" ;**-:". :': - m I Hfl I^^H ELEMENTS OF WAVE MOTION RELATING TO SOUND AND LIGHT. ELEMENTS WAVE MOTION RELATING TO SOUND AND LIGHT. A TEXT BOOK PREPARED EXPRESSLY FOR THE USE OF THE CADETS OF THE UNITED STATES MILITARY ACADEMY, WEST POINT. BY PETER S. MICHIE, LATE PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE U. S. MILITARY ACADEMY ; AND BREVET-LIEUTENANT-COLONEL U. S. ARMY. THIRD NEW YORK : JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1904. Copyright, 1882, by D. Van Nostrand, ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORX. PREFACE. rriHIS text-book, as is stated on the title-page, has been prepared expressly for the use of the Cadets of the United States Military Academy, and this specific object has therefore wholly controlled its design and restricted its scope. It is thus in no sense a treatise. Because of the limited time allotted to the sub- jects of sound and light in the present distribution of studies at the Academy, the problem of arranging a fundamental course of sufficient strength, to be something more than popular, and yet to be mastered within the allotted time, has been somewhat perplex- ing. The basis of this arrangement is necessarily the mathematical attainments of the class for which the course is intended. In this respect, the class has completed the study of elementary Mathe- matics, as far as to include the Calculus, and has had a four months' study of the application of pure Mathematics, in a course of Analytical Mechanics. With these elements to govern, this text-book has been designed for a seven weeks' course, including advance and review. The fact of being able, through the discipline of the Academy, to exact of each student a certain number of hours of hard study on each lesson, is of course an important element necessary to be stated. The study of the text is supplemented by lectures, in which the 271630 vi PREFACE. principles of Acoustics and Optics are amply illustrated by the aid of a very well equipped laboratory of physical apparatus. Carefully written notes of the lectures are submitted by each student to the instructor on the following morning for revision and criticism. Important errors of fact and misinterpretation of principle are thus at once detected, corrected, and hence prevented from obtaining a lodgment in the mind of the pupil. Another element in this matter of instruction, of sufficient importance to be mentioned, is the opportunity freely exercised by each student of making known the difficulties that he has encountered, before being called upon to exhibit his proficiency in the lesson of the day. It is required that these difficulties shall be clearly and exactly stated, in order that the instructor may, by a judicious question or a concise expla- nation, enable the student to clear up the difficulty as of himself, and thus complete the elucidation. The author believes that this method of instruction, taken as a whole, is in sufficiently intimate accord with the text as to gain the following advantages, viz. : 1, the tasks are of the requisite strength to demand all the study-time allotted to his department of instruc- tion, and thus is secured the invaluable mental effort and discipline due to a specified number of hours of hard study ; 2, while the daily tasks are progressive, they are based on fundamental princi- ples which require the exercise of a rational faith, and develop a continual growth of confidence in the mind of the pupil, and a belief in his own ability to overcome each difficulty as it arises ; 3, when the course is completed, the student finds himself equipped with a satisfactory knowledge of the essential principles of the physical science, to which he may add by further individual study, without the necessity of reconstructing his foundation. PREFACE. vil The elements of character developed in the student by this sys- tem of instruction, viz., confidence in his powers, reliance on individual effort, and capacity to appreciate truly his sources of information, are of essential importance in a career where he may be called upon in emergencies to exercise self-control, and to meet manfully unforeseen difficulties ; and they offer a sufficient reason for the importance given to these studies in the curriculum of the Academy. Text-books are generally compilations. The subject-matter of this text has been gathered by the author from whatever source appeared to him best for the purpose in view. And as it is often desirable to refer to original treatises, for a better conception of the subject under discussion, a list of authors is appended to this Preface. In the arrangement of the matter, the author has been governed alone by the necessities of the case and the restrictions of the course. It has therefore seemed advisable to arrive at the deduc- tion of Fresnel's wave surface as expeditiously as possible, and on the way to establish all of the essential principles of undulatory motion common to sound and light. Sufficient theoretical atten- tion is paid in the text to the wave surface, and a study of its model in the lecture-room makes clear its important properties and those of its special cases. Acoustics is briefly treated, and is indeed made subsidiary to Optics, by utilizing its numerous illustrations in vi- bratory motion, so that the laws of this motion may be the more clearly apprehended in the subject of light. In Optics, while the essential principles of the deviation of light by lenses and mirrors, the construction of optical images and the principal telescopic com- binations, are carried only to first approximations, and are some- viii PREFACE. what more condensed than is usual, nothing essential to the Academic course of Astronomy has been omitted. The part relating to physical Optics is very concise, but the experiments performed and illustrations given in the lecture room, especially in diffraction, dispersion, and polarization, largely remedy this defect. The figures throughout the text were drawn by Lieut. Arthur Murray, 1st U. S. Artillery, Acting Asst. Professor of Philosophy, U. S. M. A., to whom I desire to acknowledge my great indebted- ness. P. S. M. WEST POINT, N. Y., May, 1882. LIST OF AUTHORITIES. 1. AIKY. UNDULATORY THEORY OF OPTICS. 1866. 2. ANNALES DE CHIMIE ET DE PHYSIQUE. Tome XXXVI. 3. ARCHIVES DE MUSEE TEYLER. Vol. I. 1868. 4. BARTLETT. ANALYTICAL MECHANICS. 1858. 5. BILLET. TRAITE D'OPTIQUE PHYSIQUE. 1859. 6. CHALLIS. LECTURES ON PRACTICAL ASTRONOMY. 1879. 7. CODDINGTON. SYSTEM OF OPTICS. 1829. 8. COMPTES RENDUS. Vol. LXVI. 1863. 9. DAGUIN. TRAITF, DE PHYSIQUE. Vols. I and IV. 1879. 10. DONKIN'S ACOUSTICS. 1870. 11. ENCYCLOPEDIA BRITANNICA. Ninth Edition, Vol. VII. 1877. 12. EVERETT'S UNITS AND PHYSICAL CONSTANTS. 1879. 13. FRESNEL. CEUVRES DE. Vols. I, II. 14. HELMHOLTZ. TONEMPFINDUNGEN. 1866. 15. HELMHOLTZ. PHYSIOLOGISCHE OPTIK. 1867. 16. JAMIN. COURS DE PHYSIQUE. Vol. III. 1866. 17. LAME. THEORIE MATHEMATIQUE DE L'ELASTICITE. 1852. 18. LLOYD. WAVE THEORY OF LIGHT. 1873. 19. PARKINSON'S OPTICS. Third Edition. 1870. 20. POTTER'S OPTICS. 1851. 21. PRICE. INFINITESIMAL CALCULUS. Vol. IV. 1865. 22. ROOD. MODERN CHROMATICS. 1879. 23. THOMSON AND TAIT. NATURAL PHILOSOPHY. Vol. I. 1879. 24. VERDET. COURS DE PHYSIQUE. Vol. II. 1869. 25. VERDET. LEQONS DE L'OPTIQUE PHYSIQUE. Vols. I and II. CONTENTS. PART I. WAVE MOTION. AST. PAGH 1. General Equation of Energy , 17 2. Relation of Hypothesis to Theory 17 3-9. Molecular Science 18 ELASTICITY. 10-15. Elasticity, General Definitions 19 16-20. Origin of Theory of Elasticity 20 21-22. Elastic Force denned 22 23-24. Elasticity of Solids 23 25-27. Fundamental Coefficients of Elasticity 25 28-37. Analytical Expression of Elastic Forces developed in the Motion of a System of Molecules subjected to Small Displacements.. . 27 38-43. Surfaces of Elasticity 35 WAVES. 44-51. General Definitions and Form of Function 38 52. Simple Harmonic Motion 40 53-55. The Harmonic Curve 41 56-61. Composition of Harmonic Curves 42 62-64. Wave Function 45 65-66. Wave Interference 46 67-68. Interference of any Number of Undulations 48 69-71. The Principle of Huyghens 49 72. Diffusion and Decay of Kinetic Energy 52 73-74. Reflection and Refraction 53 75-76. Diverging, Converging, and Plane Waves 53 CONTENTS. xi ABT. PAGB 77-80. Reflection and Refraction of Plane Waves 54 61-87. General Construction of the Reflected and Refracted Waves 56 88-92. Utility of Considering the Propagation of the Disturbance by Plane Waves 59 93-96. Relation between the Velocity of Wave Propagation of Plane Waves and the Wave Length in Isotropic Media 63 97-102. Plane Waves in a Homogeneous Medium of Three Unequal Elas- ticities in Rectangular Directions 66 103-105. The Double-Napped Surface of Elasticity 69 106-110. The Wave Surface 71 111-112. Construction of the Wave Surface by Means of Ellipsoid (W) 75 113-115. Relations between the Directions of Normal Propagation of Plane Waves, the Directions of Radii -Vectores of the Wave Surface, and the Directions of Vibrations 77 116-125. Discussion of the Wave Surface 79 126-128. Relations between the Velocities and Positions of Plane Waves with respect to the Optic Axes 84 129-134. Relations between the Velocities of Two Rays which are Coinci- dent in Direction and the Angles that this Direction makes with the Axes of Exterior Conical Refraction. . . 86 PART II. ACOUSTICS. 135-140. Acoustics defined ; Description of Ear ; Curve of Pressure 89 141-142. Propagation of a Disturbance in an Indefinite Cylinder 91 143-149. Curve of Pressure due to a Tuning-Fork 93 150. Illustrations of Motion transmitted 96 151-154. Properties of Sound ; Intensity, Pitch, Quality 95 155-158. Curves of Pressure for Noises 97 159-161. Curves of Pressure for Simple Tones 99 162-165. Properties of Audition ; Definitions of Simple, Single, and Musi- cal Tones 101 166-167. Musical Intervals 103 16&-169. Musical Scales 104 170. Perfect Accords. . . 106 Xll CONTENTS. ART. PAGE 171-174. The Diatonic Scale ; Harmonics 106 175-178. Sympathetic Resonance 109 179-182. Separation of Vibrating Bodies into Component Parts Ill 183-184. Velocity of Sound in any Isotropic Medium 112 185-189. Velocity of Sound in Gases '. . . . 11$ 190. Pressure of a Standard Atmosphere 116 191. Height of the Homogeneous Atmosphere 116 192-194. Final Velocity of Sound in Air 117 195. Formula for Velocity of Sound in any Gas 118 196-199. Velocity of Sound in Air and other Gases as Affected by their not being Perfect Gases 119 200. Velocity of Sound in Gases independent of the Barometric Pres- sure 121 201-203. Velocity of Sound in Liquids 122 204r-209. Velocity of Sound in Solids 124 210. Reflection and Refraction of Sound 126 211. Consequences of the Laws of Reflection 127 212. Refraction of Sound 128 213-221. General Equations for the Vibratory Motion of a Stretched String 128 222-225. Application of Equations to the Longitudinal Vibration of a Rod. 136 226-228. Vibrations of Air Columns ; 1, Closed at One End 138 229-230. 2. Open Air Columns 140 231-232. Relative Velocities of Sound in Different Material 142 233-234. Transversal Vibration of Elastic Rods 143 235. Harmonic Vibrations of Elastic Rods 145 236. Tuning Forks 146 237. Vibration of Plates 147 238. Vibration of Membranes 14& 239-240. Subjects referred to by Lecture 149 PART III. OPTICS. 241-242. Light and Optics defined ; Geometrical and Physical Optics ISC' 243-245. Luminif erous Ether and its accepted Properties 150 246-247. Definitions 151 248-249. Properties of Bodies in Regular Reflection 152- CONTENTS. xiil ABT. PAGE 249-250. Medium defined ; Opaque and Transparent 153 251-252. Shadows and Shade 153 253-256. Photometry 154 257. Velocity of Light by Eclipse of Jupiter's Satellite 157 258. Velocity of Light by Aberration 158 259-261. Velocity of Light by Actual Measurement 158 GEOMETRICAL OPTICS. 262-263. Index of Refraction ; Radiant and Focus 160 264-265. Deviation of Light by Plane Surfaces 161 266-267. Refraction by Optical Prisms 163 268-270. Deviation of Light by Spherical Surfaces 165 271. Application of Eq. (305) to Reflection at Plane Surfaces 168 272. Multiple Reflection Parallel Mirrors 169 273. Multiple Reflection by Inclined Mirrors 169 274-275. Angular Velocity of the Reflected Ray 170 276-277. Deviation of Small Direct Pencils by Spherical Surfaces 170 278. Lenses 172 279. Principal Focal Distance of a Lens 173 280. Discussion of the Properties of a Lens 173 281. Relative Velocities of the Radiant and Focus 175 282-284. Discussion of Spherical Reflectors 175 285. Power of a Lens 177 286-287. To find the Principal Focal Distance of a Lens and a Reflector.. . . 178 288. Deviation of Oblique Pencils by Reflection or Refraction at Spher- ical Surfaces 180 289. Circle of Least Confusion 182 290-291. The Positions of the Foci 182 292. Particular Cases of Caustic Curves 184 293. Critical Angle for Refraction 186 294-295. Spherical Aberration 188 296. Optical Centre 191 297. Focal Centres of a Lens 192 298. The Eye 193 299. Vision 195 300-303. Optical Images 196 Xiv CONTENTS. ART. PAGE 304. Optical linages discussed for a Concave Lens 199 305-306. Optical Images discussed for a Convex Lens 200 307. Optical Instruments 201 308. Camera Lucida 201 309. Camera Obscura 202 310. Solar Microscope 202 811. Microscopes 208 312. Magnifying Power of a Microscope 204 313. The Field of View of a Microscope 205 314. Oculars 206 315. Compound Microscope 208 316. Telescopes 209 317-319. The Astronomical Refracting Telescope 209 320. The Galilean Telescope 211 321. The Terrestrial Telescope 212 322. Reflecting Telescopes 212 323. The Newtonian Telescope 213 324. The Gregorian Telescope 213 325. The Cassegrainian Telescope 214 326-328. The Magnifying Power of any Combination of Lenses and Mir- rors 215 329. Brightness of Images 218 330. Magnifying Power of the Principal Telescopes 219 PHYSICAL OPTICS. 331-333. Solar Spectrum 219 334. Dispersion 221 335. Frauenhofer's Lines 221 336. Normal Spectrum 222 337-338. Absence of Dispersion in Ether of Space 222 339. Irrationality of Dispersion 223 340-341. Color 224 342-345. Absorption 225 346. Emission 227 347. Spectrum Analysis 228 348. Spectroscopes and Spectrometers 228 CONTENTS. XV ABT. PAGE 349-350. Absorption Spectra 229 351. Color and Intensity of Transmitted Light 229 352-353. Color of Bodies ; Constants of Color ; Mixed Colors 230 354. Phosphorescence and Fluorescence 231 355-356. Achromatism 231 357. Achromatism of Prisms 232 358. Table of Refractive Indices 234 359. Achromatism of Lenses 234 360-362. Chromatic Aberration of a Lens 235 363-369. Rainbow Cartesian Theory 236 370. Rainbow Airy's Theory 240 371-378. Interference of Light 242 379-381. Colors of Thin Plates 248 382-388. Diffraction 250 389. Table of Wave Lengths of Fixed Lines in the Normal Spectrum. 255 390-392. Polarized Light 256 393-396. Polarization by Double Refraction 257 397. Polarization by Reflection 260 398. Polarization by Refraction .260 399-400. Contrasts between Natural and Polarized Light 261 401-402. Analytical Proof of the Trans versality of the Molecular Vibra- tions 262 403-404. Distinction between Natural and Polarized Light 265 405. Construction of Refracted Ray in Isotropic Media 266 406. Uniaxal Crystals 267 407-409. Construction of Refracted Ray in Uniaxal Crystals 268 410. Biaxal Crystals 269 411. Interior Conical Refraction 269 412. Exterior Conical Refraction 270 413. Mechanical Theory of Reflection and Refraction 270 414. Reflection of Light polarized in Plane of Incidence 271 415. Reflection of Light polarized Perpendicular to Plane of Incidence 272 416. Intensity of Reflected and Refracted Light 273 417. Reflection of Natural Light . Brewster's Law 273 418. Change of Plane of Polarization by Reflection 275 419. Elliptic Polarization 276 420-421. Reflection of Polarized Ray at Surface separating Denser from Rarer Medium . 277 xvi CONTENTS, ART. PAGE 432. Fresnel's Rhomb 279 423-424. Plane, Elliptical, and Circularly Polarized Light by Fresnel's Rhomb 280 425-427. Interference of Polarized Light 282 428-429. Colored Rings in Uniaxal Crystals 284 430. Colored Curves in Biaxal Crystals 286 431. Accidental Double Refraction 287 432. Rotatory Polarization 287 433. Transition Tint 287 434. Cause of Rotatory Polarization 288 435. Saccharimetry 288 Conclusion . 288 PART I. WAVE MOTION. 1. Equation (E) of Analytical Mechanics (Michie), expresses in mathematical language the law that the potential energy expended is equal to the kinetic energy developed. Every analytical discussion of the action of force upon matter must be founded upon this general equation. For the complete solution of every problem of energy, it is necessary to know the intensities, lines of action, and points of application of the acting forces, the masses acted Upon, and to possess a perfect mastery of such mathe- matical processes as are necessary to pass to the final equations whose interpretation will make known the effects. These difficul- ties, which, in Mechanics, limit the discussion to the free, rigid solid, and to the perfect fluid, are, in Molecular Mechanics, almost insuperable; since we neither know the nature of the forces which unite the elements of a body into a system, nor the constitution of the elements themselves. 2. But the faculty of observation, being cultivated and logically directed, has enabled scientific men to originate experiments which, because of our inherent faith in the uniformity of the laws of na- ture, have resulted in certain hypotheses as to the nature of sound, light, heat, and other molecular sciences. When an hypothesis not only satisfactorily explains the known phenomena of the science in question, but even predicts others, it then becomes a theory, and its acceptance is more or less complete. An hypothesis is related to a theory as the scaffolding to the structure, the latter being so proportioned in all its parts as to be in the completest harmony, while the former may be modified in any way to suit the ever- varying necessities of the architect. 18 ELEMENTS OF WAVE MOTION. While there are many matters concerning which a reasonable doubt may be entertained, because of insufficient data, the progress of scientific thought and the fertility of scientific research have, within recent times, estabi shed certain facts that are now univer- sally accepted. 3. Molecular Science* Molecular science is a branch of Mechanics in which the forces considered are the attractions and repulsions existing among the molecules of a body, and the masses acted upon are the indefinitely small elements, called molecules, of which the body is composed. It embraces light, heat, sound, elec- trics, and, in one sense, chemistry. 4. From the facts of observation and experiment, it is assumed that all matter, whether solid, liquid, or gaseous, is made up of an innumerable number of molecules in sensible, though not in actual contact ; that these molecules are so small as not to be within range of even our assisted vision ; and that they are separated from each other by distances which are very great compared with their actual linear dimensions. 5. The molecular forces, which determine the particular state of the matter, are either attractive or repulsive. When the attrac- tive forces exceed the repulsive in intensity, the body is a solid; when equal to the repulsive, a liquid ; and when less, a gas. The relative places of equilibrium of the molecules are determined by the molecular forces called into play by the action of extraneous forces applied to the body. Thus, when a solid bar is subjected to the action of an extraneous force, either to elongate or to compress it, the molecules assume new positions of equilibrium with each increment of force, and, in either case, the aggregate molecular forces developed are equal in intensity, but contrary in direction, to the extraneous force applied. In general, where rupture does not ensue, the extraneous forces applied are much less than the molecular forces capable of being called into play. 6. While we are ignorant of the true nature of force and mat- ter, our senses enable us to appreciate the effects of the former upon the latter. Our whole knowledge of the physical sciences i& based upon the correct interpretation of these sensuous impressions. Observation teaches that if a body be subjected to the action of an extraneous force, the effect of the force is transmitted throughout RELATING TO SOUND AND LIGHT. 19 the body in all directions, and since the body is connected with the rest of the material universe, there is no theoretical limit to the ultimate transfer of this effect throughout space. 7. Among the appreciable effects of force are the changes of state with respect to rest and motion. These can be transferred from an origin to another point in but two ways, viz. : 1. By the simultaneous transfer of the body, which is the de- pository of the motion. 2. By the successive actions and reactions between the consec- utive molecules along any line from the origin. In the molecular sciences, the latter is assumed to be the method of transfer, and the object of the succeeding discussion is to inves- tigate the nature of the disturbance, the circumstances of its progress, and the behavior of the molecules as they become involved in it. 8. While the initial disturbance is perfectly arbitrary, the molecular motions produced through its influence in any medium are necessarily subjected to the variable conditions which result from the action of the forces that unite the molecules into a mate- rial system. The problems are then those of constrained motion. 9. Among the physical properties of bodies, elasticity is of such great importance, that a complete knowledge of its mathematical theory is essential to the thorough elucidation of many of the phe- nomena of molecular science. The limits of this text permit but a passing allusion to its more important laws. ELASTICITY. 10. A body is said to be homogeneous when it is formed of similar molecules, either simple or compound, occupying equal spaces, and having the same physical properties and chemical com- position. In such a body, a right line of given length I and deter- minate direction is understood to pass through the same number n of molecules wherever it is placed ; the ratio - will vary with the W/ direction of I. In crystalline bodies, considered as homogeneous, - varies with the direction ; in homogeneous non-crystalline bodies, such as glass, the ratio varies insensibly, or is independent of the 20 ELEMENTS OF WAVE MOTION. direction. This supposition requires n to be very great, however small I may be. 11. That property, by which the internal forces of a body or medium restore, or tend to restore, the molecules to their primitive positions, when they have been moved from these positions by the action of some external force, is called Elasticity. 12. The elasticity is said to be perfect when the body always requires the same force to keep it at rest in the same bulk, shape, and temperature, through whatever variations of bulk, shape, and temperature it may have been subjected. 13. Every body has some degree of elasticity of bulk. If a body possess any degree of elasticity of shape, it is called a solid ; if none, a fluid. All fluids possess great elasticity of bulk. While the elasticity of shape is very great for many solids, it is not perfect for any. The degree of distortion within which elasticity of shape is found, is essentially limited in every solid ; when the distortion is too great, the body either breaks or receives a permanent set ; that is, such a molecular displacement that it does not return to its original figure when the distorting force is removed. 14. The limits of elasticity of metal, stone, crystal, and wood are so narrow that the distance between any two neighboring mole- cules of the substance never alters by more than a small proportion of its own amount, without the substance either breaking or expe- riencing a permanent set. In liquids, there are no limits of elas- ticity as regards the magnitude of the positive pressures applied ; and in gases, the limits of elasticity are enormously wider with respect to rarefaction than in either solids or liquids, while there is a definite limit in condensation when the gas is near the critical temperature. 15. The substance of a homogeneous solid is called isotropic when a spherical portion exhibits no difference, in any direction, in quality, when tested by any physical agency. When any difference is thus manifested, it is said to be celotropic. 16. Origin of the Theory of Elasticity. In Mechan- ics, by supposing the bodies perfectly rigid, and the distances of the points of application of the extraneous forces invariable, how- ever great the forces, the -problems are much simplified, without affecting their generality. But this ignores the law by which the RELATING TO SOUND AND LIGHT. 21 reciprocal influence is transmitted from point to point of the body, and by which the action of one force is counterbalanced by the actions of others. In reality, the body undergoes deformation, and when the limit is reached, rupture ensues. The mathematical theory has arisen from the necessity of a knowledge whereby these permanent deformations and rupture may be avoided. This theory has been extended to the determination of the laws of small motions, or, in general, to the vibrations of elastic media. 17. The initial state of a homogeneous body is considered to be that in which it is perfectly free from all extraneous forces, to be, indeed, that of a body falling freely in vacuo. Such a body is then the geometrical place of an innumerable number of material points, which are distinguished from the rest of space by several mechani- cal properties. Each of these material points is called a molecule. 18. When such a body is subjected to the action of an extra- neous force, either a tension or a pressure, a motion of its surface particles ensues, and this disturbance is propagated to the interior molecules ; the body becomes slightly distorted, and soon takes a new state of equilibrium. When the external forces are removed, the internal forces are again balanced, and the original condition is restored, provided there is no permanent set. All changes of form of a solid, or any variation of the relative distances of its material points, are ever accompanied by the development of attractive or repulsive forces between the molecules. These variations and forces begin, increase, decrease, and end at the same time, and hence are mutually dependent. 19. The properties of a solid body depending only upon those of its material points, they alone are the foci whence emanate these interior forces. 20. Let an extraneous force be applied to a body, and consider its effect upon any two molecules sufficiently near each other to be mutually affected by their changes of position. Should one of the molecules, on account of this exterior action, approach the other, a, mutual repulsion takes place, which, in time, overcomes the motion of the first molecule, and causes the second to take its new position of equilibrium with respect to the first. The reverse is the case when the first molecule withdraws from the second, and an attrac- tive force is developed between them. If r represent the primitive distance, Ar may represent the displacement. Then the intensity 22 ELEMENTS OF WAVE MOTION. of the attractive or repulsive force developed between the molecules may be represented by /(r, Ar). This function becomes zero when Ar is zero, whatever r may be ; it decreases rapidly when r has a sensi- ble value, whatever Ar may be, since all cohesion ceases between two parts of the same body separated by an appreciable distance. Assuming that the intensity of the molecular forces varies directly with the degree of displacement, this limitation embodies only the cases where the changes of form are very small, whether the extra- neous forces are extremely small or the bodies considered have great rigidity. Hence, f (r, Ar) is limited to the product of a function of r and the first power of Ar, which becomes infinitely small when Ar becomes infinitely small. 21. Elastic Force defined. From any molecule M in the interior of a solid, with a radius equal to the greatest distance be- yond which /(/) is insensible, describe a sphere. This volume will embrace all molecules that influence the molecule M, and may be called the sphere of molecular activity. Pass a plane through M, di- viding the sphere into the two parts SAC and SBC. Normal to KN" and having for its base a differential surface <>, con- ceive a cylinder in the hemisphere SBC. p , ure { When the equilibrium is disturbed, the molecules in SAC will act on the molecules of the cylinder. The resultant uE of all these actions is called the elastic force exerted by SAC upon SBC, referred to the infinitesimal sur- face w. Integrating this function with respect to the plane, we obtain the elastic force referred to the circle SMC. The resultant uE will, in general, be oblique to the plane element w. If it is normal to this element and directed towards the hemisphere SAC, it will be a traction ; if normal and directed toward SBC, it will be a pressure; if parallel to the plane SMC, it will be the tan- gential elastic force. Similarly, if the cylinder is situated in the hemisphere SAC, the resultant elastic force exerted upon the molecules of the cylinder by the molecules in SBC is represented by uE', referred to the same elementary surface w. If the body, slightly changed in form, RELATING TO SOUND AND LIGHT. 23 is in equilibrium of elasticity, the two elastic forces uE and should be equal in intensity, but contrary in direction. Both, however, will represent either pulls, pressures, or tangential forces ; that is, if one is a pull, the other will be a pull directly opposed to it. The elastic force uE, considered with reference to the element planes w drawn parallel to each other through all points of the body, will vary in intensity and direction from point to point ; and at the same point M will vary with the orientation of the element plane w. 22. The direction of the planes w may be determined by that of their normals. Using the angles and V to designate the latitude and longitude of the point where the normal pierces the surface of the sphere of activity, and representing by x, y, and z the co-ordi- nates of this point referred to the co-ordinate axes, we have x =. cos (^ cos V>, y = cos sin i/>, z = sin 0. Representing the orthographic projections of uE by wJT", wl 7 ", and uZ upon the co-ordinate axes, we see, in the case of equilibrium of elasticity, that <*>E will be a function of the five variables x, y, z, 0, and i/> ; and if the motion be progressive, the variable t will also enter. X, Y, and Z can be determined from uE, (f>, and V> ; and, reciprocally, the latter from the former. X, Y, and Z are, how- ever, usually determined, and are, in general, functions of the six variables (x, y, 2, 0, i/>, ), and which being found according to the special circumstances that cause the deformation of the body, would enable us to ascertain, at each instant and at each point of the body, the direction and intensity of the elastic force exerted upon every element plane passing through the given point. In brief, the determination of these functions and the study of their properties are the principal objects of the mathematical theory of elasticity. 23. Elasticity of Solids. Experiment has shown that, when a solid bar is subjected to small elongations, or those within elastic limits, the following laws are verified, viz. : 1, the elonga- tions are directly proportional to the length of the bar; 2, they are inversely proportional to the area of cross section ; 3, they are directly proportional to the intensity of the elongating force ; 4, 24 ELEMENTS OF WAVE MOTION. they are variable for bars of different materials, mental laws can be expressed by the equation, 1 PI These experi- (i) in which I is the length of the bar unloaded, s the area of cross- section, P the intensity of the stretching force, M a coefficient varying with the nature of the material, and A is the correspond- ing elongation. Making s = 1, A I, we get, from the above equation, P = M . If, therefore, the law of the elongation should remain true for all intensities, M would be that intensity which, applied to a bar of unit area in cross-section, would make the elongation equal to the original length. Such an hypothesis gives us the value of the coefficient M, which can be used within the limits of experiment. M is called the coefficient or modulus of longitudinal elasticity, or Young's modulus. While we cannot experiment over such wide limits in longitudinal compression, because of the liability to flexure, the same laws are held to be applicable, with the same limitations. Taking the metre for the unit of length, the square centimetre for the unit of area, and the gramme for the unit of intensity, the moduli of longitudinal elas- ticity for the principal metals are, according to Wertheim, as follows: Lead, . 177 xlO 6 Gold, ...... 813 xlO 6 Silver, 736 xlO 6 Zinc, . . . 873 xlO 6 Copper, 1245xl0 6 Platinum, .... 1704 x 10 6 Iron, 1861 xlO 6 Steel, 1955 xlO 6 The coefficient of elasticity decreases with increase of tempera- ture between 15 and 200 0. 24. An isotropic solid has, in addition to the modulus of longi- tudinal elasticity, a modulus of rigidity ; tho former relating to the elasticity of bulk or volume, and the latter to that of shape. If a bar be of square cross-section before elongation, it will be found afterwards to have undergone deformation in its angles, although the diagonals of the cross-section may still be at right angles. The numerical ratio of the intensity of the force applied, to the deforma- tion produced is the modulus of rigidity. The deformation is measured by the change in each of the four right angles, in terms of the radian (57. 29) as unity. RELATING TO SOUND AND LIGHT. 25 Let to the 25. Fundamental Coefficients of Elasticity. there be a rectangular parallelopipedon AH, subjected at first action of equal and opposite normal pressures on the two bases AD and EH. The vertical edges will, by the laws of elongation, shorten, and the horizontal edges increase in length ; and the relative changes in length will be proportional to the quotient of the normal pressures by the area AD ; that is, to the pressure on the unit of area. Let be the relative shortening of the Figure 2, vertical edges, ft the relative increase of the horizontal edges, and P the pressure on the unit of area, then a mP, = nP, m and n being coefficients to be determined only by experiment. If Q be the pressure applied to the unit area on the faces AF and CH, the edge AC will be shortened ', and the edges AB, AE lengthened /3', and we will have a' = mQ, f3' = nQ. If R be the pressure on the unit area of the faces AG and BH, the edge AB will be shortened ", and the edges AE and AC elon- gated /3", and we will have a" = mR, ft" = nR. If now the three pairs of pressure, P, Q, 7?, act simultaneously, their effects will be superposed, and, representing by e, e', e", the relative variations of the lengths of the edges AE, AC, and AB, we will have e =a -_(0' + 0") = mP-n(Q + R), ) e' = a' - (ft + ft") = mQ-n(P + R), \ (2) B" = "-(0 +0') = mR-n(P+Q)i ) from which we readily deduce, P = He + K(e' + e"), ) Q = He' +K(e-+e"), V (3) R = He" + Ke + e' ) 26 ELEMENTS OF WAVE MOTION. , . , rr m n in which H = m (m n) K = n I m (m n) 2n 2 J Hence the pressures exerted upon the faces of the volume, and therefore the elastic reactions, can be expressed as linear functions of the relative variations of the length of the edges by means of two constant coefficients. These two coefficients, H and K, are funda- mental in the theory of elasticity. They can only be determined by experimental investigations ; -once determined for any body, the problems of elasticity become those of rational mechanics. Exact analysis of the conditions of equilibrium in the interior of a solid elastic body shows that, in each point of the body, there exist three rectangular directions, variable from one point to an- other, such that the elements perpendicular to these directions support normal pressures or tractions. An infinitely small parallelopipedon, having its edges parallel to these three directions, is in the condition of that discussed above; and it suffices to express, in a general manner, the relations which exist between the pressures which it sustains and the changes of length of its infinitely small dimensions, to obtain the differential equations of the problem under consideration. 26. Equations (3) can be written, p = (H- K)e + K(e + e' + e"), } Q = (H K) e' + K(B + e' + e"), [ (5) R = (H K) e" + K(e + e' + e"). J Calling the relative variation of the volume, or cubic dilata- tion, we may, because of the small values of the deformations, write 6 = e + e' + e". (6) Placing H K = 2/* and K = A, we have Q = A0 + tyie', V (7) Each of the tractions or pressures is then the sum of a term pro- portional to the cubic dilatation and of a term proportional to the linear dilatation parallel to the pressure considered. RELATING TO SOUND AND LIGHT. 27 27. A liquid parallelepiped on can be in equilibrium only when the pressures exerted on its six faces are equal ; and we know be- sides that the increase of density or negative increase of volume of the liquid is proportional to the pressure. We will then have P = Q = R = M. (8) The same general theory thus comprises both liquids and solids, in admitting the coefficient 2{i of the former to be zero. The varia- tion of this coefficient from zero marks the departure of the body from the perfect liquid state and its approach to that of the solid. 28. Analytical expression of the elastic forces developed in the motion of a system of molecules, solicited by the forces of attraction or repulsion, and subjected to small displace- ments from their positions of equilibrium. Let x, y, z, and # + A#, y + ky, z-fAz, be the rectangular co- ordinates of the two molecules of the system, whose masses are respectively m and ^ and whose distance apart is r. The intensity of the reciprocal action of the molecules, being exerted along the right line joining them, is f(r) being an undetermined function of the distance. If the sys- tem is in equilibrium, we have the relations, (9) Az At a certain instant, let us suppose that the molecules of the system are displaced from their positions of equilibrium by a very small distance, and let , T/, , be the projections of the displace- ment e of the molecule m on the axes; let -f-A& 77 + AT/, +A, be the projections of the displacement of the molecule fi on the axes; and r-f p the new distance between the molecules. Kepre- senting the components of the elastic force parallel to the axes exerted upon the molecule m by all the molecules fi within the 28 ELEMENTS OF WAVE MOTION. sphere of molecular activity, by Xe, Ye, Ze, so that JT", Y. Z, are the components of the elastic force for a displacement unity in the same directions, we have Xe = (10) 29. Developing f(r + p), and neglecting the terms of a higher order than those containing p, since the displacements are regarded as very small, we obtain, recollecting that A|, A??, A, are of the same order of magnitude as p, while A#, A?/, Az, may be of any order whatever, X, = we also have r 2 = Az 2 + Ay 2 + Az 2 , A# A + Aty A?? + Az A (r) A? which give the values of the component elastic forces developed in any molecule of the medium, when the displacements are small. 30 ELEMENTS OF WAVE MOTION. 30. If the displacement is only in the direction of each axis i succession, we have the following groups of equations. Of*: r) + V (r) A?, Ofy: F 2 = m (20) Of : = m - A?, (21) 31. Combining the above equations, we have = r, + F, + (22) From Eqs. (19) we see that the total intensity of the elastic force developed is proportional to the relative displace- ment A, and since the axis has been assumed arbitrarily, it can be said, in general, that the total intensity, e\/X 2 -f F 2 + ^ 2 , devel- oped, is directly proportional to the general relative displacement, RELATING TO SOUND AND LIGHT. 31 From Eqs. (22) we conclude that the component intensity of the elastic force developed in the direction of any axis, due to any dis- placement, is equal to the sum of the three component intensities developed by three successive displacements along these axes, equal to the respective projections of the general displacement on these axes. 32. Of the nine coefficients of A A??, A, given in Eqs. (18), BIX only are distinct. Representing these by C = 4>(r) (23) we can write Eqs. (22), Xe = Ye = Ze = B AT? + D A (24) from which we conclude that the component elastic force developed along any axis, x, ior example, by a displacement e along any other axis y is equal to the component elastic force developed along the axis y by an equal displacement along the axis x. 33. From Eqs. (19-21) we see that when a displacement is made in any direction, the resulting elastic force is not, in general, in the same direction. To find whether we can refer the system to rectan- gular co-ordinate axes, so that when a displacement is made along such an axis, exceptional elastic forces will be developed, whose total resultant will be in the direction of the displacement, let , ft, y, 32 ELEMENTS OF WAVE MOTION. be the angles which the direction of the displacement makes with the axes; A, //, and v, the angles which the resultant elastic force makes with the same axes ; then we have cos A = cos t = cos v = cos = X + r 2 + z* Y + + cos (3 = + Af + AT? + cos y = (25) Since the resultant intensity of the elastic force is proportional to the displacement, we may let K represent the intensity of the elastic force corresponding to a displacement equal to unity. K varying with the direction of the displacement, we can then place AT? 2 + AC 2 for its representative, e V^T 2 + Y*+Z\ and the first of Eqs. (25) will become Xe cos /t = COS [I = cos v = Ye + (26) Substituting the values of X, Y, Z, A A??, AC, derived from these equations in Eqs. (24), after omitting the common factor e, we have JTcos A = A cos a + ^ cos f3 + Fcosy, } Kcosp = E cos a + B cos + Z> cos y, > (27) ^ cos v = ^ cos + Z> cos j8 + (7 cos y. ) RELATING TO SOUND AND LIGHT. 33 Applying the conditions A = , /* = we have the equations of condition, (A K) cos <* + ^ cos + .Pcos y = 0, \ E cos a + (5 .AT) cos + Z> cos y = 0, > (28) + D cos j3 + ((7 A") cos y = 0; ) together with cos 2 a + cos 2 -f cos 2 y = 1, (29) which make four equations containing the four unknown quantities a, (3, y, and jf. 34. In order that Eqs. (28) may be true for the same set of values of cos a, cos ft, cos y, we must have the determinant (30) Multiplying Eqs. (28) respectively by D, F, and E, we get (AD KD) cos a + DE cos ft + Z>^cos y = 0, j <* + (#.F FK) cos + Z^cos y = 0, > (31) a + Z)^ cos ft + (C# ^^) cos y = 0. ) Placing AD EF = aD, } BF DE = a'F, (32) CE - DF = a"E, ) we have (a JT)Z>cos -f JHFcosa + DEcosft + DFcosy = 0, j (0 f JT)J?ooBj3 + EFcosa H- DEcosft + DFcosy = 0, 1(33) (a ;/ JE") j^cosy + EFcoscc -{- DEcosft + DFcosy = 0; j from which (a K)D cos = (a' ^T) ^ cos /3 } , , = (a" -^)^ cosy = P; j ( } whence, C os = ^ ^ (35) 34: ELEMENTS OF WAVE MOTION. Substituting these values in the first of Eqs. 33, we obtain EF DE DP Clearing of fractions, we have - -2^2 (K-a") (K-a) = 0. (37) -D*F* (K-a) (K-a') } If DEF be positive, supposing 1, that a < a' , we obtain 00, -E*F*(a' -a) (a!' a), a" -a') (a 1 -a), a" -a) (a" -a 1 ), which, since there are three variations in the signs, shows that Eq. (37) has three real roots, one lying between a and a', one be- tween a' and a", and the third between a" and oo . Similarly, if DEF be negative, the real roots will be found as above. 2. If two of the quantities , a', a", are equal, as, for example, a' r~. a", Eq. (37) reduces to (K- a') \[EF(K- a')] {D(K- a) - EF} ~ which gives a real root between a and a', a second equal to a', and a third greater than a'. 3. If the three quantities a, a', a", are equal, Eq. (37) reduces to ( K - a^ [DEF (K-a)- E*F* - D*E* - D*F Z ] = 0, (39) giving two real roots, each equal to a, and one greater than a. Each of these real roots of K, being substituted in one of Eqs. (35), will enable us to find values for each of the cosines between -f- 1 and 1, and hence a given direction for each value of K, or in all three directions. RELATING TO SOUND AND LIGHT. ' 35 35. We therefore conclude, that the total elastic force de- veloped by any displacement is not in general in the line of direction of the displacement, bat oblique to it ; that there are three directions at right angles to each other, and, in general, only three, along which, if the displacement be made, the resultant elastic force developed will be in the direction of the displacement. 36. These three directions are called principal axes. They are not specific lines in a body, but simply mark directions along which the above property exists. 37. The angle which the direction of the displacement and the resultant elastic force make with each other is given by cos U = and, if the displacement be equal to unity, we have U= Xti = X cos a + Y cos j3 + Z cos y. ) 38. Surfaces of Elasticity. If now distances which are proportional to the elastic forces developed by a constant displace- ment, equal to unity, for example, in each direction, be laid off in all directions from any point of the medium, the extremities of these lines will form a surface which may be called a surface of elasticity. But, as for each direction there are two things to con- sider, viz., the intensity of the elastic force and the angle which its direction makes with the displacement, we cannot, in general, con- struct a surface which would unite these two particulars. 39. It will be shown, hereafter, upon what grounds we can dis- regard, in optics, that component of the elastic force, 7Tsin U 9 which is perpendicular to the displacement, and consider, as alone effective, the component whose intensity is represented by K cos U 9 parallel to the displacement. 40. Assuming then, for the present, that the effective elastic force caused by a displacement equal to unity is given by Eq. (41), and substituting the radius vector r for the first member, and the 36 ELEMENTS OF WAVE MOTION. values of JT, Y, Z, from Eqs. (24), and for A A??, A cos , cos ft cos y, their values for a displacement unity, we get r = A cos 2 a 4- 2.Z? cos cos /3 + 2^ cos cos y j + B cos 2 13 + W cos cos y + C cos 2 y, ) * ' the polar equation of a surface of elasticity of the medium. or // % Substituting for cos a, cos (3, cosy, their values -, -, -, and for r its equal V% 2 + y 2 + z 2 , Eq. (42) becomes 8 . . - 2 [A& + % 2 + Cfe* + 2^ + tFxz + Wyz\. (43) 41. Assuming that the radius vector is proportional to the square root of the elastic force, the equation takes the form + 2Dyz, (44) which is the equation of Fresnel's Surface of Elasticity. 42. By assuming each radius vector proportional to the recipro- cal of the square root of the elastic force, Eq. (42) becomes 1 =. Ax* + Bf + Cz* + 2Exy + 2Fxz + ZDyz, (45) ivhich is the equation of what has been designated as the inverse ellipsoid of elasticity, or the first ellipsoid, and is called the ellip- soid E. 43. Surfaces of Elasticity referred to Principal Axes. Principal axes are those along which, if the displacement be made, the resultant elastic forces developed will be wholly in the same direction- We have seen that, in any homogeneous medium, there are in general three, and only three, such directions. Making A?/, A^; A, AC; A, A??, respectively equal to zero in Eqs. (24), and placing A, B, C, equal to 2 , 2 , c 2 , respectively, we have . E = F = D = 0, and Eq. (44) reduces to (z 2 -t- f + z 2 ) 2 = a 2 x* + % 2 + &# (47) and Eq. (45) to V + % 2 + c 2 z* = 1. (48) RELATING TO SOUND AND LIGHT. 37 FresneFs surface of elasticity, Eq. (47), is of the fourth order, its equation being of the fourth degree. Figure (3) represents one- quarter of the principal section made by the plane ac, turned about the axis b through ,an angle of 90. Taking the axes to be a = 1.53, = 1.32, c = 1.00, we may, by Eq. (47), readily construct the principal sections. Thus, since r 4 = a?x* + TPf + &z\ .'. a 2 cos 2 a + & cos 2 (3 + c 2 cos 2 y = r 2 ; we have for the intersection by the plane ac, (3 = 90, and = 2 cos 2 a 4- c 3 cos 2 y = r ' 3 " 2 r' = a cos , and r" = c cos y = c sin Figure 3, Therefore r is equal to the hypothenuse of the right-angled tri- angle on r' and r" ; hence, describe semicircles on a and c ; draw any right line from 0, and lay off on it a distance equal to the hypothenuse on the intercepts of the two circles, and this will be a point of the curve. Three such points are constructed in tha 38 ELEMENTS OF WAVE MOTION. figure. The curve CMA is the intersection of FresnePs surface with ac\ the curve CNA is that of the ellipsoid whose semi-axes coincide with and are equal to those of the surface of Fresnel ; 00 and 00' are the traces of the cyclic planes which contain the axis b of the surface of elasticity and of the ellipsoid respectively. The principal elasticities in crystals never differ so much as those assumed above, and therefore, in many cases, the departure of the surface from the ellipsoid is negligible. .WAVE S. 44. The elastic forces of the medium, developed by the assumed arbitrary displacement of a molecule, will propagate the motion in all directions from the point of initial disturbance. As an ever- enlarging volume becomes involved in this disturbance, each mole- cule takes up a motion exactly similar to that of its predecessor, which it transmits in turn to the next molecule. This transfer is complete when a single pulse traverses the medium, and is both complete and continuous when these pulses are successively con- tinuous. In this latter case the exciting cause acts for a definite portion of time. Representing by a series of dots, a , the position of Fi^dre 4, a file of molecules in their condition of stable equilibrium and con- sidering alone the simple case of rectilineal displacements, the arbitrary displacement of the molecule m will give rise to the suc- cessive displacements of the others, and cd and ef will represent the relative positions of these molecules at the end of a given subse- quent time t, equal to the periodic time of vibration ; the former, when the displacements are parallel to the direction of disturbance propagation, and the latter, when at right angles to this direction. While, therefore, any molecule m is describing its orbit, the dis- turbance is being propagated in all directions, and, at the instant the orbit of m is completed, the disturbance will have reached RELATING TO SOUND AND LIGHT. 89 another molecule m', on the same line of direction, which will then, for the first time, begin to move ; and the molecules m and m' will, thereafter, always be at the same relative distances from their origins. 45. While this undulatory motion is being propagated, mole- cules will be found between m and m', with all degrees of displace- ment, both as to amount and direction of motion, consistent with the dimensions and shapes of their orbits. If the velocity of wave propagation be constant in all directions, the form assumed by the bounding surface containing the disturbed molecules will be spheri- cal ; but if the velocity vary, the form will depend upon the law of its variation. 46. This continuous transmission in any given direction of a relative state of the molecules, while the motion of each molecule is orbital, is characteristic of an undulation. 47. The term phase is used to express the condition of a mole- cule with respect to its displacement and the direction of its motion. Molecules are said to be in similar phases, when moving in parallel orbital elements and in the same direction; and in opposite phases, when moving in parallel orbital elements and in opposite directions. More generally, similar phases are those in which the anomalies of the molecule are the same, and opposite phases those in which the anomalies differ by 180. (By anomaly is meant the angular dis- tance from an assumed right line.) 48. A wave is the particular form of aggregation assumed by the molecules between the nearest two consecutive surfaces in which .similar phases simultaneously exist throughout. t A -wave front is that surface which contains molecules only in the same phase ; it is generally understood to refer to the surface upon which the molecules are just beginning to move. The veloc- ity of a wave front will always be that of the disturbance propaga- tion. A wave length is the interval, measured in the direction of wave propagation, between the nearest two consecutive surfaces upon which the molecules have similar phases. The amplitude of the undulation is the maximum displacement of the molecule from its place of rest. 49. From a consideration of the nature of an undulation, we .see at once that, if A be the wave length, r the periodic time, and y the velocity of wave propagation, we will have 40 ELEMENTS OF WAVE MOTION. - ' '. ; V=\, (49) and the values of F, A, and r are each, theoretically, independent of the amplitude. 50. To find an expression for the displacement of a molecule at any time during the transmission of an undulation, let x be the dis- tance of the molecule from the origin of disturbance, t the time from the epoch, r the periodic time of the molecule, A, the wave length, and V the velocity of wave propagation. Now, whatever be the displacement 6 of the molecule x, at the time t, an equal dis- placement (neglecting the loss due to increased distance from the origin) will exist for another molecule at a distance x -\- Vt' , at the time t + t'. This condition gives, whatever be the value of t', d = (x,t] = ( x + Vt',t + t'). (50) x + Vt' is the distance from the origin to the wave front at a time t subsequent to the instant at which it was at x. Hence the molecule x is behind the wave front a distance Vt x, and the dis- placement, (x, t), may be replaced by ( Vt x) ; therefore we have d = 4>(x,t) = 0(F*-a;), (51) as the form of the function. 51. We have implicitly assumed the medium to be in a state of stable equilibrium -during the passage of the undulation, and, there- fore, the molecule will necessarily describe a closed orbit about its place of relative rest. This orbit may, from the circumstances of the case, be of the most varied character, and, after the energy due to the disturbance has been dissipated, the molecule will resume its original place of relative rest, until again displaced by some new disturbance. It is necessary, in this discussion, to consider those disturbances alone which are regular and periodic, and to consider the orbit after it has become determinate. We therefore limit the discussion to that of the regular periodic disturbance, and the orbit to that of the ellipse or any of its particular cases, such as the ellipse, the circle, or the right line. 52. Simple Harmonic Motion. If a point a (Fig. 5) move uniformly in a circular orbit, tlie distance of its projection JFTC RELATING TO SOUND AND LIGHT. 41 Tom the centre, upon the vertical diameter, can always be found From the equation /o~/ \ (52) Figure 5. y a sin ^ + J, in which y is the required displacement at the time t, a is the am- plitude or maximum displacement, r the periodic time, and a the angle included between the horizontal diam- eter and that passing through the origin of motion. 2rrt The angle -- - + is called the phase of the vibration, and may be made of any value by changing the arbitrary arc #, the time t, or both together. The same value will apply to motion along any diameter. Such mo- tions are called simple harmonic motions. It may easily be shown that any two simple harmonic motions, in one line and of the same period, may be compounded into a single simple harmonic motion of the same period, but whose amplitude is equal to the diagonal of a parallelogram constructed on the am- plitudes of the components inclined to each other by an angle equal to their difference of phase. 53. The Harmonic Curve. If the motion of a point be compounded of a rectilineal harmonic vibration, and of uniform motion in a straight line perpendicular to the vibration, the point will describe a plane curve, which is called the harmonic curve. Let the vibration be along the axis of y, and uniform motion along the axis x ; we will then have for the ordinates, and . font y = a sm ( - x = Vt (53) (54) for the abscissas, due to the uniform motion. Combining these equations, eliminating #, and replacing VT by its equal A, Eq. (49), page 40, we have, for the equation of the harmonic curve, y = a sfn (-^- + a) ; \ A / (55) 42 ELEMENTS OF WAVE MOTION. in which A. is the wave length. Substituting for x, x i^, the value of y remains the same for all integral values of i. The curve, therefore, consists of an infinite number of similar parts, which are symmetrical with respect to the axis of x. Figure 6, 54. To construct the curve by points, divide the circumference into any number, as twelve, equal parts ; lay off on the axis of ab- scissas twelve equal distances, corresponding to the positions of the point in uniform motion, erect ordinates at these points and make them equal to the corresponding displacements at the given times, and we have the curve as follows : \6 1 & 9 10 11 18 55. The varying velocities of a point of a simple pendulum in motion can be represented by the ordinates of the harmonic curve ; and because of this analogy all vibrations represented by these curves are called simple or pendular vibrations. The vibration is taken to be the complete oscillation, from the time at which the moving point was in one position until it returns to the same posi- tion again. By this definition, the duration of the vibration of a second's pendulum would be two seconds, and not one second. 56. Composition of Harmonic Curves. Let y' = a sin (~- + ), (56) \ A. / y" = I sin (^-- + ft), (57) \ A / b'e the equations of any two harmonic curves, having the same wave length, but different amplitudes. The resultant value of y will be RELATING TO SOUND AND LIGHT. 43 y = c sin (^- + r), (58) which is the equation of another harmonic curve, of equal wave length, but of different amplitude from either of the components. The values of c and y are given by c cos y = a cos a -f b cos 0, (59) c sin y = a sin a -f sin 0, (60) c = a* + + 2a cos (a 0). (61) From the last equation we see that c may have any value be- tween the sum and difference of a and Z>, depending upon the value of the difference of phase, a 0, of the components. By a similar process, it can be shown that any number of com- ponent harmonic curves, of the same wave length, may be com- pounded into a single resultant harmonic curve having an equal wave length, but whose amplitude and phase differ in general from those of any of its components. 57. If the component curves have different wave lengths, they oannot be compounded into a single harmonic curve ; but when their wave lengths are commensurable, they can be compounded into a periodic curve, whose period is the least common multiple of their several periods. Thus, in the first case, where the wave lengths are unequal and incommensurable for the resultant ordinate, /%TTX \ /2nx J\ /ZTTX \ , y = a sin 1-y,- + a) + I sin i-^r + 0) + c sin h + y I +...., (62) in which the period is infinite, or the curve is non-periodic. In the second case, let m, n, r, being integers ; then the above equation becomes /2nrx y = asn _ + a . S m r-- c sml-y- (64) which, although not admitting of reduction to a simpler form, gives 44 ELEMENTS OF WAVE MOTION. constantly recurring, values of y when for x we substitute x -f- A. The wave length of the resultant curve is therefore A, and the curve is periodic. 58. The forms of the component curves depend only upon the wave lengths and amplitudes ; but their positions on the axis de- pend on the values of the phase , (3, y, etc. By assigning arbitrary values to these, we may shift any curve along the axis any desired part of its wave length. Any such shifting for any one or more of the component curves will necessarily alter the form of the result- ant curve, but will not change its wave length. 59. If the wave length of the resultant curve be assumed, the wave lengths of its components may be all possible aliquot parts of A, and the number of the possible components is therefore unlimited. Therefore every possible curve of wave length A, which could be so constructed from such component curves, would be found among those produced by placing, along the same axis, an unlimited num- ber of harmonic curves, as components, with wave lengths A, -JA? |A, etc., . . . By varying the amplitudes of the components and shifting them arbitrarily along the axis, an infinite number of resultants can be produced, all having the same wave length A. Fourier's theorem demonstrates that every possible variety of periodic curve, of given wave length A, can be so produced, provided that the ordinate is always finite and that the moving point is assumed to move always in the same direction. 60. A periodic series is one whose terms contain sines or cosines of the variable, or of its multiples ; thus, A ! cos x + A 2 cos %x -f A 3 cos 3x -+- . . . . A n cos nx -j- is a periodic series. This series goes through a succession of values as the arc increases from to %TT ; for, every term has the same value at the end and at the beginning of that period, and this con- tinuously, so that whatever n may be, the period of the function is 2n. 61. Fourier's Theorem has for its object the determination of the unknown constants, A Q , A 19 A 2 , , . . .B^ B%, B 3 , . . . ., and the determination of the conditions by which any given function, y = /(#), can be expressed in the form of RELATING TO SOUND AND LIGHT. 45 f( x ) = A o + A i cos a + .4 8 cos + B sin a; + # sin ) 1 The non-periodic term J is introduced to make the theorem conform to the most general case. If the function is capable of expression in periodic terms only, then A Q = 0; this fact can only be determined by considering each special case. The equation which expresses the mathematical statement of Fourier's Theorem is y = y + S 1 , 1 : Cl sin ( + ,), (66) in which / is the mean value of y, and each of the variable terms represents, by itself, a harmonic vibration of which the period is an aliquot part of the whole period T. 62, Wave Function. Kesuming Eq. (51), we see that, since the displacement 6 passes through all of its values while the undulation advances a distance equal .to its wave length A, it has the properties of simple harmonic motion, and, therefore, may be written 6 = ccsin ~(Vt x). (67) This is called the wave function. By making t vary continu- ously through all values from t = ^ to t = ~- , d will increase from zero to + , decrease then to a, and finally return to zero, during the time ^, which is evidently the interval of time required for the undulation to pass over the wave length A. Again, supposing t to remain constant and x to vary through all values from Vt A to Vt, we obtain again all possible values of the dis- placement, which values will evidently belong, at the same instant, to all molecules in the wave length. The following diagram illus- trates the two cases : Figure 8. 46 ELEMENTS OF WAVE MOTION. By the addition of an arbitrary arc we can cause the displace- ment to take any one of its values, at any time t, and thus change- our origin at pleasure. 63. The corresponding expression for the velocity of the mole- cule in its rectilineal orbit, sometimes called the velocity of the wave element, in contradistinction to the velocity of wave propagation, is given by ^ n u = a cos -y- ( Vt x). (68) x/ 64. The principle of the coexistence and superposition of small motions is shown in Mechanics to be applicable to planetary per- turbations. It is, for similar reasons, applicable to the determina- tion of the resultant displacement of a single molecule, arising from the concurrent effect of many disturbing causes acting sepa- rately. The acceptance of this principle is equivalent to assuming* that the several displacements are so small that their products and powers higher than the first are negligible with respect to the dis- placements themselves ; and it embodies the primary supposition that the intensity of elastic forces developed varies directly with the degree of displacement. 65. Wave Interference. If we apply this principle to de- termine the displacement of a molecule by two disturbing causes, giving rise to two undulations of the same wave length, we will have for the first, 6' = a' sin p^ ( Vt - x) + -4'1 ; (69) for the second, 6" = a " sin [y ( Vt - x) + A"^ . (70) The total displacement will be 6' + (5" = (5 = (a 1 sin A' + a" sin A") cos [^ ( Vt - fc)~] i (71) + (' cos A' + a" cos A") sin y ( Vt - x)\, whicH may be put under the form = a sm (Vt -x) + A (72) by placing a cos A a cos A' + a" cos A", a sin A = a sin A' + " sin .4". S Whence, 2 = c*' 2 + ec"2 4. 2 r // cos (.4' A j fan /4 RELATING TO SOUND AND LIGHT. 47 (74) (75) ' cos ^ + cos - By Eq. (72) we see that the resultant undulation is of the same wave length as the components ; that the maximum displacement of the resultant undulation is not, in general, equal to that of either of the components, and that it does not occur at the same time nor place with either of them. 66. Taking the square root of Eq- (74), we have a = ' 2 + " 8 + 2'" cos (A'A) ; (76) from which it is seen that, when A' A" = 0, ' + "; and, when A' A" = 180, a = a' a". Hence, in Eq. (75), A = A' A" in the first case, and A = A' = 180 + A" in the second. The maximum displacement, then, of the resultant undulation may vary between the sum and difference of the maxi- mum displacements of the two component undulations, depending upon the difference of phase. If, in the two component undulations, a' = a", will be equal to 2' when A' = A", and vary from this value to zero as the dif- ference of phase A' A" passes from zero to 180. Substituting, in the expression for the displacement, A' 180 for A', we will have ' sin -(Vt-x) + A' + n = tt ' ^vt-x + A (77) which is exactly the same as when for x we put x^f - 6 Therefore, if we suppose that two undulations of the same wave length, starting in the same phase, meet after travelling over routes which differ by one-half the wave-length, there will be no displace- ELEMENTS OF WAVE MOTION. ment of the molecule at the place of meeting, and complete inter- ference will result. The diagrams of Figure 9 illus- trate the composition of two un- dulations of equal wave length, having the same phase in the first case, and opposite phases in the second and third cases. In AB, the amplitude of the resultant un- dulation a is equal to the sum of the amplitudes of the component undulations, a' and a" '; in A'B' and A"B", equal to the difference of the amplitudes. In A"B", the displacement of the molecules is zero, and the two components mu- tually destroy each other's action. Figure 9, 67. Interference of any Number of Undulations. 1 CASE. When the component undulations have the same wave length. Let 6' =' 8 m[^ (Vt - x) + A' 1, L A- f -i -^ (Vt x) -f- ^4" , A J (7g) r' = V' r fflB| ^(F^-a;) + etc., etc., , be the values of the several component displacements. By addition we have 6' + 6" + 6'" + etc. ^) L RELATING TO SOUND AND LIGHT. 49 The second member may be placed under the form of sin A cos -r- ( Vt x) -f a cos A sin ( Vt x) A. A r? ff f (80) = a sin i- ( Vt - IK) + A = At other points, where 6 is not great and cos 6 not small, the resultant displacement becomes equal to zero when 2nb cos - J- -- = TT, 2n, STT, etc. ; A 2A 3A that is, when cos = ^ , ^, ^ , etc. The greatest resultant displacement, other than that indicated above, will be found by making in Eq. (87), =1 , : ' (90) and it will be equal to ? - ; nl cos B ' and, since the intensity of the sensation is directly proportional to- 52 ELEMENTS OF WAVE MOTION. the square of the maximum displacements, we will have the rela- tion of the intensities, ! 10 [(91) 272 ** < ?r 2 ^ cos 2 6 P 47r 2 2 cos 2 71. In acoustics it will be shown that the wave lengths corre- sponding to audible sounds will vary from - - = 57' to - /cO 40000 = of an inch, and therefore there will be no point exterior to an aperture where the displacement will not occur, and hence the cor- responding sound be heard. In light, the wave lengths vary between .000026 and .000017 of an inch, and there will be, accord- ing to the 2 case, alternations of light and darkness surrounding the central line drawn from the place of original disturbance to the centre of the aperture. These zones are called Huyghens* zones, and will be again referred to in the subject of diffraction. 72. Diffusion and Decay of Kinetic Energy. The displacement of any molecule due to wave motion of a given wave length is independent of the periodic time, and, since the orbits of the molecules are described in equal times when they arise from a given periodic motion, they will be directly proportional to the dis- placements or any other homologous lines. The velocities, then, of the moving molecules being represented by v 9 their kinetic ener- gies will be represented by Then, because these energies are the intensity of the elastic force developed by the displacement d, we have (95) whence, U = - -f F 2 . (96) Hence, when a plane wave is propagated without altera- tion in a homogeneous medium, its velocity of propagation is directly proportional to the square root of the elastic force developed by the displacement of its molecules. 77. Reflection and Refraction of Plane Waves. Let the incident plane wave AC (Fig. 11) meet the deviating sur- face at all points, in succession, from A to B. Let V and A be the velocity of wave propagation and the wave length in the medium of incidence, and V and A' those in the medium of intromittance. Let AB = ds, and CB = Vdt. While the disturbance in the in- cident wave is moving from C to B, the disturbance from A as a centre will proceed in all directions in the medium of incidence, RELATING TO SOUND AND LIGHT. 55 and be found, at the instant considered, upon the hemisphere whose radius is AD = CB = Vdt, and in the medium of intromittance on the hemisphere whose radius is AD' = V dt. Each point in the line AB will, in like manner, become in succession a new centre of dis- turbance, sending secondary waves into the media of inci- dence and of iutromittance, whose radii will, at the instant the incident wave reaches B, be Figure n, equal to V and V multiplied by the interval of time elapsing between the instant of arrival of the wave front at the centre considered and that of its arrival at B. The surface through B, which is tangent to all the reflected pulses, may be taken as the front of the reflected wave, for it will contain more energy than any other surface of equal area in the incident medium. Similarly, 'the surface through B tangent to all the refracted pulses will con- tain more energy than any other of equal area in the medium of intromittance, and may be taken as the front of the refracted wave at this instant. These surfaces are readily seen to be planes ; hence, denoting the angle CAB = ABD by 0, and ABD' by 0', we will have ds sin = Vdt, ds sin 0' = V dt ; (97) from which we obtain sn = --, sin ' = p sin (98) which is known as SnelPs law of the sines ; [i is called the index of refrac- tion. 78. The angles and ' made by the wave fronts with the deviating sur- face are, respectively, equal to the angles made by the normals to the in- cident and refracted waves with the normal to the deviating surface, and Figure 12, 56 ELEMENTS OF WAVE MOTION. are called angles of incidence and refraction. The angles of inci- dence and refraction are measured from the normal to the deviating surface on the side of the medium of incidence to the normal of the incident wave, and to that of the refracted wave produced back into the medium of incidence. The angle of reflection is measured from the normal to the de- viating surface to the normal to the reflected wave front, and is therefore negative. In the reflected wave, since the velocity of wave propagation is unchanged, \i is equal to unity, and Eq. (98) becomes sin = sm '. (99) 79. These principles may, in ordinary cases, then be summa- rized as follows : 1. The planes of incidence, reflection, and refraction are coin- cident. 2. The sine of the angle of incidence is equal to the index of refraction or of reflection multiplied by the sine of the angle of re- fraction or of reflection. The modifications which take place in polarized light will be referred to hereafter in physical optics. 80. We see from Art. 77 that the reflected and refracted waves are plane when an incident plane wave meets a plane deviating sur- face. It is evident also, from the construction, that the reflected rays are all normal to a plane NN' symmetrical with MM' with refer- ence to OX ; and that the incident and reflected rays are directed from their corresponding planes towards the deviating surface. The refracted rays are normal to a plane ER' on the same side of the deviating surface as the incident wave, and are also directed towards that surface. 81. General Construction of the Reflected and Refracted Waves. Let the deviating surface AB (Fig. 14) be any whatever, and the rays proceed from any origin ; take, in RELATING TO SOUND AND LIGHT. 57 the medium of incidence, any spherical surface SS', with centre at 0, as the incident diverging wave ; then, from all points I, I', I", etc., of AB, describe spheres, whose radii are equal to the intercepts of the rays between SS' and AB. If, now, tangent planes be drawn to the deviating surface at I, I', I", etc., and to the surface SS' at the corresponding points s, s', s", etc., each pair of tangent planes will determine, by their intersection, a right line, through which if a plane be passed tangent to the cor- responding sphere on the other side of the deviating surface, it will be symmetrical with the in- finitesimal surface of SS' at s with respect to that of AB at the point I; and similarly for the other points. By continuity, these points of tangency may be considered as forming the envelope of the re- flected wave. The direction of the reflected rays is found by joining these points with I, I', I", etc., and extending the lines toward and beyond the deviating surface. 82. By the proper modification of the radii due to the value of p, the index of refraction, the envelope of the refracted wave and the direction of the refracted rays may be constructed. 83. Considering the reflected wave as a new incident wave, the new reflected wave, by another deviating surface, can be constructed by an application of the above principles ; and since reflection may be considered as refraction whose index is 1, the principle may be generally stated, that any number of reflections and refractions may be replaced by a single refraction at a supposable deviating surface with a properly modified index of refraction. 84. Let DEF (Fig. 15) be any incident wave whose rays are not necessarily parallel ; MNP any deviating surface. At some subse- quent time t the incident wave will occupy some position such as ABG, FG being equal to EB = DA = 'vt. By the principle Figure 14. 58 ELEMENTS OF WAVE MOTION. established above, abg will be the enveloping surface of the reflected wave corresponding to ABG-, and a^b^g^ that of the refracted wave, and both will be concurrent, that is, the phases of the molecular motions on them will be similar ; #PGr', #NB', MA' will be the re- flected, and ^jPGrj, Z'jNBj, tfjMAj the re- fracted rays. 85. Prolong the con- secutive rays of either the reflected or refract- ed waves, say the re- flected wave abg, until they meet two and two ' they will be tangent to the surface j3y, which is the evolute of abg. Since the reflected rays are all normal to abg, this evolute will corre- spond to any other po- sition of the reflected Figupe I5| wave, also. The surface of which a(3y is a generatrix is in optics called the caustic surface. It is evident that the points of this caustic are not concurrent, because their distances, being equal to the radii of curvature of abg from the reflected wave, are themselves unequal; and points, in order to be concurrent, must be at equal distances from the wave surface. Whether the caustic be real or virtual, the displacements of its molecules being either due to that of two rays, or apparently so, the energy of the molecules, and hence the resulting sensation, will be greater than that due to but one ray. 86. When the evolute j3y is known, the various possible posi- tions of the reflected wave can readily be determined. In the ordi- nary cases considered in optics, the surfaces abg are those of revolution ; the caustic is then also a surface of revolution. Sup- pose abg to be one of the generatrices of the reflected wave, consid- ered as a surface of revolution, and a(3y to be its evolute ; then, by RELATING TO SOUND AND LIGHT. 59 the property of the evolute, if the tangent aaa' be caused to roll on 4/3y, each point of this tangent will describe one of the sections of the reflected wave. Thus, a'b'g', a"(3g", and abg are such sections; the second of these being of two nappes, tangent to each other and normal to the evolute at the point (3. 87. The principle that the rays, after the wave has been sub- jected to any number of reflections and refractions, are all normal to a theoretically determinable surface, and consequently to a series of surfaces, of which any two intercept the same length on all the rays, is principally applicable to the determination of caustic sur- faces, and to the formation of optical images, and will therefore be further discussed in that branch of the subject. 88. Utility of Considering the Propagation of the Disturbance by Plane Waves. In a homogeneous medium, the arbitrary displacement of a molecule gives rise to elastic forces whose intensities depend on the degree and the direction of the displacements, and whose directions are not, in general, those of the displacements. In Art. (35) we have seen that the displacements must be made only in exceptional directions, in order that the elas- tic forces varying directly with the degree of the displacement should be wholly in those directions. Should the orbit of the dis- placed molecule be curvilinear, it is evident that, at each point of its path, the elastic forces developed would vary both in direction and intensity, and thus the general problem becomes one of extreme intricacy. 89. If, however, it be possible to limit the discussion to that of molecules in the same plane, all actuated by equal and parallel dis- placements, the variation as to direction of the elastic forces may, perhaps, be eliminated. It has been shown, Art. 76, that when a plane wave is propagated without alteration in a homogeneous me- dium, the velocity of propagation is directly proportional to the square root of the elastic force developed by the displacement. Hence the importance of deducing from the general equations (18) the corresponding equations applicable to the vibratory motions propagated by plane waves. 90. At the time t let r be the distance of the plane wave, in a homogeneous medium, from the origin of co-ordinates ; e the dis- placement of the molecules whose co-ordinates are x, y, z ; |, 77, , 60 ELEMENTS OF WAVE MOTION. the projections of e on the rectangular co-ordinate axes ; and , ft y, the angles made by the displacement with the axes, respectively. We then have e = 6 sin -^ ( Vt r) ; o__ =. 6 cos a sin -y- ( Vt r) ; A 77 = (5 cos /3 sin -r- (Vt r) ; = 6 cos y sin -y- ( F^ r). (100) (101) Let r + Ar be the distance of the plane at a subsequent instant from the origin, and I, m, n, the angles made by the normal to the plane with the axes, then r = x cos I + y cos m + z cos n, (102) Ar = A# cos I 4- A/ cos m + Az cos w. (103) From Eq. (101) we have -f- A| = d cos sin -^- ( Vt r Ar) = d cos a Li in .(Vt-r) cos ~ Ar A 277 . 27T cos ~Y ( V\ t r) sin Ar ; from which, and similarly for the axes y and z 9 we have A = 6 cos sin ( Vt r) (cos -y- Ar - 1 j (104) 27T , _, . . 2n A cos -y- ( F# r) sin -r- Ar I, A A 2rr , T7 x / 2rr A AT; = d cos j3 sm -r- ( Fif r) ^cos Ar cos -T- (Vt r) sin Ar L A A, A = d cos y sin -r- ( F^ r) (cos -y- Ar 1 ) A \ A / * 2?r . x . 2rr cos -v- ( FP r) sin Ar . A A J ., (105) RELATING TO SOUND AND LIGHT. 61 Substituting these values in Eqs. (18), and, since the medium is homogeneous, the sums arising from the substitution of the second part of the values of A, A??, A and which are of the form n (r) sin ~ Ar, A, ~ sin - Ar, (106) , . A?/ Az 2n r) -= - sin -r- Ar, r 2 A , x A# Aw . 2rr (r) 3-^ sin Ar, r A . - 2 - sm Ar, r) -t sin - Ar, r A all reduce to zero, because they are formed of terms which, two and two, are equal, with contrary signs ; for, to the values of A#, ky, Az, equal, with contrary signs, correspond values of Ar which are also equal and have contrary signs. Then, replacing cos -r- Ar by its At * equal, 1 2 sin 2 ~ Ar, and 6 sin -^ ( Vt r) by its equal e, A A Eqs. (18) become, for plane waves, = ~ = cos - cos {3 Zptf) (r) - Y = ^ = cos sin 2 - Ar + cos y // A# AZ . TT r- sm 2 T r 2 A sin 2 Ar i I 4- cos j3 E^ (r) + i/> (r) -^- I sin 2 ^ Ar N Aty Az . ?. + cos y Sft T/J (r) -^-3 sm 2 - Ar, 62 ELEMENTS OF WAVE MOTION. 4- cos pzp'ip (r) -- sin 2 Ar 4- cos y Sf* If (r) + V (r) -^ \ sin 2 An (107) 91. The conditions for the propagation of the plane wave with- out change are - = ' = '- (108> cos cos j3 cos y Substituting, in Eq. (107), for Ar its equal, Az cos -f- Ay cos m -f A2 cos w = Ar, (109) and substituting in Eqs. (108) the values of X^ JT 19 Z^, thus ob- tained, we will have two relations which, with cos 2 a -f cos ? -h cos 2 y = 1, (110) will enable us to determine the angles , /?, y, which the displace- ment should make with the axes, in order that the propagation of the plane wave may be possible. 92. Because of the equality of the coefficients of cos j3 and cos a in the first and second of Eqs. (107), and of cos J3 and cos y in the third and second, and of cos a and cos y in the third and first, we can, by substitutions and reductions similar to those em- ployed in Art. 33, deduce corresponding principles, and hence determine that, for each direction of the plane wave, there corre- spond, for the molecular displacements, three rectangular directions; such that the -plane wave may be propagated without change, andi that these three directions are parallel to the three axes of an ellip- soid whose equation is, RELATING TO SOUND AND LIGHT. 2xz l(l>(r) + 1>(r)^- |sin2^Ar A^ 2 r 2 J A# Ay ^ 2 TT (r) =1. 5 sin 2 T Ar f" A ^ g sin 2 ^ Ar This is called either the inverse ellipsoid or the ellipsoid of polar- ization. Having also the relation expressed in Eq. (109), we see- that the coefficients of Eq. (Ill) depend upon the angles I, m, n, which determine the direction of the plane wave, upon certain con- stants which define the constitution of the medium, and upon the wave length. The velocity of propagation is inversely proportional to the length of that axis of the ellipsoid to which the molecular displacements are parallel. 93. Relation between the Velocity of Wave Propa- gation of Plane Waves and the Wave Length in Isotropic Media. All directions heing identical in isotropia media, we will assume the plane wave normal to the axis of x. We then have Ar = A#, and and Eqs. (107) reduce to (113) 4 ELEMENTS OF WAVE MOTION. X l = cos Sp 0. (r) + V M f- sin 2 ^ Az, Fj = cos (3 I,fi\ (r) + V (^) -|- sin 3 ^ A, t 1 I (r) -f -0 (r) sin 2 j Ao; : and the equation of the ellipsoid to ?in 2 Ao; and, since all directions perpendicular to the axis of x are identical with reference to the plane of the wave, we have Ay = A, and Eq. (114) of the ellipsoid becomes one of revolution about the axis of x. Whence, we conclude that, in an isotropic medium, a plane wave normal to a given direction can be propagated without change, whenever the molecular displacement is parallel or perpendicular to this direction. To any one direction of normal propagation in such a medium, there corresponds an infinite number of waves with transversal vibrations, having the same velocity, and but one wave with longitudinal vibrations whose velocity is different from those with transversal vibrations. 94. For the wave with longitudinal vibrations, we have a = 0, = y = 90, =X t = Z P * (r) + i> (r) TT sin* and, from Eqs. (96) and (107), we have V* = S (r) + V (r) -f] sitf \ A*. (116) RELATING TO SOUND AND LIGHT. 65 In Acoustics, it will be shown that sound is due to longitudinal vibrations of the medium. This equation will then be applicable in all cases of sound arising from such vibrations, and will be referred to, in that branch of the subject. In Optics, it will be shown that transversal vibrations only are efficacious in producing light. 95. For waves with transversal vibrations in isotropic media, the velocity is independent of the direction of the displacement We can then suppose the displacement parallel to the axis of y, ana thus have a = y = 90, = 0, r t = and F2 A 2 r A = SM 10 (r) + V (r) -- I sin 2 Aas. (117) (118) This equation is applicable in light, for the determination of wave velocity in isotropic and homogeneous media, and will be used hereafter in determining the velocity of light propagation. By developing sin 2 -r A# into a series, we find A, Substituting this in Eq. (118), we obtain 72 = * (119) in which a, b, c, ---- have for values, a= c = rf= - (120) ELEMENTS OF WAVE MOTION. These constants depend only on the constitution of the medium, and decrease very rapidly in value, for Az is always a very small quantity. If the wave length be not excessively small, if it sur- passes a certain value which observation only can determine, the terms of the second member of Eq. (119) will have very rapidly decreasing values, and we will obtain an expression approximately near to V 2 by taking only the first few terms. Hence, a must be positive, and, since observation shows that the most refrangible rays are those of the shortest wave length, and that, as a conse- quence, V decreases with A, b is necessarily negative. 96, Hence, in isotropic media, the elasticity being uniform in all directions, the form of the wave surface will be spherical, and when the displacements are longitudinal, its radius at the unit time from the epoch will be the value of V obtained from Eq. (116) ; when the displacements are transversal, the radius will be the value of V in Eq. (118). The former relates wholly to waves of sound, and the latter to those of light. The subsequent discussion will now apply to transversal vibra- tions alone, and the conclusions derived belong therefore to the transmission of light undulations. Experiment shows that the media which transmit the waves of light are not in general isotropic, and as a consequence the form of the wave surface will not be spherical. We will, therefore, now seek the form of this surface in the general case, and make use of the properties of plane waves for this purpose. 97. Plane Waves in a Homogeneous Medium of Ttiree Unequal Elasticities in Rectangular Direc- tions. In the plane wave, the following conclusions have been deduced : 1. The displacements of the molecules, in each position of the same plane wave, must be rectilineal and parallel to each other and to their original directions. 2. The elastic forces developed by these displacements must be either in the directions of the displacements or alone efficacious in these directions. 3. The propagation of the plane wave unaltered is then possible. These conclusions involve, as consequences, a constancy of ve- RELATING TO SOUND AND LIGHT. 67 locity of propagation when the plane wave is unchanged in direction, and a variation in the velocity as the direction is changed. Hence, if the elasticities of a homogeneous medium differ in all directions, and we suppose plane waves, having all possible positions, originate at any point m of an indefinite medium, these plane waves, at the end of a unit of time, will be at different distances from m. The surface which is the envelope of all these plane waves at this instant is called the wave surface. 98. Let a > I > c be the principal axes of elasticity of such a medium. Then 2 , Z> 2 , c 2 , will measure the elastic forces developed in these directions by a displacement equal to unity, and any of the surfaces of elasticity heretofore determined can be used to obtain the elastic forces devel- oped by an equal displacement in the direction of the corresponding radius vector of the surface. The velocity of wave propagation being proportional to the square root of the elastic force, Eq. (96), its value can be found when the elastic force due to the displace- ment in any direction is known. 99. Fresnel made use of the single-napped surface of elasticity whose equation is &# = r 4 ; (121) but for plane waves, the inverse ellipsoid of elasticity or first ellipsoid, 4- % 2 + cW = 1, (E) together with its reciprocal ellipsoid, r 2 ^ z 2 can be more readily used, because of its better known properties. The squares of the semi-axes of (W) and of the reciprocals of (E) are the principal elasticities of the medium. 100. There are two cases to consider: 1. The plane of the wave contains two of the principal axes, and hence is one of the principal planes of the medium. The plane cuts the ellipsoids in ellipses whose semi-axes are either two of the principal axes. Whatever be the direction and amount of 68 ELEMENTS OF WAVE MOTION. the displacement, it may be replaced by its components in the di- rection of the axes proportional to cos a and sin , a being the angle made by the displacement with either axis. Considering these separately, we see: 1, that each will commu- nicate to- the molecules in the adjacent plane analogous rectilineal motions which will be propagated without alteration of direction; 2, that the elasticities, and hence the velocities of propagation which belong to these two, are different, and that after a time there will be two series of molecules situated in parallel planes, parallel also to the primitive plane, which will contain all of the original energy ; 3, that the vibrations of the molecules in these two plane waves will be at right angles to each other. 101. 2. The plane wave is any whatever. The sections of the ellipsoids will be ellipses, but will not in general contain either of the axes of the ellipsoids. There will then be no direction of the displacement that can give a resultant elastic force in the direction of the displacement. It is, therefore, essential for a rec- tilineal oscillation of the molecule and for a consecutive transmis- sion of this oscillation, that there should be no tendency of the rectilineal displacement to be deflected on either side, but that the line of the resultant force should "be projected upon the displace- ment. As it is not in general in the plane, but oblique to it, it can fte resolved into two components : one normal to the plane, which is not effective in light undulations ; and the other, which is alone efficacious, in the direction of the displacement. In each elliptical isection there are two such directions, which are named singular 'directions^ and which are perpendicular to each other. Assume any plane section through the centre of (E) ; the elasticity measured by the squares of the reciprocals of the radii-vectores is the same to the right and left for the two axes of the section, and is the same only for them. Through either of the axes pass the normal plane to the section ; it will cut all the parallel plane sections in their homolo- gous axes. With reference to this normal plane, the radii-vectores, .and therefore the elasticities of each section, are symmetrical. Hence, if the displacement be along one of the axes of the section, the total elastic force will be in the normal plane, and will be pro- jected on the axis of the ' section. And since the ellipsoid semi- diameters are inversely as the velocities of propagation, the recipro- RELATING TO SOUND AND LIGHT 69 cals of the axes will measure the velocities of wave propagation. Hence is established the fact that for each section there are two of these singular directions, and that they are rectangular. These two singular directions perform the same function for the vibrations of the plane wave as do the axes of elasticity themselves when the plane wave contains them. Each vibration is replaced by two others in the direction of the singular directions, and these two compo- nents proceed in the medium without change of direction, but with different velocities, so that there are then, in the general case, two plane waves parallel to each other and to the original plane wave. If a be the angle made by the displacement with one of the axes, the component displacements will be proportional to cos and sin , and the elastic intensities to cos 2 cc and sin 2 . Whatever may be the direction of the original supposed vibration in the plane wave, the two plane waves which replace it are always the two above designated. 102. If the plane of the wave coincides with either of the circu- lar sections of the ellipsoid, the plane wave will be propagated without alteration, whatever be the direction of the displacement, with a velocity equal to #, the reciprocal of the mean semi-axis of the ellipsoid. 103. The Double-Napped Surface of Elasticity. If through the centre of (E) we pass any plane, and on the normal to the section at the centre set off distances inversely proportional to the semi-axes of the section, the locus of all these pairs of points is called the double-napped surface of elasticity. For, each radius vector measures the velocity of propagation of one of the plane waves, arising from a displacement in the plane of section, and the square of each of these normal velocities is the measure of the elas- tic force developed by the component displacement along the axes of the section. 104. If through each of the points so determined planes be passed parallel to the corresponding plane of section, the envelope of all these planes will be, by definition, the wave surface. Hence, the latter can be constructed by points from this surface of elasticity. 105. To get the polar equation of the latter surface, let us take for co-ordinate axes the principal axes of the medium ; let Z, m, n^ 70 ELEMENTS OF WAVE MOTION. be the angles made by the normal to the plane wave with these axes, x, y, z, respectively ; , ft y, those which OLC of the axes of the ellipse of section make with the same axes ; then we have cos a cos I + cos )3 cos m + cos y cos n = 0. (122) The elastic force developed by a displacement parallel to the axis of section is projected on the plane of the wave parallel to this displacement, and its components are X a 2 cos a, Y = & cos ft Z = c 2 cos y. (123) The cosines of the angles which this elastic force makes with the axes are then proportional to these values. An auxiliary right line perpendicular to the direction of the elastic force and to the dis- placement will lie in the plane of the wave, and if u, v, w, be the angles which it makes with the axes, we will have a 2 cos a cos u -\- b 2 cos j3 cos v + c 2 cos y cos w 0, \ cos a cos u + cos (3 cos v + cos y cos w = 0, I (124) cos I cos u -f cos m cos v + cos n cos w = 0. ) Representing the velocity of propagation of the plane wave by V, we have V 2 = a 2 cos 2 + W cos 2 j3 + c 2 cos 2 y. (125) Combining the above equations, and eliminating the quantities , ft y, u, v, tv, we will have an equation containing V, I, m, n, which will be that of the surface required. To eliminate u, ?;, w, we will make use of the method of indeterminate coefficients ; thus, multiply Eqs. (124) by B, A, and unity, respectively, add the three resulting equations, and from the conditions for B and A that the coefficients of cos v and cos w shall reduce to zero, we will have (A -f Ba) cos a -f cos I = 0, J (A + BP) cos + cos m = 0, V (126) (A 4- Be 2 ) cos y + cos n = 0. ) Multiply these by cos , cos 3, and cos y, respectively, add, and reduce by Eqs. (122), (125) ; we will have Q. (127) RELATING TO SOUND AND LIGHT. 71 Substitute this value of A in Eqs. (126), and we have cos I = B ( F 2 a 2 ) cos a, } cos m = B ( F 2 2 ) cos j3, V (128) cos w = ^ ( F 2 c 2 ) cos y. ) From which we get cos / cos m cos n F 2 cos cos ]3 cos y cos 2 Z cos 2 m cos 2 f / T7*O 19\O "T a 2 ) 3 T ( F 2 J 2 ) 2 ^ ( F 3 c 2 ) 2 J (129) Replacing cos , cos 0, cos y, in Eq. (122), by their propor- ,. , .... cos I cos m cos n tional quantities, , -, 8 we have wL/O V _ V/Vk^ ffV _ -WVKJ IV ^ y . ^ _ V (IdU; the polar equation of the double-napped surface of elasticity, in which F is any radius vector. 106. The Wave Surface. Through any point of the sur- face of elasticity pass a plane perpendicular to the radius vector at that point, and let r, A, p, v, be the polar co-ordinates of any point of the plane. The equation of the plane will be cos I cos A + cos m cos fi -j- cos n cos v = (131) T "We have also, as equations of condition, cos 2 1 4- cos 2 m 4- cos 2 n = 1, (132) cos 2 ? cos 2 m cos 2 n /IQQ\ The wave surface is the enveloping surface of the planes given by Eq. (131), and its equation can be determined by eliminating F, Z, ??i, n, and finding an equation between r, A, p, v. To do this, differentiate Eqs. (131), (132), (133), regarding cos I and cos m as independent variables, and we will have 72 ELEMENTS OF WAVE MOTION. , d cos ft 1 d V cos A -f cos v - 7 cos ? -f cos n 6? COS I d cos ft *? COS I r d cos I' COS Z COS ft (? COS ft (134) 72 _ 2 d cos cos 2 cos 2 m cos 2 ft cos m cos ft cos n ^ cos ft 1 dV COS JU -f- COS V -7 - = j . d cos m r d cos m d cos ft cos w + cos ft -j - = 0, d cos m cos 2 m cos 2 (J72_#2)2 + (F2~~ rfF ( 72 _ 02 + (135) i /\w m T i u cos n a cos n **, , , ,,. 107. To eliminate -= 7, -= , ,, , multi- d cos I d cos m d cos I d cos m ply Eqs. (134) by 1, A, and 5, respectively, add the resulting equations together, and perform the same operations on Eqs. (135). Supposing the indeterminate quantities A and B to have such d cos ft d cos ft 6? F values as will make the coefficients of -j , equal to zero, we will have dcos m cos A -f- A cos I = B cos \i + A cos in = ^ cos J ' d cos m' whence, we have A V r ' Y B = (r 2 F 2 ). (139) T Substituting these values in Eqs. (136), we obtain r cos /I Fcos I i (140) r cos jlfc F COS 7ft r cos i r cos 2 1 v F cos n C 2 ~' Y% C Z ' cos 2 m cos 2 f& ... F 2 [_( F 2 a 2 ) 2 (F 2 ^) 2 (F 2 c*)J f cos 2 A COS 2 jLt COS 2 V The first three equations (140) can be placed under the form, cos A - cos Z =(, 2 _ cos \i , ,_.., cos cos u cos m = (r 2 F 2 ) --s %i A \ / At2 A55 r*os v cos v cos n = (r 2 F 2 ) - r / r a _ C 2 Adding these, after multiplying tnem (141) Adding these, after multiplying tnem tf cos A, cos p, cos v,. respectively, and reducing by Eq. (131), we have 74: ELEMENTS OF WAVE MOTION. whence, dividing by r 2 F 2 , we have cos 2 A cosy cos 2 i; + the polar equation of the wave surface. 108. A more advantageous form for discussion can be obtained by subtracting the identical equation, 1 _ COS 2 X COS 2 [I COS 2 V fiAA\ ^ - ~fi I -- ^2" "I -- ^T~> from the equation above, by which there results a 2 cos 2 A # 2 cos 2 i c 2 cos 2 v 109. To obtain the equation of the wave surface in rectangular co-ordinates substitute for cos A, cos \i, cos v, and /*, their equals, -, -, -, and A/# 2 + y 2 + z 2 ', whence, we have T T T - P (c 2 , = 0. j ^ > This equation being of the fourth degree, the surface is of the iourth order, and, as will be shown hereafter, consists of two dis- tinct nappes, having but four points in common. 110. If two of the velocities become equal, as, for example, Jb c, the equation gives a* + 02 + s? = fl, (147) aW + Z> 2 (y 2 + z 2 ) = aW, (148) which shows that the wave surface, under this supposition, consists -of a spherical surface and that of an ellipsoid of revolution tangent to the sphere at the extremity of its polar axis. RELATING TO SOUND AND LIGHT. 75 Finally, if the three principal velocities become equal, or a = 5 = c, as in isotropic media, Eq. (146) becomes y* + if + 2 _. ^ ( U9 ) and the wave surface becomes spherical, as has been heretofore shown. 111. Construction of the Wave Surface by Means of the Ellipsoid ( W). Let us suppose that the ellipsoid (W), x* y 2 z 2 jjl + 5 + 31-l, (150) be cut by any plane through its centre, and that distances be laid off on the normal equal to the semi-axes of the elliptical section. Referring to the construction of the double-napped surface of elas- ticity by means of the ellipsoid (E), ## = 1, (151) we see that in designating the polar co-ordinates of the points con- structed by the aid of (W) by r, k, p, v, the equation of their loci can be obtained from the equation of the double-napped surface, cos 2 m by substituting for F 2 , 2 , J 2 , c 2 , I, m, n, respectively, - 2 , - 2 , ^, -, A, j^, v. We thus obtain = o, (153) 2 COS 2 A J* COS 2 |W , C 2 COS 2 V -^zr^ + tstrp + TJ^T? = ' which is the equation of the wave surface. Hence, points of the wave surface can be constructed from the ellipsoid (W) in precisely the same manner as points of the surface of elasticity from the ellipsoid (E), except that in the former, distances equal to the semi- axes are laid off on the normal, and in the latter the distances are equal to the reciprocals of the semi-axes. 76 ELEMENTS OF WAVE MOTION. 112. Direction of the Vibration at any Point of the Wave Surface. Let us consider any plane wave tangent to the wave surface; the displacement propagated by this plane wave makes the angles a, ft y, with the axes ; the radius vector of the wave surface at the point of tangency makes with the axes the angles A, //, v therefore the angle between these two lines will de- termine the required direction. Eliminate in Eq. (129) the angles I, m, n, by means of the first three of Eqs. (140), which, with the last of Eqs. (136), will give cos A cos // cos j3 cos v cos y / cos 2 A I / 9 19\9 I cos 2 v cos 2 m cos 2 n ~~^2\2 ~*~ I 172 _ A2\2 "*" 7 F2 />S F; / i_ i_ "" r \ BVr ~ ^^72' (155) whence, cos A ^^"S cos // COS V cos a cos j3 cos y (156) Substituting in Eq. (143) of the wave surface, we have / F 2 cos a cos A -f- cos )3 cos \i -f- cos y cos v = \J 1 - (157) In the figure, let M be any point of the wave surface, OM the radius vector, and OP the perpendicular F on the tan- F o gent plane to the wave surface ; - - is Figure 16. RELATING TO SOUND AND LIGHT. 77 / F* then the cosine of POM, and A / 1 2 - is its sine ; hence, OMP is complementary to POM, and therefore the vibrations at M are directed along the line PM. We conclude, therefore, that the direction of the vibrations of the molecule at any point of the wave surface is along the projection of the radius vector on the tangent plane at that point. When the tangent plane is normal to the radius vector, as is the case at the extremities of the axes, this determination is not appli- cable, but the direction is in these cases easily found. The plane OMP, which contains the radius vector and the direction of the corresponding vibration, is called the plane of vibration. 113. Relations between the Directions of Normal Propagation of Plane Waves, the Directions of Radii- Vector es of the Wave Surface, and the Direc- tions of Vibrations. By the preceding theorem we have seen that, in any plane wave whatever, the normal to this plane, the direction of the vibrations in this wave, and the radius vector drawn to the wave surface at the point of tangency, are all contained in the same plane. Besides, for each normal direction of propagation of a plane wave, there correspond for the vibrations, two directions parallel to the axes of the elliptical section of the ellipsoid (E) made by a parallel plane. These directions, therefore, being per- pendicular to each other, the planes which contain, at the same time, the same direction of normal propagation, the two vibrations, and the two corresponding radii-vectores, are rectangular. 114. Since the wave surface has two nappes, each radius vector will give two directions for the vibrations. We will now show that the planes which contain a radius vector and the directions of the two corresponding vibrations are also rectangular ; and for this pur- pose we shall show that the two vibrations which correspond to the same radius ve'ctor are contained in the two planes passing through this radius vector and the axes of the elliptical section, that a plane perpendicular to the radius vector cuts out of the ellipsoid (W). 115. Let 0, i/>, %, be the angles made by one of the axes of this elliptical section with the co-ordinate axes ; it is then necessary to demonstrate that the three lines (, ft y), (A, p, v), ( A > P> v > ~ 2 > Jp> #> respectively, which will give Let the auxiliary right line defined by the angles (A, B, C) be drawn perpendicular to the two right lines (A, \L, v) and ($, V>, %) we will have cos A cos A -f cos B cos p + cos C cos v = 0, | . . cos ^4 cos < + cos B cos V + cos C cos % = 0. j Replacing, in the last equation, for cos , cos V>, cos x> the quantities proportional to them in Eq. (158), we have a 2 cos A ff cos \i D c 2 cos v n n /1 nx -g g cos ^4 -\ g ^- cos .6 H g 2 cos ^ = 0. (160) Adding this to the first of Eqs. (159), we have cos A cos 11 cos v and recollecting that the relations cos /I cos cos (162) cos a cos ]3 cos y exist, we have finally cos cc cos A + cos ]3 cos 5 + cos y cos (7 = 0. (163) Hence, the three right lines (, /3, y), (A, /*, v), (0, ^, %), being perpendicular to the right line (A, B, C), are all contained in the RELATING TO SOUND AND LIGHT. same plane, and therefore we conclude that the planes which con- tain, at the same time, the same radius vector, the two vibrations, and the two corresponding directions of the normal propagation, are rectangular. 116, Discussion of the Wave Surface. Eq. (146), Resuming & (a 2 + 2 ) 2 2 + W = 0, and making in succession x = 0, y = 0, 2 = 0, we get for the sections made by the co-ordinate planes yz, xz, and xy, respectively, (if + z 2 - 2 ) (% 2 + cW - bW) = 0, (164) = 0, (165) = 0. (166) Remembering that a > & > c, we see that the section in the plane yz will be a circle whose radius is , entirely outside of an ellipse whose semi-axes are b and c. The section in the plane xy will be a circle with radius c, entirely within the ellipse whose semi- axes are a and b. That in the plane xz will be a circle with radius Figure \7, #, intersecting at four points the ellipse whose semi-axes are a and c. The axis of. x pierces the surface at distances equal to 5 and c from the centre, that of y at distances of a and i c, and that of 2 at & and #. 117. The surface of elasticity of two nappes cuts the axes in the same points. These principal axes of elasticity have in turn repre- sented the square roots of the elastic forces developed along the 80 ELEMENTS OF WAVE MOTION. three axes of elasticity, the principal velocities of wave propagation, the axes of the ellipsoids, and now serve to fix points on the surface of elasticity and on the wave surface. 118. Let Eq. (146) be represented by L = 0, and the angles which a tangent plane to the surface at any point makes with the co-ordinate planes xy, xz, yz, by A, B, C, respectively ; then we will have 1 dL 1 dL ~ 1 dL ,.,,,-A cos A = 7-, cos J? = - -T-, cos C = - - -j- , (167) o> dz t* dy u dx ' in which, - = = . (168) w /(dirt IdL^ IdLV v- W) +%)+(*) Taking the differential coefficients of L with respect to x, y, z, we will have (169) For y = 0, the point of tangency is in the plane xz, and we have (170) + c 2 * 2 ) + &te (a: 2 + z* - a* - S 2 ), I the second equation showing that the tangent plane is normal to the plane xz. For y = 0, the equation of the surface gives a? + ^2 _ j 2 = o, 2^ + ^2 _ ^2^2 _ ; (171) whence, for the co-ordinates x and z s we have RELATING Tp SOUND AND LIGHT. 81 x * - (172) which are real so long as a > b > c. There are then four points of intersection in the plane xz. Substituting these values in Eqs. (167), we obtain cos A = -, cos = cos =* (173) 119. The interpretation of these indeterminate values of the cosines is, that at the points considered, a tangent plane to the wave surface may have any position whatever with respect to the co-ordinate planes. This property shows that these points are the vertices of conoidal cusps, each having a tangent cone. These points, called utnUlics, belong to the exterior and interior nappe of the wave surface, just as the vertex of a cone is common to its upper and lower nappes. 120. The equation of the right lines joining these points, 01, 01', through the centre in the plane xz is z = which shows that the lines are normal to the circular sections of the ellipsoid ( W). The lines themselves are called axes of exterior conical refraction. 121. If tangent lines be drawn to the ellipse and circle, as MN, M'N ; , they will be parallel to each other, two and two, and symmetrically placed with respect to the axes OX and OZ. The equations of these lines can easily be shown to be (174) = xl Figure 18, -c 2 ' (175) 82 ELEMENTS OF WAVE MOTION. and hence the equation of the line drawn from perpendicular to the tangent to be '-\/rir^ x ' (176) which shows that these lines are normal to the circular sections of the ellipsoid (E). 122. From' the properties of this ellipsoid, we see that a plane wave perpendicular to one of the right lines MMj, M'M/, and at the same time perpendicular to xz, can be propagated without altera- tion, whatever may be the direction of the displacement in its plane, and that the velocity of propagation of this plane wave is independ- ent of the direction of the displacement. The lines MM,, M'M/, are called the optic axes of the medium, or axes of interior conical refraction. 123. We see, by comparing Eqs. (174) and (176), that the lines 01 and OM differ by the factor - in their tangents. This ratio is always very nearly unity, and therefore the lines have nearly the same direction. 124. The planes drawn through the four tangents MN, M'N', etc., perpendicular to the plane xz, are tangent to the wave surface along the circumferences of circles, which are projected in the lines MN, M'N', etc. To show this, let F(x, y,z) = (177) be the equation of the wave surface ; then, for points in the plane perpendicular to xz, we have J -pj j- = y (W + b"f + ) x 2 -f Wf + (b 2 + c 2 ) z* # 2 (a 2 + c 2 ) =0. (180) The first of these equations gives the points of contact in the plane xz ; the second represents an ellipsoid. If we combine the equation of the ellipsoid (180) with the equation of the wave sur- face, eliminating # 2 , the resulting equation will be the projection on the plane xz of the intersections of these surfaces, and since the co- RELATING TO SOUND AND LIGHT. 83 ordinates of the points projected satisfy the condition, -=- - = 0, all y the points of the wave surface in the tangent plane which is per- pendicular to xz, will be obtained from this intersection and projec- tion. The resulting equation after reduction can be put in the form of . = 0. (181) This equation can be satisfied by placing each factor separately equal to zero, and each will then be the equation of a plane passed through one of the tangent lines MN", M'N', M,N"i, M/N/; hence, each of the four planes touch the surface in those points determined by its intersection with the ellipsoid, and it is readily seen that these curves are the circular sections of the ellipsoid, Eq. (180). The four planes are called the singular tangent planes of the wave surface. 125. The circles are, in fact, the edges of the conoid al or umbilic cusps, determined by the surface of the tan- gent cones, reaching their limits by becoming planes in the gradual in- crease of the inclination of their ele- ments, as the tangential circumference recedes from the cusp points. It thus appears that the general wave surface consists of two nappes, the one wholly within the other, ex- cept at four points, where they unite. FJEure 19. 84 ELEMENTS OF WAVE MOTION. Fig. 19 represents a model of the wave surface, with sections made by the co-ordinate planes, so as to show the interior nappe. 126. Relations between the Velocities and Posi- tions of Plane Waves with respect to the Optic Axes* For each direction of normal propagation, two plane waves travel with different velocities, determined, as we have seen, by the equation of the surface of elasticity, cos 2 1 cos 2 m cos 2 n . F 2 a* + F 2 ^ + T 2 "^ 2 ~ This equation can be put under the form cos 2 1 + ( 2 + c 2 ) cos 2 m + (a 2 + 2 ) cos 2 n] F 2 , , -f W cos 2 1 + a?c 2 cos 2 m + aW cos 2 n = ; ^ and representing the two square roots by F' 2 and F" 2 , we have F' 2 + F" 2 = (& 2 + c 2 ) cos 2 1 + (a 2 + c 2 ) cos 2 m ) , , + ( 2 4- & 2 ) cos 2 , ) ^ ' F' 2 F" 2 = W cos 2 ? + a 2 ^ cos 2 m + ^ 2 cos 2 n. (185) Let 0', 0", be the angles that the direction of normal propagation makes with the optic axes, and and 180 the angles that the optic axes make with the axis x, the axis of greatest elasticity, then we have cos = A / sin = \ I (186) V Cr C 2 V Cr C* cos 0' = cos cos I 4- sin cos n, ) , . cos 0" = cos cos ? 4- sin cos w; ) whence, we get 7 cos 0' cos 0" cos 0' cos 0" cos I = - = s 2 cos 2 cos 0' 4 cos 0" cos 0' 4- cos 0" / 2 c 2 /1QQ , COS W r ; z^: A / -75 5* (loi/j 2 sm 2 V * ^ Substituting these values of cos I and cos ft in Eqs. (184) and (185), and replacing cos 2 m by 1 cos 2 1 cos 2 ft, we obtain RELATING TO SOUND AND LIGHT. 85 F'" = * (190) F"F" 2 = aW - ~- (cos 6?' - cos 0") 2 _ = aW + ^ (cos 2 0' + cos 2 0") (191) whence, = ( F' 2 + F" 2 ) 2 4 F' 2 F" 2 = (a 2 + 6 12 ) 2 + (a 2 c 2 ) 2 cos 2 0' cos 2 0" (02 _ 6 3)2 ( C0g 2 (9' + C0 g2 <9" (fl8 _ ^2)2 (! _ Cog 2 0') (! _ Q< gi . (192) and finally, F' 2 F" 2 = (a 2 c 2 ) sin 6' sin 0". (193) This equation establishes the relation between the velocities of the two plane waves which belong to the same direction of normal propagation, and the angles that this direction makes with the optic axes, 127. The directions of the two vibratory motions can be deter- mined by means of the optic axes. These directions are parallel to the axes of the elliptical section of (E) made by the plane normal to the direction of propagation ; but the elliptical section is cut by the planes of the two circular sections of the ellipsoid in two equal diameters of the ellipse, since they are equal to the radius b of the circular section ; they are therefore equally inclined to the axes of the ellipse. The optical axes being normal to the circular sections, are projected on the plane of the ellipse in two diameters whio-h are 86 ELEMENTS OF WAVE MOTION. perpendicular to those just spoken of, and are therefore also equally inclined to the axes of the ellipse. But these projections are the traces of the planes containing the directions of the normal propa- gation and each optic axis. We therefore conclude, that the bi- secting planes of the diedral angle formed by the planes containing the direction of any normal propagation and each of the optic axes, are tine planes of vibration of the two plane waves corresponding to this normal propagation. 128. The plane xz being the plane of the optic axes, any direc- tion of normal propagation in this plane will make the diedral angle and 180, and hence the planes of vibration will be the principal plane xz and a plane containing y and the direction of propagation. 129. Relations between the Velocities of Two Kays which are Coincident in Direction, and the Angles that this Direction makes with the Axes of Exterior Conical Refraction. The expressions * - & and being the cosines of the angles that the optic axes make with the axes of x, z, and making use of the analogy existing between the ellipsoid (E) to the surface of elasticity, and the ellipsoid (W) to the wave surface, we will have, by substituting for a*, tf, c 2 , in the 111 above, 2 , ^, -^, the expressions V and for the cosines of the angles that the axes of exterior conical refrac- tion make with the axes of x, z. If then r and r', the two coincident radii- vectores of the wave surface, represent the ray velocities propagated in the same direc- tion, and u' and u" be the angles made by this direction with the two axes of exterior conical refraction, a discussion in every way analogous to that above for the optic axes will determine the re- quired relation. This relation may be at once determined by replacing V, V", & , 6", in Eq. (193), by *,, ~ , u', u", respec- tively ; we then have RELATING TO SOUND AND LIGHT. 87 -To -- 773 = ("I) -- 9) Sm u ' r 2 r 2 \a 2 c 2 / 130. The axes of exterior conical refraction being normal to the circular sections of ellipsoid (W), by a similar course of reasoning as in Art. (127), we will arrive at the theorem, that the bisecting planes of the diedral angle, formed by the planes containing any radius vector of the wave surface and each of the axes of exterior conical refraction, are the planes of vibration of the two rays corresponding to this radius vector. 131. Thus, from the wave surface we can determine: 1. The position of the refracted plane waves by its tangent planes. 2. The direction of the two corresponding rays by the points of contact of the two parallel tangent planes. 3. The velocities of the two rays by the lengths of the radii- vectores drawn to the points of contact. 4. The velocities of the two plane waves by the normals from the centre upon the tangent planes. 5. The interior directions of the molecular vibrations by the projection of the radii-vectores on the tangent planes. 6. The plane of vibration by the plane of the normals and vi- brations. 132. We have now shown that when any arbitrary displace- ment is made in any homogeneous medium, a disturbance is propagated in all directions from the origin, and that it is materially affected and controlled by the elastic forces developed. In accept- ing the conclusions which result, the limitations which have been primarily established must be kept in mind, to avoid the danger of accepting these results other than as exceptional and governed by the admitted hypotheses and by the accuracy of the mathematical processes employed. Observation and experiment are essential to ascertain to what extent the corresponding physical phenomena conform to these deductions. They are to be used, when at vari- ance, to modify the hypotheses, and ultimately through this modi- fication to approach nearer and nearer the true theories of the physical science. 133. The fundamental hypotheses upon which the foregoing discussion is in part based are as follows : 1. The admission of such a constitution of the medium that 88 ELEMENTS OF WAVE MOTION. while it is variable around any molecule, it is similarly variable around all the molecules. The propagation of the disturbance without change of direction of the vibrations, when the latter are excited along the singular directions depends on this assumption. This inequality of elasticity is unquestionably exhibited in the phe- nomena of crystallization. 2. That the excursions of the displaced molecules are so small that the resultant elastic forces in any direction are proportional to the displacement. This implies that the distances separating the adjacent molecules are very great in comparison with their dis- placements. 3. The principle of the coexistence and superposition of small motions, by which any vibration can be replaced by others equiva- lent, to it which are rectilineal. 4. The inefficacy of the longitudinal component of the elastic force in light undulations, and the fact therefore of transversal vibrations. (The grounds of this assumption are to be given sub- sequently.) 5. The correlation of the total intensity of the elastic force to certain velocities, and its identity with that expressed by the equation 6. The principle of interference, by which the motion is en- tirely destroyed everywhere, except upon certain surfaces, which may be regarded as the loci of first arrival. 134. The agreement of the results obtained by experiment and from observation with the deductions from the theory is almost complete, while the crucial test of prediction in several noted in- stances leaves but little doubt of the truth of the undulatory theory. The utility of the determination of the wave surface and of its thorough discussion is thus happily verified, by its almost complete capability of satisfactorily explaining most, if not all, of the phe- nomena of physical optics. While in the limited course prescribed for the Academy we are unable to undertake the complete solution, we have, in the short and elementary discussion here presented, obtained sufficient data to prosecute the study of sound and light to the extent necessary for our purposes, and in this study we will have frequent occasion to refer to the foregoing analysis. PART II. ACOUSTICS. 135. The investigations of physical science show that all sensa- tion has its origin in the state of relative motion of the molecules of some medium with which the organ of sensation is in sensible contact. Each sensation has its peculiar organ, which, with its nerve system, receives and transmits molecular kinetic energy to the brain, where it is transformed into sensation. The motions of the molecules are, in general, vibrations, which are conveyed by undulations from the source of disturbance in all directions through- out the medium. 136. Acoustics is that branch of physical science which treats of sound. The sensation of sound usually arises from the commu- nication of a vibratory motion of the tympanic membrane of the ear, due to the slight and rapid changes of the air pressure upon its exterior surface, the vibratory motion of the air being caused by the vibration of other bodies. 137. The ear consists essentially of two parts, one being in com- munication with the external atmosphere, the other with the brain. The first consists of an irregularly formed tube, beginning at the orifice of the external ear and ending at the pharynx. Nearly midway, the tympanic membrane, or drum-skin, of the ear crosses this tube obliquely, separating the external portion, called the meatus, from the part immediately within, called the tympanum. That portion of the tube leading from the tympanum to the pha- rynx, or cavity behind the tonsils, is called the Eustacliian tube. The orifice of this tube at the tympanum is generally closed ; but the act of swallowing opens it, whereupon the air on both sides of the tympanic membrane becomes uniform in density. These three portions of the first part of the ear generally, however, contain air differing in density. In the meatus the air responds to all changes, 90 ELEMENTS OF WAVE MOTION. however slight and rapid, taking place in the external atmosphere; while the air in the tympanum and Eustachian tube is not so affected, unless communication with the external atmosphere be made as above described. 138. The other part, sometimes called the internal ear, is sur- rounded by bone, except in two places, called the round and oval windows. The cavity thus formed is called the bony labyrinth. The windows are closed by membranes which separate the tympa- num on the one side from the fluid contained in the labyrinth on the other. Connecting the tympanic membrane with the oval win- dow is a series of small bones, whose function appears to be to transfer the vibrations from the former to the latter. The laby- rinth is filled with liquid, having suspended in it many membrane- ous bags, also filled with liquid. Upon the surface of these bags are spread the terminal fibres of the auditory nerves, which, by special arrangements, are enabled to take up the energy communicated to the liquid in the labyrinth. The membrane of the round window readily yields to the pressure of the liquid, moving out and in as the oval window is moved in and out by the transfer of motion through the bones of the ear. Thus the energy communicated to the air in the external ear is conveyed from the tympanic membrane, through the series of small bones in the tympanum, to the membrane of the oval window, thence to the liquid of the labyrinth, and finally to the auditory nerves. How this energy is transformed into sensation is unknown. 139. To represent, graphically, the variations of air pressure, we will make use of the curve of pressure, in which the abscissas correspond to the times and the ordinates to the excess of the pres- sure above its mean or average value. The pressure of the air, at any point, is assumed to be measured by the pressure of air of the same density and temperature upon a unit of area. Then take to represent any curve of pressure as o b xy Figure 20, in which y = represents the standard or mean pressure, and RELATING TO SOUND AND LIGHT. 91 y = p, a pressure above or below the standard pressure. When- ever the pressures are strictly proportional to the corresponding densities, as by the law of Mariotte, the same curve may also repre- sent the curve of density. If we now assume that a curve similar to the above represents the slight and rapid changes of pressure of the air in contact with the tympanic membrane while the sensation of a particular sound exists, we see that these changes do not in general affect the average pressure of the air, for the areas above and below the axis of the curve are equal. A curve is said to be periodic when it consists of equal and like parts continuously re- peated. The wave length of a periodic curve is the projection upon the axis of the smallest repeated portion. 140. The ear clearly distinguishes between a musical sound and a noise. The former is a uniform and sustained sensation, unac- companied by any marked alteration, save that of intensity ; while the latter is more or less varied and ununiform. When a sonorous body is sounding, the most ordinary examination is sufficient to show that it is in a state of vibration. The vibrations or oscilla- tions of its parts set in corresponding motion the adjacent air-parti- cles, which in turn transmit similar motions to the next following particles, and so on. The air, then, is ever passing through alter- nate states of condensation and rarefaction. When these vibrations are regular, periodic, and sufficiently rapid, the resulting sound is uniform in character and is called a musical tone. If the resulting sound arises from vibrations which are non-periodic, it is called a noise. Ordinary observation shows that few, if any, noises are per- fectly unmusical ; and few, if any, sounds are absolutely unmixed with noise. 141. Propagation of a Disturbance in an In- definite Cylinder. Let us suppose the indefinite cylinder MN filled with air, and at the origin a piston p, capable of rapid to-and-fro motion. In the first place, let the piston be moved a distance ds from p to p', in the time dt. If the air were incompressible, it would be Figure 21, moved bodily over the distance ds. But being compressible, the air yields to the motion of the piston, and at the end of the time dt the compression will have reached a posi- 92 ELEMENTS OF WAVE MOTION. tion m, so that the stratum of air, being condensed from pm to p'm, will exert an elastic force in excess of that due to its normal state. Call this excess 6. The increased elasticity of p'm will cause it to expand in the only direction possible, towards the next stratum mn, which in turn becomes compressed. This second stratum reacts in both directions ; on the side towards mp' it brings the molecules of mp' to rest, their acquired velocity having a tendency to cause them to pass beyond their positions of equilibrium : and on the side nr it compresses the next stratum, increasing its elas- ticity ultimately by d. In this manner the compression is trans- mitted from stratum to stratum, throughout the whole length of the cylinder. 142. Let V be the velocity of propagation of the condensation, v the velocity of the piston ; then we have pp' = ds = vdt, pm = ds' = Vdt, pm pp' = p'm = (V v) tit. Supposing Mariotte's law applicable, and P to represent the normal pressure, we have P : P + 6 ::p'm: pm :: (Vv)dt : Vdt', or d = p-JL_. (196) Let p' now return to its primitive position p in the next succes- sive dt. The first layer of the stratum will be dilated, occupying the new space p'p, and its pressure P will become P 6. The elastic force of the next layer P will become, by its expansion to the left, P -, increasing that of the first also to P - But A <> the velocity acquired by the molecules of the second layer will cause them to pass beyond their positions of equilibrium, so that its elastic force will diminish until it becomes P <5, at the instant the elastic force of the first layer, continually increasing, becomes P, its normal value. The third layer will, in turn, act on the sec- ond as the second has acted on the first, so that the dilatation cor- responding to d will travel the distance pm in the time dt, during which the piston is retracing its path p'p. The magnitude of <* will evidently depend on the value pp' and the time dt. If dt be constant and 6 be varied, the condensations will vary with d. The RELATING TO SOUND AND LIGHT. 93 analysis shows that the compressions and dilatations are propagated with equal velocities, and that these velocities are independent of tha degree of condensation or of rarefaction, when the medium is the same and the amplitude is very small. 143. Let the prong of a tuning-fork p. . . .p' (Fig. 22) be dis- placed a very small but finite distance from its neutral position a. By its elasticity it will vibrate with equal displacements on each side of its position of equilibrium. Its velocity increases from zero at p to a maximum at a, and decreases in an exactly reverse manner to zero from a to p'. Let the duration of its motion from p to p' be divided into equal parts, each represented by dt, the epoch cor- responding to the position p. From p the prong describes unequal but increasing distances during the successive dt's to the position a, and unequal but decreasing distances from a to p'. Each corre- sponding compression can be found from Eq. (196) by the substi- tution of the proper value of v, and these compressions or conden- sations will be propagated with a constant velocity V. While the prong is returning from p' top, the rarefactions will increase from p' to a, and decrease from a to p, and their values may be deter- mined from the same equation. The condensations will be sym- metrically distributed with reference to the maximum condensation, neglecting the very small amplitude vp'. Likewise the rarefactions will be symmetrically distributed with respect to the maximum rarefaction. Figure 22. 144. The positive ordinates of the curve p'rs represent the suc- cessive condensations, 8 being the position of the layer reached by the first condensation when the prong has arrived at p' ; and the negative ordinates pr's will represent the successive rarefactions when the first condensation has reached the position u, and the prong has returned to its primitive position p. The ordinates of the other curves represent either condensations or rarefactions, as indicated in the figure corresponding to the particular state and position of the prong. 94 ELEMENTS OF WAVE MOTION. 145. In the figure, pu, the length of the wave is the distance traveled by the disturbance while the prong is making a complete vibration, and hence we have, n being the vibrational number and T the periodic time, A = - = Vr. (197) n 146. The mean velocity of the air molecules is evidently the same as that of the vibrating prong, and therefore this will vary with the vibrating body. In the example given, the mean velocity of the molecules is 2 mm. x 256 = 0.512 m. The actual velocity of the air molecules continually varies, and at any time is propor- tional to the ordi nates of the curve a quarter of a wave length in advance of the molecules considered. When the vibrating body has simple harmonic motion, the molecular velocity is given by v = aco82n (198) 147. The value of , the amplitude of the vibration, diminishes (Art. 72) according to the law of the inverse distance from the cen- tre of disturbance ; and for each value of a taken as constant within the wave length we have, by the above equation, sensibly exact values for the molecular velocity at any time. 148. When the vibrations of the body are sufficiently frequent during the unit of time, and of sufficient amplitude, the sensation of sound arises in the ear, which, however, we unconsciously refer to the vibrating body. A sonorous wave comprises the series of condensations and rarefactions arising from one complete vibration of the sounding body. 149. The sum of all the condensations in the condensed portion of the wave is represented by the area of the curve p'rs, and if it be divided by the duration of half the vibration, the mean condensa- tion will result. Thus, take the amplitude of the oscillation of the tuning-fork, making 128 vibrations per second to be 1 mm., and the velocity of propagation to be 340 m. ; then, from Eq. (196), we will have 1 . P * _. p ~ 340000 1- ' ^340000-256- 1327' (199) RELATING TO SOUND AND LIGHT. 95 Heuce, the change of density in the air, measured by the barometric height due to the mean condensation, is not greater than that due to 0.0226 inches of mercury, when a sound corresponding to 128 vibrations per second, and caused by the fork under the sup- posed conditions, is passing. 150. From the preceding discussion we see that we can neglect, in general, the absolute displacements of the air molecules, and consider the change in pressure and density as being alone propa- gated. Therefore, a file of elastic balls transmitting motion prac- tically illustrates the state or condition of a series of air molecules during the propagation of a sonorous wave. An excellent illustra- tion is also given by means of a chain cord. If it be attached at one of its extremities to a fixed point, and be held stretched at the other, the successive rings or spirals will assume positions of stable equilibrium with respect to each other, determined by the tension. These rings, for the purpose of illustration, may be taken to repre- sent the contiguous air strata or particles in an indefinite tube, or upon any line along which sound is supposed to be propagated. If any ring be plucked, it will, when released, ^oscillate about its place of rest while the disturbance is being propagated in both directions to the points of support. Upon reaching these points the dis- turbance will be divided, a part proceeding in the new medium, and the remainder, being reflected, will retrace its path, to be again subdivided at the other end. This will continue until the whole energy of the original disturbance has been dissipated. By increas- ing the tension the disturbance will be more quickly propagated, and conversely. Now suppose, from the point of plucking, lines be drawn in all directions, and the same phenomena occur on these,, then the behavior of each ring and the progressive motion of the disturbance illustrates what takes place in air during the passage of a sound wave along every right line drawn from the origin of the sounding body. In an isotropic and homogeneous medium, the disturbance moves with constant velocity, and the volume whose surface bounds the disturbed particles at any instant is a sphere whose radius is Vt. 151. The general properties of any sound are intensity, pitch, and quality. Intensity is that property by which we distinguish the relative loudness of two tones of the same pitch and quality. We can also, 96 ELEMENTS OF WAVE MOTION. in general, determine which of two tones of different pitch and quality has the greater intensity. The air particles have but small displacements from their positions of relative rest, when the dis- placement is caused by the passage of a sound wave. The forces which urge them back to their positions of rest are assumed to vary directly with the degree of displacement. In Analytical Mechanics, it is shown that the periodic time of the air particle depends only upon its mass and the intensity of the force of restitution; and therefore, in the same medium, with given pressure, density, and temperature, for the same exciting cause, the periodic time will be constant, but the mean velocity of the air particle will vary with the size of the orbit. The kinetic energy in the moving particle, varying as the square of the velocity, will therefore, for the same exciting cause and the same medium, under the same circumstances of pressure, density, and temperature, vary directly as the square of the maximum displacement. By the law of the decay of energy, the intensity of the sound will therefore vary inversely as the square of the distance from the origin of the exciting cause. (Art. 72.) 152. Pitch is that property by which we distinguish the posi- tion of two tones in the musical scale, and thereby recognize which is the more acute and which the more grave. The pitch depends upon the frequency of the vibration ; the greater the number of vibrations produced by a sounding body in a given time, the more acute will be the resulting sound. The siren is an instrument used to illustrate this fact. It consists essentially of a disk pierced with a number of equidistant holes, through which air is forced when it is put in rapid rotation. As the rotation increases, the sound grad- ually rises in pitch, and as it diminishes the pitch falls correspond- ingly. If a coin with a milled edge, or a cogged wheel, be put in rotation, and a card be held against it, the same changes in pitch will be observed. In these cases the single puff, or stroke of the card against the coin, or wheel, is essentially a noise, and when these strokes are multiplied sufficiently in a given time, the result- ing effect is a note of definite pitch. So that a clearer distinction than that heretofore given should be made between a noise and a musical tone. To this distinction we will again refer. 153. The quality of a musical tone is that property by which we can distinguish whether two sounds of the same pitch, of either equal or unequal intensities, arise from the same or different sono- RELATING TO SOUND AND LIGHT. 97 rous bodies. This property enables us, within certain limits, to distinguish voices and the various sounds peculiar to different musi- cal instruments. We have as yet only exacted that a musical sound shall be periodic and regular ; that is, that during any vibration the successive states of motion of the particle shall recur in the same order as in each of the previous vibrations. But it is evident that we may have an infinite variety of periodic motion, and it will be shown that the quality of the sound will varj with each variation of the periodic motion, the wave length remaining constant. 154. Every one has experienced the fact that more than one sound can be heard at once. Our attention can be, for the mo- ment, fixed upon any one of the many sounds that are constantly occurring, and at the same time we may be conscious of the exist- ence of the others. Therefore, the meeting of sound waves in the external ear does not, in general, result in mutual destruction, or in essential modification; while, at the same time, we must ac- knowledge that the air in contact with the tympanic membrane, at any given instant, can possess but one determinate pressure and density. The changes in pressure and density due to many exciting causes must, then, result from the superposition and coexistence of those arising from each separate cause, and, in general, without destruction or modification. We have here the application of the principle enunciated in Art. 204, Mechanics. The more general statement of the law of the composition of displacements would be that demonstrated in the principle of the parallelogram of forces, but when the displacements are infinitely small, we can take, rigorously, the resultant displacement to be the algebraic sum of the component displacements. The diameter of the meatus at the tympanic membrane does not exceed 0.25 inch, and 'therefore, for sounds whose sources are at ordinary distances, the wave fronts at the position of the tympanic membrane coincide sensibly with their tangent planes, and the changes of density and pressure may be compounded by the law of small motions, without appreciable error. 155. Let the broken line, in the following diagram, represent the changes of pressure upon the tympanic membrane while a con- tinuous noise, in which the ear recognizes no definite pitch, is sounding for a small part of a second, and let the dotted line repre- sent another noise of the same duration. 7 98 ELEMENTS OF WAVE MOTION. Figure 23, Then, if both noises sound together, the resultant variation of pressure will be represented by the full line obtained by joining the extremities of the ordinates found by taking the algebraic sum of the ordinates of the separate curves. These two noises do not, in general, unite into one, but are heard distinctly and simultaneously, except in the case where the two sounds are nearly alike, and the two curves nearly similar. Again, there is nothing in the resultant curve to suggest to the eye the nature of the two component curves. Hence, the ear possesses the property of separation ; while the eye, according to this method of combination and representation, does not. 156. Let the component curves be periodic, two periods of O'X' being equal to three of 0"X". o\ Figure 24, The resultant curve OX will be a periodic curve, whose repeated portions are represented above. An examination of this curve by the eye gives no clue to its components, and we may resolve it into an indefinite number of pairs of components, but one of which would represent the two notes which sounding together will give us the resulting effect upon the ear. But if the ear resolves the com- posite note represented by OX, it must resolve, in like manner, O'X' and 0"X". Observation confirms this deduction. 157. The only note the ear is incapable of resolving is that of the simple musical tone, and this incapability arises from the fact that such a tone is in reality perfectly simple, and not compound. The tones which are ordinarily called simple, are, in reality, com- pounded of a series of simple tones theoretically unlimited in num- ber. Very few of them have sufficient intensity to be heard ; but RELATING TO SOUND AND LIGHT. 99 these few form a combined note which is always the same under the same circumstances, and we habitually associate them together, and perceive them as a single note of a special character. But it is possible, with certain appliances, to partially analyze the composite note by an attentive study of the separate constituents. Whenever two sounding bodies give notes whose tones form con- sonant combinations with each other, the difficulty of analysis is increased; when the combinations are dissonant, the analysis is less difficult. 158. A noise may therefore be defined to be a combination of musical tones, too near in pitch to be separately distinguished by the unassisted ear, or to be a combination of noises, each of which is made up of sounds so near each other in pitch as to be undistin- guishable ; the separate noises may be near or far apart in pitch. It is so complex, that its analysis is beyond the power of the un- assisted ear. A simple musical tone, on the contrary, is incapable of resolution by reason of its absolute simplicity. Hence, strictly speaking, only simple tones have pitch. A simple musical tone will have a single determinate pitch. The pitch of a musical note must then be taken to mean the pitch of the gravest simple tone in its combination. If the higher simple tones be successively stopped out, the pitch, as defined, will remain unaltered, but the quality of the note will undergo variations until the single musical simple tone corresponding to the gravest tone is reached, beyond which no further modification can take place. 159. We will hereafter assume, as the fundamental simple tone, that component of any note which corresponds to the regular pe- riodic curve of the given pitch. This distinction is important ; for it is evident that there may be many periodic curves of the same pitch, and each may correspond to musical notes differing in quality. Figure 25. 100 ELEMENTS OF WAVE MOTION. The preceding curves (Fig. 25) represent notes of the same pitch, but of different quality. Helmholtz has shown that while every different quality of tone requires a different form of vibration, the converse is not necessarily true; i. e., that different forms of vibra- tion may not correspond to the same quality. 160. We have seen, page 45, that any physical condition, such as density, pressure, velocity, etc., which is measurable in magnitude or intensity, and which varies periodically with the time, may, by Eq. (195), be expressed as a function of the time. Hence, every periodic disturbance of the air, and particularly such disturbances as excite the sensation of a musical tone, can be resolved into its harmonic vibrations. A single simple tone being represented by the simple harmonic curve y' = a' sin (~ + '), (200) \ A I and another of half wave length by 9.-JT/JT \ (201) y" = a"sml~ + a"\ \ 2 / the resultant curve will be represented by ? + a'\ + a" sin (- + "), (202) y = a sn ~ which has the same wave length, but a different amplitude and phase. This change in the amplitude and phase may be varied at pleasure, by conceiving the second curve to be shifted along the axis any distance from zero to A, and again to pass through all values of the amplitude between any two limits. The resultant curve, in all cases, will, however, be a periodic curve of constant wave length. 161. Considering the simple musical tones which they represent then to be sounded together, with the same modifications, it has been found that the ear can distinguish the components when the attention is cultivated and directed to this effect. With a variation in phase only, the effect on the ear is constant and invariable, and hence we see that many different resultant curves may represent RELATING TO SOUND AND LIGHT. 101 Phase diff 'era 90 at O Figure 26. essentially the same sensation. Thus, the two curves above, repre- sent the same compound tone made up of the two simple tones, although the forms of the curves are quite different. The resultant tones are the same, both in quality and in pitch, but differ in inten- sity. By combining in the same way other simple tones of one- third, one-fourth the wave length, and so on, the quality will be changed, without affecting the pitch, as can be seen from the graphical construction, and heard by audible experience. In all these cases the untrained ear, by the aid of certain appliances, can always analyze the resultant sound into its component simple tones, and when trained, often without this assistance. When but one simple vibration of sufficient frequency and intensity to produce sensation alone exists, no such analysis takes place. 162. The investigations of Helmholtz have shown that the ear possesses the property of analysis of a single musical tone into its simple musical tones, each of which is distinctive in character, but which blend harmoniously into the single tone when sounded to- gether. The wave lengths of these components are aliquot parts of the wave length of the fundamental, and the simple tones are called the upper partials of the fundamental or prime tone. Hence, from Art. 64 and these facts, we conclude that, when several sounding 102 ELEMENTS OF WAVE MOTION. bodies simultaneously excite different sounds, the variations of air density and the resultant displacements and velocities of the air particles in contact with the tympanic membrane are each equal to the algebraic sum of the corresponding changes of density, the dis- placements and the velocities which each system of waves would have separately produced, had it acted alone. 163. This analysis by the ear clearly shows, then, that the sep- arate effects of the simple vibrations are, in general, neither modi- fied nor destroyed, but actually exist, and it remains to be proved that such is really the case, independent of the peculiar sensation which is the result of their action upon the ear. Since Fourier's Theorem mathematically demonstrates that any form of vibration, no matter how varied its shape, can be expressed as the sum of a series of simple vibrations, its analysis into these simple vibrations is independent of the capacity of the eye to perceive by examining its representative curve whether it contains the simple harmonic curves or not, and if it does, what they are. All that the curve indicates is that the more regular its form, the greater the effect of its deeper or graver tones in. comparison with its upper partials. Before proceeding to show that these component vibrations actually exist together, and that each can affect the ear or other sensitive vibrating body, let us now establish clearly the definitions pertain- ing to the subject. 164. A simple or pendular vibration is that which corresponds to the complete oscillation of a simple pendulum, and is graphically represented by the simple harmonic curve. A simple musical tone is that effect produced upon the ear when a sonorous body is executing simple vibrations only, of sufficient frequency and amplitude to be heard. According to this definition, simple tones do not in reality exist ; but in the vibrations of such bodies as tuning-forks, the component vibrations which simulta- neously exist with that of the gravest period, are generally non- periodic with it, and so deficient in intensity that their influence is negligible, and we may regard such bodies as producing simple vibrations alone without sensible error. 165. A single musical tone may be either simple or compound. When compound, it is made up of its fundamental simple tone, together with its upper partial simple tones, each of which has a frequency of either twice, three times, or so on, that of its funda- RELATING TO SOUND AND LIGHT. 103 mental. It is due to the vibration of a single sonorous body which, during its motion, vibrates as a whole, and divides also into parts which vibrate twice, three times, and so on, as rapidly as the whole. One or more of these upper partials may be wanting during the vibration ; when this occurs, the quality of the single musical tone is correspondingly affected. A composite musical tone is composed of two or more single musical tones. 166. Musical Intervals. The extreme range of the hu- man ear lies between 20 and 40000 simple vibrations per second. The corresponding wave lengths are obtained by dividing the veloc- ity of sound by these numbers, and are approximately 54.6 feet and 0.0273 feet respectively, assuming the velocity of sound to be 1092 feet at C. The ordinary sounds heard by the ear have a much less range ; their vibrational numbers lie between 40 and 4000, cor- responding to wave lengths of about 27.3 feet and 0.273 feet, respectively. When a stretched wire is put into vibration, and the tension continuously undergoes variation, the pitch of the sound passes by continuity from lower to higher, or the reverse, and we therefore experience the sensation of a musical interval between any two limiting tones. We may, then, define a musical interval by the ratio of the vibrational numbers of the two limiting tones. Thus, if the two tones correspond to the vibrational numbers 256 and 384, the name of the interval is the fifth, and it is expressed by the o fraction - Considering the simpler ratios that lie between two a tones whose vibrational numbers are as 1:2, we obtain the follow- ing musical intervals : Consonant. Unison, Minor third, Major third, Fourth, . . Fifth, . . Major sixth, Octave, . . Dissonant. Major -second, . . . 9:8 ' - Minor second, . . . 10 : 9 - = ^ One-half major tone, 16 : 15 = ^-f One-half minor tone, 25 : 24 = f Comma, 81 : 80 = The first are called consonants, because the effect is pleasing to 104 ELEMENTS OF WAVE MOTION. the ear when the tones of either of these intervals are sounded to- gether. All other intervals within range of the octave are called dissonants. 167, The measure of the musical interval represented by the ratio - is the log - This arises from the fact that if we consider q 5 q any three tones whose vibrational numbers are p, q, and r, the musical interval between p and r must be equal to the sum of the two intervals between p and q, and q and r. If the ratios of the vibrational numbers were taken to measure the intervals, we would have, for the same interval, the expressions r , q , r and - H p p q which are not equal to each other. But since *- = q ~ x T -, (203) P P V we have log *- = log J + log T -, (204) and we may therefore take the logarithm of the ratio of the vibra- tional numbers as the measure of the musical interval. The name of any interval, then, is the ratio of the vibrational numbers, and its measure is the logarithm of that ratio. The logarithms are usually taken in the common system. 168. Musical Scales. A series of tones at finite intervals is called a musical scale. If the vibrational numbers are in the proportion of the natural numbers, the musical scale is called the harmonic scale. When two tones whose interval is that of an octave are sounded together, we are conscious of a certain sameness of sen- sation, which is absent in all other intervals except multiples of the octave. We may then assume this interval as a natural unit, since it gives a periodic character to the scale. Whatever properties are found with regard to the tones in any octave, occur in the other octaves of a higher or lower pitch. The vibrational numbers of the tones of the harmonic scale, starting with a fundamental tone whose vibrational number is 128, will be as follows: 128 : 256 : 384 : 512 : 640 : 768 : 896 : 1024 : 1152 : 1280 : etc. RELATING TO SOUND AND LIGHT. 105- 169. Examining these numbers, we see that each interval in any octave is divided, in the succeeding octave, into two intervals which can be obtained from the equation 7M;JL _ 2n + 1 2^ + 2 n 2n * 2^M' n being the natural number which marks the position of the first tone of the lower interval in the harmonic scale. Thus we see that the interval 128 : 256, or the octave, is divided in the next octave . , , 2n + 1 3 384 , 2n + 2 into two intervals represented by = - = - - and - 2n 2 256 2n + I = - = -- . The first interval, 256 : 384, in the second octave is 9 O- 1 divided into the two intervals corresponding to - - = - and 9 9 K - = = in the third octave ; the second interval, 384 : 512, in An -f- 1 o the same octave, is in like manner divided into - ^ = - and 2n , o 8 2ra 6 - = ^ in the third. The first interval, 512 : 640, in the third + . 9 10 octave, is subdivided in the fourth octave into ^ and , and so o y on. Arranging all the intervals, with their corresponding subdi- visions in the next higher octave, we have 2 1st octave, 128 : 256, interval -, subdivided in 2d octave into 256 : 364 = | and 384 : 512 = ; /c o o 2d octave, 256 : 384, interval -, subdivided in 3d octave into /c 512 : 640 = | and 640 : 768 = | ; 4. 2d octave, 384 : 512, interval ^, subdivided in 3d octave into o 768 : 896 = and 896 : 1024 = ; 106 ELEMENTS OF WAVE MOTION. 3d octave, 512 : 640, interval j, subdivided in 4th octave into n 10 1024 : 1152 = 5 and 1152 : 1280 = ^ ; n 3d octave, 640 : 768, interval -, subdivided in 4th octave into 11 12 1280 : 1408 = ^ and 1408 : 1536 = Thus every interval in the harmonic scale is divisible into two other intervals, whose ratios are those of consecutive numbers in the next higher octave. 170. Perfect Accords. A perfect accord is a series of three tones, called a chord, which, sounded simultaneously, give a partic- ularly pleasing sensation to the ear. The perfect major accord consists of the three tones called the tonic, the middle, and the dom- K q mant, whose intervals are a major third and a fifth, or - and - n The perfect minor accord is composed of a minor third, - , and a fifth, |. A 171. The Diatonic Scale. The tones of this scale are usually designated by letters or symbols, as follows : C:D:E:F:G:A:B::d: etc. ut or do : re : mi : fa : sol : la : si : do : re : etc. Forming the perfect major accord on C as a tonic, we will have C : E : G, 4 2 Forming similar chords with C and G, by making a dominant and G a tonic, we will have RELATING TO SOUND AND LIGHT. 107 F! : A, : C, G : B : d, 25 3 15 9 36 2 ~S 4* Arranging these three chords in order of their pitch, we find Fj : A! : C : E : G : B : d, 25 5 3 15 9 36 4 2 T V which is a musical scale of seven notes, rising one above another by alternate major and minor thirds. Replacing in this scale F x , A t , by their higher octaves, and d by its lower octave, which is permissible, and arranging in order, we have C:D:E:F:O:A: B : c, 9543515 84323 8 w^hidi is known as the diatonic scale. The names of the intervals heretofore used are now seen to come from the position of the notes g in this scale with reference to the tonic ; thus, the interval ^ is a o major second, the interval - a major third, ^ a fourth, - a fifth, and so on. The first tone in the scale is called the tonic, the fifth the dominant, and the fourth the subdominant. Taking the vibra- tional number of the tonic C to be 24, we have the corresponding vibrational numbers of the diatonic scale, C : D : E : F : G : A : B : c, 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48. 172. The vibrational numbers of the other octaves are obtained from these by constantly doubling or halving them, according as we ascend or descend, the letters being properly accented to indi- cate in which octave the series is taken. Theoretically, the tones of the diatonic scale above belong to the harmonic scale, whose fun- damental tone has one vibration per second. This fundamental 108 ELEMENTS OF WAVE MOTION. 32 /2\ 5 tone is five octaves below the subdominant ; for = ( ) We will hereafter take the octave whose tonic corresponds to 256 vibra- tions for that of comparison, because Scheibler's tonometer, which we use in illustration in the lectures on this subject, is based on that tonic. 173. The relation of the successive tones of the harmonic scale to any tone assumed as a fundamental is as follows ; taking as the prime that whose vibrational number is 256, we have Prime or fundamental, 256 vibrations, or c ; 1 Harmonic, 512 " " c r , octave; 2 " 768 " " g', fifth in 1st octave ; 3 " 1024 " " c", second octave ; 4 " 1280 " " e", maj. third in 2d oct.; 5 " 1536 " " g", fifth of 2d octave; 6 " 1792 " " a" + , lying between 6th and 7th of 2d oct. ; 7 " 2048 " " c'", third octave; and so on. These harmonics are called overtones or upper partials, and, as seen above, bear a close relationship to the prime. When the prime is sounded and the upper partials exist at the same time, the resulting tone will have a determinate quality. And if the par- tials be successively stopped out, the quality will undergo a change, until we reach the simple tone due to the prime alone. The suc- cessive curves which represent these tones graphically will approx- imate gradually to that of the harmonic curve of the wave length of the prime, which it ultimately reaches when all of the partials are wanting. The wave lengths of the above curves are each equal to that of the prime. 174. It can be experimentally shown that a stretched cord, when plucked from its position of rest, will give a compound tone, which is made up of its fundamental united to some of its overtones. The educated ear can readily distinguish the existence of these simple tones, which, sounding together, determine the quality of the compound tone. But to demonstrate to the untrained ear the existence of these partial tones, it is necessary to make use of cer- tain appliances called resonators, whose action depends on the RELATING TO SOUND AND LIGHT. 109 principle of sympathetic resonance. These consist of metal or other hollow bodies, generally spherical in form, closed except at two places; one of the openings is to permit the mass of air within to be affected by the vibration of the air without, and the other to per- mit the air within to be brought into near contact with that in the aperture of the ear. 175. Sympathetic Resonance. If a body capable of taking up an oscillatory motion of definite period be subjected to a series of periodic impulses, whose period is the same as that of the body considered, the aggregate effect will in time become sensible, however weak the impulses may be. But if the period of the im- pulses be even slightly different from that of the body, the resultant effect will, in general, never become appreciable; for, while the kinetic energy is increased by the elementary quantities of work due to the impulses applied, soon the succeeding impulses will be deliv- ered in a direction contrary to the motion of the body, and the kinetic energy will be correspondingly diminished. The maximum energy can then never exceed a small definite quantity, and in reaching this state the body will pass through alternations of rest and motion. To determine the effect of any periodic impulse upon a body capable of being put into vibration, we have the following rule, due to Helmholtz: Resolve the periodic motion of the impulse into its component simple pendular vibrations ; if the periodic time of any one of these vibrations is equal to the periodic time of the body acted upon, sensible vibration will result, and not otherwise. 176. Now consider the mass of air within the tube AB, while a simple vibratory motion, due to a sim- ple tone, occurs in the external air. ~ \ / Let V be the velocity of wave propaga- tion in the air under consideration, and n the vibrational number of the body. Then, during the first semi-vibration, Figure 27. the molecules at B describe half their orbits while undergoing condensation, which is transmitted through the intervening molecules to A and back to B, provided 110 ELEMENTS OF WAVE MOTION. During the second semi-vibration, the rarefaction at B will be trans- mitted in the same manner, and the orbits at B will be completed. V Should BA be either > or < , the second impulse would reach MI B after or before its molecular orbits had been completed. Under these circumstances, succeeding impulses would in a short time reduce the displacements of the molecules to zero, and never permit them to attain an appreciable value, and therefore the vibration of the air column would not give a sound of appreciable intensity. But if, on the contrary, the impulses were of the same periodicity as the air molecules, each successive impulse would add to the first displacement, and this addition would continue until the work of the resistances developed was exactly equal to the increment of energy caused by each impulse. The displacements of the mole- cules would then have attained their maximum value, and the resulting sound a fixed intensity. 177. Each confined mass of air has a particular periodicity, and each of the resonators of Helmholtz is carefully contrived to respond to a given periodicity of vibratory motion. If, then, by the rule above given, any composite sound exist, and one of these resonators be applied to the ear, the resonant effect will indicate whether the simple tone corresponding to the resonator is present or absent in the composite sound. This and analogous experiments show that sympathetic vibration is not due to any property peculiar to the ear, but that it is a mechanical effect separate and distinct from the sense of audition. 178. The energy of motion depending upon the mass and ve- locity, we see clearly that of two sounding bodies, vibrating with the same amplitude, the smaller mass will more quickly give up its energy to the surrounding air and sooner cease sounding. Tuning- forks being generally made of steel, will, when put into rather strong vibration, continue sounding for a reasonable length of time. When mounted upon their resonant boxes, the latter containing a mass of air capable of vibrating in unison with it, they affect larger masses of air than when not so mounted, and come more quickly to rest; but the sound will have greater intensity, and can the more readily be used to study the phenomena of sympathetic resonance. If such a tuning-fork be in the vicinity of a vibrating sounding body whose sound contains the tone of the fork, the latter will in RELATING TO SOUND AND LIGHT. Ill time indicate the fact by coming into sympathetic vibration. The analysis, then, of any composite note can be practically made by means of a sufficient number of such forks, whose vibrationul num- bers embrace all the simple notes of the composite sound. Con- versely, the synthesis of a composite note can be effected by setting in vibration all the forks, with proper amplitudes, which the analy- sis indicates belong to the note in question. 179. When plates, bells, strings, etc., are put into vibration, they may either vibrate as a whole, or separate into parts which vibrate two, three, four, or more times as rapidly ; or both of these conditions may occur simultaneously. Each of the simple periodic vibrations has an actual existence, and corresponds to a single musical tone of definite pitch, which may be recognized as above described. 180. In listening for any simple tone in the composite note, it is important to clearly fix the attention upon the special tone whose existence is to be determined, and for this purpose the tone should be sounded alone before listening for it in the composite note. When sufficiently practiced in this manner, the ear can readily acquire the faculty of detecting them without the use of resonators. 181. By means of the monochord, which consists essentially of a string stretched over two bridges on a sounding-box, we can verify the simultaneous existence of the prime and upper partials, and estimate the influence of the latter in affecting the quality of the sound. The theory of vibrating strings shows that the frequency of vibration of the same string under the same tension is inversely proportional to its length. Plucking the string at its centre, the resulting tone will be that of its prime, modified by some of the upper partials, those of the latter being absent that require the middle point as a point of rest. By a movable bridge, the string can be divided into its aliquot parts, which being set in vibration, will give the upper partials in succession. Becoming thus acquaint- ed with these simple tones, we can verify their presence or absence in each special case. For example, if the string be plucked at one- fourth its length, theory requires the presence of the first upper partial with the prime, and the fact will be made manifest by damp- ing the string at the middle point immediately after plucking, when the octave will sing out, no longer encompassed by the prime. 112 ELEMENTS OF WAVE MOTION. 182. These and the facts of sympathetic resonance show that the analysis of all resonant motion into simple pendular vibrations is real and actual, and that any other analysis is highly improbable. The analogous property of the ear is expressed by the law of G. S. Ohm, viz., that the human ear perceives pendular vibrations ^alone- as simple tones, and resolves all other periodic motions of the air into a series of pendular vibrations, hearing the simple tones which correspond to these simple vibrations. We may therefore conclude, that in all cases whenever any motion of the air caused by a sounding body contains a simple vibration of .the same periodicity as that of any other body, the latter will in time take np a vibratory motion which, if of sufficient intensity, will affect the ear with a simple musical tone of a definite pitch ; .and the mechanical effect of vibration will ensue, whether it be of sufficient amplitude to produce a sonorous effect or not. 183. Velocity of Sound in any Isotropic Medium. The air is the medium of transfer to the ear of the vibratory mo- tion of a sounding body. Under a given temperature and density, its elastic force is constant in all directions, and it is therefore an isotropic medium. Being compressible, the motions of its mole- cules, during the passage of a sound wave, are to and fro along the line of wave propagation. They are then longitudinal vibrations, and Eq. (116), for the velocity of wave propagation, for waves with .such vibrations in an isotropic medium, is sin2 184, The wave lengths of sound in air can never be greater than 54.6 ft., nor less than 0.027 ft. ; for the usual sounds the limits are 27.3 ft. and 0.273 ft., at 0. In the above equation, Az is the distance separating two adjacent molecules, and without knowing its absolute value for any degree of pressure, we may say that A, even in the minimum sound wave, is very great with respect to A#. Therefore the arc is approximately zero, and may be substituted for sin-- - without appreciable error. We then have RELATING TO SOUND AND LIGHT. 113 (207) = !*[* Hence, with the supposition of small displacements, etc., the velocity of wave propagation of sound in air is theoretically inde- pendent of the wave length, and all sounds, whether grave or acute, will travel, in air of constant pressure and temperature, with equal velocities. Omitting the term containing A.T 4 as being small compared with that of which Az 2 is a factor, and replacing (r) by its equal ^ ^ , we have -i f i \ r F 2 = i I,p 12 AZ*. (208) /v /* 185. Let E represent the modulus of longitudinal elasticity of air, P the barometric pressure, / the length of the air column with- out pressure, and A the compression due to P. Then, by Eq. (I), we have E={P. (209) Since, if the pressure P be removed, the expansion would be indefinitely great, the compression A is sensibly equal to I, and therefore E = P ; (210) that is, the elastic force of the air is that due to the barometric pressure on the unit area. 186. In Eq. (208), i ^f(r) is the acceleration due to the a aggregate elastic forces developed in the molecules \i by the arbi- trary displacement of the molecule m, and reciprocally is the elastic acceleration of m ; hence we have, by multiplying by m, the inten- sity of the elastic force acting on m, 8 114 ELEMENTS OF WAVE MOTION. Aic - is the cosine of the angle made by this force with the axis of x, which, since the medium is isotropic, is equal to unity; multiply- ing the elastic intensity on m by the factor ^, we have the elastic intensity on the unit area, or whence, Zp/(r) = - (212) Substituting in Eq. (208), we have V = ***; (213) m Arz 8 is the volume of the molecule, and replacing it by its equal yr, and extracting the square root, we will have, finally, for the velocity of wave propagation in air or any gas, subjected to the law of Mariotte, 4 or directly proportional to the square root of the ratio of the elas- ticity of the medium to its density. 187, This conclusion is deduced on the hypothesis of the direct ratio of the elastic force to the density, and if the law of Mariotte were true for all circumstances of pressure, temperature, and den- sity, this theoretical velocity and the actual velocity determined by experiment would perfectly accord. But this relation is true only for a perfect gas and for constant temperature. 188. The relation of these two parameters of air, considered as a perfect gas, are given by the following formulae : p'd=pd', p'd = pd' (I + 0), < 217 > in which p and p' are respectively the old and the new pressures or RELATING TO SOUND AND LIGHT. 115 the corresponding elastic forces, d and d' the old and the new den- sities; the coefficient of expansion, a constant, and equal to ^-y for Centigrade scale ; and 0, degrees of temperature Centigrade. The first of these equations is the mathematical expression of Mariotte's law ; the second, of that of Charles or Gay-Lussac ; and the third, of that of Poisson. The gas to which these equations are applicable is supposed to be a perfect fluid, devoid of friction, and to have the pressure at each point uniform in all directions. The temperature is supposed constant during all changes of pressure and density in Mariotte's formula, while in that of Charles the gas takes the pressure and density determined by the change of temperature. The formula of Poisson supposes the gas subjected to sudden changes of density, and that the heat developed, whether considered positively or negatively, is not conveyed by radiation or conduction to other bodies, or, in other words, that the quantity of heat in the gas is constant. Remembering that sudden condensa- tion in air or gas produces heat, and sudden rarefaction cold, and assuming that these alternations are so rapid that neither the heat nor the cold is conveyed to the other particles, within the volume considered, much beyond the point at which they originate, we see that this heat and cold will produce an elastic force of greater in- tensity than that in either of the other two cases; therefore the value of the velocity of propagation will be greater than that given in Eq. (214), which was deduced under the supposition of the sim- ple ratio of the elastic force to the density expressed by ^- It CL might be supposed that the influence of the heat produced in the condensation of the sound wave would be neutralized by the cold produced in the rarefaction, and that therefore the resultant effect would be zero. This, however, is not the case; for the heat in the condensation has increased the difference of elastic force between the condensed stratum and the one in its front, and hence has in- creased the velocity, while the cold in the rarefaction has caused an equal difference between the rarefied stratum and the one in rear, and has thus added an equal increment of velocity to this portion of the wave. This is true for each stratum affected by the sound wave. Hence the disturbance passes each stratum of the condensed and rarefied portions with the same velocity, and this may be re- garded as the velocity of the wave. 116 ELEMENTS OF WAVE MOTION. 189. Since the vibrational number of sound waves varies be- tween 20 and 40000, for the extreme limits, the alternate condensa- tions and rarefactions occur with sufficient rapidity to necessitate the application of the formula of Poisson for the determination of the velocity of sound in air and in other gaseous media. 190. Pressure of a Standard Atmosphere. Let p be the pressure of the atmosphere when the barometric column cor- responds to 76cm., the mercurial density being 13.5962, and g 981 dynes ; we then have p = 981 x 13.5962 x 76 = 1.01368 x 10 6 dynes, (218) as the corresponding pressure of the atmosphere upon a square cen- timetre. But since the density of mercury, referred to the stand- ard at the same locality, is independent of the locality, and hence independent of g, we may assume as the standard atmosphere that whose pressure on the square centimetre at all localities is equal to 10 6 dynes. Hence, p = g x d m x li = 10 6 dynes. (219) By substituting in this equation the value of g for the latitude of the place, and solving with reference to h, we will determine the barometric height corresponding to the standard atmosphere at that locality; g varies from 978.1 dynes at the equator to 983.11 dynes at the pole. 191. Height of the Homogeneous Atmosphere. If the atmosphere be supposed replaced by an atmosphere of uni- form density D, as that of standard dry air at C., and height IT, exerting the same pressure, H may be obtained from the equation p = g-D-II 10 6 dynes; (220) from which we have = 7989.40 m. = 26212.18 ft., which is constant at the same locality, for the same temperature RELATING TO SOUND AND LIGHT. 117 and barometric height. If the temperature become 6 C., we have, by the law of Charles or Gay-Lussac, H' = (1 + 0) H = Ha T, (222) in which r is the absolute temperature, and a the coefficient of expansion. 192. Eeplacing the elastic force E by its equal, in terms of the homogeneous atmosphere, in Eq. (214), we have which is Newton's formula for the velocity of sound in air. Mak- ing r = 273, corresponding to zero Centigrade, and g = 981 dynes, we have V = V 7.9894 x 10 5 x 981 = 2.8 x 10 4 = 280.0 metres. (224) For any other temperature, we have V = V7.9894 x 10 5 x 981 x T = 280 Vl + 0. (225) 193. These values of the velocity of sound in air are about one- sixth less than those determined by experiment, the discrepancy being due to the supposition that Mariotte's law expresses the rela- tion of pressure and density. The law of Poisson is, however, appli- cable ; hence we have (D'\y P =P differentiating, dp' = yp , Whence we see that when a sound wave is passing through air, the ratio of the increment of the elastic force to that of the density is equal to the ratio of the elastic force to the density, multiplied by the constant y. The value of y can be determined from a direct observation, by accurately measuring F, a, and 6, and substituting in the equation 118 ELEMENTS OF WAVE MOTION. io 4 x 2. = y r ^7 = and solving with respect to y. Its value has been found to be, approximately, 1.41 for all simple gases not near their points of liquefaction. The final formula, therefore, is y F= 338.64 m.xVrq^ ) = 1091.35 ft. X A/1 + .003660, ) for the velocity of sound in air at the locality where g = 981 dynes, barometric height 76 cm., and temperature 6 Centigrade. 194. At West Point, assuming the barometric height to be 76 cm., and g = 980.3 dynes, we have, for the velocity of sound in air at any temperature, V = v'980.3 x 7.9894 x 1.41 x IO 5 x T = 332.3 m. x VF+~^ (229) = 1090.23 ft. x Vl + 0. Since the value of a = ^ T , we see that the velocity increases nearly 2 feet for each degree Centigrade, and hence is greater in warm than in cold weather, all other things being equal. At 60 F., we may take the velocity of sound in air to be approxi- mately 1123 feet per second. 195. The value of the velocity of sound in any gas can, in like manner, be obtained theoretically by substituting in the equation (330) for D' the density of the gas referred to that of air as unity, and for p' the value of the pressure in terms of the barometric height, y being taken as 1.41 ; or it may be obtained more simply by di- viding V = 332.3 m. x Vl + (231) by the square root of the density of the gas referred to air as unity. RELATING TO SOUND AND LIGHT. 119 At zero degrees Centigrade, we have for the theoretical value of the velocity of sound in the following gases : Air, .... ... 332 Hydrogen, 1269 Oxygen, 317 Carbon dioxide, .... 262 Carbon monoxide, . . . 337 Olefiantgas, 314 196. Velocity of Sound in Air and other Gases, as affected by their not beiny Perfect Gases. The for- mulae of Mariotte, Charles, and Poisson are only applicable to perfect gases. This condition requires the elasticities to be perfect, and the excess of the elastic force which gives rise to wave propaga- tion to be indefinitely small when compared with the elasticity of the gas in its quiescent state. A series of experiments made by Eegnault, the results of which are given in the Comptes Eendus, Vol. 66, page 209, show that these conditions are not fulfilled, and that the theoretical velocity therefore differs from the actual. The sounds were made in tubes of different cross-section, by discharging a pistol with different charges of powder. the results are grouped in the following table : Diameter of Tube, 0.108 m. Diameter of Tube, 0.3 m. Pi am. of Tube, Length, 566.74 m. Length, 1905 m. 1.10 m. Charge, 0.3 gr. Charge, 0.4 gr. Charge, 0.3 gr. Ch'ge, 0.4 gr. Charge, 1.5gr. Charge, l.OOgr. Dis- tances. ?*Tean Veloc- ities. Dis- tances. Mean Veloc- ities. Dis- tances. Mean Veloc- ities. Mean Veloc- ities. Dis- tances. Mean Veloc- ities. Dis- tances. Mean Veloc- ities. 566.74 330.99 1351.95 329.95 1905 asi.91 332.37 3810.3 332.18 749.1 334.16 1133.48 328.77 2703.00 328.20 3810 328.72 330.34 7620.6 330.43 9201 333.20 1700.22 328.21 4055.85 326.77 11430.0 329.64 1417.9 332.50 2266.96 327.04 5407.80 *323.34 15240.0 328.96 2835.8 331.72 2833.70 327.52 5671.8 asi.24 8507.7 330.87 * Un- 11343.6 330.68 certain. 14179.5 330.56 17015.4 330.50 19851.3 330.52 197. From these results we see : 1, that the mean velocity of the same wave decreases from the origin ; 2, that it is less for the same charge and route in tubes of smaller diameter ; 3, that it 120 ELEMENTS OF WAVE MOTION, decrease less rapidly in tubes of larger diameter. Regnault also, by means of sensitive diaphragms, followed the course of the waves after they became inaudible, and obtained similar results with re- spect to these. He found that a sound produced by a pistol dis- charge, of one gramme of powder, became inaudible at distances of 1150, 3810, and 9540 metres, in tubes of 0.108 m., 0.30 m., and 1.10 m. diameter, respectively, and that the waves became insensible after traveling distances of 4056, 11430, and 19851 metres respec- tively. In the tube of 1.1 m. diameter, with a charge of 2.4 grains, the wave ceased to be audible at 58641 metres, and ultimately ceased at 97735 metres. These distances of audibility are, approximately, directly proportional to the diameters of the tube. 198. The mathematical theory discusses the case of a perfect gas, and assumes that the propagation in an indefinite tube is con- tinuous. The above experiments show that this is not really the case. The assumptions made by implication in a perfect gas are : 1. That the laws of Mariotte, Charles, and Poisson are true, but it is well known that no gas obeys exactly these laws. 2. That its elasticity is unaffected by admixture with other gases. 3. That the gas offers no opposition by its inertia to wave transmission ; but experiment shows that an intense disturbance always produces a real motion of the surrounding particles, which increases the velocity, especially within sensible distances from the origin. Such is the case, no doubt, in cannon discharges, violent lightning-flashes, and other like instances. 4. Theory supposes the excess of pressure due to a vibrating body small, in comparison with the quiescent barometric pressure ; but in the cases cited above, the excess of pressure at Mie origin may be large, and hence cause an increase in the value of V near the origin. Therefore, the correction of Art. 193, called that of La Place, in such cases is not exact. 199. Regnault ascribes as the principal cause of the diminution of the intensity, the loss of kinetic energy by the reaction of the sides and ends of the tube, and confirms; this by the fact that the sounds are quite audible outside the tube during their first passage, and in a less degree at v each succeeding passage. As a secondary cause, he ascribes the influence of the walls of the tube in dimin- ishing the elasticity without affecting the density. This is con- RELATING TO SOUND AND LIGHT. 121 firmed by the fact that in the above experiments, where the waves have been produced by the same charge, and hence have* the same sensibility at the origin, they have not the same intensity after traveling over equal routes. The mean limiting velocity ought, therefore, to be the same, if the weakening is due to the loss of mv 2 on account of the sides. The experiments show that this is not the case ; hence, the sides exercise another effect on air different from that indicated as the principal cause of the diminution of the in- tensity, an action affecting the elasticity and not the density. In free air this effect would be null, and in the tube of 1.1 m. it is taken as approximately so. The mean velocity of propagation, in dry air at C., of a wave produced by the discharge of a pistol, and estimated from the origin to the point at which its sensibility can no longer be appreciated by the ear is, according to Regnault's experiments, F = 330.6 m. The mean limiting velocity, considered from the origin to the point at which its existence can 110 longer be, detected upon a sen- sitive diaphragm, is V = 330.3 m., which differs from the mean limiting velocity in the 1.1 m. tube by only 0. 32 m. 200. Velocity of Sound in Gases independent of the Barometric Pressure. Since an increase in the baro- metric pressure increases the elasticity and density in the same proportion, theory indicates that no change, due to this cause alone, will take place in the velocity. The experiments of Stampfer arid Myrbach in the Tyrol, in 1822, between two stations whose differ- ence in altitude was 1364 m., and of Bravais and Martins in Swit- zerland, in 1844, between two stations whose difference of level was 2079 m., indicated no variation in the velocity, due to the change in the barometric pressure. Regnault's experiments upon air in the tube 0.108 m. in diameter, over a distance of 567.4 m., with pressures varying from 0.557 m. to 0.838 m., and over a distance of 70.5 m., with pressures varying from 0.247 in. to 1.267 m., found no variation in the velocity, due to this cause. The theoretical ratio of the velocities of sound in gases, given by 122 ELEMENTS OF WAVE MOTION. V _ I'D T~ ~ \ ~D" was experimentally confirmed to a near degree of approximation in the cases of hydrogen, carbon dioxide, and air. The tube 0.108 m., filled for a length of 567.4 m., gave for hydrogen 3.801 m., for car- bon dioxide 0.7848 m., which differ but little from the theoretical values 3.682 m. and 0.8087 m., the velocity in air being taken as unity. Hence the formula may be taken as an expression for the limiting law. The determination of the velocity of sound in free air was made by means of reciprocal cannon discharges. There were two series of these experiments. For the first, consisting of 18 discharges, the membrane being 1280 metres distant, the mean velocity, referred to dry air at C., was found to be V = 331.37 m. For the second series, of 149 discharges, over a distance of 2445 m., during 11 days of trial, with the temperature of the air varying from 1.5 to 21.8 0., and with great variations in the wind, the mean velocity, referred to dry air at C., was V = 330.7 m., a sensible diminution of the velocity, due to the increased distance. 201. Velocity of Sound in Liquids. The value of the velocity of sound in liquids is likewise given by the general formula = V ~D^ ~ V ~'^~ X ~D (233) in which H is the arbitrary barometric height, d m the density of mercury, and g the acceleration due to gravity. The numerator is then the pressure due to the height of the barometer, and when divided by A, which is the diminution of the volume due to the increase of pressure, gd m H gives the ratio of the pressure to the corresponding compression, and is therefore the measure of the elas- tic force of the medium. The square root of this quantity, divided RELATING TO SOUND AND LIGHT. 123 l>y the square root of the density, will be the value of the velocity of sound in the liquid. 202. Colladon and Sturm made a series of experiments to de- termine the actual value of the velocity of sound in water, in Lake -Geneva, in the year 1826. The sound was caused by the strokes of a hammer upon a bell submerged one metre below the surface, and so arranged that the epoch of the stroke could be determined by a flash of powder. The instant of hearing the sound was indicated by a stop-watch to within one-quarter of a second. The distance traveled by the sound was found to be 13487 m. to within 20 m., and the time of this travel, from a mean of many experiments, was found to be 9.4 s. The temperature of the water was 8.1 C., its density at that temperature, referred to that of water at the stand- ard temperature, was unity plus a negligible fraction, its compressi- bility was taken at .0000495, and the barometric height at 76 cm. The density of mercury referred to the same temperature is 13.544, and g 9.8088. Making these substitutions in the preceding formula, we find /9. = V~ 9.8088 x 13.544 x76 006645S The actual velocity found was - - = 1435 m., differing y.4 from the theoretical value but 7 m. The latter may itself vary within wider limits, on account of the inexactness of the value of the compressibility of water, whose most probably correct value, from the experiments of Regnault, is assumed to be .00004685. 203. The principal facts derived from these experiments of Colladon are (Tome XXXVI, Annales de Chimie) that at distances beyond 200 metres the quality of the sound is changed, and the sensation is similar to the quick, brief noise 'produced by the strik- ing together of two knife-blades in air. The diminution of inten- sity with the distance is noticed, and at short distances, greater than 200 metres, it is not possible to tell whether the sound origi- nates at a near origin of weak intensity, or at a distant origin with increased intensity. The duration is less than in air ; as it should be from its value -, A being greater and V being smaller in air 124: ELEMENTS OF WAVE MOTION. than in water. When the vibrations proceeding from the sounding body reach the surface of the water at great angles of incidence, the sound does not pass into the air. At distances greater than 400 to 500 metres, the ear in air does not hear the sound originating in the water. At 200 metres the sound is readily heard. In these experiments, the bell being placed 2 metres below the surface, the angle of incidence at 400 metres is approximately 89 43' ; at 200 metres, 89 26'. Finally, the existence of a sharper acoustic shadow shows that the wave lengths are proportionally shortened in water compared with the waves made in air by the same sounding body. 204. Velocity of Sound in Solids. The ordinary solids upon which experiments have been made for the determination of the velocity of sound are glass, the various metals, and wood. In the latter, from the manner of its growth in the tree, the three di- rections, along the axis, in the direction of the radius, and normal to the plane of these two, possess necessarily different elasticities. The coefficients of elasticity also differ in different species, and in the same species, when grown in different localities, under different circumstances of soil, temperature, and moisture. Reasonably exact determinations belong then only to the particular specimen experimented upon, and mean values are usually taken for any one kind of wood in a given direction. In metals and glass, variations of the coefficients arise from the methods of their manufacture, and modifications result from every circumstance which affects their density and other physical properties. None of the solids can be said to be perfectly homogeneous ; but on the assumption that they are approximately so, different experimenters have obtained values for their coefficients which do not vary between very wide limits. 205. In solids, the sound may result either from transversal or from longitudinal vibrations. In the cases here considered, the vibrations are understood to be longitudinal, that is, the molecular displacements are in the direction of the propagation. When a solid bar, taken as homogeneous, transmits a longitudi- nal vibration, the velocity of the propagation has been found to be given by the equation (234) VI- RELATING TO SOUND AND LIGHT. 125 Substi- in which A is the elongation due to the weight of the bar. tuting for A its value in terms of Young's modulus, and making s equal to one square centimetre, I equal to one metre, and P the weight of the bar, we have (236) r=t/5' the same in form as has been found for gases and liquids. 206. Different methods have been employed to find E 9 viz., by the direct method of elongations or compressions, by flexure, by transversal and by torsional vibrations of the bar. The values given for the different metals, in Art. 23, have been obtained by Wertheim, by the method of elongations. Could we accurately determine the velocity of sound in solids by direct experiment, the value of E could be readily found by the solution of the above equation. But this velocity being very great compared with that in air, and because of the impracticability of finding sufficiently long homogeneous lengths, an accurate determination of E by this means is impossible. Biot, by a direct experiment on 951 metres in length of cast-iron pipe, found that the velocity was 10.5 times that in air; but the want of homogeneity, due to the numerous leaded joints, without doubt influenced this result appreciably. Wertheim found about the same value in wrought iron, by experimenting upon 4067.2 metres of telegraph wire. 207. Assuming the experimental values for E given in Art. 23, and taking g to be 981 dynes, the velocities of sound are, by the above formula, found to be as follows : E. D. F IN CENTIMETRES. RATIO TO FIN AIB. Lead, . . 177x981xl0 6 11.4 1.23 xlO 5 3.7 Gold, . . 813 x 981 x 10 6 19.0 1.74x108 5.3 Silver, . . 736 x 981 x 10 6 10.5 2.61xl0 5 8.0 Copper, . . 1245 x 981 x 10 6 8.6 3.56xl0 5 10.7 Iron, . . 1861 x 981 x 10 6 7.0 5.13xl0 5 15.5 Steel, . . 1955 x 981 x 10 6 7.8 4.99xl0 5 15.0 126 ELEMENTS OF WAVE MOTION. For glass, with density of 2.94, V has been found to be, by the same method, 4.53 cm. x 10 5 ; and for brass, of density of 8.47, V = 3.56 cm. x 10 5 ; or ,13.6 and 10.8 times the velocity in air, respectively. 208. The following velocities of sound in wood, deduced from the observations of Wertheim and Ohevandier (Comptes Rendus, 1846), are taken from "Everett's Physical Constants," page 65,. from which also several of the above numbers have been obtained : ALONG FIBRE. RADIAL. TANGENTIAL. Pine, 3.32xl0 5 2. 83 xlO 5 1.59xl0 5 Beech, . . . 3.34 xlO 5 3.67xl0 5 2.83xl0 5 Birch, 4. 42 xlO 5 2.14xl0 5 3.03xl0 5 Fir, 4.64xl0 5 2. 67 XlO 5 1.57 xlO 5 209. The preceding values of the velocities of sound in solids are true only when the medium is in the form of a bar of small cross-section. Wertheim has shown by his investigations, based on the theory of Cauchy, that the corresponding velocities in extended homogeneous solids are greater than the above results in the ratio of 210. Reflection and Refraction of Sound. The laws deduced in Art. 77 for the reflection and refraction of wave motion are applicable to the undulations of sound. From the equation sin = | sin 0', (237) the direction of any deviated ray or that of any deviated plane wave Y by a plane surface, can be found when -=7 is substituted for \i. If V > V, then 0' > 0, and the refracted ray is thrown from the normal ; conversely, if V < V, then 0' < 0, and the refracted ray is bent towards the normal. A ray of sound in air, incident on the surface of water, will be refracted, provided the angle of inci- dence be less than 13 26' ; for since V in water is about 1428 m. f and V in air about 332, we have RELATING TO SOUND AND LIGHT. 127 and sin = .2325, or (f> = 13 26'. For greater incidences the ray is totally reflected, and does not enter the water. 211. Consequences of the Laws of Reflection. 1. If a sound originate at one of the foci of an ellipsoid, it will be reflected to the other focus. 2. If at the focus of a paraboloid, the rays of sound will be reflected in lines parallel to the axis, and can be again collected at the focus of another similar paraboloid, with sensibly undiminished intensity. The slightest sound, as the ticking of a watch, may be employed to illustrate this case of reflection. 3. The speaking-trumpet and speaking-tube are employed to prevent the too rapid dissipation of sound. The former, partly by reflection from its sides and largely by resonance, concentrates the sound within the volume of the cone whose apex is the mouth- piece and whose section is that of the other end of the trumpet. The speaking-tube confines the energy in the narrow compass of the tube, the loss being insignificant in the ordinary lengths em- ployed. 4. When a sound is reflected by any obstacle which prevents its direct trans- mission, and the observer is at such a dis- tance that the direct and reflected sounds are not confounded, the reflected sound is called an echo. Thus, if A be the position of the observer, S the origin from which a Figure 28. sound of short duration emanates, and W the obstacle, such as a wall, then the direct sound will reach the Q A observer in the time ^-j, and the reflected sound in the time oo&A SW + WA . A0 r SW + WA SA , -3^-4 , the temperature being 0. If - ^ ^ - be sufficiently great, so that the reflected sound arrives after the ces- sation of the direct sound, then the echo will be heard, provided the intensity be of sufficient value. If the two sounds commingle, 128 ELEMENTS OF WAVE MOTION. the resultant sound will be prolonged, and partial resonance will ensue. The number of distinct impressions distinguished by the ear will determine the shortest difference of route necessary to es- 332 tablish the echo. Thus, if we take nine per second, or 37 m. is the shortest difference of route at C. 5. The conditions of interference of sound are the same as those discussed in Arts. 65-68. Hence, it is theoretically possible that two sounds affecting the ear simultaneously will result in silence, and practically it will be shown, in the lectures on this part of the course, that such an experiment is also possible. Other illus- trations of interference are also reserved for the lectures. 212. Refraction of Sound. In order that the rays of sound shall converge after deviation by refraction, we see from the Y formula that fi -=-, must be greater than unity. Then the deviated wave will, in general, become converging, and the energy accumulate on an ever decreasing surface. Examining the table, Art. 195, we see that V in carbon dioxide is 262 m., and hence, when the incident medium is air, - 332 - 1 25 ^ -262- 5 ' and sin $ = 1.25 sin '. (238) The sound lens devised by Sondhaus is a double convex lens of collodion filled with carbon dioxide, which collects the sound rays proceeding from any sonorous body and concentrates them appre- ciably at another point on the opposite side of the lens. By means of a concave lens of the same material, filled with hydrogen, V = 1269 m., it will be evident, after the study of the properties of lenses, as explained in optics, that a similar result would be effected. The slight noise produced by the ticking of a watch may be collected by this means at a point so that the noise is audible, when without this assistance it would be inappreciable at the same point. 213. General Equations for the Vibratory Mo- tion of a Stretched String. The bodies usually employed to produce musical sounds by their vibrations are strings, rods air- RELATING TO SOUND AND LIGHT. 129 columns, plates, bells, etc. When the vibrations of the particles are perpendicular to the direction of wave propagation, they are called transversal, and when in the same direction, longitudinal. We will first consider the vibrations of a perfectly elastic and flexible string, supposed to be stretched between two points whose distance apart is I, by a force which produces a tension T. Let the elongation be that given by I' -1 = ^1, (239) in which I is the natural length, I' the length after the tension T is applied, and E is the longitudinal modulus. If the displacements of the string fro in its position of rest be due to the incessant action of forces whose rectangular accelerations are X, Y, Z, these with the tension T will be the only extraneous forces considered. Let m be the mass of any element \ x, y, z, x + dx, y -f dy, z -f dz, the co-ordinates of its extremities and its length ds ; cc the area of its cross-section, and p its density ; then m = pads. Let the components of T at x, y, z, be T T^- T-- ds 9 ds ' ds' and at x -f- dx, y + dy, z -f- dz, be 2* ~ 4-^7* T^y. \ dT^- 7 7 ^4_/77 T ^ J . ds ds' ds ds 9 ds ds' The general equations of motion will then be (240) 214. These equations are simplified when we suppose that the string is arbitrarily displaced from its position of equilibrium, and 9 130 ELEMENTS OF WAVE MOTION. abandoned to itself, without the action of the forces Jf, Y, Z. It will then oscillate about its position of rest, and the only extraneous force that acts will be the tension T, whose intensity will vary be- tween known limits. Let the axis of x coincide with the string in its position of rest, and the co-ordinates of the element m, at the time t, be x -f- 1, 77, If the displacement be supposed small, , ??, and are functions of x and t, and x is independent of t, and the above equations reduce to (241) ds Let T' be the tension when the string is straight, and the string is displaced; the length of the element is in the first case dx, and in the second ds ; these are connected by the equations (242) (243) (244) (245) ds = a d& = (dx + d!;)* + drf + dt? ; from which, when dr) and d are very small, we have ds = dx -\- d%, T= T' + E- dx Substituting in Eqs. (241), we have (246) RELATING TO SOUND AND LIGHT. 131 E T' Keplacing and by u 2 and ^ respectively, we have pa pa _ dt* ~ da?-' ^n _ 2 dfy dP ~ do?' *- v * d -S. dP ~ dx* (247) The integration of these three partial differential equations give (Analytical Mechanics, Appendix IV), - vt), (248) ? = F(x+ vt) +f(x-vt). ) 215. The first equation determines the longitudinal vibrations, or those along the axis of the string, and the other two give the transversal vibrations along y and z respectively. Because of the independence of the differential equations, the three vibrations in general coexist and are wholly independent of each other, and since the differential equations are of the same form, we see that the two kinds of vibrations are subjected to the same laws. They may each be discussed separately. Each is due to a progressive motion for- ward and backward along the string. These motions may be of the most varied character, but the particular form of the motion depends on the form of the functions whose symbols are F and/. The only conditions imposed so far are that for x = and x = I, I, TJ, and are zero for all values of t. These, together with any assumed initial conditions, will enable us to determine the form of the functions F and /, and thus complete the solution of the problem. 216. Since the vibrations parallel to y and z are exactly alike in every particular, the discussion of one will do for the other, and we will consider that of y, given by the equation il = F(x + vt)+f(x- vt}. (249) 132 ELEMENTS OF WAVE MOTION. Assume the conditions" that at the epoch, or when t 0, ?? = 0(z) and jj=vil>'(x), (250) in which the functions and are supposed known, and that 0' is the derived function of -0. If t = 0, Ave have 7? = 0(3) = ^(z)+/, (251) J-J-y<)-=^-<4-/ f (*);/ (252) /. 1>(x) = F(x)-f(xj; (253) and hence, F(x ) = +M+W 9 (354) Therefore ^(z) and /(#) are known for all values of x from to I, when, as is supposed, (x) and -0 (x) are known between the same limits. For the extremities, we have, by placing x = and x = I, F(vt) +f(-vt) = 0; (256) F(l + vt)+f(l-vt) = 0; (257) whence, F (vt) and.f(vt) are equal, with contrary signs, and thus become known for all values from t = to t = . 217. The value of r\ can be expressed by means of a single func- tion by substituting vt + I x for vt in Eq. (257) ; whence, -F(M x + vt) =f(x vt) ; (258) which in Eq. (249) gives ?i F(x + vt) F (21 x + vt). (259) Again, for vt, in Eq. (257), substitute I + vt ; then .F (2? + vt) = f(vt) = F (vt) ; (260) whence we conclude that the function F takes the same value when the variable vt is increased by 21 ; and therefore by 4/, Ql, SI, ---- RELATING TO SOUND AND LIGHT. 133 or 2nl, n being a positive whole number. Therefore, if F (vt) is known from vt = to vt = 21, its value is known for all values from t = to t = oo . Replace vt by I vt, in Eq. (257), vt being less than I ; then F(2l-vt) = - (261) but f(vt) is known for all values of vt between and Z; therefore F (vt) is known for all values of vt between I and 21. Hence, the value of F (x 4- vt) is known for all values of x-\-vt from to oo ; and, similarly, the value of f(x vt) can be found for all values between and oo; and therefore the problem is completely solved. \A, Figure 29, 218. The function whose symbol is FIB subject to the following conditions, derived from Eqs. (256), (257), (260), (261), F(x) = -F(-x), F(l + x) = -F(l-x), F (x) = F (21 + x) = F (1 + x) F(x] = -F(2l-x) = -F(l-x) = (362) (263) (264) (265) From Eq. (262) we see that the curve represented by 77 = F (x) is continued in similar forms on each side of in the figure ; from Eq. (263), that the forms are similar on each side of A; from Eq. (264), that the form is repeated from 0' to 0" exactly as from to 0'; and from Eq. (265), that the form of the curve inverted is the same from 0' to A as from to A. The motion of any particle is that of oscillation about its place 21 of rest, and of which the period is This vibratory motion is gradually diminished, while the period remains unchanged, because 134 ELEMENTS OF WAVE MOTION. of the energy communicated to the air, and through the points of attachment to other bodies. The time of one complete oscillation is (366) and the number of oscillations in the unit of time is Therefore in the transversal vibrations of a string, the resulting pitch is inversely proportional to its length, directly as the square root of the tension when straight, and inversely as the square root of the density by the area of cross-section. 219. The number of longitudinal oscillations in the unit of time is whence the pitch depends only upon the length of the string and the material of which it is made, and is independent of the tension, unless the latter should be so considerable as to change the value of E. Experiment appears to indicate that the longitudinal pitch increases slightly with the tension ; but this may be accounted for in the elongation experienced, which is always accompanied with a slight diminution of density p, and should this occur, the formula indicates that the pitch should rise. The ratio of the numbers for the same string is given by ^.;:'V (269) M. Cagniard Latour experimented on a cord of 14.8 m. in length, and found and AZ = 0.05 m. Substituting in the formula, we have 188 whence, A? = 0.052 m., a sufficiently near approximation. RELATING TO SOUND AND LIGHT. 135 220. The preceding values of n and ri are the least numbers of transversal and longitudinal vibrations of the string, and therefore correspond to its fundamental tones; but we know that each of the vibrations is decomposed into any number of vibrations of equal periodicity, when the string is divided into a like number of sym- metrical parts. This can be shown more readily when the integral equation is expressed in a series which is a function of sines and cosines. Thus, it is evident that a possible solution of the differen- tial equation is given by %-nvt n . invt\ . inx cos j- BI sm -- sm -- , (271) ( when the conditions with respect to the extreme points are un- changed. In this equation, i is any entire positive number which marks the order of the term, and A ir B^ are constant coefficients depending on i and on the initial state of the string. If then this state is such that 77 is constant only for the terms for which i is a multiple of another entire number n, the string will return to the 21 same state at the end of each interval of time , which is the nv duration of its similar and isochronous vibrations. Under this sup- position, the n 1 points of the curve corresponding to distances * - - - t fi n n will be nodes, that is, will remain at rest during the whole period of the motion. Since the value of 77 is linear, every value corresponding to i = 1, 2, 3, 4, etc., will be a solution, and the sum of all the values of 77 will also be a solution of the differential equation ; hence we will have for the general integral equation . / j *7r?^ n t^lfl/l m 1/TTtJC tf^^i T) = SiSr\4i cos 7 h BI sm nrrJ sm ~r~* (272) \ i ill 221. The values of A 1 B IJ A 2 B Z , etc., are in general arbitrary, and we may suppose all to vanish up to any order n, while the rest remain arbitrary. If A^B^ are not zero, there are no actual nodes 136 ELEMENTS OF WAVE MOTION. except the fixed ends, and the first simple tone is that whose period is r and whose wave length is 21. If there is one node, the period is -, and the gravest . simple tone is that of wave length I; and, /v generally, if there are n 1 nodes, the period is - , and the ti gravest tone is the (n V) th harmonic of the fundamental tone. When the string vibrates without nodes, the series of harmonic tones is in general complete, and a practised ear can distinguish ten or more. It is also possible to make a string vibrate in such a man- ner that for any proposed value of n the coefficients A n tt n , A^ n B^ n , etc., shall disappear, so that the component harmonic vibrations whose periods are -, , etc., are extinguished. When this i& Ti &tl done, the ear does not distinguish these tones, and we may therefore conclude, from what precedes, that each component tone actually heard is produced by the corresponding harmonic vibration of the string. 222. The same general method may be applied to the longitudi- nal vibration of a rod, and the differential equation will be, as in the case of the longitudinal vibration of a string, of the form 3 -"3- ' of which the integral equation is = F (x + Vt) + f(x - Vt\ (274) and which may be put under the form of .. ITTX I A iirVt . ITT FA /rt)V K\ = x + 2 cos j- (Ai cos j h Bi sin ^ ), (275) in which is the distance from the fixed origin at an}' time t to the particles in a plane section of the rod, of which the natural distance from the end of the rod is x. The value of x therefore depends only on the particular section considered, and is independent of the origin of ; but if the vibrations cease, the periodic part of Eq. (275) would vanish, and we would have = x for all points of the rod, and therefore the periodic part gives the displacement ( x) at the time t of the section determined by the value of x. RELATING TO SOUND AND LIGHT. 13T The periodic part does not in general vanish for any value of x y so that there are in general no nodes. But there will be n nodes at sections for which x is any odd multiple of , provided At, Bi> vanish for all values of i except odd multiples of n. Thus the rod may have any number of nodes, of which those next the ends are distant from the ends by half the distance between any two consec- utive nodes. inx I ' inVt . ircVt\ / T> T' T' ltc " 2A 4A 6A and 0, --, , -j-, etc., from M, the open end of the tube, respectively. 230. These laws, which determine the positions of the nodes and ventral segments of vibrating air columns, are known as Ber- nouilli's laws. F/om them we see that the harmonics of open pipes are in the order of the natural numbers, and that those of closed pipes are as the odd numbers. Thus, the open pipe can give, by an increased pressure, the octave, the twelfth, the fifteenth, etc., while the closed pipe gives the twelfth, the seventeenth, etc. Ex- periments with organ-pipes verify the laws of Bernoulli! only approx- imately ; that is, that the nodes are not exactly at the positions defined above, nor are the nodes exactly places of rest. Organ- pipes are usually made to speak by forcing a current of air through a narrow slit, and causing it to impinge against a thin lip. Of the many vibratory motions produced in this manner, there is always one whose periodicity is such that, by the resonance of the pipe, its intensity will be raised to such a degree as to produce a marked and determinate musical sound, called the fundamental tone of the pipe. Other vibratory motions, which undoubtedly exist, are either destroyed by the interference of the reflected waves, or have so feeble an intensity as to be negligible. The wave length of the funda- mental tone is, as we have seen above, double the length of the open pipe, or four times the length of the closed pipe, approxi- 14:2 ELEMENTS OF WAVE MOTION. raately. The discrepancy between experiment and theory arises from the fact that the hypothesis is not in accord with what actu- ally occurs in the pipe. Without considering these minutely, it is sufficient to note the perturbations at the embouchure by the air current, the modifications in the pipes by the moving air, and the induced vibrations of the material of the pipe at the sides and closed end, to account for the greater discrepancies. 231. Relative Velocities of Sound in Different Material. Since in any medium, we have A Vr = V-, in which n is the vibrational number for a note of definite pitch, A the corresponding wave length in the same medium, and V the velocity of sound, it is readily seen that if free rods of different material be taken, of such lengths as to give the same note when put into longitudinal vibration, we will have A = V-, A' = V-, A" = V"-, etc.; n' n n whence A: A': A" :: V: V : V". A A' A" But as ^, , , etc., are the lengths of free rods that give the r 4 : 16 :1 ' and therefore generally 4 16 For d we have 21 4? 2n l' 4^ 3' n l Substituting the value of n taken from the latter in the former, we have ~ (383) If d = ?, which corresponds to a rod supported at both ends and yielding its fundamental sound, we have N = 1. We therefore con- 10 146 ELEMENTS OF WAVE MOTION. elude that when a rod gives a harmonic,, the parts comprised between the nodes vibrate as rods whose extremities are supported and whose length is the distance between the nodes, and that the vibrational number is inversely as the square of this length. This conclusion is inapplicable to the first nodes, because they are more or less influenced by the extremities. 236. Tuning Forks. A tuning fork may be regarded as a rod or bar free at both ends. Experiment shows that in proportion as a bar free at both ends is bent or curved the extreme nodes approach each other. Thus, in the figure the bar ab if supported at the points 1, 2, one fourth the length of the bar from the extremes, will when vibrated transversely develop nodes at these points. In the forms a'V, a"b", a"'V", the length remain- ing unchanged, the nodes approach each other as indicated in the figure. The laws which govern the vibration of a fork whose section is rectangular have been experimentally found to be ; 1, that the vibrational number is independent of the width ; 2, proportional to the thickness ; 3, inversely proportional to the square of the length increased slightly. The length is taken as equal to the projection of the prongs on the medial line of the fork. For a fork of rectangular cross-section we have from the experiments of Mercadier. (284) Figure 32, in which N is the vibrational number, t the thickness, and I the length ; is a constant which for steel is found to be 818270. When the fork yields its fundamental note its method of division is shown in the figure. The over- tones of a fork correspond to vibrational numbers which are to each other, beginning with the first ; as, 3 2 : 5 2 : 7 2 : etc. The vibra- 25 ticnal number of .the first overtone is about -- T Figure 33. RELATING TO SOUND AND LIGHT. 147 that of the fundamental. Helmholtz found by experimenting on many forks, that it varied from 5.8 to 6.6 that of the funda- mental. These overtones are so high, that they are generally of short duration, and they are also inharmonic with the prime. Tuning forks are generally mounted on their resonant boxes, by which arrangement the prime tone of the fork is greatly rein- forced to the disadvantage of the overtones. The duration of the vibration of a fork although theoretically constant, is found to increase slightly with an increase of amplitude and tempera- ture, thus slightly lowering the pitch. This is, however, not appreciable to the hearing, but can be detected by any of the graph- ical methods for determining the number of vibrations in a given period. It is a matter of importance in determining the initial velocity of projectiles, by means of the Schultz chronoscope or other devices, where the vibrations of a tuning fork enter into the calcula- tion, to limit the amplitude and to take note of the temperature, in order to obtain uniform and reliable results. When the amplitude does not surpass 3 or 4 mm. and the temperature varies but little beyond the ordinary atmospheric temperature, the vibrational number may be taken as constant within .0001 of its value. 237. Vibration of Plates. Plates are rigid bodies, gener- ally of metal or glass, whose length and breadth are very great compared with their thickness. To put them in vibration, one or more points are fixed and a violin bow is drawn across an edge. The circumstances of vibration are exhibited by sprinkling fine sand over the surface and examining the nodal lines formed by the sand which seeks that part of the plate which is at rest. The parts of the plate separated by a nodal line, evidently vibrate in opposite directions, and therefore for permanent figures the number of vibrating parts must be even. When the plate yields its funda- mental tone the resulting figure is the simplest that can be formed, and as the plate separates into a greater number of vibrating parts, the figures become more complex. Chladni has given to these figures the name of Acoustic figures. As yet, from the inherent difficulties of the problem, the mathematical laws have not been deduced, but experiment has assigned the following as the laws of vibrating plates, viz. ; 1, the vibrational numbers of plates of the same form and of the same material are inversely as the squares of 148 ELEMENTS OF WAVE MOTION. the homologous dimensions ; 2, and are proportional to the thick- ness. Hence we have t_ V_ P '' I'*' n : n If a rectangular plate be so constructed that a system of nodal lines parallel to the length be formed by a sound, which gives another system of nodal lines parallel to the breadth, when it is vibrated in these two ways, then if at any of the middle points of the ventral segments it be vibrated so as to produce the same sound, these two systems will simultaneously exist and the acoustic figure will result from the combination of these two systems. The figure illustrates five such plates where the numbers of the nodal lines are in the ratios of 2 : 3, 2 : 4, 3:4, 3:5, 4:5. Other combi- nations illustrative of the vibrations of plates are reserved for the lectures. S:6 4:5 Figure 34i 238. Vibration of Membranes. When a stretched mem- brane is near a sounding body, the air transmits to it the vibratory motion. It can respond, however, only to certain sounds depend- ing on its tension, and thus enter into synchronous vibration. This fact is made evident by the acoustic pendulum, or by the nodal lines formed by sand sprinkled upon it, as in the case of the RELATING TO SOUND AND LIGHT. 149 vibration of plates. The frames upon which the membranes are stretched are generally square or circular. Experiment has con- firmed the following deductions of Poisson and Lame, with respect to the vibrations of square membranes, viz. : 1. Membranes respond only to certain sounds, separated by determinate intervals. 2. To each sound a system of nodal lines corresponds, parallel to the sides of the membrane, and whose numbers are represented by n and ri. 3. The nodal lines which correspond to the same sound form a system of figures, such that we can pass from one to the other by continuous changes in varying the mode of disturbance, without changing the sound ; but we can never pass in a continuous man- ner from the lines of one sound to those of another. Circular membranes can only give nodal lines along the, diam- eters or circumferences, either separate or combined, depending on the method of vibration and on the point or points of enforced rest. 239. Because of the limited time allotted to this part of the course, many subjects of importance are necessarily omitted in the text. Among these are, 1. The theory of beats, and resultant sounds. 2. The phenomena of interference, whose consequences, how- ever, are readily derived from the discussion in Arts. 65-68. 3. The graphical and optical methods of the study of sonorous vibrations, and that by sensitive and manometric flames. 4. The phenomena of vibrations of air columns in organ-pipes, of elastic rods, of plates and membranes, with the applications of the latter in the phonograph, phonautograph, and telephone. By means, however, of a very complete acoustical apparatus, mainly from the workshop of Koenig, the celebrated physicist of Paris, the omitted parts, as well as those treated of above, are illus- trated in the lectures, which largely supplement and complete the study of the text. 240. The nature and essential principles of undulatcry motion, as illustrated by sonorous vibrations, have received sufficient atten- tion to enable the student to prosecute understandingly the study of similar principles connected with light in the analogous subject of optics. PART III. OPTICS. 241. Light is the agent by which the existence of bodies is made known to us through the sense of sight. That branch of physical science which treats of the properties of light and the laws of its transmission is called Optics. 242. It is divided into two parts : 1. Geometrical Optics, which embraces all the phenom- ena relating to the propagation of rays, based on certain experi- mental laws, and which is entirely independent of any theory as to the nature of the luminous agent. Experiments in Geometrical Optics, however carefully made, can never accurately prove the laws of light propagation, but serve merely to establish a certain degree of probability of their truth, and which, when applied to other phenomena of the same nature, strengthen this probability in proportion as the application is more extended. 2. Physical Optics 9 which is based on the theory of un- dulations, and seeks to explain by this theory the nature of light, and of all the phenomena arising from the action of rays on each other. 243. That light is not a material substance, but is merely a process going on in some medium, is proved by the phenomena of interference, in which results of various magnitudes occur, from less to greater, or the reverse, depending upon the manner in which the interference takes place, even when the combining magnitudes are themselves constant in value. 244. The undulatory theory asserts that light is due to the transmission of energy from luminous bodies to the finely-divided parts of the optic nerve, spread over the interior concave surface of RELATING TO SOUND AND LIGHT. 151 the eye. This energy is conveyed by the optic nerve to the brain, and there transformed into the sensation of sight. The transmission of the energy is accomplished by undulatory motion in a medium called the luminiferous ether. There is no direct proof of the actual existence of the ether, and its assumption can only be regarded as an extremely probable hypothesis, supported by nearly all the known phenomena of light, and directly contra- dicted by none. Within the present century, its reality has been almost uni- versally accepted, and as a consequence the undulatory theory has entirely supplanted the rival hypothesis of the materiality of light . molecules, known as the emission theory, which had, however, held its ground for many years. 245. The accepted properties of the luminiferous ether have resulted from theoretical considerations, modified from time to time by deductions from experimental observations, and while there are several imperfections yet to be removed, nevertheless the strong array of unquestioned facts, both observed and predicted, has estab- lished these properties as a satisfactory foundation upon which modern physical optics is now constructed. The luminiferous ether is considered to be a material substance of a more rare and subtile nature than the ordinary matter affecting the senses, and to exist not only within these bodies, but through- out space. It has great elasticity, and is capable therefore of trans- mitting its particular energy over vast distances, with great velocity and with inappreciable loss. That this energy is not transmitted instantaneously has been proved by direct experiment, and con- cluded from several astronomical observations. 246. That light is propagated in right lines from the source is a fact of observation and experiment. This statement, however, while absolutely true, is subject to modification when taken in the ordinary sense of the language. Thus, we have seen that while sound is propagated in right lines from its source, it is capable of spreading around an obstacle, so that sound can be heard out of the direct line of the source ; so, in a less degree, we can see around an obstacle, as will be shown in the discussion of the diffraction of light. The acoustic shadow, however, is as much less marked than the optical shadow as the wave lengths of sound are greater than the 152 ELEMENTS OF WAVE MOTION. wave lengths of light. But for the explanation of the principles of geometrical optics it is unnecessary to consider this refinement. 247. Bodies are called self-luminous when they are themselves the sources of light, and rays proceed directly from them. They are visible because of their emanating rays. Other bodies are called non-luminous, and become visible because rays from luminous bodies are reflected from their surfaces. A luminous point, or origin of light, is a very small portion cf a luminous surface. When light emanates from a luminous point, we consider it made up of rays of light, each of which is the small- est portion of light which can be transmitted. The ray is the right line along which the undulation is propagated, and is practically a mere conception, indicating direction. A collection of parallel, diverging, or converging luminous rays is called a beam of light, and sometimes & pencil of light, the latter name being generally applied to the last two cases. The axis of a beam is the geometrical axis of the cylinder or cone of rays ; when the axis is normal to the deviating surface, the beam is direct, and when inclined to it, oblique. 248, When a beam of light is incident upon any surface, it is generally separated into three portions, viz., a part is scattered or diffused over the surface, by which the surface becomes visible, a second part is reflected, and the remainder is refracted. The proportion of the several parts depends on the polish of the surface, the angle of incidence, and the nature of the medium. A perfectly polished surface would be invisible, and the incident beam would be separated into a reflected and refracted beam alone ; of course, such a polish is not practicable. Light regularly reflected has its intensity increased with the degree of polish, while the in- tensity of irregularly reflected light is similarly diminished. The intensity of regularly reflected light from the surface of water is, at the incidences of 0, 40, 60, 80, 89i, about 1.8$, 2.2$, 6.5$, 33$, 72$. At normal incidence, water, glass, and mercury reflect 1.8$, 2.5$, and 66f $, respectively. The differences at small angles of incidence are more marked than at greater angles, since while both RELATING TO SOUND AND LIJHT. 153 water and mercury reflect the same at 89^, -the former reflects but 5^ as much as the latter at normal incidence. 249. A medium is any substance which permits the passage of light through it. Since the luminiferous ether is supposed to pervade all matter, it might be inferred that all bodies could be classed under the head of media for light. Gold, although one of the most dense of sub- stances, does permit the passage of light, when beaten into a very thin leaf ; and no doubt if other opaque bodies possessed an equal malleability, the same property would belong to them. But owing to internal reflection and consequent interference, it is assumed that an inappreciable quantity of light, if any, passes through very small thicknesses of opaque bodies. Glass, air, water, and all other matter which permit the passage of light freely, are said to be transparent. Translucency is a term applied to such bodies as permit the passage of diffused light; thus, ground glass and flint are translucent, while clear glass and quartz crystal are transparent. 250. Since light is assumed to result from undulatory motion in the luminiferous ether, all the consequences deduced in the dis- cussion of the properties of this kind of motion in Part I are at once applicable to the phenomena of light. 251. Shadows and Shade. From each luminous point considered as an origin of disturbance, undulations proceed along right lines in all directions from this origin. Therefore, whenever they meet an opaque body, this undulation will be deviated from its original direction, and the effect of light will be wanting along this direction prolonged. The absence of this effect is called the shadow of the point of the opaque body. The line of the surface of the opaque body, along which rays drawn from the luminous point are tangent, is called the line of shade. Since each point of the luminous surface is an origin of light, we see that in all actual cases the shadow of an opaque body must be indistinct near its boundary, and gradually merge into the illuminated surface surrounding the shadow, whenever the lumi- nous source is of an appreciable area. This modified portion of the shadow is due to the overlapping of the cones of rays proceeding 154 ELEMENTS OF WAVE MOTION. from each luminous point, and is called the penumbra. It is lim- ited by the space between the two cones, whose elements are tan- gent to the luminous surface and the opaque body, one having its vertex between the two, and the other its vertex on the further side of either one of the surfaces. The softness of shadows in general is due to the finite extent of luminous surfaces. 252. Every point of the luminous source emitting rays in all directions, each will carry an image of its luminous point. Thus, if a lighted candle be placed in front of a small aperture of a darkened chamber, the aperture will permit the passage of a limited number of the rays from every point of the candle, each ray, however, carrying an image of its radiant. The image, as shown in Figure 35, will be inverted. If another aperture be made near the first, a second image of the candle will be formed, overlapping the first, and, while the luminosity will be increased, the image will lose distinctness, be- cause of this overlapping. The diffused light of a room during the day is due to the overlapping images of external objects, caused by rays proceeding from each of them, thus making their individual images indistinct. A small aperture in a darkened room will per- mit the formation of an inverted image of the external scenery upon a screen placed within the room near the aperture. 253, Photometry. The eye possesses the property of dis- tinguishing color and intensity. In determining variations of intensity, the judgment is only approximate when the colors are the same, and the difficulty of this appreciation is increased when the colors differ. Equality of in- RELATING TO SOUND AND LIGHT. 155 tensity can readily be determined by the eye, while it is not possible to ascertain the numerical ratio of different intensities by direct observation. Photometry has for its object the measurement and comparison of the intensities of different lights. 254. The principle of all photometric methods is to arrive at this comparison, by the appreciation of the equality of illumination of two near surfaces, physically identical. In assuming the dis- tance of the luminous source from the illuminated surface to be great in comparison with the dimensions of the surface, and remem- bering that the intensity of the light is due to the molecular kinetic energy, we readily see, if there be no absorption of this energy dur- ing transmission through the intervening media, 1. That the intensity of the illumination on the unit area of any surface, taken normal to the direction of propagation, at a distance d from the luminous source, varies as -^- a* 2. That if / represent the intensity of any given light, and if it be. supposed to illuminate uniformly any area A, the intensity on a unit of area varies as A 3. That the quantity of light emanating from any luminous element, and hence the intensity of illumination on the unit area, is proportional to the cosine of the angle made by the normal to the element with the direction considered, and hence varies as the co- sine of the inclination, or cos i. 4. That if the area on which the light falls is inclined to the direct line of propagation, the illumination on the unit area is pro- portional to the cosine of the angle made by this line and the nor- mal to the surface, or to cos i'. 5. That the illumination on the unit area will vary with the intrinsic brightness of the source. The intensity of the illumina- tion on the unit area, parallel to the source, at the distance unity, may be taken as the measure of the intrinsic brightness. 255. Let S and 8' be the projections of the luminous and the illuminated surfaces, respectively, on a plane normal to the direc- tion of the luminous rays; B the intrinsic brightness of the 156 ELEMENTS OF WAVE MOTION. source ; d the distance apart of the two surfaces, and / the intensity of the illumination ; then, from the above principles, we have I=B 8 j^. (385) Making 8' = 1, and calling /, the total brilliancy of the source at the distance d, we have /, = *! (280) & -^ is the apparent area of the source seen from the illuminated surface, and making this equal to unity, we have / = B. (287) Therefore the intrinsic brightness of the source is the total brilliancy of the apparent unit of area of the luminous surface at the distance 1. The general method of comparison of the intrinsic brightness of two sources consists in permitting the rays from each source to fall, nearly normal, upon adjacent portions of the same surface ; then to increase the distance of the stronger light, until the eye judges the illumination to be equal. We then have BS __ B,S, ( v. ~#- ~d?> from which, by substituting the known values of d, d n S and $,, the ratio of B, to B can be determined. 256. The apparent intrinsic brightness of an object is equal to the quantity of light received from it by the eye, divided by the area of the picture on the retina. Therefore, since the apparent ' would be greater than 0. The re- Figure 38.