6), where r = (a? + y 2 )*. When there are any number of moving cylinders in the liquid, conditions (4), (5) and (6) must be satisfied at the surfaces of each of the moving cylinders. In addition to the surface conditions, -ty* must satisfy the following conditions at every point of space occupied by the liquid ; viz. ty must be a function which is a solution of Laplace's equation (1), and which together with its first derivatives must be finite and continuous at every point of the liquid. If we take any solution of (1), and substitute its value in (4), (5) or (6), we shall in many cases be able to determine the current function due to the motion of a cylinder, whose cross section is some curve, in one of the three prescribed manners. In most of the applications which follow -\Jr will be of the form - t y) .................. (7), From these equations we see that 0+^=2t/(a; + y) ..................... (9), and therefore when ty is known, can be found by equating the real and imaginary parts of (9). 40 MOTION OF CYLINDERS AND SPHERES. Motion of a Circular Cylinder. 41. Let Transforming to polar coordinates, and using De Moivre's theorem, we obtain Tjr=- Fa 2 #/r 2 ........................ (10). When r = a, ty = Vx ; equation (10) consequently deter- mines the current function, when a circular cylinder of radius a is moving parallel to the axis of y, in an infinite liquid with velocity V. By (9) the velocity potential is < = - Vtfyfr* ........................ (11). 42. Let us now suppose that the cylinder is of finite length unity, and that the liquid is bounded by two vertical parallel planes, which are perpendicular to the axis of the cylinder. In order to find the motion when the cylinder is descending vertically under the action of gravity, let /3 be the distance of the axis of the cylinder at time t from some fixed point in its line of motion which we shall choose as the origin, and let (x, y) be the coordinates of any point of the liquid referred to the fixed origin, the axis of y being measured vertically downwards; also let (r, 6) be polar coordinates of the same point referred to the axis of the cylinder as origin. By (11) Fa 2 . Fa 2 -3 and therefore since dftjdt = V, tfV . , a 2 F 2 2a 2 F 2 . d> = --- sin0+ ---- s n* ff* rv& and therefore at the surface, where r = a, Also fl*"" I j ) + I - ;> J \drj \r dvj therefore when r = a, q* = F 2 . CIRCULAR CYLINDER. 41 Whence pjp = a Fsin - F 2 cos 20 - F 2 + g (/3 + a sin 0) + C. . .(12). The horizontal component of the pressure is evidently zero ; the vertical component is Y=-a ( painOdG. J o Substituting the value of p from (12) and integrating, we obtain F = - Trpa?(V+g). Hence if cr be the density of the cylinder, the equation of motion is Tro-a 2 V = F + no-go?, or (*r+p)F = (cr-p)flr (13). Integrating this equation, we obtain Tr (o- - p) at F=t> + v /'* , ( + />) where u is the initial velocity measured vertically downwards. We therefore see that the cylinder will move in a vertical straight line, with a constant acceleration which is equal to g( Assuming F=r m , the equation reduces to ra 2 - w 2 = 0, whence m = n, and therefore the required solution is = (Ar n + Br~ n ) e in ..................... (15). In this solution n may have any value whatever, and the real and imaginary parts of the above expression will be independent solutions of (14). Let us now suppose that the radius of the outer cylinder, which is supposed to be fixed, is c; and let the inner one be started with velocity U. Since the velocity of the liquid at the surface of the outer cylinder must be wholly tangential, the boundary condition is i=0, when r = c ., ................. (16). dr At the surface of the inner cylinder, which is moving with velocity U, the component velocities of the cylinder and liquid along the radius must be equal ; whence the boundary condition at the inner cylinder is -j?=Ucos&, when r = a ............... (17), dr being measured from the direction of U. If in (15) we put n = 1, the function = (Ar + Blr)cos0 ..................... (18) 1 This transformation can be most easily effected, by forming the equation of continuity in polar coordinates. LEMNISCATE OF BERNOULLI. 43 is a solution of Laplace's equation ; if therefore we can determine A and B so as to satisfy (16) and (17), the problem will be solved. Substituting from (18) in (16) we obtain Ac*-B = 0. Substituting in (17) we obtain Aa 2 -B= Ua\ Solving and substituting in (18) we obtain Ua* ( , c 2 \ 9 = - Y , r + - cos 0. c 2 a 2 V r) If we put c= oo , we fall back on our previous result of a cylinder moving in an infinite liquid. We can now determine the impulsive force which must be applied to the inner cylinder, in order to start it with velocity U. By 25, equation (39), it follows that if a liquid which is at rest be set in motion by means of an impulse, and be the velocity potential of the initial motion, the impulsive pressure at any point of the liquid is equal to p. Hence if M be the mass of the cylinder, F the impulse, the equation of motion is = F+pal (bcosddO Jo ' UTrpa 2 (c 2 + a 2 ) = * J b , r 2 - whence since M = The Lemniscate of Bernoulli. 44. The lemniscate of Bernoulli is a bicircular quartic curve whose equation in Cartesian coordinates is (# 2 + y 2 ) 2 = 2c 2 (a? y z ), or in polar coordinates r 2 = 2c 2 cos 20. In order to find the current function when a cylinder, whose cross section is this curve, is moving parallel to x in an infinite liquid, let us put u = x + ty, v = x ly, and assume 44 MOTION OF CYLINDERS AND SPHERES. Now w 2 -c 2 = r 2 (cos20 + isin20)-c 2 ; whence at the surface where r 2 = 2c 2 cos 20, the right-hand side becomes 2C 2 cos 2 26 + ic* sin 40 - c 2 = c 2 (cos 20 + t sin 20) 2 ; u r (cos 4- fc sin 0) whence - - -r = -7^ ^- -- . ' = r (cos - t sm 0)/c. (w 2 c 2 )* c (cos 20 + i sm 20) Therefore at the surface tyx = Ur sin = E/y. The value of -fy x given by (19), is therefore the current function due to the motion parallel to x with velocity U, of a cylinder whose cross section is a lemniscate of Bernoulli. If we put ^-i it can be shown in a similar manner, that ty y is the current function, when a cylinder of this form is moving parallel to y with velocity V ; and that i/r 3 is the current function, when the cylinder is rotating with angular velocity a> about its axis. If the cross section be the cardioid r = 2c (1 + cos 0), the vajues of ty x and 1/r,, can be obtained by writing (w* c*) 2 , (v* c*) 2 for (u 2 c 2 )*, (y 2 c 2 )* in the preceding formulae ; but the value of ^ 3 can be so simply obtained. See Quart. Jour. vol. xx. p. 246. An Equilateral Triangle. 45. The preceding methods may also be employed, to find the motion of a liquid contained within certain cylindrical cavities, which are rotating about an axis. Let ty = %A {(x + iy) 3 + (x- iy} s ] = A(a?- 3#?/ 2 ) = Ar* cos 30. Substituting in (6), the boundary condition becomes A(a?-3xy 2 ) + $a)(a? + f) = C .............. (20). If we choose the constants so that the straight line x=a, may form part of the boundary, we find . G) 2&>a 2 ELLIPTIC CYLINDER. 45 Hence (20) splits up into the factors (x a) ; x 4- y V3 + 2a ; x y \/3 + 2a. The boundary therefore consists of three straight lines forming an equilateral triangle, whose centre of inertia is the origin. Hence -fy is the current function due to liquid contained in an equilateral prism, which is rotating with angular velocity &> about an axis through the centre of inertia of its cross section. The values of i/r and <, when cleared of imaginaries, are ^ = ^-r3 cos 30 rf> = -^- r 3 sin 30. 6a 6a An Elliptic Cylindrical Cavity. 46. Let i/r = \A {(x + lyf + (x - iy) 2 } = A(x*-f). Substituting in (6) we find ca 2C the equation of the boundary becomes *.. a* + b*~ ' and a>(^-60 2 (a 2 + & 2 ) v i|r is therefore the current function due to the motion of liquid contained in an elliptic cylinder, which is rotating about its axis. Elliptic Cylinder. 47. The problem of finding the motion of an elliptic cylinder in an infinite liquid cannot be solved by such simple methods as the foregoing ; in order to effect the solution we require to employ the method of Conjugate Functions. Def. If % and 77 are functions of x and y, such that + 7=/(* + y) ................... (21), then % and ij are called conjugate functions of x and y. If we differentiate (21) first with respect to x, and afterwards 46 MOTION OF CYLINDERS AND SPHERES. with respect to y, eliminate the arbitrary function, and then equate the real and imaginary parts, we shall obtain the equations dx dy' dy dx" Comparing these equations with (8), we see that < and ty are conjugate functions of # and y. From equations (22) we also see that dx dx dy dy V 2 |=0, V 2 77 = ..................... (24). Equation (23) shows that the curves = const., 77 = const., form an orthogonal system ; and equations (24) show that and 77 each satisfy Laplace's equation. If and ^r are conjugate functions of x and y, and and i) are also conjugate functions of x and y, then and ty are conjugate functions of and 77. For + n|r = F (x + ty), and f + irj =f(x + iy), whence eliminating x + iy, we have From this proposition combined with (24), it follows that if the equation V 2 -\Jr = be transformed by taking and 77 as independent variables (25) 48. We can now find the current function due to the motion of an elliptic cylinder. Let x + iy = c cos ( iij) = c cos cosh 77 + ^0 sin sinh 77, then x = c cos cosh 77} fc . r (26), y = c sin if sinn 77] whence the curves 77 = const., f = const., represent a family of confocal ellipses and hyperbolas, the distance between the foci being 2c. If a and 6 be the semi-axes of the cross section of the elliptic cylinder 77 = /3, then, a = c cosh /8, b = c sinh /3. ELLIPTIC CYLINDER. 47 If /3 is exceedingly large, sinh /3 and cosh /3 both approximate to the value ^ce^ ; and therefore as the ellipse increases in size, it approximates to a circle whose radius is |ce^. It can be verified by trial, that (25) can be satisfied by a series of terms of the form e~ nr > (A n cos n% + B n sin nf) ; and if n be a positive quantity not less than unity, this is the proper form of A|T outside an elliptic cylinder, since it continually diminishes as 77 increases. When the cylinder is moving parallel to its major axis with velocity U, let us assume \Jr a . = J.e~ 1) sin . Substituting in (4) we obtain A~P sin f = Uc sinh /3 sin + G, where 77 = /3 is the equation of the cross section of the cylinder. Since this equation is to be satisfied at every point of the boundary, we must have (7 = 0, A = Uce 1 * sinh /S; whence fa= Uce-v+P sinh ft sin f (27). When the cylinder is moving parallel to its minor axis with velocity V, it may be shown in the same manner that -fy y =- Fce-^ cosh cos (28). Lastly let us suppose that the cylinder is rotating with angular velocity to about its axis. Then a? + y 2 = c* (cos 2 f cosh 2 77 + sin 2 sinh 2 77) = c 2 (cosh 277 + cos 2). Let us therefore assume ^ 3 = Be~*> cos 2f Substituting in (6) we obtain Be-* cos 2 + &>c 2 (cosh 2/3 + cos 2f) = G, whence B = - wtfe, G = J we 2 cosh 2/3, and therefore ^s = - i *>c 2 e- 2 "-*> cos 2f (29). 49. If we suppose that /3 = 0, the ellipse degenerates into a straight line joining the foci, and (28) becomes ty y = _ Fee-" cos (30). It might therefore be supposed that (30) gives the value of the current function, due to a lamina of breadth 2c, which moves with 48 MOTION OF CYLINDERS AND SPHERES. velocity V, perpendicularly to itself. This however is not the case, inasmuch as the velocity at the edges of the lamina becomes infinite, and therefore the solution fails. To prove this, we have dty _ city dx dty dy dtj dx dij dy dij . , ..dty . dty = c smh i} cos -jr- + c cosh 77 sin g -~ , cLco ^y and -=- = c cosh rt sin -~ + c sinh 77 cos -~ . dg dx dy whence squaring and adding, we obtain c 2 (sinh 2 T? cos 2 + cosh 2 77 sin 2 f ) q* = ( ^Y + (^ Y= F 2 c 2 e-^ . . . (31). The coordinates of an edge are x = c, y = ; and therefore in the neighbourhood of an edge 77 and f are very small quantities ; and therefore by (31) the velocity in the neighbourhood of an edge is which becomes infinite at the edge itself, where t] and ff are zero. It therefore follows that the pressure in the neighbourhood of an edge is negative, which is physically impossible. Since the pressure is positive at a sufficient distance from the edge, there will be a surface of zero pressure dividing the regions of positive and negative pressures ; and it might be thought that the interpretation of the formulae would be, that a hollow space exists in the liquid surrounding the edges, which is bounded by a surface of zero pressure. But the condition that a free surface should be a surface of zero (or constant) pressure, although a necessary one, is not sufficient ; it is further necessary, that such a surface should be a surface of no flux, which satisfies the kine- matical condition of a bounding surface 12, equation (17); and it will be found on investigating the question, that no surface exists, which is a surface of zero (or constant) pressure, and at the same time satisfies the conditions of a bounding surface. The solution altogether fails in the case of a lamina. When the velocity of the solid is constant and equal to V, the easiest way of dealing with a problem of this character is to reverse the motion by supposing the solid to be at rest, and that the liquid flows past it, the velocity at infinity being equal to - V. MOTION OF A SPHERE. 49 The correct solution in the case of a lamina has been given by Kirchhoff 1 , and he has shown that behind the lamina there is a region of dead water, i.e. water at rest, which is separated from the remainder of the liquid by two surfaces of discontinuity, which commence at the two edges of the lamina, and proceed to infinity in the direction in which the stream is flowing. Since the liquid on one side of this surface of discontinuity is at rest, its pressure is constant ; and therefore since the motion is steady, the pressure, and therefore the velocity of the moving liquid, must be constant at every point of the surface of discontinuity. It may be added that a surface of discontinuity is an imaginary surface described in the liquid, such that the tangential component of the velocity suddenly changes as we pass from one side of the surface to the other. Motion of a Sphere. 50. The determination of the velocity potential, when a solid body of any given shape is moving in an infinite liquid, is one of great difficulty, and the only problem of the kind which has been completely worked out is that of an ellipsoid, which of course includes a sphere as a particular case. We shall however find it simpler in the case of a sphere to solve the problem directly, which we shall proceed to do. Let the sphere be moving along a straight line with velocity V, and let (r, 0, , hence by 7, equation (12), the equation of continuity is _0 ......... (32). dr 2 r dr r 2 dv 2 r 2 d6 The boundary condition, which expresses the fact that the normal component of the velocity of the liquid at the surface of the sphere is equal to the normal component of the velocity of the sphere itself, is (33). Equation (33) suggests that must be of the form F(r) cos 9 ; we shall therefore try whether we can determine F so as to satisfy 1 See also, Michell, Phil. Trans. 1890, p. 389 ; Love, Proc. Gamb. Phil. Soc. vol. vn. p. 175. B. H. 4 50 MOTION OF CYLINDERS AND SPHERES. (32). Substituting this value of , we find that (82) will be satisfied, provided *rw_sr_ ........... (34) dr 2 r dr r* To solve (34), assume F = r m ; whence on substitution we obtain (m-l)(w + 2)=0; which requires that m = 1 or 2. A particular solution of (32) is therefore Since the liquid is supposed to be at rest at infinity, d(f)/dr = when r = oo , and therefore .4 = 0. To find B, substitute in (33) and put r = a, and we find A = -%Va?, Va 3 cos whence = ----- ^ ..................... (35). This is the expression for the velocity potential due to the motion of a sphere in an infinite liquid. In order to determine the motion, when a sphere is descending vertically under the action of gravity, let 7 be the distance of its centre at time t from some fixed point in its line of motion, which we shall choose as the origin ; let the axis of z be measured vertically downwards and let x, y, z be the coordinates of any point of the liquid referred to the fixed origin. -D /OK\ JL Va?(z-^) *= - -.l' and therefore since 7= V, d_ _ Fa 3 cos0 F 2 a 3 3F 2 a 3 cos 2 ~di~ ^Zr* '~2r*~ 2r 3 and therefore at the surface where r = a, also 2 = and therefore p/p = G + g ( 7 + a cos 0) + % Va cos + $ F 2 (9 cos 2 - 5) . . . (36). MOTION OF A SPHERE. 51 If Z be the force due to the pressure of the liquid, which opposes the motion, f* = 2-Tra 2 p cos 6 sin Odd Jo = |7rpa 3 (^V+g) (37) by (36). If therefore a be the density of the sphere, the equation of motion is f Tro-a 3 F = - f Trpa* (%V + g) + frato-g or (cr + ^-p) F=(o- p) g (38). Hence the sphere descends with vertical acceleration In order to pass to the case in which the sphere is projected with a given velocity and no forces are in action, we must put g = 0, and we see that V= const. = its initial value ; hence the sphere continues to move with its velocity of projection, and the effect of the liquid is to produce an apparent increase in the inertia of the sphere, which is equal to half the mass of the liquid displaced. It also follows that if the sphere be projected in any manner under the action of gravity, it will describe a parabola with vertical acceleration g (a- p)/(a- + ^p). 51. Let us now suppose that the sphere is moving with constant velocity V under the action of no forces. The equation determining the pressure is p z Since d/dz and q vanish at infinity, it follows that C = TI/p, where II is the pressure at infinity, whence p U d$_ p /> dz * q ' and therefore at the surface, P P The right-hand side of this equation will be a minimum when 6 = |TT, in which case it becomes II /p f F 2 . Hence if 42 52 MOTION OF CYLINDERS AND SPHERES. the pressure will become negative within a certain region in the neighbourhood of the equator, and the solution fails. When V exceeds the critical value (8II/op)* it is probable that a region of dead water exists behind the sphere, which is separated from the rest of the liquid by a vortex sheet. 52. In discussing the motion of a cylinder, we found that if the solid were projected in a liquid and no forces were in action, the solid would continue to move in a straight line with its original velocity of projection; and we called attention to the fact that this result was contrary to experience ; and that one reason of this discrepancy between theory and observation arose from the fact that all liquids are more or less viscous, the result of which is that kinetic energy is gradually converted into heat. The motion of viscous fluids is beyond the scope of an elementary work such as the present, but a few remarks on the subject will not be out of place. Let us suppose that fluid is moving in strata parallel to the plane xy, with a variable velocity U, which is parallel to the axis of x. Let U be the velocity of the stratum AB, U + 8U of the stratum CD, and let Bz be the distance between AB and CD. If the fluid were frictionless, the action between the fluid on either side of the plane AB would be a hydrostatic pressure p, whose direction is perpendicular to this plane, and consequently no tangential action or shearing stress could exist ; if however the fluid is viscous, the action between the fluid on either side of the plane AB, usually consists of an oblique pressure (or tension), and may therefore be resolved into a normal component perpen- dicular to the plane, and a tangential component in the plane. The usual theory of viscosity supposes, that if F be the tan- gential stress on AB per unit of area, F+8F the corresponding stress on CD, then the latter stress is proportional to the relative velocity of the two strata divided by the distance between the strata, so that u^su^-u if + d* x - ^ , oz whence proceeding to the limit dU Fee-;-. dz VISCOSITY. 53 We may therefore put F '= /A-T (39), CL% where /i is a constant. The constant /i is called the viscosity ; it is a numerical quantity whose value is different for different fluids, and also depends upon the temperature. The viscosity is a quantity which corresponds to the rigidity in the Theory of Elasticity. If a shearing stress F be applied parallel to the axis of x, and in a plane parallel to the plane xy, to an elastic solid, it is known from the Theory of Elasticity, that F = ndafdz, where a is the displacement parallel to x. Whence the ratio of the shearing stress F, to the shearing strain da/dz produced by it, is equal to a constant n, which is called the rigidity. Now in the hydrodynamical theory of viscous fluids, dll/dz is equal to the rate at which shearing strain is produced by the shearing stress F; hence (39) asserts that the ratio of the shearing stress to the rate at which shearing strain is produced, is equal to a constant /*, which is called the viscosity. If the shearing stress F is applied in the plane z = c, and if U = uz/c, (39) becomes F = fjiu/c (40), where u is the velocity of the fluid in the plane z = c. Hence if u="L, and c = 1, then F=/JL. We may therefore define the viscosity as follows 1 . The viscosity is equal to the tangential force per unit of area, on either of two parallel planes at the unit of distance apart, one of which is fixed, whilst the other moves with the unit of velocity, the space between being filled with the viscous fluid. Equation (40) shows that the dimensions of p are [ ML~ l T~ *]. If we put v = fju/p, where p is the density, the quantity v is called the kinematic coefficient of viscosity. The dimensions of v are [L*T~ 1 ]. The equations of motion of a viscous fluid are known, and the motion of a sphere which is descending under the action of gravity in a slightly viscous liquid, such as water, has been worked out by myself; and I have shown that if the sphere be initially projected downwards with velocity V, its velocity at any subsequent time will 1 Maxwell's Heat, p. 298, fourth edition. 54 MOTION OF CYLINDERS AND SPHERES, be approximately given by the equation where /- about a fixed axis. Prove that the velocity potential with reference to the principal axes of the cylinder is wxy (a? 6 2 )/(a 2 + 6 2 ), and that the surfaces of equal pressure when the angular velocity is constant, are the hyperbolic cylinders 3a 2 + 6 2 36 2 + a 3 5. If =f(x, y), '\lr = F (x, y) are the velocity potential and current function of a liquid, and if we write and from these expressions find and ->/r; prove that the new values of and ty will be the velocity potential and current function of some other motion of a liquid. Hence prove that if = cc* y 2 , -^ = 2xy, the transformation gives the motion of a liquid in the space bounded by two confocal and coaxial parabolic cylinders. 58 MOTION OF CYLINDERS AND SPHERES. 6. In example 4 prove that the paths of the particles relative to the cylinder are similar ellipses, and that the paths in space are similar to the pericycloid x = (a + 6) cos 6 + (a 6) cos ( - ~\ 6, \a-bj . a-bj 7. Water is enclosed in a vessel bounded by the axis of y and the hyperbola 2 (a? 3y 2 ) + x + my = 0, and the vessel is set rotating about the axis of z. Prove that $ = 2 (3x*y - f) + xy - m (a? - f\ fy == 2 (x 3 - 3xy 2 ) + i (^ - 2/ 2 ) + 8. The space between two confocal coaxial elliptic cylinders is filled with liquid which is at rest. Prove that if the outer cylinder be moved with velocity U parallel to the major axis, and the inner with relative velocity V in the same direction, the velocity potential of the initial motion will be , TT , * IT cosn (- 17) , = uc cosh 77 cos tVc- , ; -~ smh a cos t, cosh (p a) where rj = (3, i] = a are the equations of the outer and inner cylinders respectively, and 2c the distance between their foci. 9. If in the last example the outer cylinder were to rotate with angular velocity H, and the inner with angular velocity o>, prove that initially cosh 2 (?;- a) . ,. cosh 2 (/3- 77) . .. d> = lie 2 ^-T-o7Z - \ sm 2 S - f wc i. n)a - 4. sm 2 smh 2 (/3 - a) smh 2 (ft - a) 10. The transverse section of a uniform prismatic vessel is of the form bounded by the two intersecting hyperbolas represented by the equations V2 (x 2 - vy 2 ) + x 2 + y* = a 2 , \/2 (y 2 - x 2 ) + x* + y* = b 2 . If the vessel be filled with water and made to rotate with angular velocity co about its axis, prove that the initial component velocities at any point (x, y) of the water will be ft) respectively. EXAMPLES. 59 11. In the midst of an infinite mass of liquid at rest is a sphere of radius a, which is suddenly strained into a spheroid of small ellipticity. Find the kinetic energy due to the motion of the liquid contained between the given surface, and an imaginary concentric spherical surface of radius c; and show that if this imaginary surface were a real bounding surface which could not be deformed, the kinetic energy in this case would be to that in the former case in the ratio C 5 (3a 5 +2c 6 ) : 2(c 5 -a 5 ) 2 . 12. The space between two coaxial cylinders is filled with liquid, and the outer is surrounded by liquid extending to infinity, the whole being bounded by planes perpendicular to the axis. If the inner cylinder be suddenly moved with given velocity, prove that the velocity of the outer cylinder to that of the inner, will be in the ratio 26 2 c 2 p : p (a 2 6 2 - a 2 c 2 + 6 4 + 6 2 c 2 ) + a- (a 2 - Z> 2 ) (6 2 - c 2 ), where a and 6 are the external and internal radii of the outer cylinder, a- its density, c the radius of the inner cylinder and p the density of the liquid. 13. A solid cylinder of radius a immersed in an infinite liquid, is attached to an axis about which it can turn, whose distance from the axis of the cylinder is c, and oscillates under the action of gravity. Prove that the length of the simple equivalent pendulum is a and p being the densities of the cylinder and liquid. 14. Liquid of density p is contained between two confocal elliptic cylinders and two planes perpendicular to their axes. The lengths of the semi-axes of the inner and outer cylinders are c cosh a, c sinh a, c cosh /3, c sinh ft respectively. Prove that if the outer cylinder be made to rotate about its axis with angular velocity fl, the inner cylinder will begin to rotate with angular velocity lip cosech 2 (/3 a) p coth 2 (/3 - a) + \ and -ty- be any two functions, which throughout the interior of a closed surface 8 are single valued, and which together with their first and second derivatives, are finite and continuous at every point within S ; then j! ay ay az ...... (1) ^^dxdydz ...... (2), where the triple integrals extend throughout the volume of S, and the surface integrals over the surface of 8, and dn denotes an element of the normal to 8 drawn outwards. Integrating the left-hand side by parts, we obtain 7 T ^ dx dx where the brackets denote that the double integral is to be taken between proper limits. Now since the surface is a closed surface, any line parallel to x, which enters the surface a given number of times, must issue from it the same number of times; also the ^-direction cosine of the normal at the point of entrance will be of contrary sign to the same direction cosine at the corresponding point of exit ; hence the surface integral 64 MOTION OF A SINGLE SOLID. Treating each of the other terms in a similar manner, we find that the left-hand side of (3) The second equation (2) is obtained by interchanging and i/r. 59. We may deduce several important corollaries. (i) Let -v/r = 1, and let < be the velocity potential of a liquid; then V 2 (f> = 0, and we obtain d8 ............... (4). The right-hand side is the analytical expression for the fact that the total flux across the closed surface is zero; in other words as much liquid enters the surface as issues from it. (ii) Let and i/r be both velocity potentials, then (iii) Let = ty, where is the velocity potential of a liquid; then >Y fdy} 7 7 7 ff , dd> 7L , ,-, +(j 1 +( j ) \dxdydz** ^-cSf...(6). dee/ \dy) \dz) j JJ Y dn If we multiply both sides of (6) by |p, the left-hand side is equal to the kinetic energy of the liquid ; and the equation shows that the kinetic energy of a liquid whose motion is acyclic and irrotational, which is contained within a closed surface, depends solely upon the motion of the surface. 60. Let us now suppose that liquid contained within such a surface is originally at rest, and let the liquid be set in motion by means of an impulsive pressure p applied to every point of the surface. The motion produced must be necessarily irrotational, and acyclic; also if < be its velocity potential, it follows from 25 (39) that p = p/dn = 0, and therefore whence d(j>/dx, d}dy, and d/dz are each zero, and therefore the liquid is reduced to rest. 62. In proving Green's Theorem, we have supposed that the region through which we integrate, is contained within a single closed surface, but if the region were bounded externally and internally by two or more closed surfaces, the theorem would still be true, provided we take the surface integral with the positive sign over the external boundary, and with the negative sign over each of the internal boundaries. 63. Let us suppose that the liquid is bounded internally by one or more closed surfaces S 1} S 2 &c., and externally by a very large fixed sphere whose centre is the origin. If T be the kinetic energy of the liquid, where the square brackets indicate that the integral is to be taken over each of the internal boundaries. If the liquid be at rest at infinity, the value of at S cannot contain any term of lower order than m/r, where m is a constant, whence dfdn = d/dr = m/r* ; also if dl be the solid angle subtended by dS at the origin, therefore which vanishes when r = oo . Hence the kinetic energy of an infinite liquid bounded internally by closed surfaces is (7 >- where the surface integral is to be taken over each of the internal boundaries. B. H. 5 66 MOTION OF A SINGLE SOLID. The preceding expression for the kinetic energy shows that, if the motion is acyclic and the internal boundaries of the liquid be suddenly reduced to rest, the whole liquid will be reduced to rest. > 64. When a single solid is moving in an infinite liquid, the velocity potential must satisfy the following conditions ; (i) must be a single valued function, which at all points of the liquid satisfies the equation V 2 < = 0. (ii) < and its first derivatives must be finite and continuous at all points of the liquid, and must vanish at infinity, if any portion of the liquid extends to infinity. (iii) At all points of the liquid which are in contact with a moving solid, d/dn must be equal to the normal velocity of the solid, where dn is an element of the normal to the solid drawn outwards ; if any portion of the liquid is in contact with fixed boundaries, d(j>{dn must be zero at every point of these fixed boundaries. The most general possible motion of a solid may be resolved into three component velocities parallel to three rectangular axes (which may either be fixed or in motion), together with three angular velocities about these axes. Let us therefore refer the motion to three rectangular axes Ox, Oy, Oz fixed in the solid, and let fa be the velocity potential when the solid is moving with unit velocity parallel to Ox, and let ^ be the velocity potential when the solid is rotating with unit angular velocity about Ox. Let fa, fa, %a> %s be similar quantities with respect to Oy and Oz. Also let u, v, w be the linear velocities of the solid parallel to, and ca lf o> 2 , 6> 3 be its angular velocities about the axes. We can now show that the velocity potential of the whole motion will be = Ufa + Vfa + Wfa + !%! + G> 2 X 2 + 6)3^3 (8). For if \, fi, v be the direction cosines of the normal at any point x, y, z on the surface of the solid, we must have at the surface dfa _ dfa_ dfa _ ~1 ~~~ ">> 7 M/ 7 V, an dn dn dy l dy 2 dy z -p = vy u.z, -y^ = \z vx, -p- = /mx \y. dn 9 dn dn KINETIC ENERGY. 67 Hence - = (u yw 3 + 2&> 2 ) A, + (v = normal velocity of the solid. 65. If we substitute the value of < from (8) in (7), it follows that T is a homogeneous quadratic function of the six velocities u, v, w, &>!, ft> 2 , ^ an d therefore contains twenty-one terms. If we choose as our axes Ox, Oy, Oz, the principal axes at the centre of inertia of the solid, the kinetic energy of the latter will be equal to where M is the mass of the solid, and A 1} B lt 0^ are its principal moments of inertia. Hence the kinetic energy T of the system, being the sum of the kinetic energies of the solid and liquid, is determined by the equation, 2T= Pu? + Qv* + Rw* + ZP'vw + 2Q'wu + ZR'uv + Aw? + Ba> 2 2 + (7ft> 3 2 + 2J/&> 2 &) :! + Z&tojO! + 2<7'ft)!< 2 + 2ft)! (Lu + Mv + Nw) + 2ft) 2 (L'u + M'v + N'w) + Zw s (L"u+M"v + N"w) ................................. (9). The coefficients of the velocities in the preceding expression for the kinetic energy, are called coefficients of inertia. The quantities P, Q, R are called the effective inertias of the solid parallel to the principal axes, and the quantities A,B, C are called the effective moments of inertia about the principal axes. If the liquid extend to infinity, and there is only one moving solid, the coefficients of inertia depend solely upon the form and density of the solid and the density of the liquid, and not upon the coordinates which determine the position of the solid in space. The values of these coefficients are ...(10) by (5) ; with similar expressions for the other coefficients. When the form of the solid resembles that of an ellipsoid, which is symmetrical with respect to three perpendicular planes through its centre of inertia, and the motion is referred to the 52 68 MOTION OF A SINGLE SOLID. principal axes of the solid at that point, the kinetic energy must remain unchanged when the direction of any one of the component velocities is reversed ; hence the kinetic energy cannot contain any of the products of the velocities, and must therefore be of the form ; 2T = Pu* + Qv 2 + Rw* + Ato* + B(oJ+Ca>,? (11). If in addition, the solid is one of revolution about the axis of z, the kinetic energy will not be altered if u is changed into v, and 2 2 ) + C&> 3 2 (12). Although every solid of revolution must be symmetrical with respect to all planes through its axis, it is not necessarily sym- metrical with respect to a plane perpendicular to its axis. The solid formed by the revolution of a cardioid about its axis is an example of such a solid. In this case the kinetic energy will be unaltered when the signs of u, v or a> 3 are changed, and also when u is changed into v, and w l into w 2 ; hence in this case 2T = P (w 2 + v 2 ) + Rw 3 + A (w^ + ft> 2 2 ) + <7o> 3 2 + 2Nw fa + &> 2 ). . .(13). If the solid moves with its axis in one plane, (say zx), v and eoj must be zero, and the last term may be got rid of, by moving the origin to a point on the axis of z, whose distance from the origin is N/R. This point is called the Centre of Reaction. 66. We must now find expressions for the component linear and angular momenta. Since we are confining our attention to acyclic irrotational motion, it follows from 59 that the motion of the liquid at any instant depends solely upon the motion of the surface of the solid; hence the motion which actually exists at any particular epoch, could be produced instantaneously from rest, by the application of suitable impulsive forces to the solid ; and since impulsive forces are measured by the momenta which they produce, it follows that the resultant impulse which must be applied to the solid, must be equal to the resultant momentum of the solid and liquid. Let , ij, be the components parallel to the principal axes of the solid, of the impulsive force which must be applied to the solid, in order to produce the actual motion which exists at time t; and let X, /*, v be the components about these axes, of the impulsive couple. Then , 77, % are the components of the linear COMPONENT MOMENTA. 69 momentum, and X, /*, v of the angular momentum of the solid and liquid. Let p denote the impulsive pressure of the liquid, and let us consider the effect produced upon the solid by the application of the impulse whose components are f, 77, f, X, \i, v. By the ordinary equations of impulsive motion where I, m, n are the direction cosines of the normal at any point of 8. But if be the velocity potential of the motion instantaneously generated by the impulse, which is equal to the velocity potential which actually exists at time t, it follows from 25, that p = pcf), whence since at the surface of the solid I = d 'H) > ^-i !*> v be the component impulses, which must be applied to the solid, in order to generate from rest the motion which actually exists at time t, it follows from the first proposition of 57, that But since T is a homogeneous quadratic function of the velocities, orr dT dT dT dT dT dT 2J. =u-j- +v j- + w j- + o>i j h o> 2 -j h ft> 3 j . du dv dw dw l dco 2 dw A Comparing these two equations, we obtain (14). We are now in a position to solve a variety of problems connected with the motion of a single solid in an infinite liquid. MOTION OF A SPHERE. 71 Motion of a Sphere. 68. Let us suppose that the centre of the sphere describes a plane curve, and let u, v be its component velocities parallel to the axes of a; and y. Since every diameter of a sphere is a principal axis, the axes of x and y may be supposed to be fixed in direction, whence and since on account of symmetry P Q, we have where P = M- = M+ Trpct? I cos 2 9 sin ddO Jo where M' is the mass of the liquid displaced. Whence T and therefore Let us now suppose that the sphere is descending under the action of gravity, and that the axis of y is drawn vertically downwards; we shall also suppose that the sphere is initially projected with a velocity, whose horizontal and vertical components are U and V. Since the momentum parallel to x is constant throughout the motion, = const. = (M + %M') U, whence u = U. To determine the force acting on the system, draw two horizontal planes above and below the sphere and at a considerable distance from it. Then the force on the whole system due to gravity is % + tt(Pi -P2 + 9pz} dA - Mg' + h, when p 1} p 2 are the hydrostatic pressures on the upper and lower planes, z the distance between them, and h that portion of the pressure due to the motion of the liquid. The integral vanishes ; also h vanishes when the planes are at an infinite distance from 72 MOTION OF A SINGLE SOLID. the sphere ; whence the force is equal to (M M') g. The equa- tion giving the vertical motion is therefore or Whence if a- be the density of the sphere, dv _ cr p dt~ ff + ^p ff> and the sphere will describe a parabola with vertical acceleration fj (a p)l( its angular velocity about its axis. On account of symmetry none of the products can appear, and therefore T=%(Pi? + Qv* + A< a *) .................. (15), where P, Q, A are constant quantities. Let us now suppose that no forces are in action, and that the solid and liquid are initially at rest ; and let the cylinder be set in motion by means of an impulsive force F, whose line of action passes through its axis, and an impulsive couple which produces an initial angular velocity ft. Let us refer the motion to two fixed rectangular axes x and y, the former of which coincides with the direction of F, and let 6 be the angle which the major axis of the cross section makes with the axis of x at time t. Resolving the momenta along the axes of x and y, we obtain f cos i} sin 6 = F sin 6 + 17 cos 6 0, whence since = Pw, 77 = Qt>, we obtain Pu = Fcos0, Qv = -Fsin0 ............... (16). MOTION OF AN ELLIPTIC CYLINDER. 73 Since the kinetic energy remains constant throughout the motion, it follows that if we substitute the values of u, v from (16) in (15), and put (3 for the initial value of 6, we shall obtain or Afc=A&+ F* - (sin 2 0- sin 2 y3) ...... (17). We shall presently show that Q > P ; it therefore follows that if Q-P / V APQ 3 6 will never vanish, and the cylinder will make a complete revolution. The integration of equation (17) requires elliptic functions, but without introducing these quantities, we can easily ascertain the character of the motion of the centre of inertia of the cylinder. Let (x, y) be the coordinates of the centre of inertia referred to the fixed axes of x and y ; then x = u cos v sin 0, y=usm0 + v cos (18). Substituting the values of u, v from (16) we obtain y = F(p-Q)sin0cos0. These equations show that the centre of inertia of the cross section of the cylinder, moves along a straight line parallel to the direction of F with a uniform velocity F/Q, superimposed upon which is a variable periodic velocity, and that at the same time it vibrates perpendicularly to this line. This kind of motion frequently occurs in hydrodynamics, and a body moving in such a manner is called a Quadrantal Pendulum. If which is the limiting case between oscillation and rotation, the 74 MOTION OP A SINGLE SOLID. equations of motion admit of complete integration. Putting A \P Q)' (17) becomes whence Therefore ^ = ^ 6=/sin<9 It = log tan dy IA y= dx F I A . IA . = pj cosec ~ ~p sln " F I A x = pj log tan %6 + -y cos 0. Putting IA/F = c, and eliminating 6 we obtain the equation of the path, viz. F x = pi- l g y The curves described by the centre of inertia of the cylinder in the three cases have been traced by Prof. Greenhill, and are shown in Figures 1, 2, 3 of the accompanying diagram. 70. We shall now show that for an elliptic cylinder Q > P. When the cylinder is moving with unit velocity parallel to x, we have shown in 48 that and therefore Now at the surface sinh & sin , /3 cos . = c sinh /3/cos = c 2 sinh 2 /8/ * cos* j-dg Jo P=M-pf folds =M(I + ^ whence where P. MOTION OF AN ELLIPTIC CYLINDER. 75 jf PH 70 MOTION OF A SINGLE SOLID. Similar results can be proved to be true in the case of an ellipsoid ; from which it is inferred that when any solid body is moving in an infinite liquid, the effective inertias corresponding to the greatest, mean, and least principal axes, are in descending order of magnitude. 71. If the cylinder be projected parallel to a principal axis without rotation, it will continue to move in a straight line with uniform velocity ; but if the direction of projection is not a principal axis, it will begin to rotate, and its angular velocity at any subsequent time will be determined by putting = in (17). We shall now show that if the cylinder be projected parallel to a principal axis, its motion will be stable or unstable according as the direction of projection coincides with the minor or major axis. Let us first suppose the cylinder projected parallel to its major axis, and that a slight disturbance is communicated to it. The equation determining the angular velocity is obtained by putting n = /3=0 in (17); whence and therefore differentiating, and remembering that in the begin- ning of the disturbed motion 6 is a small quantity, we obtain Since Q> P, the coefficient of 6 is negative, which shows that the motion is unstable. If the cylinder is projected parallel to its minor axis we must put /3 = ^TT ; also if ^ = \ir 6, % will be a small quantity in the beginning of the disturbed motion; Avhence (17) becomes whence A% + F* - X = Since the coefficient of % is positive, the motion is stable. It can also be shown that if an ellipsoid be projected parallel to a principal axis, without rotation, the motion will be unstable unless the direction of projection coincides with the least axis. MOTION OF AN ELLIPTIC CYLINDER. 77 We shall however presently show, that if an ovary ellipsoid be projected parallel to its axis, the motion will be stable, provided a sufficiently large angular velocity be communicated to the solid about its axis. 72. Let us now investigate the motion of an elliptic cylinder, which descends from rest under the action of gravity. Let the axis of y be horizontal, and the axis of x be drawn vertically downwards. The equations of momentum are sin 6 + 7} cos & = 0, |- ( cos - 77 sin 0) = (M- M') g, du from the last of which we obtain f cos 6 - 77 sin 6 = (M-M')gt Solving these equations and recollecting that = Pu, 77 = Qv, we obtain Pu = (M- M'} gt cos 8, Qv = -(M- M') gt sin 0. Substituting these values of u and v in (18), we obtain /cos 2 6 sin 2 Q\ , , , , ,,. \ * = \~~p~ + ~Q J ( ~ } n < 19 >- sm e cos e The equation of energy gives (M - MJ gH* + AP=2(M- M') gx. If we differentiate this equation with respect to t, we can eliminate x by means of (19) ; but the resulting equation would be difficult to deal with. We see however from the first of (19), that x is always positive, and therefore the cylinder moves downwards with a variable velocity, which depends upon the inclination of its major axis to the vertical, as well as upon the time. We also see from the second equation, that the horizontal velocity vanishes, whenever the major axis becomes horizontal or vertical ; but if the motion should be of such a character that always lies between and TT, the horizontal velocity will never vanish. 78 MOTION OF A SINGLE SOLID. Helicoidal Steady Motion of a Solid of Revolution. 73. In the figure let 00 be the axis of the solid of revolution, its centre of inertia, and let the solid be rotating with angular velocity ft about its axis. A Let the solid be set in motion by means of an impulsive force F along OZ, and an impulsive couple G about OZ, and let a be the angle which OC initially makes with OZ. Let i/r be the angle which the plane ZOO makes at time t with a plane ZOX, which is parallel to some fixed plane, and let the former plane cut the equatorial plane in OA ; also let ZOC = 6. It will be convenient to refer the motion to three moving axes, OA, OB, 00, where OB is the equatorial axis which is perpendicular to OA. Resolving the linear momentum of the system along OZ, OX and a line Y perpendicular to the plane ZOX, we obtain cos 6 + % sin 6) cos ^ 77 sin i/r = 0, cos 6 + sin 6) sin i/r + 17 cos -|r = 0, whence (20). Since the components of momentum parallel to the axes of X and Y (which are fixed in direction, but not in position because is in motion) are zero throughout the motion, the angular HELICOIDAL STEADY MOTION. 79 momentum about OZ is constant 1 , whence (21). The equation of energy gives PU* + Rw* + A (a)! 2 + 2 ) = const ............. (22), also = Pu, = Rw, and therefore by (20) and (21) this becomes ., ,/sin 2 ^ cos 2 0\ {Q + <7H (cos a - cos 0)}* .- F-(n-+^n-) + L A ,/, - 2L + Affl = const. V P R J A sm 2 = its initial value ...... (23). This equation determines the inclination of the axis. So far our equations have been perfectly general, we shall now introduce the conditions of steady motion. These are = a, ijr = fjL, 0=0 = ............... (24). Now a) l = TJr sin a = p sin a, whence (21) becomes ^/isin 2 a = # ........................ (25). Differentiating (23) with respect to t and using (24) and (25) we obtain A p? cos a -Clp + (-l F*cosa = ......... (26). This is a quadratic equation for determining p, when O and F are given. Now //, must necessarily be a real quantity, and there- fore the condition that steady motion may be possible is that C 2 W>4,F*-A cos 2 a - ............... (27), \-tt -i I and since Rw = = F cos a, the condition becomes (28). 1 It was shown by Hayward (Trans. Camb. Phil. Soc. Vol. x.) that when the origin as well as the axes are in motion, the Principle of Angular Momentum is expressed by three equations of the form dv' --vf+wf- x'0 2 + //0i = N. Since the direction of the axes of X, Y and Z are fixed, 0j = 2 =0; also since the momenta ' and 77' parallel to X and Y are zero, the equation reduces to dv'jdt = N ; which gives ' = const., when N=0. This result can of course be proved by elementary methods. 80 MOTION OF A SINGLE SOLID. If the solid of revolution is oblate (such as a planetary ellipsoid) R > P, and therefore (28) is always satisfied ; but if the solid is prolate (such as an ovary ellipsoid) P > R, and therefore steady motion will not be possible unless ft exceeds a certain value. In order to find the path described by the centre of inertia of the solid in steady motion, we have, since ^ fit, / */l : 1\ , x = (u cosa + w sma) cosy = Jf I -^- =- ) sma cosa cospt, \Jt " i i\ 1 \ . y = (u cosa + w sin a) sin-Jr = F ( -^ ^ sina cosa sin/u, \xl ft r, /sin 2 a cos 2 a\ .gr = ;cosa u sma =-^(^^5 I n~ > V " Jt 1 which shows that the centre of inertia describes a helix. 74. In order to find whether the steady motion is stable or unstable, differentiate (23) with respect to t, and we obtain A'0+f(0) = Q (29), where (7ft cosa The condition for steady motion is, that /(a) = 0, which leads to (26), whence writing = a + x where ^ is small, (29) becomes and the condition of stability requires that f (a) should be positive. Now /' (a) = Ap? (1 + 2 cos 2 a) - SOft/i cosa + ~ - F n - ~ - cos2a, whence eliminating ft by (26) this becomes - ~ (1-3 cos The condition that the right-hand side should be positive is that * (^ - iY sin 2 a (9 cos 2 a - 1) > 0, which requires that a should lie between cos" 1 ^ and 0, or between TT cos" 1 and TT. CONDITIONS OF STABILITY. 81 As a particular example let the solid be projected point foremost; then = and = 0, and therefore since 6 is a small quantity in the beginning of the disturbed motion /(*) = If therefore R > P the motion is always stable, whether there is or is not rotation, and consequently the forward motion of a planetary ellipsoid is always stable ; but if P > R, it follows that since F=Rw, the motion will be unstable unless The motion of an ovary ellipsoid is therefore unstable, unless the ratio of its angular velocity to its forward velocity exceeds a certain value. 75. These results have an application in gunnery. When an elongated body, such as a bullet, is fired from a gun with a high velocity, the effect^ of the air upon its motion cannot be neglected ; and if the air is treated as an incompressible fluid, the previous investigation shows that the bullet will tend to present its flat side to the air, and also to deviate from its approximately parabolic path, unless it be endowed with a rapid rotation about its axis. Hence the bores of all guns destined for long ranges are rifled, by means of which a rapid rotation is communicated to the bullet before it leaves the barrel. The effect of the rifling tends to keep the bullet moving point foremost, and to ensure its travelling along an approximately parabolic path in a vertical plane. Moreover when a bullet is moving with a high velocity, the effect of friction cannot be neglected ; and it is obvious that when the bullet is moving with its flat side foremost, the effect of frictional resistance will be much greater than when it is moving point foremost, and therefore the bullet will not travel so far in the former as in the latter case. The hydrodynamical theory therefore explains the necessity of rifling guns. Motion of a Cylinder parallel to a Plane. 76. We have thus far supposed that the liquid extends to infinity in all directions ; we shall now suppose that the liquid B. H. 6 82 MOTION OF A SINGLE SOLID. is bounded by a fixed plane, and shall enquire what effect the plane boundary produces on the motion of a circular cylinder. Let the axis of y be drawn perpendicularly to the plane, and let the origin be in the plane, and let (x, y) be the coordinates of the centre of the cylinder, (u, v) its velocities parallel to and perpendicular to the plane. The kinetic energy of the solid and liquid must be a homo- geneous quadratic function of u and v, but since the kinetic energy is necessarily unchanged when the sign of u is reversed, the product uv cannot appear. We may therefore write T = % (Ru 2 + R'v*) ..................... (30). The coefficients R, R' depend upon the distance of the cylinder from the plane, and are therefore functions of y but not of x ; and as a matter of fact their values are equal. It will not however be necessary to assume the equality of R and R', since our object will be attained provided we can show that R and R' diminish as y increases. In order to produce from rest the motion which actually exists at time t, we must apply to the cylinder impulsive forces whose components are X, Y; and we must also apply at every point of the plane boundary an impulsive pressure, which is just sufficient to prevent the liquid in contact with the plane from having any velocity perpendicular to the plane. The work done by the impulsive pressure is zero, whilst the work done by the impulses X, Y is %(Xu+Yv) ........................ (31), which must be equal to T. Now (30) may be written in the form ) whence comparing (31) and (32) we see that *-*(+ X= C ^ = Ru, Y = ~ = R'v (33), du dv and therefore y=ir +*) ( 34 )- \K H / The last equation gives the kinetic energy in terms of the impulsive forces X, Y applied to the cylinder. Let us now suppose that the cylinder instead of being at a distance y from the plane, is at a distance y 1} where y-^ >y ; and EFFECT OF A PLANE BOUNDARY. 83 let R l} RI be the values of R, R' at y 1 . Then if the cylinder were set in motion by the same impulses, the work done would be (33) - Now the effect of the plane boundary is to produce a con- straint, and the effect of this constraint evidently diminishes as the distance of the cylinder from the plane increases, and therefore by Bertrand's theorem, 1\ > T. Hence 4-4,1 + F' JT/ ~ ~rv ) R 1 H J is positive for all values of X and Y, which requires that R > R! , R' > RI ', hence R, R' diminish as y increases, and consequently their differential coefficients with respect to y are negative. 77. We can now determine the motion of the cylinder. The momentum parallel to x is equal to dTjdu and is con- stant ; whence Ru = const. = X (36). Since the kinetic energy is constant, we have Ru 2 + R'v* = const. = 2T (37). Differentiating (37) with respect to t, and eliminating dufdt by (36), we obtain J TV J T)\ GLft CLl\ dy dy) ' From this equation we can ascertain the effect of the plane boundary ; for if the cylinder is projected perpendicularly to the plane, u = 0, and dR 2R' dy ' Now dR'/dy is negative, and therefore v is positive; whence it follows that whether the cylinder be moving from or towards the plane, the force exerted by the liquid upon the cylinder will always be a repulsion from the plane, which is equal to M& dR' 2R' dy ' Hence if the cylinder be in contact with the plane, and a small 62 84 MOTION OF A SINGLE SOLID. velocity perpendicular to the plane be communicated to it, the cylinder will begin to move away from the plane with gradually increasing velocity. This velocity cannot however increase indefi- nitely, for that would require the energy to become infinite, which is impossible, since the energy remains constant and equal to its initial value. If RJ denote the value of R' when the cylinder is in contact with the plane, v the initial velocity, and v the velocity when the cylinder is at an infinite distance from the plane, the equation of energy gives 7? ' r 2 _ 7? ',2 .n/ u ^1,^ u . The value of R^' = M + M', since the motion is the same as if the plane boundary did not exist, whence the ratio of the terminal to the initial velocity is v -= / V V -< M+M'' Let us now suppose that the cylinder is projected parallel to the plane with initial velocity ii . By (38) the initial acceleration v perpendicular to the plane is . _ M 2 dR V ~2Rfy'' and since dRjdy is negative, the cylinder will be attracted towards the plane, and will ultimately strike it. 78. All the results of the last two sections are true in the case of a sphere, and can be proved in the same manner. Moreover the motion will be unaltered, if we remove the plane boundary, and suppose that on the other side, an infinite liquid exists in which another equal cylinder or sphere is moving with velocities u, v. The second cylinder or sphere is therefore the image of the first. Our results are therefore applicable to the case of two equal cylinders or spheres, which are moving with equal and opposite velocities along the line joining their centres; or to the case in which the cylinders or spheres are projected perpendicularly to the line joining their centres, with velocities which are equal and in the same direction. These results have however a wider application, for according to the views of Faraday and Maxwell, the action which is observed to take place between electrified bodies is not due to any direct action which electrified bodies exert upon one another, but to EXAMPLES. 85 something which takes place in the dielectric medium surrounding these bodies ; and although the preceding hydrodynamical results do not of course furnish any explanation of what takes place in dielectric media, they establish the fact that two bodies which are incapable of exerting any direct influence upon one another, are capable of producing an apparent attraction or repulsion upon one another, when they are in motion in a medium which may be treated as possessing the properties of an incompressible fluid. EXAMPLES. 1. A light cylindrical shell whose cross section is an ellipse is rilled with water, and placed at rest on a smooth horizontal plane in its position of unstable equilibrium. If it is slightly disturbed, prove that it will pass through its position of stable equilibrium with angular velocity w, given by the equation 2. An elliptic cylindrical shell, the mass of which may be neglected, is filled with water, and placed on a horizontal plane very nearly in the position of unstable equilibrium with its axis horizontal, and is then let go. When it passes through the position of stable equilibrium, find the angular velocity of the cylinder, (i) when the horizontal plane is perfectly smooth, (ii) when it is perfectly rough ; and prove that in these two cases, the squares of the angular velocities are in the ratio (a 2 - 6 2 ) 2 + 46 2 (a 2 + 6 2 ) : (a 2 - & 2 ) 2 , 2a and 26 being the axes of the cross section of the cylinder. 3. A pendulum with an elliptic cylindrical cavity filled with liquid, the generating lines of the cylinder being parallel to the axis of suspension, performs finite oscillations under the action of gravity. If I be the length of the equivalent pendulum, and I' the length when the liquid is solidified, prove that ~ where M is the mass of the pendulum, m that of the liquid, h the 86 MOTION OF A SINGLE SOLID. distance of the centre of gravity of the whole mass from the axis of suspension, and a, b the semi-axes of the elliptic cavity. 4. Find the ratio of the kinetic energy of the infinite liquid surrounding an oblate spheroid, moving with given velocity in its equatorial plane, to the kinetic energy of the spheroid; and denoting this ratio by P, prove that if the spheroid swing as the bob of a pendulum under gravity, the distance between the axis of the suspension and the axis of the spheroid being c, the length of the simple equivalent pendulum is (1 + P) c + 2a 2 /5c I -pi a where a is the equatorial radius, a and p the densities of the spheroid and liquid respectively. 5. A pendulum has a cavity excavated within it, and this cavity is filled with liquid. Prove that if any part of the liquid be solidified, the time of oscillation will be increased. 6. A closed vessel filled with liquid of density p, is moved in any manner about a fixed point 0. If at any time the liquid were removed, and a pressure proportional to the velocity potential were applied at every point of the surface, the resultant couple due to the pressure would be of magnitude G, and its direction in a line OQ. Show that the kinetic energy of the liquid was proportional to ^pcoG cos 6, where w is the angular velocity of the surface, and 6 the angle between its direction and OQ. 7. Liquid is contained in a simply-connected surface S ; if w is the impulsive pressure at any point of the liquid due to any arbitrary deformation of S, subject to the condition that the enclosed volume is not changed, and c/ the impulsive pressure for a different deformation, show that 8. If a sphere be immersed in a liquid, prove that the kinetic energy of the liquid due to a given deformation of its surface, will be greater when the sphere is fixed than when it is free. CHAPTER IV. WAVES. 79. BEFORE discussing the dynamical theory of waves, we shall commence by explaining what a wave is. Let us suppose that the equation of the free surface of a liquid at time t, is y = a sin (mx nt) (1), where the axis of x is horizontal, the axis of y is measured vertically upwards, and a, ra and n are constants. The initial form of the free surface, i.e. its form when t 0, is y a sin mx, which is the curve of sines. The maximum values of y occur when mx = (2i+ ^) TT, where i is zero or any positive or negative integer ; and this maximum value is equal to a. The minimum values of y occur when mx = (2i + f) TT, where i is zero or any positive or negative integer ; and this minimum value is equal to a. As x increases from to ^7r/m, y increases from to a, and as x increases from ^7r/m to -rr/m, y decreases from a to 0. As x increases from tr/m to f7r/w, y is negative ; and when x has the latter value, y has attained its greatest negative value, which is equal to a ; as x increases from ITT/TO to 2-TT/m, y numerically decreases from a to 0. The values of y comprised between x = 2t7T/m and x = 2 (i + 1) irjm, evidently go through exactly the same cycle of changes. When the motion of a liquid is such, that its free surface is represented by an equation such as (1), the motion is called wave motion. WAVES. The quantity a, which is equal to the maximum value of y, is called the amplitude ; and the distance 2?r/?n, between two consecutive maxima values of y, is called the wave length. In order to ascertain the form of the free surface at time t, let us transfer the origin to a point = nt/m ; then if x' be the abscissa at time t referred to the new origin, of the point whose abscissa referred to the old origin is x, we have x = x' + and y = a sin (mx + m% nt) = a sin mx'. The form of the free surface at time t, is therefore obtained by making the point which initially coincided with the origin, travel along the axis of x, with velocity n/m. The velocity of this point is called the velocity of propagation of the wave. If V be the velocity of propagation, and X the wave length, we thus obtain the equations m-2-Tr/X, V = n/m (2), and therefore (1) may be written O-FO (3). If n, and therefore F, were negative, equations (1) and (3) would represent a wave travelling in the opposite direction. The position of the free surface at time t, is exactly the same as at time t + 2i7r/w, or t + 2iX/F, since n - 2-TrF/X ; the quantity X/Fis called the periodic time, or shortly the period, and is equal to the time which the crest of one wave occupies in travelling from its position at time t to the position occupied by the next crest at the same epoch. If r denote the period, we evidently have * = Vr (4), and (1) may be written in the form / *\ (5), which is a form sometimes convenient in Physical Optics. From (4) we see that for waves travelling with the same velocity, the period increases with the wave length. The reciprocal of the period, which is the number of vibrations executed per unit of time, is called the frequency. If therefore we KINEMATICS OF WAVE MOTION. 89 have a medium which propagates waves of all lengths with the same velocity, equation (4) shows that the number of vibrations executed in a second increases as the wave length diminishes. This remark is of importance in the Theory of Sound. Let us now suppose that two waves are represented by the equations y = a sin (x - Vt), \ y' = a sin (x Vt e). X The amplitudes, wave lengths and velocities of propagation of the two waves are equal, but the second wave is in advance of the first; for if in the first equation we put t + e/V for t, the two equations become identical. It therefore follows that the distance at which the second wave is in advance of the first is equal to e. The quantity e is called the phase of the wave. 80. Waves which are represented by equations such as (1), are called progressive waves ; their wave lengths are equal to 2?r/m, and their velocities of propagation to n/m. If such waves axe travelling along the surface of water under the action of gravity, they may be conceived to have been produced by com- municating to the free surface an initial displacement y = a sin mx, together with an initial velocity an cos mx. We therefore see that the wave length depends solely on the initial displacement, but that the velocity of propagation depends upon the initial velocity as well as upon the initial displacement. If we combine the two waves, which are obtained by writing n for n in (1) and add the results, we shall obtain y = a sin (mx nt) + a sin (mx + nt) = 2a sin mx cos nt (6). Such a wave is called a stationary wave. It is produced by means of an initial displacement alone, and gives the form of the free surface at time t, when the latter is displaced into the form of the curve y = 2a sin mx, and is then left to itself. In equations (1) or (6), m, which is equal to 2?r/\, is always supposed to be given, and the problem we have to solve, consists in finding the value of n, which determines the velocity of propagation. 90 WAVES. 81. In most problems relating to small oscillations, the motion is supposed to be sufficiently slow for the quadratic terms which occur in the equations of motion to be neglected. Under these circumstances, the equations become linear and usually admit of a solution in which the time enters in the form of the factor e inf . Throughout the present chapter this hypothesis will be made ; but it may be remarked that a solution of the complete equations of motion has been obtained by Gerstner, which leads to a species of trochoidal wave involving molecular rotation. The theory of waves involving molecular rotation is not of any great interest in the dynamics of a frictionless liquid ; but it is of the highest im- portance in the dynamics of actual liquids, which are viscous, inasmuch as the motion of a viscous liquid always involves mole- cular rotation. We shall now proceed to consider the irrotational motion of liquid waves in two dimensions, under the action of gravity. The solution of the problem involves the determination of a velocity potential , which satisfies the following three conditions : (i) (f> must satisfy Laplace's equation, and together with its first derivatives, must be finite and continuous at every point of the liquid. (ii) must satisfy the given boundary conditions at the fixed boundaries of the liquid. (iii) must be determined, so that the free surface of the liquid is a surface of constant pressure. To find the condition to be satisfied at the free surface, let the origin be taken in the undisturbed surface, and let the axis of a be measured in the direction of propagation of the waves, and let the axis of z be measured vertically upwards. The pressure at any point of the liquid is determined by the equation p/P + gz + + W = C (7). The equation of the surfaces of constant pressure is p = const., and since the free surface is included in this family of surfaces, and must also satisfy the kinetnatical condition of a bounding surface, it follows from 12 (17) that dp dp dp dp -TT+u-^ + v-^+w^ = (8). dt dx dy dz WAVES IN A LIQUID OF GIVEN DEPTH. 91 Substituting the value of p from (7) in (8), and neglecting squares and products of the velocity, we obtain + '" ........................ < This is the condition to be satisfied at a free surface, where 2=0. Waves in a Liquid of given Depth. 82. We shall now find the velocity of propagation of two- dimensional waves travelling in an ocean of depth h. The equation of continuity is 2+2- The boundary condition at the bottom of the liquid is ^ = 0, when z = -h .................. (11). To satisfy (10) assume = F (z) cos (mx - nt) .................. (12). Substituting in (10) we obtain the solution of which is F = P cosh mz + Q sinh mz ; whence = (P osh mz + Q sinh mz) cos (mx nt). Substituting in (11) and (9) we obtain P sinh mh = Q cosh mh, Pn 2 = Qmg ; whence eliminating P and Q, and taking account of (2), we obtain which determines the velocity of propagation. If the lengths of the waves are large in comparison with the depth of the liquid, h/\ is small, and the preceding result becomes V 2 = gh .............................. (14), which determines the velocity of propagation of long waves in shallow water. 92 WAVES. If the depth of the liquid is large in comparison with the wave length, h/X is large, and tanh 27rh/\ 1 approximately, whence F'=<7X/27r ........................ (15), which determines the velocity of propagation of deep sea waves. The last result may be obtained directly, for the value of F may be written in the form F= Ae mz + Be- mz , and since , and therefore F, cannot be infinite when z = oo , B = 0, and (9) at once gives the required result. 83. Equation (15), which determines the velocity of propaga- tion of deep-sea waves, may be written in the form n 2 mg', whence the form of the free surface at time t is y= a cos (ma; m*g*t) .................. (16), corresponding to the initial form y = a cos mx. If the initial form were y=F (x), Fourier's theorem enables us to express F (x) in the form of a definite integral involving cos ma;; hence by (16) the form of the free surface at any subsequent time can be written down in terms of these quantities ; but the difficulty of evaluating the definite integral is usually so great, that the results are rarely of much practical utility. As an example of the use of definite integrals, let us suppose that the initial form of the free surface is y = a exp (+ xjc), where the upper and lower signs are to be taken on the positive and negative sides of the origin respectively. When x is positive cos xujc . du 2 f exp(-a?/c)=- 7TJO cos mxdm ~l+mV ' whence the equation of the free surface at time t is _ 2ac f cos (mx mlg*t) dm ~~ 84. Returning to the general case, we see that is of the form = A cosh m (z + h) cos {mx nt). 93 If 77 be the elevation of the free surface above the undisturbed surface, we must have 7) = d/dz when z = (17), whence substituting the value of in (17), and suitably choosing the origin we obtain ij = Amn~ l sinh mh sin (mx nt). Let (x, z) be the coordinates of an element of liquid when undisturbed, (f, f ) its horizontal and vertical displacements, also let x = x + z = z + ; then = d(f>/dx' = Am cosh m (/ + h) sin (mx' nt) = d/dz' = Am sinh m (z' + h) cos (mx' - nt). Since the displacement is small we may put x = x', z = z' as a first approximation, and we obtain = a cosh m (z -f h) cos (mx nt) a sinh m (z + h) sin (mx nt), where Amjn = a ; whence the elements of liquid describe the ellipse 2 /cosh 2 m (z + h) + 2 /sinh 2 m ( z + h) = a 2 . When the depth of the liquid is very great we may put h = oc , and the hyperbolic functions must be replaced by exponential ones ; we shall thus obtain = Ae mz cos(mx nt) r n = Amn~ l sin (mx nt), and the elements of liquid will describe the circles 2 + 2 = (Am/n) 2 e mz . Waves at the Surface of Separation of Two Liquids. 85. Let us first suppose that two liquids of different densities (such as water and mercury) are resting upon one another, and are in repose except for the disturbance produced by the wave motion ; also let the liquids be confined between two planes parallel to their surface of separation. Let p, p be the densities of the lower and upper liquids respectively, h, h' their depths, and let the origin be taken in the surface of separation when in repose. 94 WAVES. In the lower liquid let < = A cosh m (z + h) cos (tnx ni), and in the upper let ' = A' cosh m (z h') cos (mx nt) also let 77 = a sin (mx nt) be the equation of the surface of separation. At this surface, the condition that the two liquids should remain in contact requires that drj/dt = dfyjdz = dfy'jdz, when z = 0. Whence - na = mA sinh mh = mA' sinh mh'. If Sp, Sp' be the increments of the pressure due to the wave motion just below and just above the surface of separation, then Sp + gprj + pd/dt = 0, and Sp' + gp'r) + p'dfi/dt = 0, and since Sp = Sp', we obtain g(p-p')f) = - pd(f>/dt + p'dfi/dt = n ( Ap cosh mh + A'p cosh mh') sin (mx nt) = (p coth mh + p coth mh') n^y/m, whence ' ,. (is), - m (p coth ra/i + p coth ra/i ) where m = 27r/\. 86. When X is small compared with h and A', then rw/i, mh' are large, and coth mh and coth m^' may be replaced by unity : we thus obtain If p > p, U" is negative and therefore n is imaginary ; hence if the upper liquid is denser than the lower, the motion cannot be represented by a periodic term in t, and is therefore unstable. If the density of the upper liquid is small compared with that of the lower, we have approximately U^gnr 1 (l-Zp'/p). If the liquid is water in contact with air, p'/p = '00122, hence if the air is treated as an incompressible fluid U- = ' 99756 xm- 1 . STABILITY AND INSTABILITY. 95 87. Secondly, let us suppose that the upper liquid is moving with velocity V, and the lower with velocity V\ then we may put (j) = Vx + A cosh m (z + h) cos (mx nt) ft = V'x + A' cosh m (z h') cos (mx nt). Let the equation of the surface of separation be F = i) a sin (mx nt) = 0. Then in both liquids F must be a bounding surface, and therefore by 12 equation (17), when z = 0, dF d$dF d$dF =Q dt doc dx dz dt] dF d^dF d'dF_ dt dx dx dz drj Whence an m Va + mA sinh mh = 0, an m Va mA' sinh mh' = 0. Hence if U = n/m be the velocity of propagation, A sinh mh = a ( V U) If Sp, 8p' be the increments of pressure at the surface of separation due to the wave motion Sp/p + gr} + d(j>/dt + ^ { V Am cosh mh cos (mx nt)}- = ^V-, Sp'/p + gy + dfijdt + \ { V - A'm cosh mh' cos (mx - nt)} 2 = % V-. Therefore since Bp = &p', ag (p p') = A mp (V U) cosh mh Amp ( V U) cosh mh' or g (p - p') = mp ( V- U) 2 coth mh + mp' ( V - 7") 2 coth mh' . . . (19), which determines U. Stability and Instability. 88. We shall now consider a question which has excited a good deal of attention of late years, viz. the stability or instability of fluid motion. If a disturbance be communicated to the two liquids which are considered in 85 87, the surface of separation may be conceived to be initially of the form 17 = a sin mx or a cos mx, where m is a given real quantity, whose value depends upon the nature of the disturbance. An equation of this kind does not of 96 WAVES. course represent the most general possible kind of disturbance, but inasmuch as by Fourier's theorem, any arbitrary function can be expressed in the form of a series of sines or cosines, or by a definite integral involving such quantities, an equation of this form is sufficient for our purpose. We have pointed out that the object of the wave motion problem is to determine n ; if therefore n should be found to be a real quantity, the subsequent motion will be periodic, and therefore stable ; but if n should turn out to be an imaginary or a complex quantity, the final solution will involve real exponential quantities, and therefore the motion will tend to increase with the time and will be unstable. To understand this more clearly, it must be recollected that we have neglected the quadratic terms in the equations of motion. The validity of this hypothesis depends firstly on the condition that the disturbance is a small quantity, from which it follows that the initial displacements and velocities must also be small quantities ; secondly, on the condition that these quantities remain small during the subsequent motion. If the solution thus obtained consists of periodic terms whose amplitudes are small quantities of the same order as the disturbance, the second condition is fulfilled, and the system oscillates about its undisturbed configuration ; but if the solution should contain an exponential term of the form e**, where k is positive, the system will tend to depart from its undisturbed configuration, and the solution will only represent the state of things in the beginning of the disturbed motion; and the subsequent history of the motion cannot be ascertained without finding the complete solution of the equations of motion, in which the quadratic terms are taken into account. In the former case the motion is stable, and in the latter unstable. To express this analytically, let us employ complex quantities, and assume that the initial form of the free surface is the real part of 77 = (A - iB) e inue , where A, B and m are real; and let n be of the form a -I- ift. Since the form of the free surface at any subsequent time is this becomes 77 = (4 - iB) e< STABILITY AND INSTABILITY. 97 the real part of which is i) = ?* {A cos (mx at) + B sin (mx at)} ...... (20). If therefore /3 is positive, the amplitude will tend to increase with the time, and the motion will be unstable. In such a case the two liquids will, after a short time, become mixed together, and will usually remain permanently mixed, if they are capable of mixing; but if they are incapable of remaining permanently mixed, the lighter liquid will gradually work its way upwards, and a stable condition will ultimately be arrived at. 89. If one liquid is resting upon another, equilibrium is possible when the heavier liquid is at the top, but in this case the equilibrium is unstable ; for since p > p, it follows from (18) that n- is negative and therefore n is of the form + t/3. Hence in the beginning of the disturbed motion, the free surface is of the form 77 = Ae*?* cos (mx e). If the upper liquid is moving with velocity V, and the lower with velocity V, the values of U or n/m are determined by the quadratic (19); and the condition of stability requires that the two roots of this quadratic should be real. Putting k, k' for m coth mh and m coth mh r , (19) becomes The condition that the roots of this quadratic in U should be real, is g (kp + k'p') (p - p'} - kk'pp' ( V - VJ > 0. It therefore follows that if p > p, that is if the lower liquid is denser than the upper liquid, the motion may be stable. But if p > p] or if no forces are in action, so that g=0, the motion will be unstable. 90. If no forces are in action, and both liquids are of unlimited extent so that h = h' = oo , the equation for determining U becomes p(V- C7) 2 + p'(F'-0 2 =0, the roots of which are B. H. 98 WAVES. Hence U, and therefore n, is a complex quantity, and we may therefore put U = a. i ft = n/ra, where a and /3 are determined from (21). If therefore the initial form of the free surface is 77 = ae tma; , its form at any subsequent time may be written where a'+ &' = a. If there is no initial displacement, ^ = when t = 0, in which case a'=b' = ^a. To express this result in real quantities, let a = A iB, and (22) becomes V) = {A cos m (x at} + B sin m (x at}} cosh mfit, corresponding to an initial displacement ?7 = A cos mx + B sin mx. 91. When the initial velocity is zero there are three cases worthy of notice. (i) Let p = p, V = V, so that the densities of the two liquids are equal, and their undisturbed velocities are equal and opposite ; then from (21), a = 0, /3 = V, whence 77 = (A cos mx 4- B sin mx) cosh mVt. (ii) Let p = p, V = 0, then a = F, /3 = |F, whence 77 = [A cos m (x \ Vt) + B sin m(x ^ Vt)} cosh \rnVt, hence the waves travel in the direction of the stream and with half its velocity. (iii) Let p = p', V = V. In this case the roots are equal, but the general solution may be obtained from (21) by putting V = V(l +7), where 7 ultimately vanishes. We thus obtain and therefore since 7 is small, (22) may be written 77 = e im <*- F <> {a + ^mVyt [a (l-i)- 2a 7 ]}. Putting c = ^mVy {a (1~ t) 2a'}, this becomes LONG WAVES. 99 Let a = A iB, c = C iD, then the real part is 77 = (A + Ct) cos m (x - Vt) + (B + Dt) sin m (a; - Vt), corresponding to an initial displacement 77 = A cos mx + B sin mx. If the initial velocity 7} is zero, (7 = mBV, _D = mJ.F, and 17 = (A + mBVt) cos m (x - Vt) + (B- mA Vt) sin m(x- Vt). The peculiarity of this solution is, that previously to displace- ment there is no real surface of separation at all. Hence if we have a thin surface dividing the air, such as a flag whose inertia may be neglected, it appears from the last equation that (neglecting changes in the density of the air), the motion of the flag will be unstable and that it will flap. Long Waves in Shallow Water. 92. In the theory of long waves it is assumed, that the lengths of the waves are so great in proportion to the depth of the water, that the vertical component of the velocity can be neglected, and that the horizontal component is uniform across each section of the canal. In 82 we saw that if the depth is small compared with the wave-length, then U~ gh, provided the square of the velocity is neglected. We shall now examine this result in connection with the above-mentioned assumption. Let the motion be made steady by impressing on the whole liquid a velocity equal and opposite to the velocity of propagation of the waves. Let 77 be the elevation of the liquid above the undisturbed surface ; IT, u the velocities corresponding to h and h + i) respectively. The equation of continuity gives u whence U z -u 2 = U' 2 (2/177 + V 2 )/(h + t)) 2 . If Bp be the excess of pressure due to the wave motion When t]/h is very small, the quantity in brackets is U 2 /h g ; whence if U 2 = gh, the change of pressure at a height h + 77 vanishes to a first approximation, and therefore a free surface is possible. If the condition U 2 = gh is satisfied, the change of pressure to a second approximation is 72 100 WAVES. which shows that the pressure is defective at all parts of the wave at which 77 differs from zero. Unless therefore rf can be neglected, it is impossible to satisfy the condition of a free surface for a stationary long wave ; in other words, it is impossible for a long wave of finite height to be propagated in still water without change of type. If however 77 be everywhere positive, a better result can be obtained with a somewhat increased value of U\ and if 77 be everywhere negative, with a diminished value. We therefore infer that waves of elevation travel with a somewhat higher, and waves of depression with a somewhat lower, velocity than that due to half the undisturbed depth 1 . 93. The theory of long waves in a canal may be investigated analytically as follows 2 . Let the origin be in the bottom of the liquid, h the undisturbed depth, 77 the elevation ; and let x be the abscissa of an element of liquid when undisturbed, the horizontal displacement. The quantity of liquid originally between the planes x and x + dx is hdx ; at the end of an interval t, the breadth of this stratum is dx (1 + d^jdx), and its height is h + 77, whence the equation of continuity is (I + d%fdx)(h + ' n ) = h .................. (23). Let us now investigate the motion of a column of liquid contained between the planes whose original distance was dx ; and let us suppose that in addition to gravity, small horizontal and vertical disturbing forces X and Fact. Since the vertical accelera- tion is neglected, the pressure will be equal to the hydrostatic pressure due to a column of liquid of height h + 77, whence r J y Ydy ............ (24). The equation of motion of the stratum is Now from (24), dp _ dtj v dt} C h+r >dY 1 Lord Rayleigh, "On Waves," Phil. Mag. April, 1876. 2 Airy, "Tides and Waves," Encyc. Met. LONG WAVES. 101 also in most problems to which the theory applies, the last two terms on the right-hand side of (26) are very much smaller than the first, and may therefore be neglected, whence (25) becomes Substituting the value of rj from (23) we obtain For a first approximation, we may neglect squares and products of small quantities, and (23) and (27) respectively become (28), In order to solve (29) when X = 0, assume ^ = e l(Ma; ~ nt) ,and we obtain n/m = (gh)^, which shows that the velocity of propagation is equal to (gh)*. Stationary Waves in Flowing Water 1 . 94. Let us suppose that water is flowing uniformly along a straight canal with vertical sides, and that between two points A and B there are small inequalities, and that beyond these points the bottom is perfectly level. Let a be the depth, u the velocity, p the mean pressure beyond A ; b the depth, v the velocity, and q the mean pressure beyond B : also let f be the difference of levels of the bottom at A and B. The total energy of the liquid per unit of the canal's length and breadth, at points beyond B, is Cb % v 2 b + g I ydy + w = ^ (v- + gb) b + w, Jo where w is the wave energy, and the density of the liquid is taken as unity. At very great distances beyond B the wave motion will have subsided and w will be zero. The equation of continuity is au = bv = M (30). * Sir W. Thomson, Phil. Mag. (5) vol. xxn. 353. 102 WAVES. The dynamical equation is found from the consideration, that the difference between the work done by the pressure p upon the volume of water entering at A, and the work done by the pressure q at B upon an equal volume of water passing away at B, is equal to the difference between the energy which passes away at B, and the energy which enters at A. Whence fa+f pau qbv = (^v~b + \gH z + w)v (^u 2 a + g I ydy) u, which by (30) becomes, p-q = ^- + ^gb + w/b-^-g(f+^a) ...... (31). Now p and q are the mean pressures, and therefore since the pressure at the free surface is zero, p = \ga, q=igb + w'/b, where w' denotes a quantity depending on the wave disturbance ; whence (31) becomes | M* (a 2 - 6 2 )/a 2 6 2 -g(a-b +/) + (w - w')/b - ...... (32). If we put D 3 = 2a 2 6 2 /(a + &)> M=VD-, D will denote a mean depth intermediate between a and 6, and approximately equal to their arithmetic mean when their differ- ence is small in comparison with either ; and V will similarly denote a corresponding mean velocity of flow. We thus obtain from (32) _f-(w-w')/gb - If b a were exactly equal to f, and there were no disturbance of the water beyond B, the mean level of the water would be the same at great distances beyond A and B ; but if this is not the case, there will be a rise or fall of level, determined by the formula V*flgD + (w-w')/gb l-V'/gD Let us now suppose that between A and B there are various small inequalities ; each of these inequalities will produce small waves whose nature is determined by the form of the functions w, w'; hence w and w will both be small quantities and the sign of y will be independent of that of w w'. Now / is positive or negative according as the bottom at A is higher or lower than the STATIONARY WAVES IN FLOWING WATER. 103 bottom at B. Hence if V- < gD the upper surface of the water rises when the bottom falls, and falls when the bottom rises ; and the converse is the case when V 2 > gD. Theory of Group Velocity. 95. When a group of waves advances into still water, it is observed that the velocity of the group is less than that of the individual waves of which it is composed. This phenomenon was first explained by Sir G. Stokes 1 , who regarded the group as formed by the superposition of two infinite trains of waves of equal amplitudes and nearly equal wave-lengths, advancing in the same direction. Let the two trains of waves be represented by cos k ( Vt x) and cos k' (Vt x) ; their resultant is equal to cos k(Vt-x)+ cos k' (V't-x)=2 cos \ {(k f V-kV)t-(k'- k) x} x cos {(k' V' + kV)t-(k' + k) x}. If k' k, V V be small, this represents a train of waves whose amplitude varies slowly from one point to another between the limits and 2, forming a series of groups separated from one another by regions comparatively free from disturbance. The position at time t of the middle of the group, which was initially at the origin, is given by (k'V-kV)t-(k'-k)x = 0, which shows that the velocity of propagation U of the group is U = (k'V-kV)/(k'-k). In the limit when the number of waves in each group is indefinitely great we have k' = k + 8k, V=V+SV, whence n _d(kV} ~~dk Capillary Waves. 96. Most liquids which are incapable of remaining perma- nently mixed, exhibit a certain phenomenon called capillarity 2 , 1 Smith's Prize Examination, 1876; and Lord Bayleigh, "On Progressive Waves " ; Proc. Land. Soc. vol. ix. 3 The reader who desires to study the theory of Capillarity is recommended 104 WAVES. when in contact with one another. This phenomenon can be explained by supposing that the surface of separation is capable of sustaining a tension, which is equal in all directions, and is independent of the form of the surface of separation. The surface tension depends upon the nature of both the liquids which are in contact with one another. Thus at a tem- perature of 20 C., the surface tension of water in contact with air is 81 dynes per centimetre; whilst the surface tension of water in contact with mercury is 418 dynes per centimetre. The surface tension diminishes as the temperature increases ; also a surface tension cannot exist at the common surface of two liquids, such as water and alcohol, which are capable of becoming permanently mixed. 97. We shall now consider the effect of surface tension upon the propagation of waves. . Let T be the surface tension, and let p and p + 8p be the pressures just outside and just inside the free surface of a liquid; then 8p/p + g = ..................... (33). But if we resolve the forces which act upon a small element 8s of the free surface vertically, and neglect the vertical accelera- tion, and put 8% for the angle which Ss subtends at the centre of curvature, we obtain SpSs = TSx> whence 8p=T-^. as Now -j- = cot Y, dx d*r) dy therefore -r{ = cosec 2 y -f* . dx* * ds Since ^ is nearly equal to ^TT, we may put cosec %=1, and ds = dx, whence to consult Chapter xx. of Maxwell's Heat ; and also the article on Capillarity in the Encyclopedia Britannica by the same author. A table of the superficial tensions of various liquids will be found in Everett's Units and Physical Constants, p. 49. CAPILLARY WAVES. 105 Substituting in (33), differentiating the result with respect to t, and remembering that 17 = dfyjdz, and that dty/dx 2 = we obtain _ .................. (34). p dz 3 y dz dt 2 This is the condition to be satisfied at the free surface. 98. We shall now apply the preceding result to determine the capillary waves propagated in an ocean of depth h. Let = A cosh m(z + h) cos (mx nt). Substituting in (34) we obtain Tm 3 /p + mg = n 2 coth mh, whence U* = n*/m? = (g\f^TT + 2irT/p\) tanh 2irhj\ ...... (35). Equation (35) determines the velocity of propagation corre- sponding to a given wave-length. 99. Let us now suppose that the depth h is so great that mh may be treated as infinite ; then coth mh = + 1 according as m is positive or negative. Hence it will be sufficient for our purpose to discuss the equation Tm s /p+mg = n' 2 ........................ (36), when m is positive. When ra = 0, ?i = 0; and as m increases from zero to infinity, the value of rt 2 is always positive, and consequently periodic motion is possible for all values of m, that is for all wave-lengths. If a given value be assigned to n, (36) is a cubic for deter- mining m. The real root is obviously positive ; and since the discriminant 1 of the cubic is positive, the other two roots are complex. Hence there is only one wave-length corresponding to any given period. 1 It can be shown by means of Taylor's theorem, that, if f(x) be any rational algebraic function of x, the condition that the equation f(x)=0 should have a pair of equal roots is obtained by eliminating x between f(x) = and /' (x)=0. The result of the elimination is a certain function of the coefficients which is called the discriminant of f(x) ; and it is shown in treatises on Algebra that two of the roots of a cubic will be real, equal or complex, according as the discriminant is negative, zero or positive. If the cubic be Ax 3 + Bx 2 + Cx + D, the discriminant is 3 - IS A BCD - 106 WAVES. If U be the velocity of propagation, n = mU; whence (36) becomes Tm?/p-mU 2 + g = ..................... (37). Hence U = oo , when m = and m = oo ; and therefore U must be a minimum for some intermediate value of m, which by ordinary methods can be shown to be given by the equation U 2 =2TrH/p. By means of this result, the minimum velocity of propagation and the corresponding wave-length can be shown to be given by the equations U=(4Tg/pp, \ = 2ir(T/gp} ............... (38). A wave whose length is less than the preceding critical value of X is called by Lord Kelvin a ripple 1 . Now if we write (36) in the form U* = gX/2Tr + 27rT/p\ .................. (39), it follows that when X is given by (38) both terms on the right- hand side are equal; also for long waves the first term is the most important, whilst for short waves the second is the most important. Hence the effect of gravity is most potent in pro- ducing long waves, and the effect of surface tension in producing ripples. 100. In 85 we have considered the propagation of waves at the surface of separation of two liquids, which are moving with different velocities. We shall now consider the production of ripples by wind blowing over the surface of still water. Let V be the velocity of the wind, which is supposed to be parallel to the undisturbed surface of the water, cr the density of air referred to water. Since the changes of density of the air are very small in the neighbourhood of the water, the air may approximately be regarded as an incompressible fluid, whence if the accented letters refer to the water, the kinematical conditions at the boundary give = Vx + a(U-V) + $ { V- am ( U - F) sin (mas - nt)} 2 - % V 2 = 0, or &p + acr {g + n ( U F) m ( U V )} sin (mac nt) = 0. Similarly Sp' + (g Uri) a sin (mac nt) = 0, whence (40) becomes Tm*-mU*-o-m(U-V)* + (l-a-)g = Q ......... (41). Putting U=n/m this may be written Tm*- - -"- ............ (46) - the value of n will be complex, and periodic motion will be impossible. Equation (44) gives the minimum value of W ; hence in order that wave-motion may be possible for waves of all lengths, we must have When (47) is satisfied, (46) may be written A 1 + 0-- (1 We shall now discuss this equation. 108 WAVES. Case(i). In this case one of the values of U is positive and the other negative ; hence waves can travel either with or against the wind. Moreover since the positive value is numerically greater than the negative value, waves travel faster with the wind than against the wind ; also the velocity of waves travelling against the wind is always less than W. Case(ii). V> W V(l + W; when V > 2 W this velocity is < W; and when V=ZW, the velocity of waves travelling with the wind is undisturbed. EXAMPLES. 1. A liquid of infinite depth is bounded by a fixed plane perpendicular to the direction of propagation of the waves. Prove that each element of liquid will vibrate in a straight line, and draw a figure representing the free surface and the direction of motion of the elements, when the crest of the wave reaches the fixed plane. 2. Prove that the velocity of propagation of long waves in a semicircular canal of radius a and whose banks are vertical, is 3. If two series of waves of equal amplitude and nearly equal wave-length travel in the same direction, so as to form alternate lulls and roughness, prove that in deep water these are propagated with half the velocity of the waves; and that as the ratio of the depth to the wave-length decreases from oo to 0, the ratio of the two velocities of propagation increases from to 1. 4. If a small system of rectilinear waves move parallel to and over another large rectilinear system, prove that the path of a particle of water is an epicycloid or hypocycloid, according as the two systems are moving in the same or opposite directions. EXAMPLES. 109 5. A fine tube made of a thin slightly elastic substance is filled with liquid; prove that the velocity of propagation of a disturbance in the liquid is (XO/ap)^ where a is the internal diameter of the tube, its thickness, X the coefficient of elasticity of the material of which it is made, and p the density of the liquid. 6. A horizontal rectangular box is completely filled with three liquids which do not mix, whose densities reckoned down- wards are cr l , a 2 , a s> and whose depths when in equilibrium are l-i, h, 1 3 respectively. Show that if long waves are propagated at their common surfaces, the velocity of propagation V must satisfy the equation /^ + /r, we obtain 112 RECTILINEAR VORTEX MOTION. Substituting from (2) and (4) in (1), we obtain _ __^^ = Q (5) dt dy dx dx dy) From this equation it follows that the molecular rotation is not a quantity which can be chosen arbitrarily; for ty must satisfy (5), and the corresponding value of is then determined by (4). When the motion is steady, none of the quantities are functions of t, and we obtain from (1) by Lagrange's method 2f = *"(*), where F is an arbitrary function, which agrees with 24. The pressure is determined by (31) of the same section. The current function at all points of the rotationally moving liquid is now determined by the equation V 2 ^+ J P'(^) = ........................ (6). At every point of the irrotationally moving liquid which sur- rounds the vortices, f = 0, and therefore (7) da? dy 2 Equations (4) and (7) show, that ty is the potential of in- definitely long cylinders composed of attracting matter of density /27r, which occupy the same positions as the vortices. 102. The integral fftdxdy is called the vorticity of the mass of rotationally moving liquid ; and we shall now show that the vorticity is an absolute constant. Draw any closed curve which completely surrounds all the rotationally moving liquid and does not cut any of it. Then since is zero at all points of the liquid where the motion is irro- tational, it follows that if r denote the vorticity, T = ftfdxdy, where the integration extends over the whole area enclosed by the curve. Substituting the value of from (3) we obtain by Green's theorem, = f(udx + vdy), SINGLE CIRCULAR VORTEX. 113 where the line integral extends round the closed curve. By | 27, this line integral is equal to the circulation K due to the whole of the rotationally moving liquid within the curve ; and by the same article, the circulation has been shown to be constant. Hence the vorticity is constant and equal to half the circulation. 103. We shall now consider the steady motion in an infinite liquid of a single rectilinear vortex, whose cross section is a circle of radius a, and whose molecular rotation is constant. In order that the cross section may remain circular, it is necessary that -^r should be a function of r alone. Denoting the values of quantities inside the vortex by ac- cented letters, equations (4) and (7) become ?=0 ..................... (8), dr* r dr which gives the value of ty inside the vortex, and ^ + 1^ = (9) dr* + r dr which gives the value outside. The complete integrals of (8) and (9) are and ty = G log r + D. Now T|/ must not be infinite when r= 0, and therefore A = ; also at the boundary of the vortex, where r = a, ty' = i/r, dty'/dr = d^Jr/dr ; whence B- a 2 = (7 log a + D -fa=0, and therefore C = - 0? = - O-/TT = - mjir, where cr is the area of the cross section, and ra is the vorticity of the vortex. The constant D contributes nothing to the velocity, and may therefore be omitted, whence i^ = K(a-r)-(m/ir)loga ............... (10), ty= (w/7r)logr ........................... (11). B. H. 8 114 RECTILINEAR VORTEX MOTION. Now - dtyjdr is the velocity perpendicular to r, whence inside the vortex & ........................ (12), which vanishes when r 0, and outside - d-^r/dr = m/7rr ........................ (13). Hence a single vortex whose cross section is circular, if existing in an infinite liquid, ivill remain at rest and will rotate as a rigid body. It ivill also produce at every point of the irrotationally moving liquid with which it is surrounded, a velocity which is perpendicular to the line joining that point with the centre of its cross section, and which is inversely proportional to the distance of that point from the centre. 104. Outside the vortex, where the motion is irrotational, a velocity potential of course exists. To find its value we have d(f> d-^r my d _ dty _ mx dx dy TiT 2 ' dy dx -rrr 2 ' whence d> = - \ y x ~ x y = (mlir) tan" 1 ylx .......... (14). TT J x 2 + y 2 It therefore follows that is a many-valued function, whose cyclic constant is 2m. The circulation, i.e. the line integral f(udx + vdy), is zero when taken round any closed curve which does not surround the vortex, and is equal to 2m when the curve surrounds the vortex ; whence if K be the circulation, m = $K, and the values of and -fy* may be written < = (/c/2?r) tan" 1 yjx, ty = (/c/2-Tr) log r. 105. The investigations of the last two articles are kinematical ; we shall now calculate the value of the pressure within and without the vortex. Let the values of the quantities inside the vortex be distin- guished from those outside by accented letters. Outside the vortex p/p=C--$q*, and since < = 0, and q = m/Trr = /c/2?rr, we obtain p/p = C- K 2 /87rV 2 ..................... (15), SINGLE CIRCULAR VORTEX. 115 whence if II be the pressure at an infinite distance n K 2 p p S-TrV 2 The equation of motion inside the vortex is (16). p dr r 4-7T 2 a 4 ' t) /eV 2 P whence = h (17), p 87r 2 a 4 p where P is the pressure at the centre of the vortex. At the surface of the vortex where r = a, p =p' , whence P/p = U/p - 2 /47r 2 a 2 (18), ft II 2 / 1 st \ and therefore = - (1 -(19)- p p 47r 2 a 2 V 2a 2 / Hence if II < /c 2 p/47r 2 a 2 , p' will become negative for some value of r < a, which shows that a cylindrical hollow will exist in the vortex, which is concentric with its outer boundary. When there is no hollow, equations (16) and (19) show that the pressure is a minimum at the centre of the vortex, where it is equal to II /c 2 p/47T 2 a 2 , and that it gradually increases until the surface is reached, at which it is equal to II 2 p/87T 2 a 2 , and that it then continues to increase to infinity, where its value is II. It is also possible to have a hollow cylindrical space, round which there is cyclic irrotational motion. Such a space is called a hollow vortex. The condition for its existence requires that p = when r = a, and therefore by (16) H = K 2 />/87r 2 a 2 . This equation determines the value of the radius of the hollow, when the pressure at a very great distance is given. 106. Kirchhoff has shown that it is possible for a vortex whose cross section is an invariable ellipse, and whose molecular rotation at every point is constant, to rotate in a state of steady motion in an infinite liquid, provided a certain relation exists between the molecular rotation and the angular velocity of the axes of the cross section. 82 116 RECTILINEAR VORTEX MOTION. The current function is evidently equal to the potential of an elliptic cylinder of density f/271-. Let a and b be the serai-axes of the cross section, then the value of -fr inside the vortex may be taken to be +' = D - (A* + By*)l(A + B}, where A, B, D are constants, for this value of ty' satisfies (4). Let x = c cosh 77 cos %,y = c sinh 77 sin , where c = (a 2 ft 2 )*, and let 77 = ft at the surface ; the value of ty' becomes TJT = D - fc s (A cosh 2 1] cos 2 + B sinh 2 rj sin 2 )/( J. Also let the value of ty outside the vortex be ijr = A'e~*> cos 2f + When 77 = /3, we must have tyty' = const., d-^rfdr) Therefore A 'e~* = - i fc 2 ( J. cosh 2 /3 - 5 sinh 2 yS)/(^l + J5) and A'e-v = $&* (A - B) sinh cos Whe: ^'fa - bY- - &( A <* 2(A+ Therefore Aa = Bb and tf = D- Z(bx> + ay*)/(a + b). Let a> be the angular velocity of the axes ; u, v the velocities of the liquid parallel to them, then (a + b), (a + b). The boundary condition is .dF .dF *a? + *^.-* where F = (a;/a) 2 + (y/6) 2 -1 = 0. Whence therefore w = 2ab^/(a + 6) 2 . We therefore obtain x = aooy/b, y = bwxja, the integrals of which are * = La cos (tat + a), y = Lb sin ( (^ + f) - \M log {f +(as- c) 2 } [f + (a + c) 2 } = A. 109. If two opposite vortices of vorticities ra and m are present in the liquid, the vortices will move perpendicularly to the line joining them with velocity M/2c, where 2c is the distance between them. 1 Greenhill, "Plane Vortex Motion," Quart. Journ. vol. xv. p. 20. MOTION OF TWO VORTICES. 119 In this case there is evidently no flux across the plane which bisects the line joining the vortices, and which is perpendicular to it ; we may therefore remove one of the vortices and substitute this plane for it. Hence a vortex in a liquid which is bounded by a fixed plane will move parallel to the plane, and the motion of the liquid will be the same as would be caused by the original vortex, together with another vortex of equal and opposite vorticity, which is at an equal distance and on the opposite side of the plane. This vortex is evidently the image of the original vortex, and we may therefore apply the theory of images in considering the motion of vortices in a liquid bounded by planes. 110. If there is a vortex at the point (x, y) moving in a square corner bounded by the planes Ox, Oy, the images will consist of two negative vortices at the points ( x, y}, (x, y), and a positive vortex at the point ( x, y)\ for if these vortices be substituted for the planes, their combined effect will be to cause no flux across them. Since the vortex is incapable of producing any motion of translation upon itself, its motion will be due solely to that pro- duced by the combined effect of its images ; whence, M My Mx- x = 2y 2 (a? + f) 2y (a? + y 2 ) ' M MX Mtf 2x 2(x 2 + y 2 ) 2x (x 2 + y 2 ) ' therefore x/x 3 + yjy 3 = 0, whence a 2 (x 2 + y-) = x 2 y 2 , or r sin 20 = 2a, 120 KECTILINEAR VORTEX MOTION. which is the curve described by the vortices. The curve in question is the reciprocal polar with respect to its centre of a four-cusped hypocycloid ; but it also belongs to the class of curves called Cotes' Spirals, which are the curves described by a particle under the action of a central force varying inversely as the cube of the distance. Since the vortex describes the spiral in exactly the same way as a particle would describe it, if repelled from the origin with a force 111. The method of images may also be applied to determine the current function due to a vortex in a liquid, which is bounded externally or internally by a circular cylinder. Let H be the vortex, a the radius of the cylinder, OH = c ; and let S be a point such that OS=f=a 2 /c, then the triangles SOP and POH are similar, therefore SPO = OHP, OPH = OSP, also OSP + SPA = OAP = OP A = OPH + HPA, therefore SPA = HP A. Let us place another vortex of equal and opposite vorticity at S, then the velocity along OP due to the two vortices is ~s But sin HPO sin HPO sin SPO ~ sin OHP = OH/a = HP ISP, hence u = and there is no flux across the cylinder. VORTEX IN A CIRCULAR CYLINDER. 121 Hence the image of a vortex inside a cylinder is another vortex of equal and opposite vorticity situated on the line joining the vortex with the centre of the cylinder, and at a distance 2 /c fro m the centre, and the vortex will describe a circle about the centre with a velocity The current function of the liquid at a point (r, 6) within the cylinder is r 2 + c 2 - 2rc cos 6 When the vortex is situated outside the cylinder, the image consists of a vortex of equal and opposite vorticity at H, together with a vortex of equal vorticity at 0. The latter vortex does not produce any alteration in the normal velocity at the surface of the cylinder, and its existence arises from the fact that the circulation round any closed curve which surrounds the cylinder must remain unaltered. The circulation due to vortex at H is 2irM, whilst that due to the vortex at is 2-Tr.lf, so that the two circulations cancel one another. 112. We have shown that the velocity potential due to a source is m log r ; hence if we have a combination of a source of strength m, and a vortex of vorticity m , the velocity potential due to the two is < = m log r + M tan" 1 y/x, where M = m'/Tr. Whence mac My _ my + MX ''~ ' An arrangement of this kind is called Rankines free spiral vortex. In order to find the stream lines let us transfer to polar co- ordinates, and we find dr_m d6 _M dt~r' T dt~^' whence if m/M = a, we obtain r = Ae a9 , and therefore the stream lines are equiangular spirals. 122 RECTILINEAR VORTEX MOTION. 113. We shall conclude this Chapter by proving three funda- mental properties of vortex motion. We have defined a vortex line to be a line whose direction coincides with the direction of the instantaneous axis of molecular rotation. If through every point of a small closed curve a series of vortex lines be drawn, they will enclose a volume of fluid which may be called a vortex filament, or shortly a vortex. We have shown that if the forces which act on the fluid have a potential, and the density is a function of the pressure, the motion of the fluid constituting the vortex can never become irrotational. It will now be shown that every vortex possesses the following three fundamental properties : (i) Every vortex is always composed of the same elements of fluid. (ii) The product of the molecular rotation of any vortex into its cross section is constant with respect to the time, and is the same throughout its length. (iii) Every vortex must either form a closed curve, or have its extremities in the boundaries of the fluid. To prove the first proposition, let P and Q be any two adjacent points on a vortex, co the molecular rotation at P. Then by the definition of a vortex line, PQ is the direction about which the rotation co takes place. Let P', Q' be the positions of P and Q at the end of an interval Bt ; then we have to show that P'Q' is the instantaneous axis of rotation at P'. Let x, y, z be the coordinates of P ; u, v, w the velocities of the element of fluid which at time t is situated at P. If PQ = h, the coordinates of Q are evidently x + h/co, y + hrj/co, z + h^/co ; also since u = F (x, y, z, t), it follows that if u , v l , w be the velocities of Q, u l = F(x + kg/to, y + hy/co, z + h/co, t) h / ..du du . ^du\ co \ dx dy hp co by 20. PROPERTIES OF VORTEX MOTION. 123 The coordinates of P' are x + uSt, y + v&t, 2 + w8t, and those of Q' are - - (p dt = x + u8t + hp? wp by (20), where />' is the density, and ', ?/, ' are the components of molecular rotation at P'. Hence if h' denote the length of P'Q', and X', //, v its direction cosines, then \'h' = hpg'ftop', /j.'h' = hprj'/wp', v'ti = hp^'/wp (21), whence X'/f = pf^' = i/'/f, which shows that P'Q' is the instantaneous axis of rotation at P', and therefore P'Q' is the element of the vortex line, which at time t occupied the position PQ. This proves the first theorem. To prove the second theorem, square and add (21) and we obtain h' = hpw'/wp'. But since the mass of the element is constant phcr = p'h'a', whence cos edS = 0, where e is the angle between the axis of rotation and the normal to S drawn outwards. Now if we choose S so as to coincide with the surface of any finite portion of a vortex of small section, together with its two 124 RECTILINEAR VORTEX MOTION. ends, cos e vanishes except at the two ends ; and is equal to + 1 at one end, and 1 at the other ; hence &> l rf be the angle which any element 8s makes with the axis of x, the equation of motion is pySs = ~ (T, sin ) Bs + pYSs. Now sin < = dy/ds ; also since the displacement y is small, the curvature will also be small, and we may therefore put ds = dx. The tension T^ may also be regarded as constant throughout the length of the string, whence the equation of motion becomes d^_T 1 d-y dt* ~ p dx^ If the motion does not take place in a plane, we may resolve the displacements and forces into two components respectively parallel to the axes of y and z, and we shall thus obtain a second 138 VIBRATIONS OF STRINGS AND MEMBRANES. equation of the same form as (1), in which z, Z are written for y, Y respectively. 125. Let us now suppose that the length of the string is equal to I, and that there are no impressed forces ; also let tf^TJp (2). Equation (I) now becomes - *i- B daf To solve this equation, assume y = F(x)e an Substituting in (3) we obtain the solution of which is f = sin mx + D cos mx. The solution of (3) may therefore be written in the form y=^(G sin mx + D cos moo) e*"* 1 * ............ (4), where m is at present undetermined, and C and D are complex constants. The value of m will depend upon the particular problem under consideration. We shall now suppose that both ends of the string are fixed ; in this case the conditions to be satisfied at the fixed ends are, that y and y should vanish when x = and x = I. These conditions evidently require that D = 0, sin ml = ; from the last of which we deduce m = STT/l, where s is a positive integer. Writing G = A t,B and rejecting the imaginary part, the solution becomes 00 y = S (A g cos STrat/l + B s sin STrat/l) sin siroc/l ...... (5), and therefore the period r g of the sth component is given by _2Z_2^ /p ~m~7V 2V and the frequency FREQUENCY OF VIBRATION. 139 The gravest note corresponds to s=l, and therefore its frequency is f-L fL fl ~ 21 V P From these results we draw the following conclusions. (i) The frequency is inversely proportional to the length; and therefore if the string be shortened, the pitch of the note will rise, and conversely if the string be lengthened, the pitch will fall. We thus see why it is that in playing a violin different notes can be obtained from the same string. (ii) The frequency is proportional to the square root of the tension, accordingly if the string be tightened the pitch will rise. (iii) The frequency is inversely proportional to the square root of the density ; and therefore if two strings having the same lengths, cross sections and tensions, be made of catgut and metal respectively, the pitch of the note yielded by the catgut string will be higher than that yielded by the metal string ; also the pitch of the note yielded by a thick string will be graver than that of the note yielded by a thin string, of the same material, length and tension. If s be any integer other than unity, we learn from (5) that the displacement is zero at all points for which x = rljs, where r = 1, 2, 3, ... s I ; it therefore follows that, corresponding to the sth harmonic, there are s 1 points situated at equal intervals along the string, at which there is no motion. These points are called nodes. 126. The constants A and B depend upon the initial circumstances of the motion. Now the motion of dynamical systems of which a string is an example may be produced either by displacing every point in any arbitrary manner, subject to the condition that the connections of the system are not violated; or by imparting to every point an arbitrary initial velocity, subject to the same condition. Hence the most general possible motion is obtained by communicating to every point of the string an initial displacement, and an initial velocity. We shall now show that when the initial displacements and velocities are given, the constants A and B are completely determined. 140 VIBRATIONS OF STRINGS AND MEMBRANES. Let y , yo be the initial displacements and velocities. Then it follows from (5) that y = 2 t A s sin STTX/I ..................... (6), y = ^ 1 (sTra/l) B t sin STrxfl ............ (7). ri Now the integral I sin (STTX/I) sin (s'lrxjl) dx is equal to Jo zero if s and s' are different integers, and is equal to ^l if s = s'; whence multiplying (6) by sin STTX/I and integrating between the limits I and 0, we obtain S7TX , --dx ..................... (8). 2 f l 8 = |J Similarly from (7) 2 f l . . STTX , I y sin j ax ............ ..... ( ( J ). Jo I Since y , y are given functions of x, these equations completely determine the constants. We notice that B s is zero when the initial velocity is zero, and that A g is zero when there is no initial displacement. 127. As an example of these formulae, let us suppose that a point P, whose abscissa is b, of a string fixed at A and B, is displaced to a distance 7 and then let go. From x = to x = b, y = yx/b, and therefore for this portion of the string . , 27 f b . STTX , 27 / b s-rrb I . S7rb\ A * = TT I x sm ~T dx = IT --- cos 1- + 1 -, sm -T I- bl J o I b \ S7T I S 2 7T 2 I / From x = b to x = l, y = y (I x)/(l b), whence 27 [ l . STTX . A > *fip%) > s sin STTX/I ..................... (18), where s is a function of the time which satisfies (14), and whose value is therefore determined by (17). The quantities denoted by < g are called normal coordinates ; and we shall now prove that the expressions for the kinetic and potential energies do not contain any of the products of the normal coordinates. This is the characteristic property of these quantities. If T be the kinetic energy, we have Cl [I oo . T = %p I y z dx = ^p I {S (f> s sin S7rx/l} 2 dx. Jo Jo 1 KINETIC AND POTENTIAL ENERGY. 145 Since all the products vanish when integrated between the limits, we obtain T=lpl i & ........................ (19). The potential energy is equal to the work done in displacing the string to its actual position. In order to calculate its value, let the string be held in equilibrium in its actual configuration at time t by means of a force Y applied at every point of its length. The value of this force per unit of mass is equal to _Tidfy p da?' by (1). Let SF be the work which must be done by this force in order to displace every element of the string through a space By ; then the work done upon an element Bs and therefore since Bs = 8x, the whole work done is Integrating by parts, and recollecting that By = at both ends, we obtain g r<^ ry^y 'Jo eta dx 'Jo \dooj ri /v7x + -=- Tex -3- 8x, dx\ * dxj dy \ dy) which becomes ( }> if = C7r(m 2 +w 2 )Va. The gravest note is obtained by putting m = n = 1, and corresponding to this note there are no nodes. In the next place we shall determine the nodal lines cor- responding to vibrations whose frequency is ^c^/5/a. Here V5 = V( 2 + w 2 ), which requires that m = 2, n = l or ra = l, ?i = 2; and therefore the complete vibration corresponding to this period is, w = (C sin 27rx/a sin Try fa + D sin TTX/O, sin 2,7ryfa) cos pt. In this expression C and D depend solely upon the initial circumstances of the motion, and may have any values whatever consistent with the boundary conditions. If however we suppose 102 148 VIBRATION'S OF STRINGS AND MEMBRANES. that the initial conditions are such, that the ratio C/D has an assigned value, we may obtain a variety of special cases. (i) Let D = 0. The nodal system now consists of the line x = $a, which bisects the membrane. (ii) Let (7 = 0, and we have a nodal line y=^a, similarly bisecting the membrane. (iii) Let C = D; then the value of w may be written w = 4(7 sin TTX/a sin Try /a cos \ir (x + y)/a cos TT (x y)/a cos pt. This expression vanishes when, x = a, y = a, sc + y=a, x y = a. The first and second equations correspond to the edges ; the fourth must be rejected, because it does not represent a line drawn on the membrane ; and the third represents one of the diagonals of the square. Since a nodal line may be supposed to be rigidly fixed without interfering with the motion, the preceding solution determines the frequency of the gravest note of a right-angled isosceles triangle. (iv) Let G= D, and we shall find that the nodal line is y = x, which represents the other diagonal of the square. For further examples in this branch of the subject, the reader is referred to Chapter IX. of Lord Rayleigh's treatise. 136. The motion of a circular membrane, which is the best representative of a drum, cannot be solved by elementary methods. The simplest case of all, is when the vibrations are symmetrical with respect to the centre, so that (22) becomes d?w _ 2 /d-w I dw ~ ~ ,dr 2 r dr , and if we put w = F(r) &<*, the equation for F is dW IdF dr* r dr This equation cannot be integrated in finite terms. The two solutions are called Bessel's functions 1 , from the name of their 1 During recent years the ungrammatical phrase Bessel functions has begun to creep into mathematical literature. The use of a proper noun as an adjective is a violation of one of the most elementary rules of grammar ; and in cases where it is not possible to form the corresponding adjective without introducing a cumbrous and inelegant term, the genitive case of the proper noun ought always to be employed. EXAMPLES. 149 discoverer; and the investigation of their properties constitutes an important branch of analysis. Algebraic solutions may how- ever be invented, by supposing that the density, and therefore c, is a function of r. EXAMPLES. 1. A string of length 1 + 1', is stretched with tension T between two fixed points. The linear densities of the lengths I, I' are ra, m' respectively; prove that the periods T of transverse vibrations are given by m'* tan (^Trlm^rT^ = m* tan 2. Investigate the motion of a string of length I, which is initially at rest in a straight line, each extremity of which is subject to the same obligatory motion y = k sin mat. Show that if a sufficient period be allowed to elapse for the natural vibrations to subside, the position of the nodes will be given by the equation 2m# = ml + (2i + 1) TT, where i is any integer. 3. A uniform string in the form of a circle of radius a, rests on a smooth plane under a central repulsion, whose value at distance r is ga n /r n . Show that if the string be slightly displaced, so that it is initially at rest and in the form of the curve r = a + 2 the area of the cross section at P. Also let X, Y, L be the tangential and normal components of the impressed forces and the couple at P, per unit of mass, measured in the directions of T, N, 0. The equations of equilibrium of the wire, are obtained by resolving all the forces along the tangent and normal at P, and taking moments about this point ; whence 152 FLEXION OF WIRES. T - (T + ST) cos 80 + (N + 8N) sin 8 + aa>X8s = 0, N-(T+ 8T) sin 8-(N+ 8N) cos 80 + - (1) - = \- N = acoL as 138. We must now find an expression for the flexural couple G. The curve which passes through the centre of inertia of each cross section, is called the axis of the wire. When a wire is bent in such a manner that its curvature is increased, the filaments into which the wire may be conceived to be divided, which lie on the outer side of the axis, will usually be extended ; and those which lie on the inner side will usually be contracted, whilst the axis itself undergoes no extension nor contraction. Cases of course may occur, in which the axis undergoes extension or contraction, and when this is the case, the difficulties of the problem are greatly increased, and cannot be satisfactorily discussed without a knowledge of the Theory of Elasticity. We shall therefore confine our attention to the case in which the extension or contraction of the axis is so small (if it exists), that it may be neglected. In the figure, let AE be the axis of the wire, PQ any filament, whose distance from AE is h, the centre of curvature at B; also let these points after deforma- tion be denoted by accented letters. It can be proved in a variety of ways 1 that the tension T' at P' due to the action of con- tiguous portions of the wire is equal to the product of the extension of the element PQ and a physical constant called Young's modulus. If k is the resistance to compression, n the rigidity of the substance of which the wire is made and q See Thomson and Tait's Natural Philosophy, 6823. BENDING MOMENT. 153 Young's modulus, it is shown in treatises on Elasticity that q = 9nk/(3k + ri). Hence if a- be the extension T' = q* .............................. (2). Now if p, p be the radii of curvature before and after de- formation P'Q'_p' _= _ AB p A'E'~ p' Since we assume that the extension of the axis may be neglected, AB = A'B'; whence neglecting A 2 , &c., P'Q'-PQ ,/i i and p p Accordingly where /c 2 , whence the first two of (1) become and therefore -*-. the integral of which is T= C cos $ + D sin , whence N = G sin + D cos <. Let t be the tension of the string, and a and TT a the values of $ at the two extremities, then t cos a = N= C sin a + D cos a, t sin a = T, = C cos a + D sin a, therefore (7=0, D = -t. Writing A = q/c 2 a), and remembering that p = oo , since the natural form of the wire is straight, we have G = A/p', whence the third of (1) becomes Integrating, we obtain 2A ^ tsm = E. Since G = when = a, E = t sin a, whence if A/t = a* - = _ ... (5)> d(f) (sm - sin a)* which determines the intrinsic equation to the curve. We may also integrate (4) in a different manner, for if the string be the axis of x, and its middle point be the origin, cos = dy/ds, THE ELASTICA. 155 and therefore (4) may be written ds p whence p'y = a 2 ; .............................. (6), no constant being required because p'~ l = when y = 0. AT , ds dy dy Now ==ec whence integrating (6) again y z = 2a 2 (sin $ - sin a) (7). The forms of the various curves which the wire is capable of assuming, are shown in Thomson and Tait's Natural Philosophy, Part n. p. 148. If a lies between and TT, the maxima values of y are obtained by putting = ^TT, and are therefore equal to + 2a sin (?r Ja). The form of the curve is shown in the figures 1, 2 or 3 of that work ; and if the curve be bent upon itself and the slight torsion be neglected, the forms are shown in figures 4 and 5. In all these cases except the first, in which the wire is bent into the shape of a bow, the maximum value of y is numerically equal to its minimum value. If, however, a lies between TT and 2?r, we may put it equal to TT + /3, in which case (7) becomes 2/ 2 = 2a 2 (sin < + sin /3). In this case the maximum value of y occurs when < = ^TT, and is equal to 2a cos (^TT 1-/3), and the minimum when the form of the curve is shown in fig. 7. The constants a and a are capable of being determined when the lengths of the wire and string are given ; and the equation of the curve in Cartesian coordinates can also be obtained, but to do this a knowledge of elliptic functions is required 1 . If, however, a = |7r, the integral in an algebraic form can be obtained; for since tan = dx/dy, (7) becomes # = = _ (4 a 2 _ y2)i + a log {2a/y + (4a 2 /y 2 - 1)*} + 0. 1 Greenhill, Mess. Math. vol. vm. p. 82. 156 FLEXION OF WIRES. Since y = a^2 when x = 0, C= whence x = (4a 2 - y^ - a V2 + a log It will be noticed that this curve is the same as that described by an elliptic cylinder, in the limiting case between oscillation and rotation. See page 75. 141. Equation (5) enables us to prove a theorem discovered by Kirchhoff, and which is known as Kirchhoff's kinetic analogue. The theorem is, that if a point move along the elastica with uniform velocity, the angular velocity of the tangent at that point, is the same as that of a pendulum under the action of gravity. If V be the' velocity of the moving point, (5) may be written d V j7 = 7o ( sm - Sm a ) at a\/2 If we put x i 71 " + $> a = |TT + /3, dy V this becomes -r L = TK (cos y cos p)*, at a\ '2 which is the equation of motion of a common pendulum, whose length is equal 4ga z /V 2 . Stability under Thrust. 142. When a thin straight wire or column is subjected to a pressure or thrust, which is applied at one extremity in the direction of its length, experiment shows that as soon as the thrust exceeds a certain limit the wire commences to bend. There are several methods by which this limiting value can be obtained, but the following is perhaps the simplest. Let a thrust P be applied at one extremity of the wire ; and let P be the limiting value of P which is just sufficient to produce bending. Then if P is less than P , no bending will take place ; but if P is slightly greater than P , the wire will assume a sinuous form which differs very little from a straight line. We must therefore solve the equations of equilibrium on the hypo- thesis that the wire is not absolutely straight in its configuration of equilibrium, but assumes a slightly sinuous form ; and we shall find that our solution leads to a certain equation of the form P = a, where a is a quantity upon which the sinuous curve assumed STABILITY UNDER THRUST. 157 by the wire depends. Now P may have any value we please ; if therefore we assign a value to P which is less than the least value of a, the equation P = a cannot be satisfied, which shows that equilibrium in the sinuous form is impossible. Hence the minimum value of a is the limiting value of P, which is just sufficient to produce bending. 143. Let -or denote the curvature of the wire when slightly bent ; then -sr = p /-1 , p~ l = 0,0 = Ats, and equations (1) become -*- <*> T S +T -= Q < 9 >' A + N =0 . ...(10). ds From (8) and (10) we obtain dT d _ ds ds whence rn T) i A 2 /I 1 \ J_ = JD -k -O.TS ^11^. Since -or is a very small quantity throughout the length of the wire, the constant B may be put equal to P, where P is the thrust applied to the end of the wire; accordingly (11) may be written T= P ^-A^ ...(12). Owing to the smallness of -ar, we may neglect its cube ; whence eliminating N from (9) and (10) we get A d* r> A h Pv = 0, the integral of which is iy = C cos us + D sin/is (13)> where fj? = P/A. We have now three cases to consider. Case I. Let the lower end A of the wire be firmly clamped, whilst the upper end B is pressed vertically downwards by a force P, but is otherwise free ; also let I be the length of the wire, and let the arc s be measured from A. 158 FLEXION OF WIRES. At the end B, G and therefore is are zero ; whence C cos pi + D sin pi = ..................... (14). Also by considering the equilibrium of the whole wire, it follows that JV=0 at A, whence by (10) d-sr/ds = Q when s = 0, accordingly D = 0. This requires that cos pi = 0, whence which gives (15). The least value of the right-hand side of (15) occurs when n = ; and this gives the thrust P which must be applied to the upper end of the wire to produce an infinitesimally small deflection ; if, therefore, the thrust is less than this quantity, no deflection will" take place and the wire will remain straight. Whence the condition of stability is that P<^A/l 2 ........................ (16). Case II. Let the wire be pressed between two parallel planes which are perpendicular to its undisplaced position. If the planes were perfectly hard, smooth and rigid (a condition which can only be approximately realized in nature), the ends of the wire would tend to slip on the slightest pressure being applied; we shall therefore suppose that the ends are in contact with mechanical appliances which will prevent any such slipping taking place, but are otherwise free. Under these circumstances the terminal conditions are or = when s = and s = I. Whence by (13) (7=0, sin pi = 0, pi = TT, and the condition of stability is that PAIP ........................... (17). Case III. Let both ends of the wire be clamped. The ter- minal conditions require that the values of N and or at the two ends should be equal to one another. Consequently D (1 cos fd) = C sin pi, C (1 cos pi) = D sin pi. Eliminating C and D we obtain and the condition of stability is (18). STABILITY OF A COLUMN. 159 The value of tr may now be written OT = G COS 27TS/I, which shows that there are two points of inflexion, which occur when s = %l and s = f I. The first case corresponds to a column or pillar whose lower end is cemented into a bed of concrete, whilst the upper end supports a building which simply rests upon but is not fastened to the pillar; and we see that in this case the weight required to cause the pillar to collapse is less than in the other two cases. The second case corresponds to a pillar or rod both of whose ends rest on bearings to which they are not cemented. The third case corresponds to a pillar whose ends are respectively cemented to the foundations and to the building supported. In the third case, the force required to cause the pillar to collapse is four times greater than in the second and sixteen times greater than in the first case. 144. Another interesting problem is the greatest height of a rod or column consistent with stability. Let the lower end of a vertical rod be firmly fixed; then it can be shown by experiment that the rod will bend when its length exceeds a certain limit. To find this limit, we shall sup- pose that the rod when slightly bent is in equilibrium. Since dx = ds, and p~ l = vs, equations (1) become dT -Nv= 9 dN A + N = dx From the first and third we obtain (19). where B is a constant. At the free end, where x = l, T and must vanish ; whence Substituting this value of T in the second of (19), eliminating N from the third and neglecting squares of -ar, we obtain A d ~=a= dv/dx, and since is very nearly equal to \TT, $ = dv/dx, and therefore (24) becomes d s v Ar , d 3 v ,-.. + N = J Q ^ 149. For the complete discussion of these equations we must refer the reader to Chapter vm. of Lord Rayleigh's treatise ; but one or two special cases may be noticed. If the wire is so long that it may be treated as infinite, we may neglect the conditions to be satisfied at its extremities. If therefore the vibrations consist of waves of length \, we may assume as a solution of (28) that v is proportional to Substituting in (28) we obtain, and therefore the frequency is 150. We shall now investigate the flexural vibrations of a wire 1 of length I. Taking the origin at the middle point of the wire, we may assume v = U exp (ifcbmH/l*), where U is a function of x, and m is a constant whose value has to be determined. Substituting in (28) we obtain dx*~ I* ' To solve this equation assume U exp (pmx/l), and we see that the values of p are the four fourth roots of unity, viz. 1, 1, i, i. The solution may therefore be written U = A sin mxjl + B sinh mac/I + G cos mx/l + D cosh mxjl ............... (30). 151. We have now three cases to consider. (i) Let both ends of the wire be free. The first of (29) requires that A sin mx/l + B sinh mx/l G cos mx/l + D cosh mxjl = 0, 1 Greenhill, Mess. Math. Vol. xvi. p. 115; Lord Eayleigh, Theory of Sound, Ch. vin. 112 164 FLEXION OF WIRES. when x = 1. This equation of condition may be satisfied in two different ways ; we may first suppose that C = D = ; and -A sin w-f -Bsmhra = ............... (31), or that A = B = 0, and shim = ............... (32). The first solution corresponds to the first line of (30), which is an odd function of x, and may therefore be called odd vibrations ; whilst the second solution corresponds to the second line of (30), which is an even function of x, and may be called even vibrations. We thus see that the odd and even vibrations are independent of one another. Taking the case of the odd vibrations, the second of (29) requires that A cos \m + B cosh \m = 0, and therefore by (31) m = tan |m ..... , ............... (33). For the even vibrations the second of (29) gives C sin \m + D sinh \m = 0, and therefore by (32) tanh \m = tan \m .................. (34). Equations (33) and (34) determine the values of m for the odd and even vibrations respectively, and consequently the frequency of the different notes can be found. (ii) Let both ends of the wire be clamped. In this case the conditions to be satisfied at the ends are, that U = 0, dU/dx = ..................... (35), the first of which expresses the condition that the displacement at each end should be zero, and the second that the direction of the axis should be unchanged. The solution for this case may evidently be obtained by integrating the results of case (i) wire twice with respect to x, and consequently the values of m for the odd and even vibrations are given by (33) and (34). EXTENSIONAL VIBRATIONS. 165 (iii) Let the wire be clamped at x = ^l, and free at x = ^l. When x \l equations (35) have to be satisfied ; and when x=\l, the conditions are given by (29). Taking the value of U given by (30) and writing out the four equations of condition in full, it will be found that they can be satisfied in two ways, i.e. either, B =(7 = 0, A sin \m + D cosh \m = 0, A cos \m D sinh \m = 0, which gives tanh \m = cot \m (36), or, A = D = 0, B sinh \m + C cos ^ m = 0, B cosh ra + C sin ^ m = 0, which gives tanh |ra = cot \m (37). Equations (33) and (34) are both included in the equation cos m cosh m=l (38), and (36) and (37) in the equation cos m cosh m = 1 (39). For a discussion of the roots of these equations, we must refer to Lord Rayleigh's Theory of Sound, Chapter vni. and to Prof. Greenhill's paper. Extensional Vibrations. 152. The equation of motion for extensional vibrations of a straight wire may be obtained immediately from the first of (22). In this case p = , ds = doc, and therefore dT_ d?u u being the longitudinal displacement. But T**fm&-, whence putting q/cr = 6 2 , we obtain cte*~ dp' 1(56 FLEXION OF WIRES. which is the same equation which we have obtained for the trans- verse vibrations of a string. The condition to be satisfied at a fixed end is that u = ; whilst the condition to be satisfied at a free end is that T = 0, or du/dx = 0. In the case of a wire of infinite length, which propagates waves of length \, we must put ^ _ 2ura;/A4- ipt and therefore p = 27T&/A, = - A / - . A< y O" In the corresponding case of flexural vibrations of the same wave-length, 27TK /g X 2 V0-' whence p'/p = K/\. Since X is usually very much greater than K, which is the radius of gyration of the cross section, we see that the pitch of notes arising from extensional vibrations is usually much higher than that of notes arising from flexural ones. It will be observed that when the vibrations of a straight wire are extensional, the displacement is parallel to the length of the wire ; hence such vibrations are sometimes called longitudinal vibrations. But if the natural form of the wire is curved, exten- sional vibrations usually involve the normal as well as the tangential displacement of the central axis. EXAMPLES. 1. A naturally straight wire AB, of which the end A is fixed, is lying on a smooth horizontal plane, and the other end is pulled with a force F, whose direction is perpendicular to the undis- placed position of the wire. Prove that the projection of any length AP on the undisplaced position AB is equal to (2F/A$ (V(cos ) - V(cos - cos <)}, where is the angle which the normal at P makes with AB, and $ is the value of < at the end B. EXAMPLES. 167 2. If a uniform horizontal wire, both of whose ends are fixed, be displaced horizontally, so that one half is uniformly extended, and the other half is uniformly compressed, prove that the displace- ment at time t of any particle whose abscissa is x is (4wZ/7r 2 ) 2 (2i + 1)- 2 cos (2t + 1) Trat/^l cos (2t + 1) Trx/21, where 11 is the length of the wire, the middle of which is the origin, and nl is the initial displacement of that point. 3. The extremities of a uniform wire of length I are at- tached to two fixed points distant I apart by springs of equal strength. Show that if the longitudinal displacement of the wire is represented by Pe imat sin (mxjl + a), the admissible values of m are given by the equation (my - Z> 2 ) tan m + Zmqlp = 0, where p, is the strength of either of the springs, and q the ratio of the tension to the extension in the wire. 4. An elastic wire, indefinitely extended in one direction, is firmly held in a clamp at the other end. If a series of simple transverse waves travelling along the wire be reflected at the clamp ; show that the reflected waves will have the same ampli- tude as the incident waves, but that their phase is accelerated by one quarter of a wave-length. 5. A heavy wire of uniform section is carried on a series of supports in the same horizontal plane, L r is the bending moment at the rth point of support, l r the distance between the (? l)th and the rth support, and m the mass of the wire per unit of length ; prove that Lr-Jr + 2L r (l r + l r+l ) + L r+l l r+1 = 6. Prove that if an elastic wire of length I, with flat ends, im- pinges directly with velocity V on a longer wire at rest, of length nl and of the same material and cross section, also with flat ends, the first wire will be reduced to rest by the impact ; and the second wire will appear to move with successive advances of the ends with velocity V for intervals of time 2l/a, and intervals of rest of Z(n 1) I/a, a denoting the velocity of propagation of longitudinal vibrations. 168 FLEXION OF WIRES. 7. An elastic rod of length I lies on a smooth plane, and is longitudinally compressed between two pegs at a distance I' apart. One peg is suddenly removed ; prove that the rod leaves the other peg just as it reaches its natural state, and then proceeds with a velocity equal to V (I l')/l, where V is the velocity of propagation of a longitudinal wave in the rod. 8. A metal rod fits freely in a tube of the same length, but of a different substance, and the extremities of each are united by equal perfectly rigid discs fitted symmetrically at the end. Show that the frequencies of the notes emissible, which have a node at the centre of the system, are given by xjZTrl, where 21 is the length of the rod or tube, and n is a root of the equation 2Mx = ma cot x/a + mfa cot x/a' ; where M, m, m are the masses of a disc, the wire, and the tube, and a, a' are the velocities of propagation of sound along the wire and the tube. 9. Two equal and similar elastic rods A G, EG are hinged at C so as to form a right angle, while their other extremities are clamped. One vibrates transversely and the other longitudinally ; prove that the periods are 2P/f 2 0' 2 , where 6 is given by the equation 1 + cosh 6 cos 6 + (sin d cosh Q - cos Q sinh d) (gljf z d) cot (0*f 2 /gl) = 0, where I is the length of either rod, and / g are two constants depending on the material. 10. The natural form of a thin rod when at rest is a circular arc, and the rod makes small oscillations about this form in its own plane. Assuming that the couple due to bending varies as the change of curvature, and that the tension follows Hooke's law, prove that if the arc be a complete circle the periods ZTT/P are given by the quadratic, p 4 - (6 (n 2 + 1) + an 2 (n? - 1)} p 2 + abn 2 (n 2 - 1) = 0, where n is any integer, and a, b are two constants which depend upon the moduli of stretching and bending, and on the radius of the circle. EXAMPLES. 169 11. If in the last example the arc be not a complete circle, but have both ends free and be inextensible, show that it can be made to vibrate symmetrically about its middle point by suitable initial conditions in a period 2ir/p, provided the angle 20, which the arc subtends at its centre, satisfies the equation 2 (f + 1) (V 2 ~ ?" 2 ) cot tf + , Srj be the angles of contingence and torsion at P, so that POP' 8(j), OP'O' = 877 ; let p, a- be the radii of principal curvature and torsion at P. Also let T + 8T, Ni + SN^ &c. be the values of the resultant stresses at P'; and K, "jB, %> ; 1L, Jl, ffi the components of the bodily forces and couples per unit of length of the wire. The equations of equilibrium are obtained by resolving all the forces and couples which act upon PP' parallel to Px, Py, Pz. Resolving the forces parallel to Pz, we get (T + BT) cos 8 < f>-T-(N l + &NJ cos 877 sin 80 + (N 2 + SN 2 ) sin 8< sin 877 + 5Ss = 0, or BT - N<> + 58s = 0. Treating all the other forces and couples in the same way, we obtain the following six equations : as ds cos 0', where 8&> is the rotation due to bending, FLEXTJRAL COUPLE. 173 about a line through Q perpendicular to the plane of bending. Now if C' be the centre of curvature in the plane PCQ after bending, So = PG'Q - PCQ = PQ (-, - -}, \P pJ whence the displacement of q along pq is r$ = A ( J sin tf> G v = cos d> = A ( ) cos \P P (5), .(6). Let QO, QO' be the principal normals at Q before and after bending ; 0, 0' the centres of principal curvature ; also let CQO = x, CQO' = x- Let R x , R x ', R y , R y ' be the radii of cur- vature before and after bending in planes perpendicular to Qx, Qy, and let p lf pi be the radii of principal curvature before and after bending. Then 1 1 1 1 1 ; = -, cos y , P Pi = cos y. P Pi 11. 1 _ 1 ._ ^x pi Rx~~ P i sli x *r 1 1 i i "y PI y PI Since the curvature in the plane through the tangent which is perpendicular to the plane of bending is unchanged, .(8). TORSION AL COUPLE. From the first and second of (7) combined with (8) we get 11. 1 -p-, = sin q> -- cos

Jo J by (2) and (3). The work done by torsion is o o It therefore follows that if W be the potential energy per unit of length, 2 (12). Equilibrium of Naturally Straight Wires. 158. The preceding formulae can be simplified when the wire is naturally straight. In this case the curvature in every plane through the axis of the wire is zero before deformation ; and since the change of curvature in that plane through the tangent to the deformed wire which is perpendicular to the plane of bending is zero after deformation, it follows that the curvature in the above-mentioned plane is also zero after deformation. Hence the plane of bending is the osculating plane of the deformed wire. From this it follows that whence, by the fourth of equations (1), _ ds ~ or H = const., which shows that the torsional couple is constant throughout the length of the wire. This is a very important proposition. NATURALLY STRAIGHT WIRES. 177 159. We shall now proceed to integrate the equations of equilibrium of a naturally straight wire. Since H is constant and O l = 0, it follows that if CT denote the curvature, so that -& = l/p, equations (1) become Since G 2 = Ata, we obtain from (13) and (17) (13), dN, N 2 _>___ 2+7V = o ..................... (14), dN 2 N, -ds + ^ = ..................... (I*)* (17). dT dts -j- + A-ST -=- = 0, ds ds whence T = P - \A^ (18), where P is a constant. / A \ From (16) we get N 2 =(H j (19), and from (13) and (18) N^ = -A~ (20). ds From (19) and (20) combined with (15) we get H^-A(^} = 0, ds ds V a- J A whence = 1>H + (21), <7 <&* where Q is a constant ; accordingly by (19) ..(22). To obtain a third integral, substitute the values of T, N lt N* from (18), (20) and (22) in (14) and we get A^ + ^H'-PAjv-^ + lsAW^O (23). as* TX B. H. 12 178 THEORY OF CURVED WIRES. Integrating we obtain \'= -IAW+^P-IH^^+R^-Q*.. (24), where R is another constant. 160. From (24) we see that (d^jdsf is a cubic function of or 2 , and therefore or' 2 can be expressed in terms of s by means of elliptic functions of the first kind. Let \A*Z denote this cubic function; then collecting our results from (18), (21) and (24), we have the following three first integrals of the equations of equi- librium, viz. (25). The first of (25) merely determines the tension, but the second leads to important results. If the curve assumed by the wire is a plane curve, er = oo ; whence if Q is not zero, TX must be constant, and therefore the curve is a circle. If, however, Q is zero, a is constant, and therefore the curve assumed by the wire is one 01 constant tortuosity ; and if we suppose the curve to be plane, so that (30). T sm a + iv 2 cos a = cos a a a 2 J Equations (30) show that the resultant force F which must be applied to the ends of the wire must be parallel to the axis of the cylinder on which the helix is traced, and that its magnitude is H A F= cos a 7 sin a cos 2 a (31). a a 2 The resultant couple (ffi is & 2 = # 2 + ^cos 4 a (32). The resultant force and couple are therefore to a certain extent arbitrary, since both contain the torsional couple H, the only limitation on whose value is that it must not be large enough to break the wire or to produce a permanent set. We have there- fore two special cases to consider. 162. Case I. Let H = 0; then the terminal stresses consist of a pushing force or thrust P, whose value is P = A\d? . sin a cos 2 a, together with a flexural couple G 2 , whose value is A/a . cos 2 a. The pitch of the helix is sin" 1 (Pa/6r 2 ), from which we see that in order that equilibrium may be possible Pa must not be greater than # 2 . 122 180 THEORY OF CURVED WIRES. Case II. Let F=0, then A A H= sin a cos a = (33), a a whilst (ft = A\a . cos a. The torsional couple is therefore pro- portional to the tortuosity; also since H cos a G 2 sin a = 0, H sin a + G 2 cos a = A /a . cos a, it follows that the terminal stress consists of a couple whose axis is parallel to the axis of the cylinder on which the helix is traced, and whose magnitude is A/a. cos a. Stability. 163. In 143 we investigated the stability of a naturally straight wire which was subjected to longitudinal thrust ; we shall now investigate the stability when the wire is subjected to a torsional couple as well as a thrust. In Case L, in which one end is undamped, TS must vanish at the free end; and in Case II., in which both ends are undamped, CT must vanish at each end. Now (21) may be written whence if is vanishes anywhere, the constant Q must be zero. In Case III., both ends are clamped, and the tangents at the ends of the wire are consequently parallel ; hence there must be at least two points of inflexion, and consequently vf must vanish at two points of the wire. Hence in this case also Q = 0. Writing P for P in (23), and recollecting that nr 3 is to be neglected, the equation becomes where ^ 2= & + z ........................ (34)> whence -cr = C cos /is + D sin /&s, accordingly if the right-hand side of (34) is less than the least value of /A, equilibrium in the sinuous form will be impossible, and the straight form will be stable. STABILITY OF A TWISTED WIRE. 181 Case I. Here one end is undamped, whence or = 0, when s = I ; whilst at the clamped end dur/ds, which is proportional to N 1} must also vanish; whence D=Q, and cos/i = 0, therefore yu, = \ir\l, and the condition becomes H^ P -^ 4,A 2 + A < 4,1* ' Case II. Here CT = 0, when s = and s = I, whence the con- dition is H* P 7T 2 4,A* + A < I 2 ' Case III. Here OT and d^/ds must be equal when s = and s = I, and the condition is H 2 P 47T 2 4,A* + A < I 2 ' All these results agree with our former ones, as can be seen by putting H = 0. 164. A wire whose natural form is a tortuous curve is first unbent; secondly, the wire is twisted, and thirdly, the ends are joined together; it is required to find the condition that a circle may be a possible figure of equilibrium. When a circle is a figure of equilibrium none of the quantities can be functions of s ; we therefore obtain from (1 ) (35) H = const., 6^2 = const., N^ = H/a,\ The constancy of G 2 and H requires that the changes of curvature and twist which occur in passing from the natural to the circular form shall be constant quantities. These conditions will be satisfied if the natural form of the wire is a helix, which includes as a particular case a circular coil of fine wire, the radius of whose cross-section is small in comparison with the mean radius of the coil. 165. If in the last example the natural form of the wire is a straight line, the circular form will be unstable when the twist exceeds a certain limit. To find this limit, we observe that since the natural form is straight, it follows that in the circular form G z = A/a; whilst in the sinuous form H will still be constant and 182 THEORY OF CURVED WIRES. GI zero. But T and Nj, will be small quantities depending on the change of curvature ts, also we must write Accordingly equations (1) become dT N,_ -j u, as a _ 7 T^ V > as a, and p' = a, since the difference between p'~ l and a" 1 may be neglected when multiplied by T or N. These equations therefore become, measuring u in the opposite direction, so that u and $ increase together, dT AT d^~ N = ^f-r + T =-v} (36), Na = Q d(f> + the rotatory inertia being neglected. We must now find an expression for the change of curvature due to deformation. If R, 3> be the coordinates after displacement, of the point on the axis which was initially at (a, ), then R = a + v, 4> = < + u/a. 1 Mess. Math. vol. xix. p. 184. 2 Proc. Lond. Math. Soc. vol. xxm. p. 120 ; American Journal, vol. xvii. p. 315. 3 Crelle, vol. LXIII. ; Lord Kayleigh, Theory of Sound, 233. 184 THEORY OF CURVED WIRES. Hence if P be the perpendicular from the centre on to the tangent to the deformed axis, at the point in question, we have by a well-known formula, 1 = J. dP p R dR 1 1 r 1 dR Now r* =^n + .R 2 (1 + du/ady-\ ' also the displacements and their differential coefficients are all small quantities ; whence expanding and neglecting cubes of small quantities, the above equation becomes , D /- 1 d*v\ dv ,, whence dP =1 --- 7 -^ dd>. \ a d$>) d(j> Also dR = -=-; dd>, d c 111 /d 2 v /0>7X '- = -* +V .................. (37) ' which determines the change of curvature in terms of the normal displacement. We must next find the condition that the axis undergoes no extension. The elementary arc ds' of the deformed surface is given by the equation ds' 2 = (dvj + (a + v? (d + dujaf, and since this is equal to a 2 r&/> 2 , we obtain, neglecting squares of small quantities, which is the condition of inextensibility. Substituting from (37) and (3) of 138 in the last of (36) we obtain d s v dv VIBRATIONS OP A CIRCULAR RING. 185 From the first two of (36) we obtain d /d 2 N , T \ (d z v dii U /V I = fTfl.M I - J- - _ ' W by (38), whence eliminating N we obtain \ A /or\ = ........... (39). , 9 , a a 4 \d< 2 / d* \d(f> 2 To solve this equation, assume that v oc <-pt+^ i an( j we obtain qjSs^-YT o-a 4 (s 2 + l) If the wire is a complete circle, v must necessarily be periodic with respect to , and therefore s must be an integer, unity and zero excluded. We therefore see that there are an infinite number of modes of vibration, whose frequencies are obtained by putting s = 2,3,4... in (40). If the wire is not a complete circle, s is not an integer ; its values in terms of p are the six roots of (40), but since p is unknown, another equation is necessary. This equation is obtained by considering the boundary conditions to be satisfied at the free ends, which are that T, N and G should vanish there. These conditions will furnish six additional equations, by means of which the six constants which appear in the solution of (39) can be eliminated, and the resulting determinantal equation combined with (40), will determine the frequency 1 . 1 See Lamb, "On the flexure and the vibrations of a curved bar," Proc. Land. Math. Soc. vol. xix. p. 365. CHAPTER X. EQUATIONS OF MOTION OF A PERFECT GAS. 167. WE have already called attention to the fact, that air is the vehicle by means of which sound is transmitted ; we must therefore investigate the equations of motion of a gas. The general equations of fluid motion, which we obtained in Chapter I, are of course applicable to elastic fluids such as air and other gases, as well as to incompressible fluids such as water; but in order to investigate the propagation of sound in gases, these equations require modification. In all problems relating to vibrations, the velocities upon which the vibrations depend, are usually so small that their squares and products may be neglected ; also the variation of the density of the gas is usually a small quantity. If therefore a gas, which is at rest, be disturbed by the passage of sound waves, we may write d/dt for d/dt + ud/dx + vd/dy + wd/dz, and also put p = p Q (1 + s), where s, which is called the condensation, is a small quantity. The equations of motion therefore become du di dv v 1 dp^ = A -j p ax _ Y ldp V . dt pdy r (1), dw _ I dp | dt p dzj whilst the equation of continuity, 6, equation (5), becomes ds du dv dw _ VIBRATIONS OF A GAS. 187 We shall also suppose that the bodily forces (if any) which act upon the gas arise from a potential U, and also that the motion is irrotational ; (2) therefore becomes We have already shown that when the motion is irrotational, the pressure is determined by the equation Now ( is to be neglected, also if we assume Boyle's law to hold, we shall have p=kp = kp (l and therefore = ks + C', neglecting s 2 &c. Whence (4) becomes If there were no forces in action and no motion, the first three terms would be zero ; whence C' = C, and therefore, ks+ U+ 4> = ......................... (5), or if Sp denote the small variable part of p, (5) may be written = ........................ (6). Po Eliminating s between (3) and (5) we obtain ^_V*- dt*~ ^ ~dt ......... Equation (6) and (7) are the fundamental equations of the small vibrations of a gas. 168. In almost all the applications of these equations, no impressed forces act, and therefore 7=0; accordingly (7) becomes < 8 >- Let us now suppose that plane waves of sound are propagated in a gas of unlimited extent. Let I, m, n be the direction cosines ISS EQUATIONS OF MOTION OP A PERFECT GAS. of the wave front, a the velocity of propagation of the wave. We may assume Substituting in (8) we obtain k = a? ............................... (9). This equation determines the physical meaning of k, and shows that it is equal to the square of the velocity of propagation. We may therefore write (8) in the form Let , 77, f be the displacements of an element of fluid, then ^if _ _T A trie" 1 (lx+my+nz-at) 7 , ~ ~ 7 - -iXt/tl/C , at Ax whence = - (A I /a) e llt *+'+-*) , with similar expressions for rj and We thus obtain $1 = r)/m = /n, which shows that the displacement is perpendicular to the front of the wave. This constitutes one of the fundamental distinctions between waves of sound and waves of light, for it is well known that in a wave of light the direction of displacement always lies in the wave front. It therefore follows that waves of sound are incapable of polarization ; they are, however, capable of interfering with one another and also of being diffracted, since these phe- nomena do not depend upon the direction of vibration. 169. Equation (9) enables us to calculate the velocity of sound in a gas, and we shall now show how it may be applied to obtain the velocity of sound in air. We have where p is the pressure corresponding to a given density. Now it is found by experiment that at C. under a pressure equal to the weight of 1033 grammes per square centimetre, at the place where the experiment is made (i.e. a pressure equal to 1033 g barads 1 ), the density of dry air is '001293 grammes per cubic 1 In the report of the British Association at Bath, 1888, the Committee on Units recommended the introduction of the following additional units, viz. that (i) The unit of velocity on the c. G. s. system, i.e. the velocity of one centimetre per second, should be called one kine. THERMODYNAMICS OF GASES. 189 centimetre. Hence if we employ the C. G. s. system units, and take , accordingly (15) may be written (16). 171. This equation is the analytical expression of a very im- portant but somewhat recondite law, known as the second law of Thermodynamics. For a full discussion of the second law, we must refer to treatises on Thermodynamics, but a few remarks on this subject may be useful. If one substance at a temperature S be placed in contact with another substance at a lower temperature T, heat will flow from the hot substance into the cold substance ; and this process will continue until both substances are reduced to the same temperature. It can however be shown by means of a theoretical heat-engine devised by Carnot, that it is possible to transfer heat from a cold body to a hot body by means of the expenditure of work ; and the second law asserts, that it is impossible to do this without ex- penditure of work. The law was first enunciated by Clausius in the following terms : It is impossible for a self-acting 'machine, unaided by external agency, to convey heat from one body to another at a higher temperature. Lord Kelvin states the law in a slightly different form as follows : It is impossible by means of inanimate material agency, to derive mechanical effect from any portion of matter, by cooling it below the temperature of the coldest surrounding objects. 192 EQUATIONS OF MOTION OF A PERFECT GAS. By means of the experimental law, that the intrinsic energy of a gas depends upon its temperature and not upon its volume, the second law of Thermodynamics may be dispensed with in dealing with gases ; or to put the matter more correctly, the second law can be deduced as a consequence of the experimental law. But in the case of substances which are not in the gaseous state, the first law is not sufficient to enable us to investigate their thermo- dynamical properties. Moreover, although it is always assumed that the pressure, temperature and volume are connected together by a certain relation, which may be mathematically expressed by an equation of the form F( p, v, 6) = ; yet the form of the func- tion F is not accurately known, except in the case of perfect gases. It can be shown that for all substances the second law is mathe- matically expressed by means of equation (16), and it thus leads to a certain function , which is capable of being theoretically expressed as a function of any two of the quantities p, v, 9, and which specifies the properties of the substance when it is not allowed to gain or lose heat. The function $ was called the Thermodynamic Function by Rankine ; but it is now always known as the Entropy. 172. Returning to 170, let us take p and 6 to be the independent variables; equation (13) may then be written dH=Rdp + K p d8, where K p is the specific heat at constant pressure. Substituting in (11) and eliminating dp by (12) we obtain dE = (Rpje + K p )d0-p(l + R/v) dv. Since the right-hand side of this equation must be identical with the right-hand side of (14), we must have whence K p K v = h ........................... (17). Equation (17) shows that the difference between the two specific heats is constant ; also since the specific heat at constant volume has been shown to be a function of the temperature, it follows that the specific heat at constant pressure must also be a function of the temperature. 173. The value of the specific heat of air at constant pressure has been determined by Regnault, and he finds that it is very nearly independent of the temperature, and is equal to 183'6 foot- ADIABAT1C LINES. 193 pounds 1 per degree Fahrenheit. It therefore follows from (17) that K v is also very nearly independent of the temperature. It also follows from Regnault's experiments, that the value of h for air is 53*21 foot-pounds per degree Fahrenheit ; we thus obtain K v = Kp h, = 183-6 - 53-21 = 130-4 The quantity with which we are most concerned in Acoustics, is the ratio of the specific heat at constant pressure, to the specific heat at constant volume, which is usually denoted by 7. We accordingly find 7 = ^/^ = 1-408. The specific heats of all perfect gases are so very nearly independent of the temperature, that they may be treated as constant. The value of the ratio 7, is also approximately the same for all gases. 174. We have already proved the equation dH -- dd$. The quantity (/> is called by Clausius the entropy of the gas, and is a quantity which specifies in an analytical form, the properties of a gas which expands or contracts without loss OF gain of heat ; for when this is the case dH = 0, and therefore = const. If there- fore we suppose that is expressed as a function of p and v, the curve = h\og v + K v log + const .............. (19). By (12) and (17), this may be expressed in the form = (K p - K v ) log v + K v logpv/h + const., whence pw = A**'*' ........................... ( 20 )> where A is a constant. This is the equation of the adiabatic lines of a perfect gas. i This calculation is taken from Chapter XI. of Maxwell's Heat, in which British units are employed. B. H. 194 EQUATIONS OF MOTION OF A PERFECT GAS. If p be the density of the gas, v oc p~ ] ; whence by (20) the relation between the pressure and density of a gas, which expands without loss or gain of heat, is p = k'py ........................... (21), where k' is a constant. The equation of the isothermal lines may be written pv = (K p -K v )6 ........................ (22). 175. The mechanical properties of perfect gases are specified by two quantities, viz. their densities and their elasticities. The density, as is well known, is defined to be the mass of a unit of volume ; but in order to understand what is meant by the elasticity of a gas, some further definitions will be necessary. The elasticity of a gas under any given conditions, is the ratio of any small increase of pressure, to the voluminal compression thereby produced. The voluminal compression, is the ratio of the diminution of volume to the original volume. Hence if v the original volume, be reduced by the application of pressure Sp, to v + 8v (8v being of course negative), the elasticity E is equal to The quantity E is called the compressibility by Lord Rayleigh (Chapter xv.), and is denoted by him by m. The value of dp/dp, and therefore E, depends upon the thermal conditions under which the compression takes place. The two most important cases are, (i) when the temperature remains constant, (ii) when there is no loss or gain of heat. We shall, following Maxwell, denote the elasticity under these two conditions by E e and E+. In the first case p = kp, whence dp/dp k ; accordingly E e = kp=p ......................... (24). In the second case p = Wp"*, whence dp I 'dp = k'yp^~ l ; accordingly E^ = k'ypy = yp ...................... (25). From (24) and (25) we obtain, INTENSITY OF SOUND. 195 Velocity of Sound in Air. 176. Having made this digression upon the thermodynamics of gases, we are prepared to investigate the velocity of sound in a gas. From (21) we obtain jf = 7^T^~ 1 k'y = ^ /V"" 1 + & o since p = p (1 + s). Whence (4) becomes Eliminating s from (3) we obtain and therefore the velocity of sound is equal to Now k=p n /p , and k' = p /p Q y , whence k'pj~ l = k. The velocity of sound is therefore equal to (7)* and is there- fore augmented in the ratio <\/7 : 1- In the case of air, the value of & in feet per second has already been shown to be equal to 918'49, and therefore = 1083-82, which nearly agrees with the value 1089 feet per second given above. Intensity of Sound 1 . 177. We have stated in 118 that the intensity of sound is measured by the rate at which energy is transmitted across unit area of the wave front. We shall therefore find an expression for this quantity. Let the velocity potential of a plane wave be

PO + &P be the pressures when the air is at rest and in motion respectively, the rate dW/dt at which work is trans- mitted is dW d and since Bp = p(j> = p V - sin (x Vt), A A, we obtain dW ( r^ A 27T, T7-.J 27T . 27T, sin - a; -Fsm.- dt X = - v " + periodic terms, since FT = X. We therefore see that the rate at which energy is transmitted, consists of two terms : viz. a constant term, which shows that a definite quantity of energy flows across the wave front per unit of time; and a periodic term, which fluctuates in value and con- tributes nothing to the final effect. The first term measures the intensity of sound, and shows that it varies directly as the square of the amplitude, and inversely as the product of the velocity of propagation in the medium and the square of the period. CHAPTER XL PLANE AND SPHERICAL WAVES. 178. WE shall devote the present chapter to the consideration of certain special problems relating to plane and spherical waves of sound. The theory of the vibrations of strings, which was discussed in Chapter VIL, explains the production of notes by means of stringed instruments ; but in order to understand how notes are produced by means of wind instruments, it will be necessary to investigate the motion of air in a closed or partially closed vessel. The simplest problem of this kind is the motion of plane waves of sound in a cylindrical pipe, which we shall proceed to consider. Motion in a Cylindrical Pipe. 179. Let I be the length of a cylindrical pipe, whose cross section is any plane curve, and let the fronts of the waves be perpendicular to .the sides of the cylinder. We shall suppose for simplicity, that the motion is in one dimension, whence measuring x from one end of the pipe, the equation of motion is d^_ dQ ~dV~ L da? ' where a is the velocity of sound in air. Since the motion is periodic, we may assume that = 'e tnt , whence if n/a = 2-7T/X = K ......................... (2), 198 PLANE AND SPHERICAL WAVES. (1) becomes ^ + *y = 0, the solution of which is ' = (A cos KX + Rsiu KX). If the pipe is closed at both ends, d/dtv vanishes whenever x = rl/i, where r is any integer not greater than i. Corresponding to the iih harmonic, there are therefore i 1 nodes which divide the pipe into i equal parts. The increment of the pressure due to the wave motion is given by the equation 8p = - /?, and therefore 8p vanishes whenever cos iirxjl = ; i.e. whenever x = (2r + 1) l/2i, where r is zero or any positive integer less than i. Points at which there is no pressure variation are called loops. We thus see that corresponding to the gravest note (i 1, r = 0), there is a loop at the middle point of the pipe. The loops corresponding to the overtones, occur at points x = l/2i, 3//2t. . . ; and consequently the loops bisect the distances between the nodes. The conditions that a node may exist at any point of the pipe, can be secured by placing a rigid barrier across the interior of the MOTION IN A CYLINDRICAL PIPE. 199 pipe at that point. The conditions for a loop may be approxi- mately realised, by making a communication at the point in question with the external air ; and consequently it was assumed by Euler and Lagrange, that the open end of a pipe may be treated as a loop. This supposition is however only approximately true, but the error is small provided the diameter of the pipe is small in comparison with the wave-length. Whenever a disturb- ance is excited in a pipe which communicates with the air, the external air is set in motion, and a complete solution of the problem would necessitate the motion of the latter being taken into account. 181. Let us in the next place suppose that one end of the pipe is fitted with a disc, which is constrained to vibrate with a velocity cos nt. The condition to be satisfied at the origin, where the disc is situated, is -^r- = cos nt. when x = 0. doc If therefore we assume = (A cos KX + B sin KX) cos nt, we obtain BK = 1. If the other end of the pipe is closed, d/dx = when x = I, whence cos K(I x) = r^ s ^ cos nt. K sin Kl If the other end be open, the condition is that = when as = I, whence sin K(IX) = s-z cos nt. K cos Kl The value of K is of course n/a. Reflection and Refraction 1 . 182. We shall now investigate the reflection and refraction of plane waves of sound at the surface of separation of two gases. Let the origin be in the surface of separation, let the axis of 1 Green, Trans. Camb. Phil. Soc. 1838. 200 PLANE AND SPHERICAL WAVES. x be drawn into the first medium, and let the axis of z be parallel to the line of intersection of the fronts of the waves with the surface of separation. Let i be the angle of incidence, r the angle of refraction ; also let F, V l} be the velocities of propagation in the two gases, and p, p 1 their densities when undisturbed. Then in the first medium we must have p' = p (1 + s), p' = k'p'y = k'pv (1 + ys) and in the second medium p'i = Pi (1 + *i)> P'I = fcVi* = &Vi (1 + 7*i)- Since the two gases are supposed to be in equilibrium when undisturbed by the waves of sound, we must have k' p y=k\p\ ........................... (5). Again, V* = k'ypir- 1 , V? = k'wpJ-\ whence V*p=Vfa ........................... (6). The equations of motion in the first medium are dp * ' and in the second *fe_W*fe.>to dt* ~ J {do? df) ' The boundary conditions are, (i) That the component velocity perpendicular to the surface of separation should be the same in both media. (ii) That the pressure in the two media should be equal at their surface. The first condition gives d$_d$ 1 dx~ dx " and the second gives P = Pi> which by (5), (8) and (10) gives (12). REFLECTION AND REFRACTION. 201 If we suppose that the velocity potential of the incident wave is < = 4 6 KH-&V+) ........................ (13), the velocity potentials of the reflected and the refracted waves may be written ...................... (15), for the coefficient of t must be the same in these three equations, because the periods 27r/&> of the three waves must be the same ; whilst the coefficients of y must be the same, because the traces of the three waves on the surface of separation must move together. Substituting the value of + (ft' in (7), and the value of fa in (9), we obtain G>* = F 2 (a 2 + Z> 2 ) = F 2 (a' 2 + 6 2 ) = V? (a? + 6 2 ) ......... (16), and therefore a = a. Also if X, \ be the wave-lengths in the two media a = (2-7r/\) cos i, b = (2?r/X) sin i = (27r/\ 1 ) sin r) . ^ Ox = (27T/XO cos r, a> = 2?r F/X = 2-rr F^ J ' ' From the equation a = a, we see that the angle of incidence is equal to the angle of reflection ; and from (17) it follows that (18), . sin % sin r which is the law of sines. To obtain the ratio of the amplitudes, we must substitute the values of + 0' and ^ from (13), (14) and (15) in (11) and (12); we thus obtain (A -A')a=A l a l By (17) and (18) these become (A J/)tan r A^ tant ( A + A') sin 2 T = A! sin 2 i from which we deduce tan (i + r) 2 A sin 2 r cot i /2>\ 1 ~~ sin (i + r) sin (i r)-" 202 PLANE AND SPHERICAL WAVES. The first formula is the same as Fresnel's tangent formula for the intensity of the reflected light, when the incident light is polarized perpendicularly to the plane of incidence ; and we ob- serve that the reflected wave vanishes when i + r = |TT, i.e. when 183. When light is reflected at the surface of a medium, which propagates optical waves with a velocity which is greater than that of the medium from which the light proceeds, it is well known that the light will be totally reflected, when the angle of incidence exceeds a certain value which is called the critical angle ; and that total reflection is accompanied with a change of phase. We shall now show that a similar phenomenon occurs in the case of sound. Since cos r = { 1 - ( VJ F) 2 sin 2 i}*, it follows that if F a > V, cos r will vanish when i = sin" 1 F/ V 1 , and for angles of incidence greater than this value, cos r will become imaginary; and therefore by (17), = <3>e l(ca{ , this becomes (V^ 2 ) = ........................ (25). i The remainder of this Chapter is taken from Lord Bayleigh's Theory of Sound, Vol. n. Chapter xvn. His original investigations are given in the Proc. Land. Math. Soc. Vol. rv. pp. 93 and 253. 204 PLANE AND SPHERICAL WAVES. By (12) of 7, it follows that if r, 6, is a function of r alone, (25) becomes d*tf) 2 d<& , dr 2 r dr which may be written in the form the integral of which is If the motion is finite at the origin, we must have A = B, in which case in which A may be complex. 185. This equation may be applied to determine the sym- metrical vibrations of a gas, which is enclosed within a rigid spherical envelop of radius c ; for the condition to be satisfied at the surface of the envelop is d/dr = 0, which gives K cos KG c -1 sin KG = 0, or tan KG = KG (28). Since the wave-length X = ZTT/K, and the frequency is equal to /ca/2-TT, (28) determines the notes which can be produced. The roots of (28) have been investigated by Lord Rayleigh, and he finds that the first root is KG = 1'4303 x TT. We therefore see that the frequency of the gravest note is 7151 x (a/c) ; accordingly the pitch falls as the radius of the sphere increases. This result exemplifies a general law, that the frequencies of vibration of similar bodies formed of similar materials, are inversely pro- portional to their linear dimensions. The loops are determined by the equation sin KT = 0, which gives r = mir/tc, where m is an integer. VIBRATIONS IN A CONICAL PIPE. 205 186. Since any circular cone whose vertex is the origin is a nodal cone, the above solution determines the notes which could be produced by a conical pipe closed by a spherical segment of radius c. If a conical pipe be open at one end, and we assume that the condition to be satisfied at the open end is that it should be a loop, we obtain K = nnr/c, and therefore the value of is = 2t Ar~ l tmnatlc sin mirr/c. The frequency of the gravest note is therefore ^a/c, which is less than if the pipe were closed. 187. The most general value of in the case of symmetrical waves is = Ar~ l e l * (at+r} + Br~ l e l(C (at ~ r) ............. (29), the first term of which represents waves converging upon the origin, whilst the second represents waves diverging from the origin. Let us now draw a very small sphere surrounding the origin ; then taking the second term of (29), the flux across the sphere is { ( JJ r 2 dtl = - B (1 + IKT) e^ at ~^ dtl dr when r = 0. The second term of (29) therefore represents a source of sound diverging from the pole, of strength ^irBe lKat ; similarly the first term represents a source of sound converging towards the pole 1 . 188. The general solution of (25) cannot be effected without the aid of spherical harmonic analysis, but there is one solution of considerable utility, which we shall now consider. Let .. (f> = ^e lltat cos 0, where is a function of r alone. Substituting in (25), we obtain **+?** **+*..ft dr* r dr r 2 1 The corresponding problems in two-dimensional motion cannot be investi- gated without employing the Bessel's function of the second kind Y O (KT). It is worth noticing, that certain expressions for these functions in the forms of series and definite integrals, can be obtained by means of the theory of sources of sound. See Lord Rayleigh, Proc. Lend. Math. Soc., Vol. xix. p. 504. 206 PLANE AND SPHERICAL WAVES. To solve this equation, put $> = dw/dr and integrate; we at once obtain d-w 2dw j-i + - -J- + ?W = 0, ar 2 r dr the solution of which has already been shown to be w = r- 1 (Ae lKr +B- tKr ); accordingly 4> = ^ (A?* - Be-"") - ^ (Ae r + Be-"") ...... (30). In order to find the condition that the motion should be finite at the origin, we must expand the exponentials in powers of tier, and equate the coefficients of negative powers of r to zero ; we shall thus find that A= B, whence writing A for 2iK?A, the solution becomes , <> A ( sin /cr\ . = - cos/cr -- ................ (31). KT\ KT J If gas, contained in a spherical envelop, be vibrating in this manner, the frequency is determined by the equation d<>/dr = 0, when r = c ; . 2/cc which gives tan KG = ^ -- 2 . The least root of this equation (other than zero), is found by Lord Rayleigh to be KC = '662 x TT ; and therefore the frequency of the gravest note is '331 x (a/c). This note is the gravest note which can be produced by gas vibrating within a sphere ; it is more than an octave lower than the gravest radial vibration, whose frequency has been shown to be -7151 x (a/c). Since the motion is symmetrical with respect to the diameter = 0, every meridional plane is a nodal plane ; but since d<&/d0 does not vanish anywhere except along the diameter in question, there are no conical nodal sheets. 189. We shall now consider the motion of a spherical pendulum surrounded with air, which is performing small oscil- lations. Since the periods of the pendulum and of the air must be the VIBRATIONS OF A SPHERICAL REDUCTION. 207 same, we may suppose the velocity of the pendulum to be re- presented by Ve LKat , and therefore the condition to be satisfied at the surface of the sphere is Ve tKat cos6 ..................... (32). The form of this equation suggests that = 3>e llcat cos 6, where is given by (30). Since the disturbance is propagated outwards, A = 0, and therefore < = - Br~* (1 + tier) e~ lKr . Substituting in (32), we obtain 3ncc (88)> where c is the radius of the sphere. If X be the resistance experienced by the sphere, cos IK (1 + IKG) where = Ve lKat , is the velocity of the sphere. Rationalising the denominator, and putting _2_-KK 2 C 2 K*C* P ~ 4 + * 4 c 4 ' q ~ 4T+ * 4 c 4 ' and remembering that | = t/caf, we obtain Z = M' (p% + * where M' is the mass of the displaced fluid. The first term of this expression represents an increase in the inertia of the sphere; whilst the second term represents a re- sistance proportional to the velocity, which is therefore a viscous term, and shows that initial energy is gradually dissipated into space. If M be the mass of the sphere, I the distance of its centre from the point of suspension, the equation of motion of the pendulum is {M(l 2 + fc 2 ) + M'lp] + M'faaqe + (M- M') gW = 0. 208 PLANE AND SPHERICAL WAVES. By 129, the integral of this equation is of the form and the modulus of decay is 2 [M (I* + fc 2 ) + M'Pp}/M'l* K aq. If the wave-length X, of the vibrations of the gas, is large in comparison with the radius of the sphere, KG will be of the order cf\, and will therefore be small ; accordingly the value of p will be nearly equal to ^, whilst the value of /cq, upon which the viscous term depends, will be of the order c 3 /\ 4 . We therefore see that in this case the viscous term will be very small, and the motion will die away gradually ; hence the sphere will vibrate very nearly in the same manner as if the gas were an incompressible fluid. If, on the other hand, c were large compared with \, p would be nearly equal to unity, and the apparent inertia of the sphere would be greater than when c/\ is small ; but icq would be of the order cr 1 and would therefore be small. 190. Another interesting problem is that of the scattering of a plane wave of sound by a fixed rigid sphere, whose diameter is small compared with the wave-length. Measuring 6 from the direction of propagation, the velocity potential of the plane waves may be taken to be jA _ e iic(at+x) = gix (at+rcosfl) the positive sign being taken, because the waves are supposed to be travelling in the negative direction of the axis of x. If c be the radius of the sphere, it follows that in the neighbourhood of the sphere, KT or 2?rr/\ is a small quantity, and therefore expanding the exponential and dropping the time factor for the present, we may arrange in the form of the series 1 0=1 -/eV 2 + ucrcos 6 - ^^(3 cos 2 0- 1)... When the waves impinge upon the sphere, a reflected or 1 The reader, who is acquainted with Spherical Harmonic analysis, will observe that we have arranged in a series of zonal harmonics. It can be shown that the solution of (25) can be expressed in a series of terms of the type F (r)S n , where S. A is a spherical surface harmonic. SCATTERING OF A PLANE WAVE. 209 scattered wave is thrown off, whose velocity potential may be assumed to be <' = A $> + A& cos 6 + %A 2 <& 2 (3 cos 2 d - 1) + . .. The quantities , ! are given by (26) and (30) respectively ; but since the scattered wave diverges from the sphere, we must put A = 0, and take B=l, since the constant B may be supposed to be included in A , A^.. ; accordingly 4>! = - r~ 2 (1 + tier) - With regard to 2) it can be verified by trial, that a solution of (25) is < 2 (3 cos 2 1), where 2 is a function of r alone ; it will not however be necessary to consider the form of 4> 2 , since it introduces quantities of a higher order than /c 2 c 3 , which will be neglected. The equation to be satisfied at the surface of the sphere is cto dr dr when r = c. This equation must hold good for all values of 6, whence l = , dr which determine A , A,. Substituting from (34) we obtain i 3 v- approximately, since we shall not retain powers of c higher than c 3 . We thus obtain so At a considerable distance from the sphere, the term Kc'/r 2 is small that it may be neglected, we may therefore write or 14 B. H. 210 PLANE AND SPHERICAL WAVES. Restoring the time factor and putting K = 2?r/X, we finally obtain in real quantities +f cos e)cos~(at-r) ......... (35), corresponding to the wave ..................... (36). Equation (35) accordingly gives the velocity potential of the scattered wave, corresponding to the incident wave whose velocity potential is given by (36). This expression is however only an approximate one, and the correctness of the approximation depends upon the assumption, that the radius of the sphere is so small in comparison with the wave length, that terms of a higher order than c?/\ 2 may be neglected. We have also neglected tcc 3 /r 2 , which is equivalent to supposing, that the point at which we are observing the effect of the scattered wave, is at a considerable distance from the sphere. For a more complete investigation, we must refer to Lord Rayleigh's treatise. EXAMPLES. 1. If two simple tones of equal intensity and having a given small difference of pitch be heard together, prove that the number of beats in a given time will be greater, the higher the two simple tones are in the musical scale ; and prove that the pitch of the resultant sound in the course of each beat is constant. 2. One end of a tube which contains air is open, whilst the other is fitted with a disc, which vibrates in such a manner that the pressure of the air in contact with the disc is n (1 - k sin 27r/ T ) where & is a small quantity. Find the velocity potential of the motion. 3. The radius of a solid sphere surrounded by an unlimited mass of air, is given by R (1 + a sin nat), where a is the velocity of sound in air. Show that the mean energy per unit of mass EXAMPLES. 211 of air at a distance r from the centre of the sphere, due to the motion of the latter is %n 2 a?a?R s (1 + 2n 2 r 2 )/?- 4 (1 + nlR 2 ). 4>. Prove that in order that indefinite plane waves may be transmitted without alteration, with uniform velocity a in a homogeneous fluid medium, the pressure and density must be connected by the equation where p , p are the pressure and density in the undisturbed part of the fluid. 5. Two gases of densities p, p l are separated by a plane uniform flexible membrane, whose equation is y = 0, and whose superficial density and tension are a- and T. If plane waves of sound impinge obliquely at an angle i, and the displacements of the incident reflected and refracted waves of sound and of the membrane, be represented by (i) A sin {m (x sin i y cos i) nt + a], (ii) A' sin [m (x sin i + y cos i) nt + a'}, (iii) A 1 sin [m 1 (x sin r y cos r) nt + aj, (iv) a sin (mx sin i nt), respectively; find the relations to be satisfied, and prove that the ratio of the intensities of the reflected and incident waves is equal to (Tm z sin 2 i an 2 ) 2 + (p^ sec r pm sec i) 2 (Tm? sin 2 i cm 2 ) 2 + (pm sec r + pm sec if ' 6. If sound waves be travelling along a straight tube of infinite length which is adiathermanous, and no conduction of heat takes place through the air, prove that the equations of motion may be accurately satisfied by supposing a wave of con- densation to travel along the tube, with a velocity of propagation which at each point depends only on the condensation at that point, and which for a density p is where p , p a are the pressure and density at each end of the wave. 212 PLANE AND SPHERICAL WAVES. 7. Prove that in a closed endless uniform tube of length I filled with air, a piston of mass M will perform 7?* complete small vibrations under the elasticity of a spring, if Mnnrl (v? ,\ tan mirl a = -^-f, - 1 , M a \w? / where M' is the mass of the air in the tube, and a the velocity of sound, supposing the piston to make n vibrations in a second when the air is exhausted. 8. Investigate the forced oscillations in a straight pipe, which will occur when the temperature of air in the pipe is compelled to undergo small harmonic vibrations expressed by 6 cos m (vt x), where x is measured along the axis of the pipe. 9. The greatest angle inclination of the adiabatic lines of a gas to its isothermals occurs, when the slope of the isothermal to the line of zero pressure is TT cot" 1 7 ; and the locus of all these points of maximum angle, is a straight line through the origin, inclined to the line of zero pressure at an angle cot" 1 7*. 10. A sphere of mean radius R, executes simple harmonic radial vibrations of amplitude a, in air of density p ; prove that its energy is radiated into the atmosphere in sound waves at the rate per unit of time, where X is the length of the waves propagated in air, and a is their velocity. CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. DEC 1 ?, EMS LIBRARV 52SJREGIONAI. LIBRARY A r\rln "' '"'