Physics dept. 
 
toWBt Division 
 
TEbe THniver0it\> tutorial Series. 
 
 GENERAL EDITOR: WILLIAM BRIGGS.M.A., LL.B..F.C.S., F.B.A.S. 
 
 THE TUTOEIAL PHYSICS. 
 
 ,^t. VOLUME I. 
 
 A TEXT-BOOK OF SOUND. 
 
THE TUTORIAL PHYSICS. 
 
 YOL. L TEXT-BOOK OF SOUND. 
 
 BY E. CATCHPOOL, B.SC. LOND. 
 
 3s. 6d. 
 
 VOL. II. TEXT-BOOK OF HEAT. 
 
 BY R. WALLACE STEWART, D.SC. LOND. 
 
 3s. 6d. 
 
 YOL. III. TEXT-BOOK OF LIGHT. 
 
 BY R. WALLACE STEWART, D.SC. LOND. 
 
 3s. 6d. 
 
 YOL. IY. TEXT -BOOK OF MAGNETISM AND 
 ELECTRICITY. 
 
 BY R. WALLACE STEWART, D.SC. LOND. 
 
 5s. 6d. 
 
CALIFORNIA 
 
 DEPARTMENT OK PHYSICS 
 
 1Hniversit tutorial Seriee. 
 
 THE TUTORIAL FETMC& 
 
 VOLUME I. 
 
 A ^ TEXT-BOOK OF SOUND. 
 
 WITH NUMEROUS DIAGRAMS AND EXAMPLES. 
 
 BY 
 
 EDMUND CATCHPOOL, B.Sc. LOND. 
 
 FIRST CLASS HONOURS IN PHYSICS AT B.SC., 
 SENIOR LECTURER IN PHYSICS AT UNIVERSITY TUTORIAL COLLEGE. 
 
 LONDON : W. B. OLIVE, 
 
 UNIVERSITY CORRESPONDENCE COLLEGE PRESS. 
 WAREHOUSE : 13 BOOKSELLERS Row, STRAND, W.C. 
 
 NOW PIJm.lQJJtrrt r*\/ 
 
ca 
 
 PHYSICS 
 
 
PREFACE. 
 
 In writing this text-book I have tried to keep in view the 
 following aims : 
 
 To include all the facts of which a knowledge is expected 
 in elementary examinations. 
 
 To present a clear picture of the external physical processes 
 which cause the sensation of sound. As the mathematical 
 symbols for a physical process are easily made a substitute 
 for a clear conception of the process, the mathematical 
 symbols have been avoided as far as possible. 
 
 To keep before the reader the distinction between phrases 
 which describe actual processes or conditions and phrases 
 which, while they facilitate the prediction of real processes 
 and real phenomena, do not themselves stand for any phy- 
 sical condition or event. Such phrases are used in every 
 department of physics, and are only mathematical symbols in 
 disguise. Where I have departed from, or added to , the 
 usual forms of explanation, it has usually been with the 
 intention of making this distinction clearer. 
 
 As the book is intended as a physical treatise, it is the 
 physical processes which cause the sensations of sound, and 
 not the sensations themselves, which form its subject- 
 matter. Only those peculiarities of the sensation are con- 
 sidered which throw light on the external physical processes 
 which are taking place. 
 
 The following portions of the book may be taken as a 
 suitable course of first reading : 
 
 Arts. 1-6, 11-17, 22-24, 26-35, 44-54, 58, 60-64, 
 70-72, 74, 75, 77, 79, 84-88,' 90-99, 101, 104, 105. 
 
 I desire to acknowledge the help I have received in pre- 
 paring the book from Mr. J. Dibb, B.Sc., Dr. E. W. 
 Stewart, and Dr. A. Walker. I shall be indebted to any 
 readers who are kind enough to point out errors, or portions 
 of the book which require to be more clearly expressed. 
 
 November, 1894. 
 
 G'VI 
 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 
 Vibratory Motion ... ... ... ... ... . , ... I 
 
 CHAPTER II. 
 
 Progressive Undulation ... ... ... ... ... ... 21 
 
 CHAPTER III. 
 
 Velocity of Sound 46 
 
 Examples I. ... ... ... ... ... ... ... ... 60 
 
 CHAPTER IV. 
 
 Interference ... ... ... ... ... ... ... ... 61 
 
 CHAPTER V. 
 
 Forced Vibration ... ... ... ... ... ... ... 75 
 
 CHAPTER VI. 
 Fourier's Theorem 80 
 
 CHAPTER VII. 
 
 The Ear and Hearing 93 
 
 CHAPTER VIII. 
 
 Reflection of Sound ... 109 
 
CONTENTS. Yll 
 
 CHAPTER IX. 
 
 PAQI 
 
 Stationary Undulation... ... ... ... ... ... ... 125 
 
 CHAPTER X. 
 
 Vibrations of Air in Pipes 136 
 
 Examples II. 153 
 
 CHAPTER XI. 
 
 Transverse Undulation ... ... ... ... ... ... 155 
 
 Examples III. ... ... ... ... ... ... ... 175 
 
 CHAPTER XII. 
 
 Acoustic Measurements ... ... ... ... ... ... 177 
 
 APPENDIX 
 
 (A) Harmonic Motion. . . ... ... ... ... ... 1 95 
 
 (B) Use of the term "Intensity" 196 
 
 (C) Use of the term ' ' Interference " 196 
 
 (D) Velocity of Pulses 197 
 
 Answers to Examples... 200 
 
 Index ..201 
 
EEEATA. 
 
 Students are requested to correct at least those marked with an 
 asterisk before reading. 
 
 * Page 5, line 4 For " left to right " read " right to left." 
 p. 15 1. 9 For " Chapter VII." read " Chapter XI." 
 *p. 27 1. IS For " Appendix B " read " Appendix D." 
 p. 33 Art. 19 This experiment is much better performed with 
 
 the apparatus of Art. 73. 
 
 *p. 39 11. 36, 37, 40, and p. 40, U. 5, 8, &G.For " wave -sur- 
 face" read " wave -front." 
 
 *p. 46 1. SFor "Appendix B " read " Appendix D." 
 p. 51 1. 23 For " 1-26 " read. 11 about 1-32." 
 p. 54 L 21 For "7" read " the adiabatic elasticity." 
 p. 75 1. 25 For " regulation " read " regular." 
 p. 93 1. 22 For " four " read "three." 
 *p. 104 1. 10 For "harmonic" read "harmonious." 
 *p. 128 1. 9 For " line 2 " read "line 11." 
 p. 128 1. 29 For "ventral segments," read "ventral segments 
 
 or loops." 
 p. 146 Exchange the numbers of Figs. 50 and 51. 
 
CHAPTER I. 
 
 VIBRATORY MOTION. 
 
 1. Cause of Sound. In every case in which the sensation 
 of sound can be traced to an external cause, we find that the 
 cause is something in a state of vibration. This vibration is 
 often sufficiently great to be evident to the eye ; not that the 
 movements can be actually watched, for they are always too 
 rapid for that, but they often produce the blurred indistinct- 
 ness of outline, which we know from experience to result 
 from rapid movement. This is well seen in the string of a 
 harp, or along the edge of a large bell, when these are pro- 
 ducing loud sounds. When the vibration has become too 
 pmall to be visible, it can still be detected by touching the 
 string or bell with the tip of. the finger, which is able to 
 perceive, as separate and successive, movements which follow 
 one another much too rapidly for the eye to distinguish. If 
 sound is still heard when even touch detects no vibration, we 
 find that a suspended pith ball, held so as to hang just in 
 contact with the sounding body, is driven away every time 
 it touches the surface, and we naturally attribute this to 
 blows or taps given to the ball by the sounding body, and 
 infer that there is still vibration. 
 
 The vibration of the string or bell cannot, of course, be 
 the immediate cause of the sensation of sound. It is a fact 
 so familiar as to seem an axiom that " nothing acts except 
 where it is " ; the immediate cause of the sensation must be 
 something in contact with the sense organ, the ear. If we 
 stretch a thin membrane, such as goldbeater's skin, on a ver- 
 tical ring of metal, so as to form a sort of tambourine, and 
 hang a pith ball just in contact with the membrane, it will 
 be found that the pith ball is driven away whenever the 
 arrangement, which is called an acoustic pendulum, is in any 
 
 SB. B 
 
place where sound can be heard ; by special devices for 
 detecting the movement of the ; ith ball, it is found that this 
 is true even for the faintest audible sounds. We infer that 
 sound is only heard in places where a stretched membrane 
 would be in vibration. 
 
 The vibration of the membrane depends on some special 
 condition of the air surrounding it, for if the " acoustic pen- 
 dulum" is under the exhausted receiver of an air pump, it 
 does not vibrate even when loud sounds are audible every- 
 where round the receiver. And this condition of the air not 
 only requires for its production a vibrating body ; the vibrat- 
 ing body must be in contact with the air. Thus, if one of 
 the deep-toned clocks, which strike on a coiled wire, is sus- 
 pended by pieces of cotton inside the exhausted air-pump 
 receiver, no sound is heard when the hammer strikes the 
 wire, although the wire visibly vibrates. If the clock, 
 instead of being suspended, stands on the air-pump plate, it 
 will be heard to strike, but in this case it can be shown that 
 the plate itself is vibrating, and the plate is in contact with 
 the air. 
 
 A vibrating body does not produce this condition in the 
 whole of the air at the same moment. If we stand near a 
 large clock bell, we cannot detect that there is any interval 
 between the fall of the hammer and the perception of the 
 sound, but if we are two hundred yards from the bell the 
 interval is very noticeable. 
 
 Though movements of the air cannot be seen or felt, like 
 those of solid bodies, it seems obvious that a condition of the 
 air which is produced by the vibrations of a body in contact 
 with it, which spreads from the air near the vibrating body 
 to that at a distance, and which enables the air to set in 
 vibration light membranes which expose a large surface to it, 
 must be a vibratory motion of the air itself, and this becomes 
 a certainty when we find that the velocity with which the 
 condition travels is exactly that with which it can be calcu- 
 lated that a vibratory movement would spread from one part 
 of the air to another. 
 
 This condition of vibratory movement spreads not only 
 through air, but through all known kinds of molecular 
 matter, and air is not in all cases the substance whose move- 
 
VIBRATORY MOTION. 3 
 
 ment produces the sensation of sound. Thus persons whose 
 ears are so defective in structure that vibrating air does not 
 affect them, can often hear a watch held between their teeth, 
 and in this case it is through the bones, not through air, that 
 the vibration spreads. 
 
 The sensation of sound is never produced by vibrations of 
 the ether of space, nor does the vibratory condition which 
 causes sound ever spread from one material substance to 
 another through a region which contains ether only ; this has 
 been shown in the two air-pump experiments just described. 
 It is also proved by the fact that the tremendous explosions 
 which constantly take place on the sun are quite inaudible 
 on the earth. 
 
 Though vibratory movements which are very slow, like that 
 of a pendulum, or extremely rapid, like those of the air in 
 some very small whistles, do not produce any sensation of sound, 
 yet, as such vibrations do not differ in their physical nature 
 from those which are audible, it is convenient to include all 
 in the same department of physics. The physical study of 
 sound then includes all kinds of vibration of molecular matter, 
 but not the vibrations of the ether. 
 
 2. Meaning of " Vibration." Any point is said to vibrate 
 when it goes through the same, or nearly the same, series of 
 movements at regular or nearly regular intervals. In popular 
 language the term is confined to motion backwards and 
 forwards along the same straight line, but the scientific use 
 of the term is not so limited, and a point moving in a circle 
 or a figure of 8 is also said to be vibrating. The time occu- 
 pied by the complete series of movements which constitute a 
 vibration, from the moment when the point passes any given 
 position, to the moment when it passes the same position, in 
 the same direction, to go through the same series of move- 
 ments again, is called the period of the vibration, and is 
 usually denoted by t. The number of such complete periods 
 in one second is called the frequency of the vibration, and is 
 
 usually denoted by n. Evidently n is always equal to -. 
 
 t 
 
 When a point moves backwards and forwards in a straight 
 line, its vibrations are called rectilinear. In the case of 
 
4 SOUND. 
 
 rectilinear vibrations, most French, and some English, writers, 
 count the time taken in moving from one end of the line to 
 the other as the period of the vibration ; we shall follow the 
 usual English custom, and consider it only half a period. 
 
 3. Curves of Displacement. A point may vihrate in a 
 straight line in a great variety of ways ; for instance, it may 
 move rapidly in one direction and slowly in the other, like 
 the piston of a Cornish pumping-engine ; or quickly at one 
 end of its path and slowly at the other, like an elastic hall 
 dropped on the ground and rebounding again and again ; or 
 quickly in the middle of its path and slowly at the ends, like 
 a pendulum-bob, or a ball dancing up and down at the end of 
 a piece of elastic ; and so on. These differences constitute 
 the character of a vibration. In all these cases, if we marked 
 down on paper the actual track of the vibrating body, that 
 track would be simply a straight line, and there would be 
 nothing to indicate the differences in the characters of 
 the movements. In order to show these differences graphi- 
 cally, the following device is often used : Suppose the body 
 is vibrating vertically ; imagine that a sheet of paper is 
 drawn horizontally at a uniform rate from right to left 
 behind the body, and that the body leaves a trace on the 
 paper. The form of this trace will depend on the relative 
 velocities of the body at different times of its vibration. 
 Thus, a ball dancing up and down at the end of a piece of 
 elastic would leave a trace like Fig. 1 , while a ball dropped 
 
 Fig. 1. Fig. 2. 
 
 on a level surface and rebounding, would mark out Fig. 2. 
 (In each of these cases successive vibrations are drawn 
 exactly similar in extent and period ; the gradual loss of 
 velocity due to imperfect elasticity is neglected.) Suppose in 
 this last instance the ball moved up and down a fixed line 
 
VIBRATORY MOTION. 5 
 
 YZ) Fig. 3,* Z being the point of the level surface against 
 which it rebounded, and let the ball have left a trace, shown 
 by the dotted curve, on a sheet of paper drawn uniformly from 
 deft to right behind it. A will be a point of the paper which 
 was behind Z when the ball was at F, and B will be a point 
 on the paper which was behind Z when the ball was at Y 
 
 A w -I i B 
 Fig. 3. 
 
 again after an interval , which is the period of the vibration. 
 If we divide AB into say six equal parts, and draw on the 
 paper verticals through the points of division, these verticals 
 
 t 2t 
 are lines which coincided with YZ at instants , , &c. 
 
 after the instant when A passed behind Z, and the distance 
 up any vertical from AB to the curve is the distance that 
 the ball was from Z at the moment when that vertical co- 
 incided with YZ. So that, instead of supposing the curve 
 actually traced by the moving body, we may construct it 
 geometrically in this way. Mark off any length AB on a 
 horizontal line, and divide it into, say, six equal parts. At 
 the first point of division put up a perpendicular called an 
 ordinate, equal in height to the distance of the moving body 
 from some fixed point, when it has been moving for ^ of its 
 period, and so on. The curve can then be drawn through 
 the ends of these ordinates. Or, if the movements of the 
 
 * In all curves in this book which represent the condition of the 
 same body at different times, a point to the left of another represents 
 the condition at an earlier moment. 
 
6 SOUND. 
 
 point are inconveniently long or short for such a diagram, 
 the ordinates may be made longer or shorter than the actual 
 distances in any convenient ratio ; the curve will still indicate 
 the character, though no longer the actual extent, of 
 the vibratory movement. Such a "curve, in which equal 
 distances along a horizontal line represent equal intervals of 
 time, while the distance of the curve from the line at each 
 point is proportional to the distance of the moving point from 
 a fixed one at the corresponding instant of time, is called a 
 curve of displacement. Practical methods of determining the 
 frequencies and displacement curves of points on vibrating 
 bodies will be described in Chapter XII. 
 
 4. Curve of Velocities. Another way of distinguishing 
 between rectilinear vibrations of different characters, which, 
 though not quite so simple, is of much greater value in the 
 study of sound, is to draw a curve whose ordinate at each point 
 is proportional to the velocity instead of the displacement of the 
 vibrating body at the moment represented by the position of 
 the ordinate along the horizontal line ; velocities in one direc- 
 tion being represented by heights above that line, velocities 
 n the opposite direction by distances below it. As an instance, 
 we will take again the case of a ball falling and rebounding 
 vertically, whose curve of displacements was given in Fig. 2, 
 and construct its curve of velocities. Mark off along a 
 
 horizontal line any length, such as AB (Pig. 4), to represent 
 the time of a complete vibration, and let A represent the 
 moment when the ball is at its highest point and just about 
 to begin falling, while B represents the moment when it 
 has just reached the highest point again. At the moment 
 
*', 
 
 DEPARTMENT OF PHYSICS 
 
 VIBRATORY MOTION. 7 
 
 corresponding to A the ball has no velocity, and the length 
 of the ordinate to the curve at A is zero. After the lapse of 
 \ of a complete period the hall has a downward velocity, 
 represented by FG, and this downward velocity increases 
 uniformly with the time, as shown by the straight portion 
 AD. Just before half the period is completed, at the 
 moment represented by J7", the ball touches the ground, and 
 in the short time represented by HJ, the downward velocity 
 represented by HD is changed into the upward one repre- 
 sented by JE. At the instant represented by J, the ball 
 ceases to touch the ground, and from this moment till the 
 end of the period the upward velocity uniformly diminishes, 
 as shown by the straight line EB. 
 
 The character of a rectilinear vibration is fully defined by 
 either its displacement curve or its velocity curve, and either 
 curve can be easily drawn when the other is given. The 
 advantage of the velocity curve will appear later. 
 
 5. Harmonic Vibration, There is one special kind of 
 rectilinear vibration which is very important in the study 
 of sound. If a point A (Fig. 6) moves to and fro along a 
 straight line CI) in such a way that the line from A to a 
 point JB, which is moving uniformly round a circle, remains 
 parallel to the same direction, then A is said to execute a 
 
 
 s 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 2J_ 
 
 ^X 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 9 \ 
 
 A 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 1 
 
 F 
 
 A 
 
 f\ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 J r "~ - . 
 
 
 H * t 
 
 . c 
 
 \ < 
 
 - 
 
 
 
 \ 
 
 
 
 
 
 
 V\ 
 
 O 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / \ 
 
 7 
 
 
 
 
 
 
 
 
 
 \ 
 
 1^ 
 
 ^ 
 
 / 
 
 \^ 
 
 ^s 
 
 Fig. 5. Fig. 6. 
 
 Harmonic vibration. It is also sometimes called Pendular 
 motion, because any pent of a pendulum moves very nearly 
 in this manner when its swings are short. 
 
 Suppose B is moving uniformly round the circle CGD, 
 
O SOUND. 
 
 and that the point A is at the same time moving up and 
 down the straight line CD, so as always to be on the same 
 level as B, A then vibrates harmonically. If we divide the 
 circle into, say, sixteen equal parts, so that B takes equal 
 times in moving from E to F, F to (9, and so on, we see 
 that A will take equal times in moving from to /, / to ff, 
 &c., so that it will move slowly at the ends, quickly in the 
 middle of its path. 
 
 To draw the displacement curve for this movement, we 
 may mark off: sixteen equal spaces along a horizontal line 
 JTJ, to represent the equal intervals of time occupied by A 
 in passing from to/, / to g, &c., and at these points draw 
 ordinates equal (or, if preferred, proportional) to the 
 distances of 0, /, g, &c., from some fixed point, which may 
 conveniently be the middle point of CD. The distances 
 from of the points below must be considered as of 
 opposite sign to the distances of those above, and the 
 corresponding ordinates drawn in the opposite direction. 
 If we start at the moment when A is passing through on 
 its way upwards, the curve drawn through the ends of the 
 ordinates will be like the one shown in Fig. 5. Such a 
 curve is called a hcvrmonic or sine curve, the latter name 
 being given because, as can easily be shown, the ordinate at 
 each point of HJ is proportional to the sine of the angle 
 which OB makes with OE at the instant represented by 
 that point. 
 
 Any quantity is said to vary harmonically with the time 
 when it changes so as to be at every instant proportional to 
 the distance of a harmonically vibrating point, such as A, 
 from some fixed point in CD ; so that we may have harmonic 
 velocities, pressures, or electric currents. In any case where 
 a quantity varies harmonically, if we put up ordinates at 
 equal distances along a line HJ, and make the ordinates pro- 
 portional to the value of the quantity at equal intervals of 
 time, the curve through the ends of the ordinates is a har- 
 monic or sine curve. 
 
 "We have seen that, in general, the velocity curve of a 
 vibrating point is quite different to its displacement curve. 
 Eut if a point vibrates harmonically, we can easily show (see 
 Appendix A) that its velocity also varies harmonically, so 
 
VIBRATORY MOTION. 9 
 
 that its velocity curve is also a sine curve. Its velocity at 
 any moment is, however, not proportional to its displacement 
 at that moment, but to the displacement which it will have a 
 quarter of a period afterwards. So the curve of velocities is 
 a quarter of a period behind the curve of displacement, as 
 shown in Fig. 7. 
 
 Fig. 7. 
 
 "When a point vibrates harmonically, it may be shown (see 
 Appendix A) that its acceleration is always towards, and pro- 
 portional to its displacement from, its mean position. So 
 that if the force on a material particle, when the particle is 
 displaced from its position of equilibrium, is at every stage of 
 displacement directed towards the position of equilibrium, 
 and proportional to the amount of displacement, the particle 
 will vibrate harmonically. We shall see later that, even 
 when a vibration is not harmonic in character, it may be con- 
 veniently treated as the resultant of two or more harmonic 
 vibrations. It is to this fact that the importance of harmonic 
 vibrations is due. 
 
 The distance from the middle point to either end of a har- 
 monic vibration is called the amplitude of the vibration ; it is 
 evidently one-fourth of the total distance traversed by the 
 moving point in one complete vibration. 
 
 When the force on a particle is proportional to its distance 
 from a given point, the time of vibration about that point is 
 independent of the amplitude. (See Appendix A.) So that 
 when, for instance, a steel rod is fixed in a vice, bent aside, and 
 let go, the period of its vibration does not alter as the vibration 
 itself dies away. Such vibrations are called isochronous. 
 Isochronous vibrations need not be harmonic, but harmonic 
 vibrations are practically always isochronous. 
 
10 SOUND. 
 
 6. Phase. The phase of a harmonically vibrating point 
 usually means the fraction of a whole period of its vibration 
 which has elapsed since it last passed its mean position in 
 the direction which we are counting as positive. Some 
 writers, however, count the time which has elapsed since it 
 started from the positive end of its vibration. TJpwards and 
 to the right are usually counted the positive directions for 
 vertical and horizontal vibrations respectively. If two points, 
 both vibrating harmonically in the same period, are in the 
 same phase at one instant, they are, of course, always in the 
 same phase, and if they are not, the difference of phase 
 between them may be conveniently defined by stating the 
 difference between the times at which they cross their mean 
 positions in the positive direction as a fraction of the whole 
 period of either ; it may also be stated as the corresponding 
 angle of revolution of the uniformly revolving point, either 
 in degrees or in radian measure. Thus, in Pig. 5, if we 
 suppose / and g to be positions at the same instant of two 
 points vibrating harmonically in the same period, and that 
 they are moving in the same direction, their phase -difference 
 
 may be called or -^ or 22 , as we prefer. 
 16 8 
 
 When the two vibrations are of different periods, their 
 phases change at different rates, and there is no constant 
 phase-difference between them, and there will be instants at 
 which the two vibrations are in the same phase. The 
 " phase -difference " is, in this case, usually taken to mean 
 the difference between the times at which the points cross 
 their mean positions in the positive direction, expressed as a 
 fraction of the greatest common measure of their periods, 
 but other notations are also used. 
 
 7. Composition of Harmonic Vibrations. A point may 
 execute two or more harmonic vibrations at the same time. 
 This does not mean, as it appears to, that a point A can be 
 moving, at any one instant, with respect to a given point of 
 reference such as the earth, with more than one velocity or 
 in more than one direction. A conventional meaning is 
 attached to the phrase. Harmonic motion, like all motion, 
 is, of course, relative ; thus the piston of the engine of a 
 
VIBRATORY MOTION. 11 
 
 steamer which travels up and down the cylinder in such a 
 way that its distance from the end varies harmonically, is 
 vibrating harmonically with respect to the cylinder, though 
 if the ship is rolling at the same time the movements of the 
 piston with respect to the earth may be very complex. If, 
 then, a point A is moving harmonically with respect to a 
 body B, while B is moving harmonically with respect to the 
 earth, the movement of A with respect to the earth is said to 
 be compounded, or the resultant, or the sum, of the two 
 harmonic motions, or, more shortly, A is said to be executing 
 both movements at once. This is not very accurate, but it 
 is not misleading, since there is no other sense in which a 
 point can be moving in two ways at once. 
 
 When a point is executing two rectilinear harmonic vibra- 
 tions at once, its motion need not be either rectilinear or 
 harmonic ; if the periods are incommensurable, it is not even 
 a vibration. Of this we shall have many instances. 
 
 8. Lissajous' Figures. An important case occurs when the 
 two harmonic motions are at right angles to one another ; 
 for instance, let A vibrate horizontally with respect to a 
 point -Z?, while B vibrates vertically with respect to the 
 paper. If the periods of the two vibrations are incommen- 
 surable, the motion of A will never exactly repeat itself ; but 
 if m horizontal vibrations occur in exactly the same time as 
 n vertical ones, then at the end of this period the movements 
 of A, both horizontal and vertical, will exactly repeat 
 themselves, and the point will describe the same closed curve 
 over and over again. The form of this curve will depend 
 upon, and may therefore be used to determine, the amplitudes, 
 relative periods, and phase-difference of the two vibrations. 
 
 When these are given, the curve may easily be constructed by a 
 method which will be best understood from an example. Suppose 
 we require to know what curve will be traced by a point which 
 vibrates vertically and horizontally at the same time, the periods 
 of the vertical and horizontal vibration being as 4 : 3, and their 
 amplitudes 1 and inch respectively, and the two vibrations 
 being in the same phase. Imagine that there is a sheet of glass 
 lying on the page, with a dot on it which we will call A, and 
 a horizontal line through A. If a point B moves harmonically right 
 
12 
 
 SOUND. 
 
 and left along this line, with an amplitude of } inch on each side 
 of A, and at the same time the plate of glass moves harmonically up 
 and down the page with an amplitude of 1 inch, then B is said to 
 be vibrating both vertically and horizontally with respect to the 
 paper, and the line on the paper over which B passes is the curve 
 required. Suppose that L (Fig. 8) is the point on the paper which is 
 under the dot A at the moment when A and B are both at the 
 
 z\ 
 
 middle points of their vibrations, 
 and therefore coincide. MN is the 
 line over which A moves up and 
 down, and if we draw a circle D of 
 1 inch radius on the paper with 
 the centre on the same level as L, 
 
 \ I / / and divide this circle into 16 equal 
 
 parts, and draw horizontal lines 
 to MN through the points of di- 
 vision, then LP is the distance A 
 will move in T V of its period, and 
 so on. If we draw on the glass 
 another circle E of f inch radius, with its centre directly below A t 
 and divide it into 12 equal parts, and draw vertical lines such as 
 RS on the glass through the points of division, then, as B starts 
 moving to the right from A on the glass at the same moment that 
 
VIBRATORY MOTION. 13 
 
 A starts upwards from X, B will reach the lino E8 on the glass at 
 the same moment that A is above the point P on the paper, for T ^ of 
 A's period is the same as y 1 ^ of JB's. B is therefore at this moment 
 above the point T on the paper, for B is always on the same level as 
 A; and LT is the line on the paper over which B has passed. 
 Similarly, B crosses each of the rectangles diagonally, so that it 
 moves over a curve on the paper like the one shown in the figure. 
 
 In Fig. 8 the complete lines are supposed drawn on the paper, 
 and to remain stationary, while the system of dotted lines is sup- 
 posed drawn on the glass and to be moving up and down with it. 
 
 If there is a phase-difference, say ^, between the vibrations, and 
 the horizontal vibration is in advance, then (since -J- of the Gr.C.M. 
 of the periods represented by 3 and 4 is of the shorter or T ^ of 
 the longer) at the moment when A is over L, B will have arrived 
 at a vertical line on the glass passing through Z, to a point of 
 the circumference of E from N, and will be where this line cuts 
 L K. If we divide E into 12 parts, beginning at Z, and number 
 these Z { , Z 2 , &c., then B will pass through the point on PJ, which is 
 exactly above Z b the point on QH which is exactly above Z 2 , and so 
 on, and a different curve will be traversed. 
 
 The circles might have been divided into any other numbers of 
 parts in the ratio 4 : 3, instead of into 16 and 12. 
 
 These curves are known as Lissajous 1 figures. Their forms, 
 for some of the most important ratios and phase-differences, 
 are shown in Fig. 9, where the numbers at the left hand 
 show the ratio of the period of the horizontal to that of the 
 vertical vibration. As the figures stand on the page, they 
 correspond to the phase -differences from to as indicated 
 above them ; if the page is inverted, the same figures will 
 correspond to the phase-differences from to 1, as shown by 
 the fractions which will then he above them. 
 
 If n horizontal vibrations take very nearly but not exactly 
 the same time as m vertical ones, the curve traced will be 
 very nearly one corresponding to the ratio m : n, but when 
 m vertical vibrations are finished, rather more or less than 
 exactly n horizontal ones have been performed, and the 
 phase-difference is not exactly the same as at starting. A 
 slightly different curve, corresponding to a different phase- 
 relation, is therefore traced in the next cycle, and as the 
 phase-difference alters during each repetition of the figure, 
 a series of curves corresponding to gradually increasing or 
 diminishing phase-differences are traced in turn. 
 
14 
 
 SOUND. 
 
 If the horizontal and vertical vibrations are nearly equal 
 in period and are at first in the s^rne phase, the curve traced 
 in successive vibrations will gradually change, assuming in 
 turn the forms of the first line of Fig. 9 from left to right 
 
 3:4 
 
 and back again, and returning to its original form when one 
 movement has been repeated once more than the other. This 
 gives a very accurate method of determining the difference 
 between the frequencies of two vibrations. The practical 
 details are given in Chapter XII. 
 
 9. Methods of producing Lissajous' Figures. Though 
 the connection with the subject is rather remote, it is usual 
 in text-books on sound to mention several mechanical and 
 optical devices for showing such curves as we have just 
 described. A simple device is TPheatstone 1 s Kaleidophone, 
 a good form of which consists of a straight strip of steel, 
 such as a piece of clock spring, twisted at the middle, BO 
 that the planes of the two halves are at right angles 
 
VIBRATOS Y MOTION. 16 
 
 (Fig. 10). If the lower end of the rod is fixed in a vice, 
 the upper end can vibrate either parallel to the jaws, in 
 
 Fig. 10. 
 
 which case the lower part remains straight, and the time 
 of vibration depends on the length and stiffness of the upper 
 half ; or at right angles to the jaws, and in this case only 
 the lower part bends, and the time of vibration depends on 
 the length of that part. 
 
 The vibrations of a spring fixed at one end soon 
 harmonic, as will be explained in Chapter "5H:, and 
 adjusting the position of the rod in the vice the periods 
 the two vibrations may be made to have any desired ratio. If 
 the ratio is made a simple one, and the top of the rod dis- 
 placed obliquely to the plane of the jaws, the extremity will 
 describe the curve corresponding to the ratio. If a globule 
 of mercury is attached by a little grease to the tip of the 
 rod to form a brilliant point, the whole curve traced will be 
 visible at once, by persistence of vision, if the vibrations of 
 the rod are rapid enough. 
 
 If two tuning-forks are fixed so that the vibrations 
 of one take place in a vertical, and those of the other 
 
16 
 
 SOUND. 
 
 in a horizontal, plane (Fig. 11), and if a small mirror is fixed 
 either on the side or the end of one prong of each fork, so 
 that, when the prong vibrates, the surface of the mirror is 
 tilted through a corresponding angle, and if a convergent 
 beam of light falls on one of the mirrors, is reflected from it 
 to the other mirror, and from that to a screen, then, if only 
 the first fork vibrates, the spot of light on the screen will 
 vibrate harmonically up and down ; if only the second fork 
 vibrates, the spot will vibrate harmonically right and left ; 
 if both vibrate together, the movement of the spot will be 
 the resultant of these two vibrations, which, if their periods 
 
 Fig. 11. 
 
 have a simple ratio, will be the corresponding Lissajous' 
 figure. It was in this way that Lissajous produced the 
 figures. 
 
 Instead of projecting the spot on a screen, we may watch, 
 with the eye or a telescope, the image of a bright point 
 formed by successive reflection in both mirrors ; in this case 
 the figure will be visible even when the vibrations of the 
 forks are very minute. 
 
 In these two methods, the track described by the spot is 
 
VIBRATORY MOTION. 
 
 17 
 
 only visible by persistence of vision, and the vibrations must 
 therefore be rapid. Blackburn's Pendulum is a contrivance 
 for leaving a permanent trace, so that the vibrations may be 
 much slower. A funnel, F, filled with sand (Fig. 12), is 
 suspended from one of three strings, knotted together at (7, 
 
 Fig. 12. 
 
 and the other two strings are fastened to two points, A and 7?, 
 on the same level. If the funnel is set swinging in the 
 vertical plane which contains A and B, only the part CF 
 swings, and the period is that of a pendulum of length CF\ 
 if the movement of F is perpendicular to the plane through 
 AB, the whole system of strings swings together about the 
 line AB, in the period of a pendulum of length DF. By 
 adjusting the length of CF, any desired ratio can be given to 
 these periods, and if this ratio is made a simple one, and F is 
 then displaced obliquely to the plane of AB, it will describe 
 one of Lissajous' figures. If the sand runs out slowly, it will 
 leave a sand-trace of the figure on a table placed below. 
 
 Tisley'sHar monograph is a much better device. It consists of 
 two heavy pendulums, which swing in planes at right angles 
 to each other, and whose rods extend a few inches above 
 their centres of suspension. In one form, shown in Pig. 13, 
 two horizontal strips of wood, each of which rests at one end 
 
 so. c 
 
18 
 
 SOUND. 
 
 on the pointed top of one of the pendulum rods, are hinged 
 together at the other by a glass pen, which passes through 
 
 Fig. 13. 
 
 both strips. Either pendulum, swinging without the other, 
 would give the pen a very nearly harmonic rectilinear motion ; 
 when they swing together, the pen traces one of Lissajous' 
 figures. 
 
 In all these methods, the periods of vibration in the two 
 directions are independent, and, therefore, they can never be 
 adjusted quite exactly to any simple ratio ; the curve de- 
 scribed, therefore, always changes slowly. It is easy, by 
 means of rods connected to toothed- wheels working in one 
 another, to give to a pencil harmonic motions at right angles, 
 which have exactly a simple ratio, and so to trace always 
 the same curve, but it is not usual to describe such methods 
 in works on Sound. 
 
 If the figure compounded of two vibrations at right angles 
 is given, the ratio of the periods of the vibrations can at once 
 be found by inspection; it is the ratio of the number of times 
 any vertical line is cut by the curve to the number of times 
 any horizontal line is cut by the curve. 
 
 10. Harmonic Vibrations in the same line. Another im- 
 portant case is that in which the same point executes at the 
 
VIBRATORY MOTION. 
 
 19 
 
 same time two or more harmonic vibrations in the same line. 
 Tor instance, suppose that vibrates harmonically, with 
 respect to the paper, along XY, while A vibrates harmoni- 
 cally, with respect to -Z?, in the same direction. The motion 
 of A, with respect to the paper, will be rectilinear, and its 
 character can therefore only be shown by some device such 
 as a displacement curve. If is the middle point of the 
 vibration of B (or any other point fixed with respect to the 
 paper), then the distances of B from at the successive 
 instants represented by 1, 2, &c., will be equal to the 
 ordinates at those points of some harmonic curve, such as 
 CD (Fig. 15), the length of a double bend of which repre- 
 
 Fig. 14. 
 
 sents the period of .Z?'s vibration. And the distances ^f 
 A from B at the same instants will be equal to the ordinates 
 of another harmonic curve, such as FGH, the length of a 
 complete double undulation of which represents the period 
 of A's vibration. As the distance of A from is always 
 the algebraic sum of the distance of B from 0, and that 
 of A from B, the displacement curve of A's vibration 
 with respect to the paper will be found by drawing at 
 each point ordinates equal to the algebraic sum of the ordi- 
 nates to CDE and FGH at the same point, and drawing the 
 curve through the ends of these ordinates. The dotted 
 curve in Fig. 15 shows the displacement curve of the vibra- 
 tion compounded of the two harmonic vibrations whose dis- 
 placement curves are CDE and FGH. 
 
 It is evident at once that this is not a harmonic curve, so 
 that a point which executes two harmonic vibrations at once 
 
20 SOUND. 
 
 may be vibrating non-harm onically. The importance of this 
 fact will appear in Chapter Y. 
 
 A spot of light may be maae to execute on a screen a 
 vibration which is the resultant of two or more harmonic 
 vibrations in the same direction, by successively reflecting a 
 beam of light from mirrors carried by tuning-forks in exactly 
 the same way as described for Lissajous' figures, except that 
 all the forks must vibrate so as to displace the spot in the 
 same direction, e.g., all in vertical planes. The vibrating 
 spot will, however, simply appear as a straight line of light ; 
 there will be nothing to show the character of its movement. 
 The displacement curve corresponding to the vibration may 
 be demonstrated by placing between the last fork and the 
 screen a mirror rotating continuously on a vertical axis ; the 
 beam, being reflected from this after reflection by all the 
 forks, will sweep round and round the room, and, vibrating 
 vertically while it traverses the screen horizontally, will 
 trace on the screen the displacement curve of its vibration, 
 so rapidly that the whole curve will be visible at once. 
 
 There are many mechanical methods of tracing the dis- 
 placement curves of such resultant non-harmonic vibrations, 
 but it is not usual to describe them in elementary text- 
 books. 
 
CHAPTER II. 
 
 PBOGKESSIVE UNDULATION. 
 
 11. Transmission of Condensations and Rarefactions 
 through Air. Imagine a flat plate, held straight in front of 
 you, vertically, with its edge towards you, and then suppose 
 it rather suddenly displaced a very short distance towards 
 your left. As the plate moves, the layer of air immediately 
 to the left of it will, of course, be driven to the left with a 
 velocity equal to that of the plate, and the air in that layer 
 will also be compressed, the increase in pressure and in density 
 being, in the early stages of the movement, proportional to 
 the velocity of the plate (Appendix B). But though the 
 velocity and the additional density of this layer of air both 
 depend on the velocity with which the plate is displaced, 
 these conditions are rapidly transferred from the layer of air 
 in contact with the plate to a layer a little further off, and 
 from that to the next, and so on, with a velocity which - is 
 practically independent of the velocity of the plate, and 
 depends only on the properties of the air. In just the same 
 way, a layer of air immediately to the right of the plate will 
 move with it to the left, and at the same time its pressure 
 and density will be diminished ; and these conditions of 
 motion to the left, reduced pressure, and reduced density, 
 are transferred from that layer to other layers of air more 
 and more distant from the plate with the same velocity as 
 the compressed condition on the other side. 
 
 The simplest way of explaining this process is to suppose 
 the air divided, by purely imaginary surfaces, into small 
 blocks, or particles, in contact with one another ; these 
 particles being, not molecules, but portions of air, each 
 containing many millions of molecules at least. Such a 
 particle of air behaves, as regards compression, very much 
 
22 SOUND. 
 
 like an elastic solid ; like a sponge, or ball of wool, to use 
 Boyle's illustration ; and the air of a room transmits displace- 
 ments and condensations very much as a mass of balls of wool 
 in contact might be imagined to do. The way in which a 
 series of elastic bodies transmits a condensed or compressed 
 condition to a distance, while the bodies themselves are only 
 very slightly displaced, is well illustrated by a row of railway 
 carriages, standing with their buffers in contact, when one of 
 the carriages is pushed. "We will call the carriages in order, 
 from right to left, A^ B, (7, &c. (Fig. 16, line 1), and suppose 
 that the engine suddenly moves a short distance, say a foot, 
 towards A (2). The buffer-springs of A are driven in, so 
 that the carriage (if we measure to the ends of the buffers) is 
 
 * 
 
 Fig. 16. 
 
 shortened, or, as we may call it, condensed, and the force due 
 to the compressed springs makes A move with increasing 
 velocity towards B. When it reaches the half-way point 
 between the engine and JB, the pressures at the two ends 
 of A will be equal (3), but, owing to its inertia, it will 
 not stop at that point, but will continue to move to 
 the left till the increasing pressure between it and B 
 brings it to rest. This will happen when A has moved 
 just one foot, so that the distance between it and the 
 engine is the same as at first (4). Meanwhile, B has 
 begun to move to the left, and exactly repeats the move- 
 ments of A, and when it has been displaced one foot in its 
 turn, it stops at its original distance from A (5), and the 
 buffers between A and B cease to extend. The whole of A 
 
PROGRESSIVE UNDULATION. 23 
 
 is then at rest, including the buffers, and it has its original 
 length. C repeats the movements of .#, just as B repeated 
 those of A, and so a condition of condensation and movement 
 is transferred from carriage to carriage all along the train.* 
 
 In this process several points are noteworthy. First, that 
 though a condition, that of compression or condensation of 
 the carriages, travelled continuously along the train from 
 right to left, yet each carriage moved only a short distance, 
 and then stopped, so that it is a condition, and not any mate- 
 rial substance, which travelled along the train. Secondly, 
 not only this transfer of a condition, but the existence of the 
 condition, depends on the fact that each carriage repeats the 
 movement of one next it a little later ; if they all moved 
 together there would be no condensed condition anywhere, 
 and the direction in which the condition travels is from the 
 earlier-moving to the later-moving carriages. Third, the 
 condensed condition is to be found always, and only, in car- 
 riages which are moving ; when any carriage comes com- 
 pletely to rest (buffers included) it is exactly its original 
 length. Fourth, the condition may travel much faster than 
 the carriages themselves move even at their quickest ; in our 
 example, the condensed condition is transferred from one 
 carriage to another in the time in which the carriage itself 
 has moved only a foot. In all these particulars the trans- 
 mission of a condensation through air or any elastic substance 
 resembles that through a row of railway carriages. 
 
 "We shall find, as we proceed, that many different condi- 
 tions can be transferred besides that of condensation, and the 
 four statements just given are true of all of them. In the 
 case described, we may notice also that the movement of the 
 carriages themselves was in the same direction as the move- 
 ment of the condition, but this is only true when it is a 
 condition of condensation which is transferred. 
 
 If instead of a momentary impulse the engine gives 
 to A SL push lasting for an appreciable time, several carriages 
 will have begun to move before A comes to rest, and as the 
 
 * In this description the movement of the carriages has been 
 slightly modified to make it illustrate more exactly the movements 
 of the air. The carriages would not move just as described. 
 
24 SOUND. 
 
 condition travels, there will always be several carriages 
 moving at once. But it will still be true that each carriage 
 moves exactly like the one n^xt nearer the engine, but 
 arrives at each stage of its movement a little later ; and also 
 that each carriage is compressed as long as it is moving, and 
 recovers its original length as it comes to rest. The move- 
 ment may be considered as due to a succession of momentary 
 impulses given to A one after the other, and each passed on 
 in turn. 
 
 In the case of a row of carriages, each communicates its 
 movement to one of equal mass, and the movement of an 
 elastic substance does not correspond exactly to that of the 
 carriages except when the portions which are successively in 
 motion are equal. This is not the case when a plate is 
 waved in free air, so it will be simpler first to consider a 
 column of air confined in a tube. Let AB, Fig. 17, be a 
 long tube full of air, and D a piston. We may imagine the 
 
 D B 
 
 Fig. 17. 
 
 air in AB divided by imaginary planes, transverse to the 
 tube, into equal discs or layers, and we will suppose these 
 planes to move with the air, if it moves, so that the air 
 between the same two planes is always the same air. If D 
 is displaced a short distance to the left, the first of these 
 layers, that next to D, becomes compressed and moves to the 
 left, and then the next, and so on, just like the railway 
 carriages. As with the carriages, the movement may have 
 extended to a number of layers before D comes to rest, so 
 that a number are moving at once ; but, in any case, each 
 moves exactly like the one next nearer to Z>, but begins to 
 move, and reaches each stage of its movement, a little later. 
 As with the carriages, also, every layer is condensed while it 
 is moving, and comes to rest as it is restored to its original 
 length. The only differences to be noticed (and these are 
 only apparent) are, first, that the air-discs become condensed 
 
PROGRESSIVE UNDULATION. 25 
 
 all through, instead of shortening only at the ends like the 
 carriages, and, secondly, that, however thin we make our 
 imaginary layers, each will always be a little later in its 
 movements than the one next nearer D, so that no finite 
 length of the air-column begins to move absolutely at the 
 same instant, as a whole railway carriage appears to do. 
 
 If we suppose the carriages A,B, (7, &c. all held compressed 
 between an engine to the right of A and the end of a siding 
 to the left of Z, then, if the engine moved one foot away from 
 A, allowing the buffers at that end of A to expand, A would 
 be, for the moment, longer than the other carriages, or rarefied, 
 as we may call it, and would begin to move towards the 
 engine. As this would diminish the force on the end of B 
 next A) B would follow and become rarefied, while A 
 would stop when it reached its original distance from the 
 engine, and would have been compressed to its original 
 length again when B stopped after moving one foot in its 
 turn. In this way a condition of rarefaction passes from A 
 to Z, while each carriage in turn moves a foot in the direction 
 from Z towards A. In every travelling rarefaction the move- 
 ment of the matter is in the opposite direction to the movement 
 of the rarefied condition. 
 
 Air is always in a compressed condition, for there is no 
 limit to the extent to which it expands if pressure is removed ; 
 and particles of air move like compressed railway carriages. 
 If the piston D was displaced to the right, first the air in the 
 layer just to the left of .Z), and then that in layer after layer 
 further to the left, would become rarefied and move to the 
 right, each in turn coming to rest at the same moment at 
 which it was restored to its original thickness. The air to 
 the right of D is of course condensed by the movement of D 
 to the right, and a condensation travels away on that side, 
 but for the present we consider only the air to the left of D. 
 
 "Whether we move D to the right or to the left, then, the 
 movement is repeated, first, by the air in contact with D, 
 and then, layer after layer, by the air further and further off, 
 so that the condition of being in motion travels away from 
 D in either case. "When D moves ,to wards the air on the 
 side we are considering, it condenses the air, and each layer 
 moves in turn away from -Z), and is condensed while it is 
 
26 SOUND. 
 
 moving; when D moves away from the air on that side, it 
 rarefies the air, and each layer (beginning with the one close 
 to If) moves in turn towards Z), and is rarefied while it is 
 moving. 
 
 If we move D backwards and forwards, in any manner, 
 exactly the same movement is performed by each layer, but 
 later and later the further the layer is from .Z), so that D and 
 layers near it may have begun to move one way, while 
 more distant layers are still moving the other. All the layers 
 which are moving away from D are condensed, and all those 
 which are moving towards D are rarefied. 
 
 12. Progressive Undulation. This process, in which a 
 condensed or rarefied condition is transmitted through the 
 carriages of a train, or air in a tube, is an instance of what 
 is called progressive undulation, which may be defined as the 
 continuous transference in the same direction of a condition 
 of altered relative position of adjacent particles by similar 
 movements performed successively by consecutive particles. 
 In the instances we have so far considered, whether the 
 condition transmitted is of condensation or rarefaction, the 
 movements of the particles are in the line along which the 
 condition is transferred (though they may be in the opposite 
 direction along that line). When this is the case, the pro- 
 gressive undulation is called longitudinal. In longitudinal 
 progressive undulation the condition that is transferred is 
 always one of altered distance between successive particles, i.e. 
 of condensation or rarefaction ; it is practically the only kind 
 which occurs in air and gases. In Chapter XI. we shall 
 treat of other kinds of progressive undulation in which the 
 condition transmitted is an altered relative direction instead 
 of an altered relative distance. 
 
 Since the sensation of sound is usually caused by the 
 arrival through the air of condensations and rarefactions 
 transmitted by the process of longitudinal progressive undu- 
 lation, this process is often conveniently called sound; 
 travelling condensations and rarefactions are called sound 
 waves, and the velocity with which such waves are trans- 
 mitted in any substance is called the velocity of sound in the 
 substance. The term pulse is often used to denote either a 
 
PHOGRESSIVE UNDULATION. 27 
 
 condensation or a rarefaction, and avoids the frequent repe- 
 tition of both terms. 
 
 13. Undulation and Vibration. It is important to 
 distinguish clearly between undulation and vibration . When 
 the same point repeats the same movements over and over 
 again, the point is vibrating. When different parts of the 
 same substance perform a similar movement one after the 
 other, the substance is undulating. If it happens that the 
 movement which the parts perform in turn is a vibratory 
 movement, then the substance is both vibrating and undu- 
 lating. This is what is usually happening when sound passes 
 through the air. Eut there is no necessary connection 
 between the two ; a substance may undulate without vibrat- 
 ing, like the train after a push, or it may vibrate without 
 undulating if all the parts of it move at the same time, like 
 a pendulum. 
 
 14. Eelation between Velocity and Condensation. It can 
 
 be shown ( Appendix <8^that in any longitudinal progressive 
 undulation there is a constant ratio, depending only on the 
 nature and condition of the substance conveying the undula- 
 tion, between the velocities with which different parts are 
 moving, and the amounts by which the densities of those 
 parts differ from the average density. If we call the differ- 
 ence from the average density the " condensation" or " rare- 
 faction,"*' the velocity at any point is proportional to the 
 condensation or rarefaction there. 
 
 This is, of course, true close 'to D as at any other point, 
 and the velocity of the air close to D is the velocity of D ; 
 the degree of condensation which exists close to D (and 
 therefore the excess of pressure on the side towards which D 
 is moving) is always proportional to the velocity with which 
 D is moving, and does not depend at all on how D has pre- 
 viously moved. 
 
 * Strictly, " condensation " means not the difference between the 
 actual and average densities, but the ratio of this difference to the 
 average density. But these are proportional to one another, so that 
 the velocity with which the air is moving is proportional to the 
 " condensation" in either sense. Rarefaction may be conveniently 
 included in the term condensation, and considered merely as the 
 negative variety. 
 
28 
 
 SOUND. 
 
 15. Wave-form. Suppose that when D has been moving, 
 in any manner, for some time, we draw a horizontal line, 
 OX, Fig. 18 (1), along the tub-3, and from it draw ordinates 
 at different points, proportional to the velocities of the air at 
 
 r c 
 
 (1) 
 
 (2) 
 
 RRRRRR 
 
 Fig. 18. 
 
 those points, at some given moment, drawing the ordinates 
 above OX to represent velocity in the direction in which 
 the waves are moving, and below for velocity in the op- 
 posite direction. Then' the same ordinates are also pro- 
 portional to the degrees of condensation or rarefaction at the 
 same points, ordinates above OX representing condensation. 
 So that the curve in Fig. 18(1) denotes the same condition 
 which is represented by shading in (2), and by the relative 
 heights of the letters (7andJ2 (Condensation and Raref action) 
 in (3). In (3) the relative velocities are represented by the 
 lengths of the arrows over the letters ; this is unnecessary in 
 the case of a progressive undulation, as the velocities are 
 proportional to the condensation or rarefaction, but we shall 
 find it useful in other cases. 
 
 A curve like that in Pig. 18 (1), whose ordinates, at equi- 
 distant points along a horizontal line, are proportional to the 
 amounts of condensation and rarefaction at equidistant 
 points in the undulating air, is said to show the wave-form of 
 the undulation, the wave-form meaning the way in which 
 the density of the air varies from one part to another of the 
 wave. It will, of course, be understood that waves of con- 
 densation and rarefaction in air have not anything which can 
 
PROGRESSIVE UNDULATION. 29 
 
 be termed form in the ordinary sense of the word, as waves 
 of the sea, for instance, have ; the term wave-form is not to 
 be taken literally. 
 
 The difference between the pressure at any point and the 
 average pressure of the air is proportional to the difference 
 between the actual and average density, so that the ordinates 
 of the wave-form represent this pressure-difference as well as 
 the condensation and the velocity. 
 
 In air, condensations and rarefactions are transmitted at 
 about 1,100 feet per second. So that the condition of the air 
 a foot from D now is the condition in which the air close to D 
 was TTO Q- second ago, and so on, and in velocity the air close 
 to D corresponds with D itself. The wave-form is, therefore, 
 a velocity-curve representing the past velocities of D. 
 Thus, from the curve in Fig. 18 (1), we see that D must have 
 moved first with gradually increasing velocity to the left, 
 producing gradually increasing degrees of condensation, 
 which now exist in the layers between E and F ; then have 
 stopped rather suddenly, producing, as it slackened speed, 
 decreasing degrees of condensation now existing between f 
 and G ; then have remained a short time at rest, with the air 
 in contact with it at its average density, a condition which 
 has now travelled to the region between G and H; and, 
 lastly, have moved, and be now moving, with increasing ve- 
 locity to the right, producing the increasing rarefaction in tht> 
 region between H and J. 
 
 All the conditions existing at different parts of the tube 
 travel away from D at 1,100 feet per second; if the wave- 
 form is supposed drawn along with that velocity, its ordinates 
 will at any instant represent the condition of the air in which 
 they are. Thus the part of the wave-form further from D 
 than any point, say F, shows the successive conditions which 
 have existed at F, while the part nearer to D than F shows 
 the conditions which are going to exist at F. The wave-form 
 therefore shows not only the conditions which exist at the 
 same time at different places, but the successive conditions 
 which exist at the same place at different times. 
 
 16. Wave-length. If D vibrates, or repeats the same move- 
 ments at regular intervals, the same series of conditions will be 
 
30 SOUND. 
 
 produced at its surface over and over again, and travel along the 
 tube. Suppose, for instance, that it repeats its whole move- 
 ment in TrV^ second, then the a ; r, 1 foot, 2 feet, &c., from the 
 surface of J) is in the same condition as at the surface, and if 
 we divide the tube into lengths of 1 foot, the air in any of 
 these sections, at any instant, is moving in the same way as 
 the air in any other section. One of these sections is called 
 a complete ivave, and its length is called a wave-length. It is 
 evident that the wave-length is the distance which a given 
 condition travels before D comes round to the same stage of 
 its vibration again ; in other words, it is the velocity of the 
 undulation multiplied by the period of the vibration. If V 
 denotes the velocity of sound and X the wave-length, 
 
 \= Vt. orX = -. 
 
 ft 
 
 17. Harmonic Waves. If D vibrates harmonically, the 
 wave-form of the undulation in the tube will be a harmonic 
 curve. Such waves are called harmonic waves. Two com- 
 plete harmonic waves travelling, like those in Fig. 18, from 
 right to left, are represented in Fig. 19, the heights of 
 
 
 
 cCCCc . 
 
 RR 
 
 RR 
 
 R 
 
 C 
 
 cc 
 
 Cc 
 
 1? 
 
 RR 
 
 R R 
 
 cC 
 
 Fig. 19. 
 
 the letters C and R being proportional to the condensation 
 or rarefaction. The corresponding wave-form is shown 
 below. 
 
 The way in which a condition of condensation or rarefaction 
 can move always in the same direction while the air itself 
 moves backwards and forwards, is illustrated by successive 
 rows, 1, 2, &c., in Fig. 20. In this figure the successive rows 
 represent successive stages of a movement in which each of 
 the vertical lines vibrates harmonically to right and left of a 
 fixed point, from which it never departs more than i of an 
 inch, a vibrating about a point under A, and so on ; but each 
 
DE: ^T OF PHYSICS 
 
 PROGRESSIVE UNDULATION. 31 
 
 line of a row, in order from right to left, is a little behind the 
 line to the right of it in its movement, and the result is, as 
 seen, that though the mean positions of the lines are equi- 
 
 D C B A 
 
 ) j (MM 
 
 (in j (MM! itfn'i U \ i 
 
 ffi'i iiii ( jijjjj! i\ \i * 
 I'M \ \ M I HilTi j II I I U * 
 
 j j i \ i M ijjjjit i I'l \ U.U 
 j j i i j u j j'iYj j I I \ ! Vj j'j 
 
 M M j jj(/m i I i \ I iii/ ' 
 
 j i N I | iijjjli i ! \ I ! (j//! a 
 i I I ()(()(( i i i i j i ((((I 
 
 i i j jjfi'fi \ M i \ jjljii! ' 
 
 i jni/jj i 1 1 1 i ijf/iji i " 
 
 I )((((( i i \ \ i M///(| I \ I2 
 
 h'Tiii i i i i i i iiiiii i i i 
 
 Fig. 20. 
 
 distant, the lines themselves are at any one time closer in 
 some places and further apart in others, and that these con- 
 ditions are found further to the left at each successive stage 
 of the motion. The construction for the harmonic motion of 
 a few lines is given, and it will be seen that in any one row, 
 each line is T l of its period behind the one to the right of it, 
 and that in each row all the movements are T V of a period 
 
32 
 
 SOUND. 
 
 further advanced than in the previous row. The thirteenth 
 row is the same as the first, every line having returned to its 
 original position. 
 
 If any line is traced (by the dotted lines) from one row 
 to another, it will be seen that it is when it is moving to the 
 left (with the conditions) that it is nearest to its neighbours, 
 and when it is moving fastest to the right that it is in the 
 most rarefied region. 
 
 If we imagine an equal mass of air confined between each 
 line and the next, and moving with the lines so that the same 
 air always remains between the same two lines as they move, 
 but alters in density as the distance between the lines changes, 
 this will correspond to the actual movement of the air. 
 
 Fig. 21. 
 
 18. Cr ova's Disc. A good and simple device for illus- 
 trating the movements of the air or other substance when a 
 harmonic progressive undulation is passing through it, is 
 
PROGRESSIVE UNDULATION. 33 
 
 Crova's disc (Fig. 21) ; this the student can easily construct, 
 and should do so if he finds any difficulty in realizing the 
 motion. 
 
 In the middle of a circle of cardboard, A, 8 inches in diameter, 
 draw a small circle of i inch radius, and divide its circumference into 
 twelve equal parts. With each of these points of division in turn as 
 centre, describe a circle in ink, making the radius of the first f- inch, 
 of the second inch, and so on, increasing ^ inch each time. When 
 you get to the last of the centres go on with the one you used first, 
 and so round the circle again, making twenty or more circles in all. 
 In another piece of card cut a slit, EF, 3 inches long and inch wide, 
 and make a pinhole at G in a line with EF and 1 inch from E. Push 
 a pin exactly through the centre of the disc, and put its point 
 through G ; then, holding D in one hand, rotate the disc behind D by 
 means of the pin. 
 
 The portions of the circles visible through the slit are 
 practically straight lines, close in some places, and wider 
 apart in others, like those in Fig. 21 ; as the disc rotates 
 these condensations and rarefactions will he seen to travel 
 continuously along EF, but if the disc is rotated slowly and 
 the motion of any one of the lines carefully followed, it will 
 be seen that it simply moves harmonically to the right and 
 left, and never departs more than \ inch from its mean 
 position. We can also easily verify the fact that where the 
 lines are near together they are moving in the same direction 
 as the condensations and rarefactions, but not so fast, but 
 that where the lines are far apart they are moving in the 
 opposite direction, and that the quickest motion of the lines 
 is to be found where there is the greatest degree of conden- 
 sation and rarefaction. 
 
 (Sujdfa^ O 
 
 19. Spiral Wire. The same motion can also be well shown 
 by means of a long spiral wire. Coil a long brass wire, 
 about 20 gauge, 600 times round a tube or round ruler, 
 from 1 inch to 1 inches in diameter, winding the coils close 
 together; then slip the coil off and fasten one end of it to a hook 
 in the wall, 3 or 4 feet fronf the floor, and stretch the coil out 
 till it is about 12 feet long, holding the other end in your 
 hand ; the spiral will hang in a curve from your hand to the 
 hook, and the turns of wire will be about | inch apart. If 
 you now move your hand rather suddenly about 2 inches 
 
 sp. D 
 
34 SOUND. 
 
 further away from the hook, some of the turns of wire 
 close to your hand will be stretched wider apart ; a visible 
 rarefaction will be produced, ard this rarefied condition will 
 travel all along the spiral away from your hand, each turn 
 of wire successively moving towards your hand. This shows 
 very clearly an important point : the rarefaction originally 
 produced in the part of the spiral close to your hand, does 
 not spread so as to include more and more of the spiral ; the 
 rarefied condition transfers itself along the spiral and dis- 
 appears entirely from the region in which it was produced ; it 
 everywhere occupies the same length of the spiral, and both 
 before and behind it the turns are at their original distance 
 apart. 
 
 If you move your hand towards the hook, instead of 
 away from it, a compressed portion will be formed and will 
 travel along the spiral in just the same way, each turn of 
 wire moving successively away from your hand. 
 
 20. Illustrations of Travelling Rarefaction. The appa- 
 rent paradox, that the movement of a rarefaction in one 
 direction is always due to a movement of the rarefied ma- 
 terial in the opposite direction, is illustrated not only by the 
 figures and experiments just described, but by the movements 
 of many creeping animals. A worm, for instance, first 
 stretches forward and rarefies its front segments, and then 
 pulls the other segments forward in turn ; the stretched 
 condition of the segments begins in front and travels back, 
 though the segments themselves move only forwards. The 
 legs of a centipede show the same thing ; those in front move 
 forward first, and then those further and further back, and 
 waves of rarefaction, or unusually great distance between 
 consecutive legs, run continually along each row of legs, from 
 head to tail. If a slug, crawling on a window, is watched 
 from the other side of the glass, lighter and darker coloured 
 bands will be seen following each other alternately with con- 
 siderable velocity along the foot from head to tail, looking 
 very much like waves on water, but, if we look attentively, 
 we see that no part of the substance of the foot ever moves 
 back, but that the head moves forward with a varying velocity, 
 a.nd that each variation in its velocity is followed by a similar 
 
PROGRESSIVE UNDULATION. 35 
 
 variation in the velocity of each portion in order from head to 
 tail. No part of the substance ever moves forward nearly as 
 fast as the hands run hack. 
 
 21. Energy Transmitted by Progressive Undulation. 
 "We have already pointed out that in progressive undulation 
 there is no continuous transference of anything material, but 
 it would perhaps hardly be correct to say that nothing except 
 a condition travels from one place to another. Usually, pro- 
 gressive undulation is a process by which energy is transferred 
 from one place to another, and it is now becoming quite usual 
 to speak of energy as a thing, rather than a condition. Strictly 
 speaking, progressive undulation need not be accompanied by 
 a transfer of energy ; for instance, the movement of the lines 
 in Crova's disc (Fig. 21) was progressive undulation, though 
 no energy was transferred along the slit ; and so is the move- 
 ment of the stalks of corn in a field when a gust of wind 
 passes across and bends them down in turn ; in this case, also, 
 no energy passes from corn-stalk to corn-stalk. But such 
 eases are quite exceptional. When a succession of particles 
 perform similar movements in turn, it is nearly always because 
 the movement of each causes the movement of the next, and 
 one body can cause the movement of another only by trans- 
 ferring to that other some of its own energy. 
 
 In the case of air, through which waves of condensation 
 and rarefaction are travelling, we can easily see that the 
 wave as a whole the condensed and rarefied portions taken 
 -together contains more energy than an equal volume of 
 undisturbed air. For both the air in the rarefied portion 
 and that in the condensed portion is moving relatively to the 
 earth, and work could be done by stopping this motion. 
 When this energy was exhausted, and all the air at rest, we 
 could still get work done by allowing the air in the condensed 
 portions to expand through a suitable engine into the rarefied 
 portions ;* this work would be done by using part of the heat 
 of the compressed portions. The air would then be in its 
 
 * It is not, of course, meant that this is a practicable experiment ; 
 that it is an imaginable one proves the existence of energy in the 
 wave. 
 
36 SOUND. 
 
 ordinary condition, so that it must, while undulating, have 
 contained more than the usual amount of energy. It is 
 usual to distinguish the part of this energy which depends 
 on the motion of the air relatively to the earth as its kinetic 
 energy, and the part which depends on the differences of 
 pressure in different parts as the potential energy. If we 
 make this distinction, it can he shown that half the energy 
 of a complete wave is kinetic and half potential. The 
 kinetic energy is greatest in the most condensed and most 
 rarefied portions of the wave ; the regions where there is 
 average density are also at rest with respect to the earth, and 
 do not differ in any way from undisturbed air, so that there 
 is no extra energy in them. 
 
 We shall see later that this is rather an artificial way of 
 looking at the energy of a sound-wave in air, since the 
 whole energy is really kinetic energy of the movements 
 of the molecules. Eut it is convenient for the present to 
 regard the air simply as an elastic suhstance, and to divide 
 the extra energy due to its undulation into kinetic and 
 potential portions, as we are obliged to do in the case of 
 solids and liquids where we do not know the real nature of 
 the ''potential" part of the energy. 
 
 It will be seen that though sound is a special mode of 
 transmission of energy, it is not a distinct kind of energy, for 
 part of the energy of a sound-wave is ordinary kinetic energy 
 of moving air, and the other part is heat. 
 
 22. Sound Waves in Free Air. Even in a tube, where the 
 movement of each layer is passed on to another of equal mass, 
 it is not exactly true that the movement as it travels is un- 
 altered ; a little of the energy is converted by friction into 
 heat, and so the amplitude of the vibration is less and less in 
 successive layers, and the character of the movement also 
 changes a little from layer to layer, for a reason to be ex- 
 plained in the next chapter. But these changes are very slow, 
 and in a wide tube there may be very little difference between 
 the vibration when it starts at the source and when it has 
 reached the air a mile away. But if the air is not enclosed 
 in a tube, the result is different. Suppose, for instance, as 
 before, that D, Tig. 22, is a piston vibrating close to the end 
 
PROGKESSIVE UNDULATION. 37 
 
 of the tube AB* but this time we will consider the free air 
 to the right of D. First it must be noticed that in this case a 
 slow vibration of D is practically not transmitted to a distance. 
 The first slight movement of D to the right produces, as in 
 confined air, a slight condensation of the air to the right of 
 
 Fig. 22. 
 
 D, which is transmitted in the way already described, but if 
 the movement of D in that direction continues, the air simply 
 flows away along the surface of D to equalize the pressure, 
 and the condensed condition is destroyed instead of being 
 passed on. It is only when the movement of D is reversed 
 
 * A moving plate surrounded on all sides by free air would pro- 
 duce condensation on one side and rarefaction on the other, at the 
 same time, and these would spread through the same air, and the 
 resulting effect depends on principles to be explained in Chapter IV. 
 The plate is, therefore, described as moving at the end of a tube so 
 that only the effect of the condensations and rarefactions produced 
 on one side of it need be considered. 
 
38 SOUND. 
 
 at very short intervals, so that the air has not time to now 
 away, that waves travel away through the surrounding air ; 
 and the smaller the plate the shorter must he the intervals, 
 so that to produce very long waves in free air a very large 
 moving surface would be necessary. Thus the quite in- 
 visible vibrations of the sides of a tumbler struck with the 
 finger, may transmit to the walls of the room far more energy 
 than the most vigorous waving of a fan, which only disturbs 
 the air in its immediate neighbourhood. Even when D does 
 vibrate rapidly, it communicates to the air much less energy 
 at each vibration than it would do to air confined in a tube. 
 Secondly, when the condensations and rarefactions succeed 
 each other rapidly enough to be transmitted, they spread 
 through the air in all directions, so that when the piston has 
 finished a few vibrations, the air near it will be in the con- 
 dition represented in Fig. 22, alternate spherical shells of air 
 being condensed while the others are rarefied. As in the 
 tube, wherever the air is condensed it is moving away from 
 the piston, and wherever it is rarefied it is moving in towards 
 the piston ; the movement of the air is everywhere at right 
 angles to the shells. The air in each shell keeps transferring 
 its condition to the next shell outside it, so that the con- 
 densed and rarefied conditions alternately spread continuously 
 outwards from shell to shell of air, like the circular wave- 
 crests produced when we throw a stone into a pond. Thirdly, 
 as the movement and energy of one shell of air are always 
 communicated to a larger one, the extent of the movement 
 and the degree of condensation and rarefaction become less 
 and less as we go outwards from the piston, or the wave-form 
 becomes flatter. Fourthly, in the immediate neighbourhood 
 of the piston very complex actions occur, with the result that 
 the movement of the air a little distance from the piston 
 may differ considerably in character, as well as in amplitude, 
 from the previous movements of the air close to the piston, so 
 that the wave-form changes as the wave spreads, and at a 
 little distance is not the same as the velocity-curve of the 
 movement of the piston. Eut the change in character is not 
 very great unless the piston is small in its dimensions com- 
 pared with a wave-length ; and it is almost confined to the 
 first wave-length from the . piston, beyond which the move- 
 
PROGRESSIVE TJNDTJLATION. 39 
 
 ment of successive shells, though always diminishing in 
 extent, changes only very gradually in character. 
 
 The same is, of course, true of the waves produced by any 
 surface vibrating in air, such as a spring, or tuning-fork, or 
 stretched string. 
 
 Though the movement of the air near a vibrating body 
 such as a spring is a vibration similar to that of the spring 
 itself, the causes of these similar movements are very different. 
 When we bend a spring and let it go, it returns of itself to 
 its original position, passes it, returns again, and so on, each 
 movement being a consequence of its previous movement ; 
 it vibrates in a particular way, and with a particular period. 
 But if we displace a part of the air, either in an infinitely 
 long tube, or in open space, it shows not the slightest tendency 
 to return to its original position. The air, therefore, cannot 
 be made to vibrate, like a spring, by simply displacing a part 
 of it and then leaving it free ; each movement that it makes 
 is the result, not of its own previous movements, but of a 
 previous movement of something else, transmitted to it 
 through the air between. The air vibrates only as a pump- 
 handle vibrates; it requires a separate push from outside 
 for every movement. It therefore vibrates just as easily 
 in one way as another, and in one period as another ; in 
 fact, it simply copies the movements of some other vibrating 
 body. 
 
 "We shall see later that in a tube of finite length the air 
 may vibrate merely in consequence of having been displaced, 
 without any other vibrating body to cause each movement. 
 
 At different distances from the piston the air will be in 
 different stages of vibration, but it will be possible to draw 
 round D continuous surfaces, in each of which all the air is 
 moving in the same way at the same moment ; these will be, 
 approximately, spherical surfaces, having D as their centre. 
 Such a continuous surface, at every point of which the air 
 is in the same stage of its vibration at the same moment, 
 is called a wave-surface. In the case of the waves produced 
 by a piston in a tube, the wave-surfaces were planes trans- 
 verse to the tube ; in the open air, at a distance from the 
 source, they are generally roughly spherical, as far as they 
 are complete. The form of these wave-surfaces must not be 
 
40 SOUND. 
 
 confused with, or called, the wave-form, a term which, as 
 explained above, is used in an entirely different sense. 
 
 We may now define a wave-length rather more accurately 
 as the distance, measured at right angles to the wave- 
 surfaces, between two consecutive wave-surfaces in which 
 the air is at the same stage of its movement. If the move- 
 ment is harmonic, the wave-length is the distance between 
 two surfaces in which the air is in the same phase. 
 
 23. Law of Inverse Squares. If, through any point in the 
 air, we suppose an imaginary surface drawn at right angles to 
 the line along which the air is vibrating, then the amount of 
 energy which passes, in each second, through a square centi- 
 metre of this surface round the point, is called the intensity 
 of sound at that point. It is not usual to measure it in abso- 
 lute units, but only to compare the intensities of the same 
 
 I 
 
 Fig. 23. 
 
 sound at different places. If sound is travelling outwards 
 in all directions from a source P, the intensities at two 
 points $ $2 must be inversely as the squares of their dis- 
 tances from P. For, suppose imaginary spherical surfaces drawn 
 through 2 (Fig. 23). Then, if R R 2 are the distances of 
 $!, $ 2 from P, the areas of these spherical surfaces are as J2j 
 to -#2- ^ n ^ ^ ne sound energy which starts from P in (say) a 
 
PBOGRESSIVE UNDULATION. 41 
 
 second, takes a second in passing 8 l 8 l 8 lt and also a second 
 in passing $ 2 S 2 8 y Therefore, the amounts of energy which 
 pass in a second through one square centimetre of S 1 8^S 1 and 
 $ 2 $ 2 $ 2 respectively must be inversely as the areas of these 
 
 surfaces, that is, as is to . 
 
 1 2 
 
 In this proof we have assumed that all the sound-energy 
 which starts from P passes S 1 S 1 S 1 and $ 2 $ 2 $ 2 unchanged in 
 form. This is nearly true for moderate distances if the 
 sound spreads in all directions through air in which there 
 are no sudden changes of density ; the energy of each shell is 
 passed on to the next almost unaltered in total amount for a 
 very long distance, and though a very little remains behind 
 (in the form of heat) in every part of the air through which 
 the wave has passed, yet, many miles from the source, by far 
 the greater part of it is still travelling on. It is because of 
 the diminution of the intensity of the vibration, due to the 
 distribution of the energy of over larger and larger 
 shells of air, rather than because it has been converted into 
 heat, that we cease, at a great distance, to be able to detect 
 the vibration ; the whole of the energy is, however, ulti- 
 mately converted into heat. When the waves are trans- 
 mitted along a tube, there is friction between the vibrating 
 air and the walls of the tube, and the energy is converted 
 much more rapidly into heat than in the open air, but as the 
 still untransformed remnant of energy is always contained in 
 the same amount 'of air as it travels on, instead of in an ever- 
 increasing volume, the intensity of the undulation diminishes 
 much less rapidly in this case than in the open air, and along 
 a tube of suitable size (6 to 10 feet in diameter) the sound 
 of ordinary conversation can be heard for miles. Yery 
 small or rough tubes, however, rapidly convert the undula- 
 tory energy into heat, especially if the waves are long. 
 
 24. Intensity and Amplitude. At any two points in the 
 air, where the air is vibrating with the same frequency and 
 in the same manner, but with different amplitudes, the 
 intensities are proportional to the squares of the ampli- 
 tudes. It is not possible to prove this by elementary 
 methods, but it is connected with the fact that the average 
 
42 SOUKD. 
 
 velocities of the air at the two places are proportional to 
 these amplitudes (since at each place the air moves four 
 times its amplitude in each vibration, and the number of 
 vibrations per second is the same at the two places), and that 
 the average kinetic energies of equal masses, moving similarly, 
 are proportional to the squares of their average velocities. 
 
 Since when sound spreads in all directions the intensity 
 varies inversely as the square of the distance from the source, 
 and since, whether the sound spreads in all directions or not, 
 the intensity is directly proportional to the square of the 
 amplitude, it follows that when sound spreads in all direc- 
 tions the amplitude of vibration of the air must vary 
 inversely as the distance from the source, so that at 
 100 yards from a bell the air is moving backwards and for- 
 wards half as far as at 50 yards. 
 
 It should be understood that the intensity at a point does 
 not depend on the distance of the point from the source as 
 well as on the amplitude of vibration ; it depends (for waves 
 of the same form and frequency in air of the same density) 
 only on the amplitude, and it is only when the amplitude 
 diminishes with increasing distance that the intensity does 
 so. In a tube, where successive layers move with nearly 
 the same amplitude, the intensity diminishes very little with 
 distance. 
 
 The illustrations given to explain the nature of progressive 
 undulation, being chosen because of the obvious movement 
 of the undulating substance, are liable to suggest an exag- 
 gerated idea of the extent of the movements of air conveying 
 sound. There is no theoretical limit either to the largeness 
 or smallness of the movements which can be passed on through 
 the air, all are passed on in the same way, and (unless very 
 large) with practically the same velocity. But the backward 
 and forward swingings of the air by which ordinary conver- 
 sation is transmitted to us are of very small extent, usually 
 less than a thousandth of an inch, and often less than a 
 millionth. Thus, though the movement of the air is reversed 
 hundreds or thousands of times per second, its velocity, even 
 at its quickest moments, may be very small ; the sound of 
 a whistle is quite audible at a place where the air has at no 
 instant a velocity of 2 inches an hour. The velocity with which 
 
PROGRESSIVE UNDULATION. 43 
 
 the conditions travel may thus be millions of times as great 
 as the velocity of the movements of the matter which 
 transmits them. The difference in density between the con- 
 densations and the rarefactions is correspondingly small. 
 
 25. Doppler's Principle. The velocity with which a pulse 
 travels relatively to the air, when it has once been produced, 
 is the same whether the vibrating body which produced it 
 was at rest in the air or moving through it, and the same 
 whether the air is moving relatively to the earth or not. 
 Suppose that a source of sound, say a whistle, produces n 
 condensations per second, and let it be moving in some 
 direction, which we will call northward, with a velocity of 
 a centimetres per second through the air, which we will consider 
 at rest. Let V centimetres per second be the velocity with 
 which pulses travel through the air. When the whistle begins 
 to sound, a condensation starts away from it in every direction 
 
 with a velocity F; at the end of second, when the whistle 
 is just producing the next condensation, the part of the 
 
 first which has travelled northward has travelled centi- 
 
 n 
 
 metres through the air, and the whistle has followed it 
 
 ~^L 
 centimetres ; the distance (on the north side of the whistle) 
 
 between two successive condensations is therefore 
 
 n 
 
 centimetres, and this is the length of the waves which travel 
 northward. 
 
 If a point P, to the north of the whistle, is moving northward 
 through the air with a velocity I centimetres per second, a total 
 length F b of successive waves will pass P in each second ; 
 
 F a 
 
 as the length of each of these waves is - the number of 
 
 n 
 waves which pass P per second is 
 
 i-v i\ V~ a IV b\ 
 
 <r-)+ _(_). 
 
 If a = b, or the whistle and Pmove in the same direction 
 with the same velocity, and so keep the same distance, the 
 
44 SOUND. 
 
 number of waves passing P per second becomes n, tbe same 
 as that produced by the whistle, as was, perhaps, obvious, 
 since the waves cannot be stored up anywhere between. 
 
 If the whistle is at rest, and P moves away from it with a 
 velocity v, the frequency with which the waves pass P 
 
 becomes n ( j . If P moves towards the whistle with 
 
 velocity #, this becomes n I ^- j , since a velocity v south- 
 wards is the same as a velocity v northwards. 
 
 If P is at rest and the whistle moves towards it with a 
 velocity 0, the frequency with which the waves reach P 
 
 becomes n ( j , or n ( ) if the whistle moves away 
 
 from P with velocity v. 
 
 "We see then that, if P keeps the same distance from the 
 whistle, the frequency with which the waves pass P is simply 
 n, whether P and the whistle are at rest in the air or move 
 with the same velocity through it, or, which is the same 
 thing, if the wind blows past both of them. But if the dis- 
 tance between P and the whistle is diminishing, the frequency 
 at P will be greater than n, and, if the distance is increasing, 
 less than n. In this case, however, as the formulae show, the 
 frequency at P depends, not only on the rate at which its 
 distance from the whistle is increasing or diminishing, but on 
 the velocity of both P and the whistle relatively to the air ; 
 it is not the same if the whistle is at rest in the air and P 
 moves towards it with a velocity v as it is if P is at rest in the 
 air and the whistle moves towards it with the velocity v. 
 The two frequencies are nearly the same if v is small com- 
 pared with F, but the reason of increased frequency is quite 
 different in the two cases ; movement of the whistle through 
 the air alters the length of the waves ; movement of P 
 through the air only alters the rate at which P is overtaken 
 by or meets them. The difference between the frequency at 
 P and that of the whistle only exists while P continues to 
 move, if due to the movement of P through the air, or till 
 the last of the waves produced while the whistle was moving 
 has reached P, if due to the movement of the whistle. 
 
PROGEESSIVE UNDULATION. 45 
 
 It is not implied in the above formulae that the air need be 
 at rest relatively to the earth, but only that F, 0, and b are 
 the velocities relatively to the air. 
 
 The frequency with which waves arrive at the ear de- 
 termines the pitch of the sound heard, greater frequency 
 corresponding to higher pitch. If our distance from a source 
 of sound is increasing, we therefore hear a sound of lower 
 pitch, and if our distance is diminishing, one of higher pitch, 
 than we should do if the distance was constant. The change 
 in the pitch of the sound heard when a whistling locomotive, 
 which has been approaching us, passes us and begins to recede, 
 is a familiar instance of this. 
 
 The fact that the frequency of the arrival of waves at a 
 point is affected by movements of the point, or of the source 
 of the waves, is called Doppler** Principle. 
 
 26. Sound Waves in Liquids and Solids. So far we have 
 spoken only of sound waves travelling in air. But conden- 
 sations and rarefactions are transmitted through liquids and 
 solids in exactly the same way, by exactly similar movements. 
 If one end of a rod of wood or metal is pushed or pulled, suc- 
 cessive layers move, and are compressed or rarefied, in turn, 
 just like the layers of air in a tube, and if any point of a 
 solid block is made to vibrate, condensation and rarefaction 
 spread through it in widening spheres, as they do through air. 
 
 It must not be supposed that this handing- on of condensa- 
 tions and rarefactions by successive movements of different 
 portions is a process specially connected with audible sound ; 
 it is the only way in which a pull or a push can be trans- 
 mitted at all. When a signalman moves a distant signal by 
 pulling at a handle, which is connected with the signal by a 
 rod or wire, the movement of the signal appears to be simul- 
 taneous with the movement of the lever, but this is only 
 because of the great velocity with which pulses travel in iron, 
 not because any pull or push can be transmitted in a different 
 way or with a different velocity from sound. When the man 
 begins to move the lever, he produces an elongation or rare- 
 faction of a piece of the wire at his end, and this state of 
 rarefaction travels along the wire at the rate of about three 
 miles a second, and it is only when the rarefaction reaches 
 the other end of the wire that the signal begins to move. 
 
CHAPTER III. 
 
 VELOCITY OF SOUND. 
 
 27. Velocity of a Pulse along a Eod. If we take a long 
 uniform rod AB of any elastic substance, and apply at one 
 end A a force of / dynamical units, directed towards .Z?, a 
 portion of the rod close to A shortens, till it exerts a force / 
 on the next portion, and so the compressed condition extends 
 along the rod, each original unit length of the rod shortening 
 by a certain amount I. If m is the mass of a piece of the 
 rod of unit length, it can be shown (Appendix B) that the 
 velocity with which the compressed condition extends along 
 
 the rod is \/~. This is true whatever the circumstances 
 
 V m 
 
 under which the compression takes place, and whether the 
 "rod" is solid or fluid, but only if/ is measured in the 
 dynamical units based on the units of length, mass, and time 
 employed in the other measurements. Thus, if I and m are 
 in centimetres and grammes, / must be in dynes, and the 
 velocity will be in centimetres per second ; or, if I and m are 
 in feet and pounds, /must be \npoundals. 
 
 For moderate forces, I is proportional to/, so that \ ~ 
 
 V wi> 
 
 is the same whether the force is smaller or larger, if not very 
 large. Also, if we double the sectional area of the rod, we 
 require twice the force to produce the same change in the 
 length of a centimetre, but a centimetre has twice the mass, 
 
 so that \ I ~ is the same for a thick rod as for a thin one of 
 
 the same material. We do not need, therefore, to know 
 either the force applied or the diameter of the rod to calculate 
 
VELOCITY OF SOUND. 47 
 
 the velocity with which the condensed condition extends 
 along the rod, for it would be the same for any diameter and 
 any force. If we suppose the rod of unit sectional area, / is 
 numerically equal to the pressure* applied, and if a rod of 
 
 unit area is free to expand sideways, 3- is equal to the 
 
 quantity called Young's Modulus for the substance of the 
 rod, which is the ratio of a change of stress applied to the 
 ends of a rod of that substance to the change it produces in 
 each unit of length of the rod, when the rod is free to 
 expand sideways. Also, if the rod is of unit sectional area, 
 each unit of length of it contains unit volume, and m is 
 numerically equal to the density of the substance. So that 
 for such a rod, and therefore for any rod, 
 
 -, ., /Young's Modulus 
 
 velocity = \/ . 
 
 V density 
 
 If the force at A is a pull instead of a push, the velocity 
 
 is still \ / 3 where I is the increase in length of each 
 
 V m 
 
 centimetre produced by the force /. If / is small, this 
 increase is equal to the decrease produced by an equal push, 
 so that the stretched or rarefied condition extends at the same 
 rate as the condensed condition. 
 
 The change of length produced by a given change of stress 
 is nearly the same whatever the actual stress may be, and 
 the density of a solid is only slightly altered by any ordinary 
 stress. The velocity of a condensation or rarefaction along a 
 rod or wire, therefore, depends hardly at all on whether the 
 rod or wire is stretched or not. 
 
 28. Velocity in a Fluid in a Tube. If instead of a solid 
 rod we had a fluid column contained in a tube, and applied a 
 force of / dynes to a piston at one end, the condensation 
 
 * In this book, " pressure " always means stress, i.e., the ratio 
 of the force to the area on which it is exerted. Some writers call 
 the force itself the " pressure," and the ratio of force to area the 
 " intensity of pressure," 
 
as 
 
 48 SOUND. 
 
 would extend along the tube with a velocity \l j~ just 
 
 before. If our tube were of unit sectional area, we should 
 be applying an increase of pressure f not only to the end, 
 but to every side, of the condensed portion of the fluid, and 
 I would be not only the shortening in each unit length, 
 
 but also the diminution in each unit of volume of the fluid 
 
 p 
 column. So ~ would be the ratio of a change of hydrostatic 
 
 I 
 
 pressure to the change produced by it in each unit volume of 
 a fluid ; a quantity which is called the volume elasticity* of 
 the fluid. The velocity V with which a condensed or 
 rarefied condition spreads along a liquid or gaseous column is 
 
 ., n /volume elasticity 
 
 therefore \ / rr- * 
 
 V density 
 
 density 
 
 29. The two ways of measuring Elasticity. The volume 
 elasticity of a fluid depends on the conditions under which 
 the change of volume takes place. If we could have a cubic 
 centimetre of air confined under a piston (7, Fig. 24, in a tube 
 AB of unit sectional area, both piston and tube being of very 
 low thermal conductivity, and applied to the piston a very 
 small downward force of / dynes, the piston would instantly 
 descend, diminishing the volume of the air by an amount x. 
 
 * Strictly, Young's Modulus is defined not as the ratio of change 
 of stress to change produced in unit length, but as the limit which 
 this ratio approaches when the change of stress is indefinitely 
 diminished ; and, similarly, the volume elasticity is the limit of the 
 ratio of change of hydrostatic pressure to change produced in each 
 unit of volume, when the change of pressure is indefinitely 
 
 diminished. So that the formula V = ^Young's Modulus and 
 
 V density 
 
 v = /volume elasticity are on] absolutel tme for indefi . 
 
 V density 
 
 nitely small changes of stress. But, even for much larger changes 
 of stress than those which occur in sound waves, the ratio of change 
 of stress to change of length or volume is practically the same as for 
 indefinitely small changes of stress, 
 
TELOCITY OF SOUND. 
 
 49 
 
 At the same time a quantity of heat equal to the work done 
 in the descent of the piston would he produced in the com- 
 pressed air, which would raise its temperature. As the air 
 cooled to its original temperature, the piston would descend 
 
 
 r 
 
 cjf ! 
 
 *'{ 
 
 ? 
 
 
 1 
 
 further, till the volume of the air had heen diminished 
 altogether by a volume x. Similarly, if we applied an 
 upward pull / to the piston, it would instantly rise till the 
 volume of the air under it had heen increased hy #, and the 
 temperature of this air would fall ; then, as the air returned 
 to its original temperature, the piston would rise still further 
 
 till the total increase of volume was 
 
 The ratio -7- is 
 
 called the adialatic elasticity (x heing measured before any 
 
 heat has entered or left the air), and -*- is called the isothermal 
 
 x' 
 
 elasticity. As x is greater than #, the isothermal elasticity 
 is smaller than the adiabatic. 
 
 The experiment as described above is impracticable, as the 
 gas loses or gains heat much too quickly to allow of deter- 
 mining #, but when a pulse travels along a tube, the air in 
 each centimetre of the tube changes from its original to its 
 altered value in less than 30 ^ Q- of a second, which would be 
 too short a time for it to lose or gain much heat, even if air 
 were a good conductor and radiator instead of an extremely 
 bad one, so that the change of each centimetre length of the 
 air column is the same as the instantaneous movement which 
 would be produced by the same force applied to the piston of 
 
 SD. E 
 
50 SOUXD. 
 
 l ~~f~ 
 
 our imaginary experiment. The quantity \l-j- in such a 
 
 \ Dili 
 
 . ,-, p /adiabatic elasticity m, 
 
 case is therefore equal to \ -*. The same 
 
 V density 
 
 is true for a liquid. , 
 
 /""/" 
 
 It will be .seen that the quantity \l -- is greater in the 
 
 V wi 
 
 case either of * rarefaction or a condensation than it would 
 be if no change of temperature took place. This is some- 
 times expressed by saying that the rise of temperature pro- 
 duced by condensation and the fall of temperature produced 
 by rarefaction both increase the velocity of the sound. That 
 is not a good way of putting it, because it suggests that if 
 these changes of temperature did not occur the sound would 
 
 T .,T IT -n T .j /isothermal elasticity 
 
 travel with the smaller velocity \ I ^ ; in 
 
 V density 
 
 reality it would not travel at all. Indeed, we shall see later 
 that the change of temperature and the movement of the air 
 are different ways of expressing the same fact. Kewton, who 
 was the first to calculate the velocity of sound from dynamical 
 principles, did not know of the changes of temperature, but 
 supposed that the velocity would be equal to 
 
 /isothermal elasticity g y 
 
 " density 
 
 it was Laplace who first pointed out that the adiabatic elas- 
 ticity should be used. * 
 
 l~7~ 
 
 Similarly, in the formula V == \ I ~- for the velocity of 
 
 V Itn 
 
 pulses along a rod, I should strictly be the change in unit 
 length instantly produced by a force /, before any heat has 
 left or entered, and therefore 
 
 Y ~ /Young's Modulus 
 
 ^V density 
 
 is not exactly true unless the adiabatic value of Young's 
 Modulus is taken. But in solids and liquids the difference 
 between adiabatic and isothermal changes is very small. 
 
VELOCITY OF SOUND. 51 
 
 30. Indirect Formulas for Velocity. There is no simple 
 method of measuring the adiabatic elasticity directly, but it 
 can be shown theoretically that it bears a constant ratio to 
 the isothermal elasticity, and that this ratio is the same as 
 that of the specific heat at constant pressure to the specific 
 heat at constant volume (Text-look of Heat, 42), a ratio 
 
 usually denoted by y. So 
 
 v _ ly x isothermal elasticity 
 
 density 
 
 This is true for all substances, and as the isothermal elasticity 
 can be directly measured, and y can be calculated, by thermo- 
 dynamic methods, from the specific heat at constant pressure 
 and the coefficient of expansion, this formula can be used to 
 find V. For most gases a simpler formula is very nearly 
 true. If any gas exactly obeyed Boyle's law, it can be shown 
 that its isothermal elasticity would be equal to its pressure, 
 and this is very nearly the case with all gases which are far 
 from their condensing temperatures. So that for such gases 
 
 r= Jtjr- nearl y- 
 
 The value of y depends on the number of atoms in a molecule 
 of the gas : for one-atom molecules, like mercury gas, it is 
 1-66; for two -atom molecules, like oxygen and nitrogen, it 
 is 1'41 ; and for three-atom molecules, like water vapour, it 
 is 1*26. 
 
 The velocity of sound really depends, of course, on the 
 changes which actually take place in the air as the wave 
 travels through it, not on the changes which would take 
 place under quite different circumstances. The isothermal 
 elasticity and the specific heats have therefore no direct con- 
 nection with the velocity, for the air is neither compressed 
 without change of temperature nor heated without change of 
 pressure, nor, heated without change of volume, when a 
 sound wave passes through it. But the physical properties 
 of gas on which the velocity of sound depends (the chief of 
 which, as we shall shortly see, is the velocity of its mole- 
 cules) are also factors in determining its elasticities and 
 specific heats, and therefore it is quite possible to express the 
 velocity of sound in terms of these quantities. 
 
52 SOUND. 
 
 31. Velocity in Free Air. The velocity of sound along a 
 rod which is free to expand sideways is much less than it is- 
 
 when the transverse expansion is prevented, the ratio ~ being 
 
 I 
 
 much larger in the latter case. In a column of air which 
 was free to expand sideways sound would not be propagated 
 
 at all, as in that case the ratio -y would be zero. Eut in 
 
 open air the velocity, except within a very short distance 
 from the source, is the same as along the air in a tube. For 
 as the condensed condition travels outwards in all directions 
 from the source, the condensed air does not move except in 
 the line of propagation of the sound, since in all directions at 
 right angles to this line it is in contact with air condensed as 
 much as itself. The velocity with which the condensation 
 travels is therefore the same as if transverse expansion were- 
 prevented by a rigid tube. For the same reason the velocity 
 of a sound wave through a large mass of a solid substance, a 
 cliff for instance, is much greater than along a rod. 
 
 32. Velocity Independent of Pressure. Let D be the 
 density of a gas at 0C. and any standard pressure JJ dynes per 
 
 sq. cm. ; then is a constant for the gas, and does not de- 
 pend on what the pressure H was at which D was measured. 
 We will denote : by d ; it is the density which the gas 
 
 would have at C. and a pressure of 1 dyne per sq. cm* 
 Then the density of the gas at a pressure P dynes and a tem- 
 perature t C. is Pd( J* 3 \ ; see Text-book of Heat, 32. 
 
 So the velocity of sound in the gas at this pressure and 
 temperature is 
 
 (273 + 
 
 273 \ V 273d 
 
 It is therefore independent of the pressure, a fact which may be 
 explained by saying that the adiabatic elasticity and the density 
 
VELOCITY OF SOUND. 53 
 
 are both proportional to the pressure, and that therefore their 
 ratio, on which the velocity depends (29), is the same for all 
 pressures ; but this is a mathematical rather than a physical 
 explanation, and the physical reason will appear later. 
 
 In the formula) just given it must be noticed that d is not 
 the density of the gas actually conveying the sound, but the 
 density it would have at C. and at a pressure of 1 dyne, or 
 the constant ratio of its density to its pressure at C. 
 
 33. Velocity and Temperature. If V^ T 2 are the velo- 
 cities of sound in the same gas at two different temperatures 
 
 V 2 
 
 V 3 
 
 278 + *, 
 
 /y (273 + * 8 ) V273 + 2 
 
 273^ 
 
 or the ratio of the velocities is the square root of the ratio 
 of the absolute air-thermometer temperatures. (Text-look 
 <of Heat, 35.) This is true whether the pressures are the 
 same or not. 
 
 34. Velocities in different Gases. If V r , V y are the ve- 
 locities of sound at the same temperature in two different 
 : gases, X and Y, which have the same number of atoms to th* 
 molecule, and therefore the same value of y 
 
 /rI273+l) 
 
 V 2734 
 
 '& 
 
 ,_ 
 _ M j 
 
 * x 
 
 or the velocities are inversely as the square roots of the 
 densities of the gases at C. and 1 dyne pressure, and 
 therefore inversely as the square roots of the densities at any 
 pressure and temperature (the same for both). These den- 
 sities are proportional to the atomic weights, so the velocities 
 #re inversely proportional to the square roots of the atomic 
 weights. This is true whether the pressures are the same 
 or not. 
 
54 SOUND. 
 
 For instance, if X is hydrogen and Y oxygen, 
 
 so that sound travels four times as fast in hydrogen as in 
 oxygen at the same temperature. 
 
 Sound travels more quickly in water vapour than in air, 
 for, though y is smaller for water vapour, d is also smaller, 
 
 and ^- is greater for water vapour. The presence of water 
 
 vapour in the air therefore increases the velocity of sound 
 slightly. 
 
 The density of dry air at C. and a million dynes per sq. 
 cm. pressure (a il c.g.s. atmosphere ") is '0012759, so that d' 
 for air is -0000000012759. The velocity of sound in air at 
 any temperature t C. is therefore 
 
 1-41(273 +7)~ 
 273 (-000000001275)' 
 
 1.41 
 
 If t is small, \/ 1 + ^ = 1 H nearly, and the velo- 
 
 V 2io 54o 
 
 city is 33,240 ( 1-f ], or 33,240 + 60^ approximately. 
 V 546/ 
 
 This agrees very closely with experimental determinations. 
 
 35. Experimental Determination of Velocity. The velo-^ 
 city of sound can he determined by direct experiment, and) 
 this is, in fact, one of the most accurate methods of finding jjf,-' 
 which can be calculated from the formula given for V if V 
 is known independently. A simple method of finding the 
 velocity is for two observers, at a distance of some miles, each 
 to fire a cannon, as nearly as possible at the same time, andl 
 for each to notice the interval between seeing the flash and' 
 hearing the report of the distant cannon. The intervals will 
 not be equal unless the night is absolutely windless, for the 
 sound travels relatively to the air with the same velocity 
 whether the air is moving relatively to the earth or not, SO' 
 
VELOCITY OF SOUND. 55 
 
 that the velocity of the sound relative to the earth, which is 
 what is measured in the experiment described, is greater than 
 the velocity relative to the air when the sound travels in the 
 direction of the wind, and smaller in the opposite direction. 
 The average of the two velocities found by dividing the dis- 
 tance between the observers by the observed intervals will be 
 free from this error. 
 
 Another error arises from the fact that we do not perceive 
 either the flash or the report exactly at the moment when 
 the light and sound reach us, the processes of perception 
 requiring appreciable times, which are unequal for sight and 
 hearing. This can be avoided in several ways, of which the 
 best is perhaps to dispense with the observer altogether, and 
 let an electric current which fires the cannon mark the 
 moment of its occurrence on a revolving cylinder, while the 
 arrival of the sound at the distant station makes another 
 mark on the same cylinder by setting in vibration an acoustic 
 pendulum (1). For this purpose the membrane and pith -ball 
 are gilt to make them conducting, and form part of an electric 
 circuit ; the breaking of this circuit on the arrival of the 
 sound releases the armature of an electro-magnet, which 
 marks the cylinder. 
 
 By this and similar methods the time taken by the sound 
 in travelling from one station to the other can be very exactly 
 ascertained, but as it is not possible to determine at all accu- 
 rately the average temperature of the air through which the 
 sound passed, and as the velocity depends greatly on tempera- 
 ture, there is not much value in such determinations. 
 
 36. Relation of Velocity to Molecular Structure. So far 
 we have considered the air as a continuous elastic substance, 
 of which every portion exerts a pressure on all the sur- 
 rounding portions. This is not an incorrect way of regarding 
 the matter, but it is rather a superficial one, for the transmis- 
 sion of sound through a substance is closely connected with its. 
 molecular structure. It is well known that air, for instance, 
 is not a continuous substance, but consists of very minute 
 detached bodies, called molecules, of which there are, very 
 roughly, a million billions in a cubic centimetre. These fly 
 about in every direction with an average velocity which 
 
56 SOUND. 
 
 depends on the temperature (more accurately, perhaps, the 
 temperature depends on the average velocity), but which at 
 ordinaiy temperatures is about 50,000 cm. (1,700 feet) per 
 second. Though the space actually filled by the molecules 
 is certainly less, and probably much less, than y^Vo^ ^ ^ no 
 whole space, the number of molecules is so great that the 
 average distance traversed by a molecule before coming into 
 collision with another is only 200 1 000 of a centimetre, and, 
 though there is of course no regularity in these collisions, it 
 must be a very rare thing for a molecule to move -^ ^^ -$ of a 
 centimetre (y^ ^ a hair's-breadth) without an encounter 
 which quite changes its direction. Thus, though the mole- 
 cules move as fast as cannon-balls, they do not make rapid 
 continuous progress, and most of those which are in a par- 
 ticular cubic centimetre of the air at one moment would still 
 be found in a compact group a second or two later, though 
 the position of this group, relatively to the earth, may have 
 changed considerably. A " particle" of air is thus like a 
 swarm of bees, which may be at rest, or moving in a constant 
 direction, though every bee is moving much faster, and 
 continually changing its direction. Where we have spoken 
 in the preceding chapter of the velocity of the air, it is this 
 swarm-velocity that is meant, and it is equal to the average 
 velocity of the molecules reckoned algebraically, velocities 
 in one direction being counted as of opposite sign to those in 
 the other. 
 
 To simplify the explanation of the way in which the trans- 
 mission of sound is related to molecular movements, we will 
 suppose that all the molecules have the same mass (which is 
 not the case in a mixed gas like air), and have the same 
 velocity, which we will take as 50,000 cm. per second. We 
 will also suppose that they all move in horizontal lines run- 
 ning east and west, and that all the collisions which happen 
 are direct, so that two molecules still move after collision in 
 the same line as before, and that the molecules behave like 
 perfectly elastic equal balls in direct collision, so that each 
 moves after the collision in the same direction and with the 
 same velocity as the other did before collision. Suppose a solid 
 plate in the air moving from east to west, with a velocity of 
 10 cm. per second. Then all the molecules on the west side 
 
VELOCITY OF SOUND. 57 
 
 of the plate which strike it do so with a velocity, relative to 
 the plate, of 50,010 cm., and rebound with the same relative 
 velocity ; that is, they move westwards with a velocity of 
 50,020 cm. Each of these soon encounters a molecule 
 coming eastwards at 50,000 cm. per second, and sends this 
 molecule off westwards at 50,020 cm. per second, while it 
 returns eastwards at 50,000 cm. per second, till it meets the 
 plate again, and so on. Meanwhile the molecule to which 
 it has passed on its velocity passes it on to one still further 
 west, and returns eastwards at 50,000 cm. per second. Thus 
 in the air near the plate all the eastward-moving molecules 
 Tiave a velocity of 50,000 cm. per second, while the west- 
 ward-moving ones have a velocity of 50,020 cm. per second, 
 and this condition of the air is spreading westwards with the 
 velocity of the molecules themselves. The swarm-velocity of 
 this air is the average of +50,020 and 50,000, or 10 cm. 
 per second, the same as that of the plate, and it is in the 
 same direction as the condition is spreading. It is a con- 
 densed condition, for there is all the air between the plate 
 and the furthest point to which the condition has extended 
 westwards that there would have been if the plate had been 
 at rest, and the distance is smaller than it would have been 
 in that case. Also it is a condition of raised temperature, for 
 the relative velocity of the molecules moving eastwards to 
 those moving westwards is 100,020, while in the rest of the 
 air it is 100,000, and the temperature of a gas is an expres- 
 sion for the relative velocity of its molecules among them- 
 selves. Thus all that we have previously learnt about a 
 travelling condensation is seen to be a natural result of the 
 molecular constitution of the air. It will be obvious how 
 the description must be altered to make it apply to a rare- 
 faction. 
 
 Roughly speaking, then, the velocity of sound in a gas is 
 the velocity of the molecules, but in an actual gas the con- 
 densed or rarefied condition spreads outwards only about two- 
 thirds as fast as the average velocity of the molecules. This 
 is chiefly because most of the collisions which happen are 
 oblique, so that each molecule rebounds in a different direc- 
 tion to that in which the other one was going before the 
 collision, and thus the extra velocity communicated to one 
 
58 
 
 SOUND. 
 
 molecule by the moving plate is handed on in a zig-zag- 
 manner. There are also other reasons, which cannot be 
 explained satisfactorily in an elementary book. 
 
 The reason why the velocity of sound is independent of 
 pressure may now be more clearly explained. The tempera- 
 ture of a gas is an expression for (though not simply propor- 
 tional to) the average kinetic energy of each of the molecules, 
 and this depends on their velocity and mass, not on their 
 number. So that for any one gas the velocity of its mole- 
 cules, and therefore the velocity of sound in it, is always the 
 same at the same temperature, however other conditions may 
 vary. 
 
 It will now be seen that the two statements that " the air 
 in a condensed region is moving in the same direction as the 
 waves," and that " the air in a condensed region is at a 
 higher temperature than the undisturbed air," are two ways, 
 each incomplete, of stating the fact that in a condensed 
 region the molecules which are moving in the direction of 
 the sound are moving with more than usual velocity, while- 
 those which are moving in the opposite direction have their 
 ordinary velocity. It is only this particular kind of mole- 
 cular movement which transfers itself from one part of the- 
 air to another, so that raised temperature is an essential part 
 of the condition of travelling condensation, and not merely a 
 
 Fig. 25. 
 
 cause increasing the velocity with which it travels. The- 
 same is of course true of the lowered temperature of a 
 travelling rarefaction. 
 
 Prom the molecular point of view it seems as if the con- 
 
VELOCITY OF SOUND. 
 
 densed condition ought to travel a little faster than the 
 rarefied one, for each condition is handed on hy the molecules 
 which travel in the same direction as the condition, and these 
 have more than the average velocity in a condensation, less 
 in a rarefaction. The formula of Art. 33 suggests the same 
 thing, since the condensations are at a higher temperature 
 than the rarefactions. The matter is not nearly so simple a& 
 it appears, and the real explanation cannot he given here, but 
 it is a fact that condensation does travel faster than rarefac- 
 tion. So that, in a wave-system, the wave-form gradually 
 changes as the waves get further from the source, just as- 
 sea-waves change in form when, reaching shallow water, 
 the crests travel perceptibly faster than the troughs. Some 
 stages in the transformation of waves whose wave-form was 
 originally harmonic are shown in Fig. 25 ; the waves, like 
 those in previous diagrams, are supposed to be travelling 
 from right to left. The greater the degree of condensation 
 and rarefaction the greater the difference in the velocities,. 
 and the more rapid the change in wave-form. 
 
 CALCULATIONS. 
 
 The most important formulae for the velocity of sound are 
 (1) 7- 
 
 where t is the centigrade temperature, and d the quotient obtained 
 by dividing the density of the gas at C. and any pressure by that 
 pressure expressed in dynes per sq. cm. 
 
 (2) For the same gas at different temperatures, t lt f 2 , 
 
 (3) For different gases, both having the same number of atoms to- 
 the molecule, at the same temperature, 
 
 = . /atomic weight of y 
 Vy V atomic weight of x 
 
60 SOUND. 
 
 EXAMPLES I. 
 
 ELEMENTARY. 
 
 1. At what temperature is the velocity of sound in nitrogen 
 (atomic weight, 14) equal to its velocity in oxygen at 10 C. (atomic 
 weight, 16) ? 
 
 2. At what temperature is the velocity of sound in air twice as 
 great as in air at 20 C. ? 
 
 3. The velocity of sound in air is 34,000 centimetres per second 
 when the thermometer is at 13 C. and the barometer at 78 centi- 
 metres. What will be the velocity if the thermometer rises to 22 
 and the barometer falls to 72 centimetres ? 
 
 4. The velocity of sound along a tube full of air is found to be 
 1100 feet per second at 0. What will be its velocity in the same 
 tube (1) if the air in the tube is raised to 20 without letting any 
 escape from the tube ; (2) if the air in the tube is raised to 20, and 
 the pressure kept constant ? 
 
 ADVANCED. 
 
 5. A whistle makes 200 waves a second. What number of waves 
 will a person receive per second (a) if the whistle comes towards 
 him with the velocity of sound ; (b) if the whistle goes away from 
 him with the velocity of sound ; (c) if he goes towards the whistle 
 with the velocity of sound ; (d) if he goes away from the whistle 
 with the velocity of sound ? 
 
 6. A cylindrical rod is 2 metres long when laid down, and 2 centi- 
 metres in diameter, and weighs 2000 grams. When it stands 
 upright, with one end on the ground, it is -^ millimetre shorter than 
 when laid down. What is the velocity of sound in the substance of 
 the rod ? 
 
 7. Calculate the velocity of sound in hydrogen at 20 C. and 
 1,100,000 dynes pressure, given that the density of hydrogen at 
 and 1,000,000 dynes pressure is '0000884, and that the ratio of the 
 specific heats of hydrogen is 1*408. 
 
 8. Given that the velocity of sound in air at J is 33,240 cm. per 
 second, and that the density of air at and a pressure of 10* 5 dynes 
 is '001275, find the adiabatic elasticity of air at a pressure of 10 6 
 dynes. 
 
CHAPTER IV. 
 
 INTERFERENCE. 
 
 37. Principle of Interference. Suppose that we have two 
 sources of sound waves, two tuning-forks, for instance ; we 
 -will call them A and B. Consider a region of air (7, 
 so situated that waves of quite small amplitude pass 
 through it if either A or B vibrates alone. Let A and B he 
 vibrating together, but suppose that we know what the 
 condensation and velocity of the air would be at each point 
 of the region (7, at a given moment T, if A was vibrating as 
 it actually is, but B was not vibrating, and that we also 
 know what the condensation and velocity at each point 
 would be, at the same moment T, if B was vibrating as it 
 actually is, but A was not vibrating. Then it can be shown 
 from the principles of dynamics that the actual condensation 
 and velocity of the air at each point of the region (7, at the 
 given moment T, are very nearly the same as would be found 
 by adding the condensation and velocity which there would 
 be at that point if only A was vibrating, and the condensation 
 and velocity which there would be if only B was vibrating. 
 
 In this addition, rarefaction and condensation must be 
 counted of opposite signs and added algebraically, and 
 velocities must be added in the only way in which quantities 
 which have direction as well as magnitude can be added, by 
 finding the diagonal of the parallelogram whose sides repre- 
 sent in magnitude and direction the velocities whose sum is 
 required. 
 
 Similarly, the displacement of any particle of the air from 
 its mean position is the same as would be found by adding 
 the displacement which it would have if A was vibrating 
 alone and the displacement which it would have if B was 
 vibrating alone, displacements being added like velocities. 
 
62 SOUND. 
 
 The principle that, when the condensation, velocity, and 
 displacement of the air are small, they are equal to the sums 
 of the condensations, velocities, and displacements which 
 there would be if each of the sources was vibrating without 
 the others, is called the principle of interference. It is a 
 singularly ill-chosen name, because the fact which is especially 
 to be noticed is that no interference occurs, but that each 
 source of sound produces the same difference in the condition 
 of the air that it would do if the other sources were absent. 
 As Lord Rayleigh says, " if this is interference, it is difficult 
 to see what non-interference would be." But the term is 
 firmly established, and must be used. 
 
 In the ordinary physical sense of the terms, neither the 
 condensations and rarefactions which A would produce by 
 itself, nor those which B would produce by itself, exist in 
 the region C when A and B vibrate together. Tor instance, 
 at a point where there would be a certain degree of conden- 
 sation if A was vibrating alone, and an equal degree of rare- 
 faction if was vibrating alone, the principle of interference 
 shows that the air is at its average density, and is neither 
 condensed nor rarefied. But the easiest way of finding the 
 actual condition of the air in the region C is to imagine the 
 condensations and rarefactions which there would be there if 
 A was vibrating alone, and then those which there would be 
 if was vibrating alone, and then add these imaginary con- 
 ditions to find the real condition of the air. It is usual, and 
 convenient, to speak of these imaginary wave-systems as 
 " existing " at the same time in the region (7, and to call its 
 actual condition their resultant. This should not cause any 
 confusion, because there, cannot, in any physical sense, be 
 two wave-systems at once in the same air, any more than 
 there can be two different winds at once at the same place ; 
 the same air cannot have two densities at once. But 
 it is important to remember that, whenever we speak of 
 two wave-systems in the same air, these wave-systems are 
 imaginary, and that what really exists in the air is one 
 wave-system different from either of the imaginary ones. 
 
 When the imaginary wave-systems in a given region travel 
 in directions which cross each other, the real movements of 
 the air are very complicated, each particle moving in curves 
 
UNIVERSITY Ol 
 
 DE^ CS 
 
 INTERFERENCE. 63 
 
 something like Lissajous' figures. It is more important to 
 consider the real condition of the air when the imaginary 
 wave-systems travel along the same line, either in the same 
 or in opposite directions. 
 
 38. Wave-systems in the same Direction. Suppose, as 
 before, that two bodies, A and B, are vibrating together. 
 Let A and B be so situated that, if either of them vibrated 
 alone, waves would travel in the same direction through a 
 given region, say the direction XO (Fig. 26). Let A' A A 
 
 Fig. 26. 
 
 be the wave-form of the waves which would be travelling 
 through the given region if A was vibrating alone ; B'BB' 
 that of the waves which would be travelling through the 
 same region if B was vibrating alone. Then at any point, 
 such as D, the actual condensation is proportional to the 
 algebraic sum of the ordinates from D to A A A and 
 B'BB ', so that, if we draw a curve EEE whose ordinate at 
 each point is the algebraic sum of the ordinates to A A A 
 and B'B'B ', the ordinates of EEE will be proportional to 
 the actual condensations. 
 
 Since the two wave-systems which there would be if 
 A and B vibrated separately would both travel in the direc- 
 tion XO, ordinates to either curve above the axis OX indi- 
 cate velocities which the air would have in the direction XO, 
 and ordinates below the axis indicate velocities which the air 
 would have in the direction OX. The actual velocity of the 
 air at any point is therefore proportional to the algebraic sum 
 of the ordinates to A A A' and B'BB' at that point, i.e., it 
 is proportional to the ordinate to EEE, and it is a velocity in 
 the direction XO where the ordinate to EEE is above the 
 axis. 
 
64 SOUND. 
 
 The actual condition of the air, then, is a condition in- 
 which both the condensation and the Telocity are propor- 
 tional to the ordinates to EFE, and are therefore propor- 
 tional to each other. Now, this, as we have seen, is the 
 distinctive feature of progressive undulation. The actual 1 
 condition of the air, then, is a progressive wave -system,, 
 travelling in the direction XO, and having a wave-form 
 EEE. A progressive system having this wave-form might 
 be produced by suitable movements of a single body, such as- 
 the piston in Chapter II., and there would be no physical 
 difference whatever between such a wave-system and the one 
 which A 1 and B 1 produce when they vibrate together. 
 
 Whenever there are two or more vibrating bodies, each of 
 which, vibrating by itself, would produce in a given region, 
 a progressive undulation in the same direction, there is in the 
 given region a progressive undulation whose wave-form is 
 the sum of the wave-forms of the undulations which there- 
 would be if the bodies vibrated separately, and such a pro- 
 gressive undulation bears no physical trace of having origi- 
 nated from several sources, but is exactly like an undulation 
 which might be produced by a single vibrating body. 
 
 If, on the other hand, the undulations which the bodies 
 would produce separately would travel in different directions- 
 through any region, the actual condition of the air in that 
 region is one in which condensation and velocity are not 
 proportional to each other, and is therefore a condition 
 which could not be produced by any possible vibrations of a 
 single source. 
 
 39. Harmonic Waves of Equal Length. If the imaginary 
 wave-systems (those which would actually exist if the 
 different sources vibrated separately) are all harmonic, and 
 all in the same direction, it does not follow that the actual 
 wave-system is harmonic, for the sum of two harmonic curves- 
 is not always a harmonic curve (Art. 10). But it can be 
 shown that the curve obtained by adding the ordinates of 
 harmonic curves whose lends are of the same length is always 
 a harmonic curve with bends of that length, whatever the 
 relative positions of the bends. Since harmonic curves whose 
 bends are of the same length are the wave-forms of harmonic 
 
INTERFERENCE. 
 
 65 
 
 wave-systems of equal wave-length, it follows that, where 
 there are several imaginary harmonic wave-systems of equal 
 wave-length, travelling in the same direction, the actual 
 undulation is a harmonic one, and of that wave-length. 
 There are several important special cases of this. 
 
 ( 1 ) If the imaginary wave-systems are in the same phase 
 (that is, if the wave- system which would exist at a given 
 moment if one source was vibrating alone has its maximum 
 condensations in the same positions as the maximum conden- 
 sations of the wave- system which would exist at the same 
 moment if another source vibrated alone), the maximum 
 ordinates of the actual wave-form are the sums of the 
 maximum ordinates of the imaginary wave-systems, and the 
 actual maximum velocity is therefore the sum of the imagin- 
 ary maximum velocities. The amplitudes of harmonic vibra- 
 tions of equal frequency are proportional to their maximum 
 velocities (Appendix A), so that the actual amplitude of 
 vibration is the sum of the amplitudes of the vibrations 
 which the air would execute if the sources vibrated separ- 
 ately. 
 
 This case is shown in Tig. 27, where the continuous curves 
 
 Fig. 27. 
 
 are the wave-forms of two imaginary wave-systems in the 
 same phase, and the dotted curve, obtained by adding the 
 ordinates of the others, is the wave -form of the actual con- 
 dition of the air. 
 
 (2) If the imaginary wave-systems are in opposite phases 
 (that is, if the maximum condensations of one imaginary 
 system coincide with the maximum rarefactions of the other), 
 the actual amplitude of the vibrating air is the difference of 
 
 SD. 
 
66 
 
 SOUND. 
 
 the amplitudes which it would have if the sources vibrated 
 separately. (Fig. 28.) 
 
 (3) In case 2, if the imaginrry systems are equal, the real 
 wave-form is a straight line ; the air is of uniform density, 
 
 Fig. 28. 
 
 and there is no real wave-system. The condition of the air 
 is exactly as if the sources were not vibrating. 
 
 (4) If the condensations of one imaginary system do not 
 coincide in position either with the condensations or with the 
 rarefactions of the other imaginary system, the amplitude of 
 the real vibration is less than the sum and greater than the 
 difference of the amplitudes of the imaginary vibrations. 
 
 In all these cases the intensity is of course proportional to 
 the square of the amplitude. 
 
 40. Illustrative Experiments. In order to verify by 
 experiment the principle of interference in these different 
 
 Fig. 29. 
 
 cases, we shall require some instrument for roughly com- 
 paring the intensities of sounds the most convenient is the 
 
INTERFERENCE. 67 
 
 ear. All that we require to know about its indications at 
 present is that vibrations of the air of the same frequency but 
 different intensity cause sensations of different loudness, the 
 more intense vibrations producing the louder sensations. 
 
 All the cases may be illustrated by means of an instru- 
 ment, shown in section in Fig. 29, which is slightly altered 
 from a form devised by Prof. Quincke. A tube A divides 
 into two branches B and (7, which re-unite at D. The 
 length of C can be altered by drawing out the sliding portion 
 E (on the trombone principle). D ends in a flexible pipe, 
 or, better, in two flexible pipes, one of which is placed in 
 each ear. In the branch B is a sliding door by which the 
 branch can be closed. 
 
 If the branch CE is adjusted till it is the same length as 
 j?, and a vibrating fork held opposite the opening at A, it is 
 evident that the condensations which there would be in D if 
 waves travelled by one branch would be in the same places as 
 the condensations which there would be, at the same instant, 
 if waves travelled only by the other branch. The actual 
 vibration in D is therefore greater than if waves travelled 
 only by one branch, as is shown by the diminution of loud- 
 ness when the sliding door in B is closed. 
 
 The same occurs if the difference between the lengths of 
 ABD and ACECD is made any exact number of wave- 
 lengths of the waves from the fork. 
 
 If A CECD is made half a wave-length longer than B, the 
 condensations of the waves which there would be in D if 
 waves travelled only by CE would be in the same places as the 
 rarefactions of the waves which there would be if waves tra- 
 velled only by -Z?, and the amplitudes of these two systems are 
 nearly equal. There is, therefore, little vibration of the air 
 in D when the sliding door in B is open, and hardly any 
 sound is heard. When the sliding door is closed, the sound is 
 heard clearly. 
 
 The same occurs if the difference between the lengths of 
 B and CE is made any odd number of half wave-lengths. 
 
 This apparatus may evidently be used to determine the 
 length of the waves from a fork, but it is not a good method, 
 as there is no point of absolute silence, and the points of 
 faintest sound are not easy to determine exactly. For abso- 
 
68 SOUND. 
 
 lute silence it would be necessary, not only that the maximum 
 condensations of one imaginary system should exactly coincide 
 with the maximum rarefaction^ of the other, but that the 
 condensation of one system should be equal at every point to 
 the rarefaction of the other. If the wave-forms of the two 
 systems are alike (as in the experiment just described), this 
 requires that the wave-form should be exactly symmetrical, 
 the part below the axis being exactly like the part above, 
 inverted. Harmonic waves would, of course, fulfil this con- 
 dition, but neither perfectly harmonic waves nor waves of any 
 other perfectly symmetrical form are easy to produce. 
 
 If a whistle is fitted into the middle of the bottom of a box 
 lined with felt (to prevent reflection from, and transmission 
 through, the sides of the box), and two holes are made in 
 the top of the box at equal distances from the whistle, simi- 
 lar pulses start from these openings simultaneously when the 
 whistle is sounded. At points in the external air above the 
 box, which are equidistant from the openings, a membrane 
 stretched on a ring, and held horizontally, is strongly 
 affected, and sand scattered on it is instantly thrown off, 
 but it is easy to find points, nearer to one hole than to the 
 other, where the membrane is very little affected ; these are, 
 of course, points whose distances from the two holes differ 
 by an odd number of half wave-lengths. 
 
 In these instances the two imaginary wave-systems are 
 produced in the same phase, and the difference in their 
 phases, at the points where the air is undisturbed, is due to 
 the different distances they have travelled. When the 
 imaginary wave-systems in a region are in opposite phases, it 
 is usually either for this reason, or because they were origin- 
 ally produced in opposite phases by parts of the same vibra- 
 ting body which move in opposite directions at the same 
 time. This is a very common cause, for nearly every vibra- 
 ting body has parts which move simultaneously in opposite 
 directions, like the prongs of a tuning fork or opposite sides 
 of a bell ; and, even in the simple case of a spring held in a 
 vice, one side of the spring is producing a condensation while 
 the other side is producing a rarefaction. A good illustra- 
 tion is furnished by a circular plate of glass or metal (A, 
 Pig. 30) fixed in a horizontal position to a stand by a screw 
 
INTEKFEKENCE. 
 
 69 
 
 through the middle, but free everywhere else. "When a 
 violin bow is drawn across the edge of such a plate, the plate 
 vibrates in an even number of sectors four if it is not 
 touched anywhere except by the bow of which at a given 
 
 Fig. 30. 
 
 moment half are moving upward and the alternate ones down- 
 ward, and we have, therefore, condensations starting from 
 one set and rarefactions from the other at the same moment, 
 and at a little distance above the plate, where the effects due 
 to each of these wave-systems would be about equal, we have 
 practically no changes of density at all. If we cut a circle 
 of cardboard of the same size as the plate, divide it into the 
 same number of sectors that the plate is vibrating in, cut 
 away alternate sectors nearly to the middle, and hold the 
 remaining part of the card B so as to cover, but not to touch, 
 alternate sectors of the plate, we shall find that the sound at 
 a few feet above the plate is much louder than before, since 
 the waves from one set of sectors are absorbed by the card 
 and those which pass through the openings are all in agree- 
 ment. 
 
 The number and position of the sectors of the plate may 
 be ascertained by scattering on it a little sand, which quickly 
 collects on the lines which divide the oppositely vibrating 
 sectors. 
 
 Hopkins* s Forked Tube is used in conjunction with a 
 
70 
 
 SOUND. 
 
 vibrating plate to illustrate the principle of interference. It 
 is a branched tube shaped like a letter Y The ends of the 
 branches are open, but a piece of gold-beater's skin is 
 stretched over the end of the main stem. The tube is held 
 in an inverted position (^) so that the membrane is at the 
 top, and a little sand is scattered on the membrane. If a 
 plate like that shown in Pig. 30 is made to vibrate, and the 
 tube held so that the two openings are over sectors which 
 vibrate in the same direction, the sand is thrown off ; but, if 
 the openings are symmetrically over sectors which move in 
 opposite directions, the sand is only slightly disturbed. 
 
 The prongs of a tuning-fork (Fig. 31) move outwards 
 together ; they therefore condense the air outside them while 
 
 Fig. 31. 
 
 they rarefy the air between them, and condensation starts off 
 in all directions from the outer surfaces A, A' at the same time 
 that rarefaction is starting from the inner surfaces B, B'. The 
 air in the direction of the line joining the prongs is, to a 
 large extent, shielded from the waves from B< B by the prongs 
 themselves, while along a line at right angles to this the 
 effect of the waves from the outer surfaces A^ A is small, but 
 in four directions which make angles of about 45' with the 
 line joining the prongs the waves from the outer surface 
 A of one prong are almost exactly balanced by the waves 
 from the inner surface B' of the other prong, and in these 
 four directions very little sound is heard. The simplest way 
 
UrTERFERENCE. 71 
 
 to show this is to strike the fork, and then to twist the stem 
 of it round between your fingers so as to make the fork pre- 
 sent each side to your ear in turn ; the sound will be heard 
 to swell out and die away four times in each rotation of the 
 fork. 
 
 If, when your ear is in one of the four directions from the 
 fork in which the sound is faintest, you hold a paper tube so 
 as to enclose, but not to touch, one of the prongs, the sound 
 becomes much louder, showing that the waves produced by 
 the two prongs vibrating together are of less intensity than 
 either would produce by itself. 
 
 For reasons given above, it is very difficult to arrange an 
 experiment in which the imaginary wave-systems are so 
 exactly opposite that there is complete silence. A very near 
 approach to success can be made, however, by mounting two 
 equal organ pipes, open at the upper ends (as well as at the 
 lower, or mouthpiece, ends), close side by side on the same 
 wind-chest. When these are blown, they always start in 
 opposite phases, so that a condensation leaves the mouth of 
 one at the same moment that a rarefaction leaves the mouth 
 of the other, and, as the waves from open pipes are almost 
 perfectly harmonic, the condensation of one imaginary system 
 is almost exactly equal to the rarefaction of the other at 
 every point in the surrounding space. If there is any vibra- 
 tion of the air at all due to the two pipes, it cannot be heard 
 among the rushing sound from the air at the mouth- 
 pieces. Yet, if we stretch a thin membrane on a ring, and 
 lower this horizontally, with a little sand on it, into either of 
 the pipes, the sand is at once thrown off, showing that the 
 air-columns are vibrating. 
 
 41. Energy of Waves from two Sources. When the 
 condensations of two equal imaginary wave-systems agree in 
 any region, the actual amplitude of vibration should, on the 
 principle of interference, be twice as great as either of the 
 sources would produce separately, and, therefore, the actual 
 intensity four times as great as the intensity of the sound 
 which either of the sources would produce separately. There 
 is, therefore, four times as much energy passing per second 
 through the region as there would be if only one source 
 
72 SOUND. 
 
 vibrated. This looks at first like a creation of energy, but 
 of course that never occurs. If the sources of the two 
 imaginary systems are several ^ave-lengths apart, there are 
 some regions in the surrounding space where the imaginary 
 wave-systems are in the same phase, and some regions where 
 they are in opposite phases ; in the former the intensity is 
 greater, and in the latter less, than the sum of the intensities 
 of the imaginary systems. The total energy travelling away 
 from the sources is the sum of the amounts which would be 
 travelling away if the sources vibrated separately, though it 
 is differently distributed; in some directions more energy 
 travels away, and in other directions less, than the sum of 
 the amounts which would travel in those directions if the 
 sources vibrated separately. 
 
 If the sources are very close together, and vibrate in the 
 same phase, the imaginary wave-systems are in the same 
 phase in every part of the space round the sources ; in this 
 case either the actual amplitude must be less than the sum of 
 the imaginary amplitudes (so that the principle of interfer- 
 ence does not hold), or the sources must give out energy 
 faster than they would do if they vibrated separately. 
 Usually it is the latter that occurs ; each vibrating body, 
 moving in air which is already moving in the same direction, 
 gives up energy to it faster than it would to undisturbed air, 
 just as a man pushing at a truck gives energy to it much 
 faster when it has acquired some speed than he does when it 
 has only just started and is moving very slowly. 
 
 42. Waves of unequal Length. So far we have consi- 
 
 Fig. 32. 
 
 dered only cases in which the imaginary wave-systems are of 
 the same wave-length. Fig. 32 illustrates the actual con- 
 
INTERFERENCE. 73 
 
 dition of the air when there are in it two imaginary wave- 
 systems, both harmonic and travelling in the same direction, 
 but of different wave-lengths. As in previous figures, the 
 complete lines are the wave-forms of the imaginary wave- 
 systems, and the dotted curve, obtained by adding the 
 ordinates of the others, is the wave-form of the actual wave- 
 system.* Its bends are not harmonic in form, and their 
 maximum ordinates are greater in some places than in others, 
 so that in some regions the condensations and rarefactions are 
 intense, in others only slight. As the conditions indicated 
 by this wave-form pass any point the air there moves back- 
 wards and forwards through a distance which alters at each 
 movement, being greatest when the most intense regions of 
 the wave-system are passing. 
 
 In this case the actual wave-system contains as much 
 energy as the two imaginary systems would contain, but 
 differently distributed ; it is nearly all in the regions where 
 the condensations and rarefactions are intense. f 
 
 The sound heard by an ear at which such a wave-system 
 arrives presents peculiarities which will be discussed later, 
 but the most obvious is the alternation of increased and 
 diminished loudness as more intense and less intense regions 
 of the travelling system arrive at the ear. 
 
 43. Beats. These changes in loudness are called leat$. 
 They are readily heard when two tuning forks, whose fre- 
 quencies differ by three or four, are sounded at the same 
 time, or two organ-pipes, which have nearly the same fre- 
 quency. 
 
 If V is the velocity of sound, and n l9 n 2 the frequencies 
 of two sources of sound, then there are n complete waves of 
 one imaginary system, and n 2 complete waves of the other, in 
 
 * For greater clearness, the imaginary and real wave-forms are 
 here drawn on separate axis-lines, instead of as in the previous 
 diagrams, but the three curves denote conditions of the air in the 
 same region. 
 
 t The name " interference " is, by some writers, reserved for the 
 cases in which the actual energy is very differently distributed from 
 that of the imaginary systems; that is, for the cases where the 
 imaginary systems are of equal, or nearly equal, wave-lengths. 
 
74 SOUND. 
 
 
 
 a distance F", and, therefore, n 1 n 2 places in that distance 
 where the phases of the imaginary systems agree ; these are 
 the regions of maximum intensity. The distance between 
 
 77" 
 
 two regions of maximum intensity is, therefore, , and 
 
 ! n% ^ 
 
 the number of such maxima which pass any given point in a 
 second is n l n v The number of beats heard per second is, 
 therefore, the difference of the frequencies of the sources. 
 
 These beats or variations in the intensity of the vibration 
 at a point must be carefully distinguished from the conden- 
 sations or variations of density at the point. The beats in 
 fact are variations in the amount of variation of density ; the 
 mean density is the same in the more intense parts of the 
 wave-system as in the less intense parts. 
 
 The principle of interference is very nearly true when the 
 condensations and rarefactions are slight ; but in cases where 
 the density -differences are not very small, compared with the 
 mean density of the air, the actual wave-form is different 
 from the one found by adding the ordinates of the wave-forms 
 of the wave-systems which the sources would produce if 
 they vibrated separately. Such cases will be considered in 
 Chapter VI. 
 
CHAPTER V. 
 
 FORCED VIBRATION. 
 
 44. Free and Forced Vibration. As the series of condi- 
 tions which constitutes a wave-system travels through the 
 air, the density at any fixed point increases and diminishes 
 alternately, and the pressure changes proportionally to the 
 changes of density. If there is a solid body stationary in 
 the air, the pressure on its surface rises and falls as the 
 waves pass it, varying harmonically .(Art. 5) if the waves 
 are harmonic, or, in any case, rising and falling propor- 
 tionally to successive ordinates of their wave-form. As all 
 bodies yield more or less to pressure, the body will vibrate. 
 Such vibrations, caused by continuous changes of external 
 pressure, are called forced vibrations, to distinguish them 
 from the free vibrations which take place when a body is dis- 
 placed from a position of equilibrium and then left to itself. 
 In most cases it is not possible to determine by elementary 
 methods the exact character of the forced vibrations which a 
 given wave-system will cause in a given body ; we can only 
 say that, as the differences of pressure in different parts of an 
 ordinary sound wave are very small, the vibration produced, 
 if it depends on the changes of pressure actually taking 
 place at the moment, will usually be very small also. 
 
 45. Resonant Forced Vibration. But if the successive 
 waves are all exactly equal and similar, so that the pressure- 
 change^ is exactly periodic (i.e., repeats itself exactly at 
 regulars intervals), and if the period of this pressure 
 change is related in a certain way, to be explained presently, 
 to one of the periods of free vibration of the body (it may 
 have several, according to the way in which it is displaced), 
 the effects of a number of successive changes of pressure 
 
76 SOUND. 
 
 may be added together, and so vibratory movements of the 
 body may be produced, which so greatly exceed in amplitude 
 those which would be caused by the pressure-changes 
 actually occurring at the moment, that the latter may be neg- 
 lected in comparison. Such forced vibration, greatly exceed- 
 ing in extent that which would be due to the pressure-changes 
 actually occurring, and resulting chiefly from past pressure- 
 changes, is called resonant forced vibration. We shall some- 
 times call it simply resonant vibration, though this term is 
 often applied also to free vibration when caused by a previous 
 sound. 
 
 46. Experiments on Kesonant Vibration. We have next 
 to determine under what circumstances such resonant forced 
 vibration is produced, and for this purpose we may try some 
 simple experiments. 
 
 If we hang up two heavy weights, A and B, by strings 
 of equal length, and set them swinging, we shall find it easy 
 by slightly altering the length of one string to get them to 
 swing in almost exactly equal times, so that if started in the 
 same direction they will keep time with each other for some 
 hundreds of consecutive swings. Now stop B, and tie a fine 
 thread to it, and every time A passes the middle point of its 
 swing in one direction, say from right to left, give a very 
 slight momentary pull to the thread, not enough to produce 
 any visible movement of B. After a few seconds, we shall 
 find that B is swinging very perceptibly, and at the end of a 
 minute or two we shall have it swinging through quite a 
 large arc. The weight swings to a distance on each side of 
 its mean position much greater than the displacement which 
 would be due to the actual pull of the thread ; often to 
 a greater distance than the thread could pull it aside without 
 breaking. 
 
 The same thing happens in many other cases. The regular 
 tramp of soldiers crossing a bridge will break the bridge 
 down if its period of oscillation agrees with the interval 
 between the steps ; and the vibrations of the air caused by 
 an organ (it is said even by the voice) will break a pane of 
 glass in a window if the pane is of such a size that it vibrates 
 with the same frequency as the waves reach it. 
 
FOBCED VIBRATION. 77 
 
 47. Different Effects of Harmonic and Non-harmonic 
 Pressure-Changes. If we alter the length of A's string a 
 few inches, so that the times of swing are slightly unequal, 
 and repeat the experiment, no large vibration of B will be 
 produced. But if we lengthen As string till A makes only 
 one vibration to two of B (or, which is simpler, if we make 
 A swing with the same period as B, but pull the thread only 
 at the end of every two double swings of A] we shall 
 gradually increase the movement of B, till it becomes nearly 
 as large as before. The same occurs if we pull the thread 
 every third time A passes its lowest point from right to left, 
 or at any regular intervals which are exact multiples of .Z?'s 
 free-vibration period. So that in some cases periodic 
 impulses can cause a resonant vibration whose period is an 
 exact submultiple of their own. But there is a very im- 
 portant exception to this. If the periodic impulses, instead 
 of being forces which begin and stop suddenly, like the pull 
 of the thread, are forces which increase and diminish har- 
 monically, they will not set in resonant vibration a body 
 whose natural period is an exact submultiple of their own. 
 This can easily be shown by a slight variation of our 
 experiment. Hang up A and B (still by strings of equal 
 length) a long distance, say 20 feet, apart, and tie one end of 
 a piece of very thin elastic, about 2 feet shorter than this 
 distance, to B. Hold the other end of this elastic in your 
 hand, exactly under A, so that it is slightly stretched. Set 
 A swinging, with an amplitude of about a foot, along a line 
 passing through B, and move your hand as A swings, so as 
 to keep the end of the elastic exactly under A. The pull of 
 the elastic on B will then increase and diminish harmonically, 
 or very nearly, and it will be found that B will begin to 
 vibrate, and keep increasing its vibration, if its free-vibra- 
 tion period is the same as that of A. But if we lengthen 
 the string of A till it makes only one swing to two of J?, and 
 then keep the end of the elastic under A as it swings, no 
 considerable movement of B will be produced. B does move, 
 of course, but the movement does not increase beyond that 
 due to a single pull. Now, we saw that a slight pull on a 
 thread tied to B, occurring once in every two swings of J5, 
 would produce a large vibration. The difference between 
 
78 SOTTED. 
 
 the two cases is this : The pull of the thread is a force 
 which, at regular intervals, increases very suddenly from 
 zero to its maximum, and, aftor lasting a very short time, 
 falls very rapidly to zero. In other words, the thread exerts 
 on B a force which varies periodically but not harmonically, 
 while the pull of the elastic varies harmonically. We con- 
 clude that a periodic force which increases and diminishes 
 harmonically does not cause resonant vibrations of periods 
 which are submultiples of its own, though a force which 
 varies non-harmonically may do so. 
 
 48. Period and Amplitude of Forced Vibrations. Now 
 adjust the lengths of the strings till A swings nearly, but 
 not exactly, with the same frequency as B. Suppose 30 
 swings of A require the same time as 31 of . If we repeat 
 the experiment with the elastic, we shall find that B will 
 still be set vibrating, and that its amplitude of vibration will 
 increase for a time, but will soon reach a limit which is much 
 smaller than when the periods agree more exactly. "We 
 shall find also that the vibration of B agrees in period 
 exactly with that of the pulls given to the elastic, and does 
 not depend on its own natural period of vibration. If the 
 period of A's vibration is 2 seconds, the forced vibrations of 
 B will have a period of 2 seconds also, whatever the natural 
 period of B) but, unless that natural period is about 2 seconds, 
 the vibration of B never becomes very much greater than 
 that which one of the pulls would produce. 
 
 Though the vibration produced in a body by a periodic 
 force, which only nearly agrees in period with the free 
 vibrations, is always of smaller amplitude than it would be 
 if the periods agreed exactly, the extent of the difference 
 depends very much on the mass of the vibrating body and 
 the amount of friction resisting its motion. If the mass is 
 large and the friction small, as in the case of a heavy 
 pendulum or tuning fork, the resonant vibration produced by 
 impulses of nearly the natural period of the vibrating body 
 is very small compared with that produced by equal impulses 
 of exactly the right period. Eut if the mass is small and the. 
 friction large, as in the case of a boat rolling in the waves, 
 or air vibrating in a narrow tube, impulses of nearly the right 
 
FORCED VIBRATION. 79 
 
 period will produce nearly as strong resonant vibration as 
 impulses of exactly the right period. 
 
 Practically, when we have to do with a body which is not 
 very light, and periodic pressure-changes which are not very 
 violent, we may assume that no perceptible movement will 
 be caused by a harmonic pressure-change of any period which 
 is not almost exactly one of the periods in which the body 
 can vibrate freely.* So harmonic waves do not cause strong 
 vibration of bodies which they pass, unless the waves arrive 
 with frequencies which are nearly free -vibration frequencies 
 of the bodies. 
 
 We have not yet discovered what are the conditions under 
 which non-harmonic waves cause resonant vibration, though 
 we have seen that a non-harmonic rise and fall of pressure, 
 such as non-harmonic waves would produce, sometimes causes 
 resonant vibration where harmonic waves, arriving with the 
 same frequency, would fail. To determine what non-har- 
 monic waves will cause resonant vibration of a given body, 
 we must resort to a mathematical device, explained in the 
 next chapter. 
 
 * Under exceptional circumstances this may be untrue ; an 
 instance is given in a later chapter. 
 
CHAPTER VI. 
 
 FOURIER'S THEOREM. 
 
 49. Components. We saw in the last chapter how the 
 rise and fall of pressure at any point can be represented by 
 a curve. Suppose that the dotted line A CBDZ (Fig. 33 or 
 34) represents the rise and fall of pressure at some point P, 
 on a scale of, say, an inch to a second, the ordinate A' A 
 representing the amount by which the pressure is below the 
 average at a given moment T\ an ordinate \ inch to the 
 
 Fig. 34. 
 
 right of A A representing the difference from the average 
 pressure | second later, and so on. Now, there 
 
 s no 
 
 , . , 
 
 physical difference between a force / and two forces, acting 
 together, whose sum is equal to /. So that, instead of saying 
 that the pressure -difference* at the moment T is A' A, we 
 
 * The term "pressure-difference" will be used to mean the 
 difference between the actual pressure of the air at a given moment 
 and its average pressure. 
 
FOURIER'S THEOREM. 81 
 
 can say that it is the sum of pressure-differences represented 
 by AF and A'J. So that, if FGHY and JKLX are any 
 two curves whose ordinates everywhere add up to those of 
 A CBDX, there is no difference between a rise and fall of 
 pressure represented by A CBDX and two pressure-changes, 
 happening at the same time, represented by FGHY and 
 JKLX. 
 
 A dynamical proposition, too difficult for this book, shows 
 that if the force on any body, due to a displacement from its 
 position of rest, is proportional to the displacement (which is 
 usually very nearly the case in the small displacements which 
 we are considering), then the movement of this body produced 
 by several pressure-changes happening together is the sum 
 (in the sense explained in Art. 7) of the movements which 
 the same pressure-changes would produce separately. 
 
 Pressure-changes such as those represented by FGHY &&& 
 tTJTLX, which add up to a given pressure-change such as 
 that represented by A CBZ, are called components of the 
 pressure-change represented by A CBZ. 
 
 In Figs. 32, 33, FGHY and. JKLX are drawn as harmonic 
 curves, and so represent harmonic pressure-changes, but the 
 proposition just given is just as true for any other kind. 
 And we could evidently draw as many sets of curves as we 
 pleased whose ordinates would add up to those of A CBZ, so 
 that any pressure-change has an unlimited number of sets ~of 
 components, and the dynamical proposition just given shows 
 that, to predict the movements which will be produced by 
 any pressure-change such as that represented by A CBZ, we 
 may first find what movements would be produced by the 
 members of any set of components separately, and then add 
 these movements. 
 
 All this is true whatever kind of pressure-changes the 
 components are. But, unless the components are harmonic 
 pressure-changes, the device of components does not help us, 
 for it is no easier to tell what movements they will produce 
 than it is to tell what movements the pressure-change repre- 
 sented by A CBZ will produce. 
 
 If we could find harmonic pressure-changes which would 
 add up to the pressure-change whose effect is to be found, 
 then, as we saw in the last chapter, it would be very easy to 
 
 SD. G 
 
82 SOUND. 
 
 say whether any of these, separately, would produce resonant 
 vibration of a given body whose free-vibration frequency is 
 known. It would not be easv to tell exactly what move- 
 ments the others would produce separately, but we can be 
 sure that they would be very small compared with the 
 resonant vibrations. So that practically, if any of these 
 harmonic pressure-changes would produce resonant vibration 
 of the given body, we may neglect the effect which the 
 others would produce. 
 
 50. Fourier's Theorem. Figs. 15, 32, and 33 show that 
 there are some non-harmonic curves whose ordinates are the 
 sums of the ordinates of harmonic curves, so that if a rise and 
 fall of pressure corresponds to one of these non-harmonic 
 curves, its effect in producing resonant vibration can be easily 
 calculated by finding the effects of pressure-changes corres- 
 ponding to the harmonic curves. A very important theorem, 
 due to Fourier, shows that this can be done not only in some 
 cases, but in all. 
 
 For our purpose we may state Fourier's Theorem in this 
 way. Let there be any given line, such as the dotted line 
 ACB in Fig. 32 or 33, which does not overhang anywhere, 
 so that no ordinate cuts it more than once, and let A, B be 
 two points on it. Then it is always possible to find sets of 
 harmonic curves whose ordinates will add up to those of the 
 line ACB. And (which is the important point for our 
 purpose), if A and B are points whose ordinates are equal, 
 there is, among the sets of harmonic curves whose ordinates 
 add up to those of the line A CB, one set (and only one) 
 which consists entirely of harmonic curves whose lengths 
 (the length of a double bend measured along the axis) are 
 contained an exact number of times between the ordinates to 
 A and B. ' V 
 
 Figs._ 32,~$3 illustrate this. The ordinates at A and B 
 are equal, and the ordinates at every point of A CB are the 
 sums of the ordinates of the two harmonic curves FGH and 
 JKL, whose lengths are each contained an exact number of 
 times in the length A'B'. We could find other sets of 
 harmonic curves which would add up to A CB, but there is 
 only this one set of harmonic curves whose lengths are con- 
 
FOURIER'S THEOREM. 83 
 
 tamed an exact number of times between A and B ', which will 
 add up to A CB. 
 
 To simplify the diagram, we have illustrated a case in 
 which this set of harmonic curves consists of only two. 
 Usually there are more than two, and the number may be 
 infinite. 
 
 51. Application to Sound. Fourier's Theorem is purely 
 a mathematical one. Its connection with sound arises from 
 the fact that, as the pressure of the air cannot have more 
 than one value at any one point at the same moment, any 
 possible rise and fall of pressure can be represented by a 
 curve which does not overhang anywhere, and that if this 
 rise and fall of pressure is periodic, or exactly repeats itself 
 at equal intervals, this curve may be divided into lengths 
 which are all alike, so that the first and last ordinate of each 
 length are equal. Suppose the dotted line A CBZ represents 
 a periodic rise and fall of pressure, and that A CB represents 
 one complete cycle of changes of pressure, so that the rest of 
 the curve consists of repetitions of A CB. Let FGH and 
 JKL be the set of harmonic curves whose lengths are con- 
 tained an exact number of times between A and B ', and 
 which add up to A CB. Then, if FGH, JKL are continued 
 to the right, they repeat themselves in the next period B'D' 
 and these portions add up to the same curve as before 
 that is, to the form BD. So that, however far we 
 continue ACBLZ, the harmonic curves FGHX, JKLY, 
 also continued, will everywhere add up to the curve 
 ACBDZ. 
 
 As the harmonic curves FGH, JKL repeat themselves 
 exactly an exact number of times in the length ABf, they 
 represent harmonic pressure-changes which repeat themselves 
 an exact number of times in the period represented by A'B', 
 that is, they are harmonic pressure-changes whose frequencies 
 are exact multiples of the frequency of the non-harmonic 
 pressure-change represented by A CBZ. In the figure, FGH 
 represents a harmonic rise and fall of pressure occurring once, 
 and JKL a harmonic rise and fall of pressure occurring three 
 times, in one complete period of the non-harmonic pressure- 
 change represented by the dotted line. 
 
84 SOUND. 
 
 52. Harmonic Components. If there is a non-harmonic 
 pressure-change repeating itself with frequency n, then 
 Fourier's Theorem shows us thpt we can always find a series 
 of harmonic pressure - changes, of frequencies which are 
 exact multiples of n (n itself being included), such that, 
 occurring together, they are the given non- harmonic 
 pressure - change. These harmonic pressure -changes are 
 called the harmonic components of the non-harmonic pressure- 
 change. 
 
 It should be clearly understood that the statement that a 
 non-harmonic pressure-change of frequency n is always the 
 sum of harmonic pressure-changes of frequencies which are 
 multiples of n, does not at all imply that any actions are 
 taking place which could cause pressure-changes of these 
 frequencies, or that these pressure-changes have any separate 
 existence. It is only true in the same sense that anything 
 5 feet long is the sum of two parts, one 3 feet and the other 
 2 feet ; it does not mean that there is any physical distinction 
 between the parts. For instance, if a man rows a boat, 
 making (we will suppose, for convenience, that it is possible) 
 one double stroke per second, he exerts on the oar a push- 
 and-pull whose frequency is 1 per sec. This push-and-pull 
 is not harmonic ; instead of gradually rising to a maximum 
 and then gradually diminishing, it is nearly uniform till near 
 the end of the stroke. Fourier's Theorem shows us that it 
 is the sum of a number of harmonic pushes and pulls whose 
 frequencies are 1, 2, 3, 4, 5, &c., per second, happening 
 together. This does not mean that the man is really giving 
 pushes and pulls at these rates ; he pulls continuously for 
 more than half-a-second at a time. But if we replaced the 
 man by a number of men, of whom one pushed and pulled 
 harmonically once a second, one twice a second, one three 
 times a second, and so on, then, if the intensities of these 
 harmonic pushes and pulls, and the moments of beginning 
 them, were rightly chosen, the push-and-pull on the oar 
 would be the same as the non -harmonic push-and-pull exerted 
 by the actual rower. And this is all that is meant by calling 
 these harmonic pushes-and- pulls the components of his non- 
 harmonic push-and-pull. 
 
 The proposition of Art. 49 shows that the effect of a non- 
 
FOURIER'S THEOREM. 85 
 
 harmonic change of pressure in producing vibration is the 
 sum of the effects which its harmonic components (or any 
 set of components) would produce separately. And we saw 
 in the last chapter that a harmonic rise and fall of pressure 
 produces resonant vibration in a body if the frequency of the 
 pressure-change is one of the free-vibration frequencies of 
 the body. Putting these facts together, we see that a non- 
 harmonic rise and fall of pressure on a body will set it in 
 resonant vibration if, and not unless, one of the harmonic 
 components of the non-harmonic pressure-change has a 
 frequency which agrees with one of the free-vibration 
 frequencies of the body. 
 
 Of course, any periodic pressure-change on a body sets it 
 in vibration. The statement just given relates only to 
 resonant vibration. The changes of pressure, as sound-waves 
 pass, are so small that, unless the effects of a number of 
 changes are added by resonance, we may generally neglect 
 the movements produced. 
 
 We now see why, in the experiment described in Art. 46, 
 a momentary pull every 4 seconds sets a pendulum, whose 
 free-vibration frequency is 2 seconds, in resonant vibration. 
 The pull of the string on the pendulum rises and falls non- 
 harmonically once in 4 seconds, and one of the harmonic 
 components of this non-harmonic variation of force is a pull 
 rising and falling harmonically twice in 4 seconds. We do 
 not actually pull twice in 4 seconds, but a harmonic pull 
 whose frequency is twice in 4 seconds, occurring at the same 
 time with other harmonic pulls whose periods are contained 
 exactly in 4 seconds, would be the non-harmonic pull (in- 
 creasing and diminishing again very rapidly once every 
 4 seconds and then remaining zero the rest of the period) 
 which the string really gives to the pendulum. 
 
 53. Fundamental and its Harmonics. If a non-harmonic 
 pressure-change has a frequency n, then a harmonic pressure- 
 change of frequency n is called the fundamental pressure- 
 change of the non-harmonic one, and harmonic pressure- 
 changes whose frequencies are 2w, 3w, 4w, &c., are called the 
 2nd, 3rd, 4th, &c., harmonics of that fundamental. The 
 harmonic components of a non-harmonic pressure-change are 
 
86 SOUND. 
 
 all included among its fundamental and the harmonics of 
 that fundamental. If the fundamental is one of the 
 harmonic components, it is called the fundamental com- 
 ponent. 
 
 The same method and the same terms may be applied to 
 anything which varies with the time, and which can there- 
 fore be represented by a curve. For instance, a non-harmonic 
 vibration of a point of frequency n is always the sum (Art. 7) 
 of harmonic vibrations of frequencies which are multiples of 
 n, and these harmonic vibrations are called the harmonic 
 components of the non -harmonic vibration. 
 
 54. Harmonic Components of a Wave-System. We may 
 
 also apply the same theorem to the wave-system which causes 
 the rise and fall of pressure. Instead of representing a 
 pressure-change, suppose now that the curve A CBZ (Fig. 32 
 or 33) is the wave-form of any non-harmonic wave-system, 
 so that A CB is the wave -form of one wave-length. Then 
 Fourier's Theorem shows us that we can always find a set 
 of imaginary harmonic wave-systems whose wave-forms, 
 FGHY and JKLX, add up to the wave -form of the non- 
 harmonic system, and whose wave-lengths are contained an 
 exact number of times in one wave-length of the non -harmonic 
 system. These imaginary harmonic wave-systems are called 
 the harmonic components of the real non-harmonic wave- 
 system. Each wave-system would produce, as it passed any 
 point, a rise and fall of pressure represented by a curve which 
 is the same as the wave form of the system, so that the effect 
 of any non-harmonic wave-system is the sum of the effects 
 which its harmonic components, if they were real wave- 
 systems, would produce separately. 
 
 A non-harmonic wave-system, therefore, sets, a body in 
 resonant vibration if any one of the harmonic components of 
 the non-harmonic system is one whose waves, if they were 
 real, would arrive with a frequency corresponding to one of 
 the free -vibration frequencies of the body. 
 
 A given non-harmonic wave-system is often said to " consist 
 of" its harmonic components. This is rather misleading, 
 because it suggests the idea that this particular set of com- 
 ponents has a real existence in some different sense to other 
 
FOURIER'S THEOREM. 87 
 
 sets of components, harmonic or other, whose wave-forms 
 would also add to that of the actual wave-system. This is 
 not the case ; we could always find sets of non-harmonic 
 waves, and even sets of non -harmonic waves all of the same 
 kind, whose wave-forms would add up to the wave-form of 
 any given wave-system, and the real waves " consist of" any 
 of these sets of non-harmonic components exactly in the same 
 sense as they " consist of " the harmonic ones. All " com- 
 ponents," harmonic or other, are imaginary, and the only 
 reason why we do not usually refer to any other set except 
 the harmonic ones is that the effect of a succession of similar 
 harmonic waves in producing resonant vibration is so easy to 
 predict. It is only when we want to know what resonant 
 vibrations a given wave-system can produce (and in one 
 other case which will be considered later) that there is any 
 advantage in Fourier's device of adding the effects which the 
 imaginary harmonic components would produce separately, to 
 find the effect of the real waves. 
 
 We shall see later that, owing to the nature of the 
 mechanism of hearing, a separate sensation corresponds to 
 each of the harmonic components of a system of non-harmonic 
 waves. This of course does not in any way indicate that the 
 harmonic components have an external physical existence, 
 any more than the seven distinct colour sensations which we 
 receive from the spectrum indicate that there are seven 
 physically different kinds of light. It only shows that the 
 mechanism of hearing is such that each distinct sensation 
 depends on the resonant vibration of a distinct body, so that 
 the sensations which a wave-system can cause depend on the 
 bodies it can set in resonant vibration. We shall see in the 
 next chapter how this mechanism works. 
 
 54. Harmonic Analysis. As the harmonic components of 
 a wave-system, and the harmonic components of the rise and 
 fall of pressure produced by it, are represented by the same 
 harmonic curves, to find the harmonic components of the 
 pressure-change is equivalent to finding the harmonic com- 
 ponents of the wave-system. If we knew the character of 
 the rise and fall of pressure exactly, so that we could draw 
 the curve representing it, we could find by geometrical 
 
88 SOUND. 
 
 methods the harmonic curves which add up to that curve, 
 and these represent the harmonic components of the pressure- 
 change. Eut it is not practical]^ possible to determine either 
 the wave-form of a series of waves or the rise and fall of 
 pressure (represented by the same curve) which the waves 
 produce. It is much easier to determine the harmonic com- 
 ponents of the pressure-change by finding experimentally 
 what bodies the pressure-change will set in resonant vibration. 
 It is not necessary to have bodies of every possible free- 
 vibration period for this. The harmonic components of a 
 pressure-change are all pressure-changes whose frequencies 
 are exact multiples of the frequency of the actual rise and 
 fall of pressure. If, for instance, the complete cycle of 
 changes of pressure is repeated 100 times per second, the 
 harmonic components of the pressure -change are all pressure- 
 changes whose frequencies are multiples of 100. So that, if 
 we expose to this rise and fall of pressure bodies whose free- 
 vibration frequencies are 100, 200, 300, 400, &c., some of 
 these will be set in resonant vibration, and the frequencies of 
 these are the frequencies of the harmonic components of the 
 actual rise and fall of pressure. 
 
 For this experiment it is evidently desirable that the 
 bodies used should be bodies which have only one free- 
 vibration frequency. There are very few such bodies ; the 
 best for the purpose is the air contained in a globular or 
 cylindrical vessel which has an opening much narrower than 
 the greatest diameter of the vessel. Such a vessel is called 
 a " resonator." The details of the experiment are given in 
 Chapter XII. 
 
 The determination of the harmonic components of a 
 pressure-change, vibration, wave-system, &c., is called the 
 harmonic analysis of it. 
 
 55. Relation of Harmonic Components to Source of 
 Waves. In some cases, as shown in Art. 39, a non-harmonic 
 wave-system is produced by two or more sources, each of 
 which alone would produce a harmonic system. In this 
 case, as we saw, if the vibrations are of small amplitude, the 
 wave-form of the actual waves is the sum of the harmonic 
 wave-forms of the wave-systems which the sources would 
 
FOURIER'S THEOREM. 89 
 
 produce separately. These harmonic wave - systems are 
 therefore the harmonic components of the actual wave- 
 system. 
 
 Even where there are not any sources which really vibrate 
 harmonically, the harmonic components of a wave-system, 
 though quite imaginary, are often similar to real harmonic 
 wave-systems which might be produced by the actual source. 
 It often happens that a body vibrates so that every part of it 
 moves harmonically in the same period ; in this case it usually 
 produces nearly harmonic waves. If there are several 
 different ways in which the body could vibrate harmonically, 
 with different periods, there are several harmonic wave- 
 systems of different wave-lengths which it could produce. If 
 such a body vibrates in any other way, it is usually one in 
 which its movement is the sum (Art. 7) of two or more of 
 the harmonic vibrations which it could perform ; in this case 
 the waves from it have a wave-form which is the sum of the 
 wave-forms of the harmonic waves which these vibrations 
 would produce, so that these harmonic waves are the harmonic 
 components of the actual wave-system. 
 
 In these cases the harmonic components of the waves 
 produced are similar to real waves which might be produced 
 by the same sources. It must not be supposed that this is 
 always the case, still less that the possibility of finding 
 imaginary harmonic systems whose wave-forms would add up 
 to that of the waves from a given source depends at all on 
 whether the source of sound could vibrate so as to produce 
 such harmonic systems. In many cases there is no relation 
 between the harmonic components of a wave-system and any 
 real harmonic waves which the source could possibly produce. 
 For instance, if air is blown from a pipe against a revolving 
 disc having 100 holes drilled in a circle at the same distance 
 from the axis as the end of the pipe, a puff of air will come 
 through each hole as it passes 100 puffs per second if the 
 disc rotates once in a second. At the disc, every process 
 which occurs at all is repeated 100 times a second, and not 
 oftener; there is no operation being repeated 200 or 300 
 times a second. A hundred similar waves, each about 1 1 feet 
 long, start from the disc in each second. These waves are 
 far from harmonic in character, the pressure of the air on the 
 
90 SOUND. 
 
 side of the disc away from the pipe rising very suddenly as 
 a hole comes opposite the pipe and falling very suddenly 
 when the hole has passed, instead of the gradual harmonic 
 rise and fall. But the wave-form of these waves, as Fourier's 
 theorem shows, might be made by adding the ordinates of the 
 wave-forms of harmonic waves 11 feet, -y-feet, -^-feet, &c., 
 in length. So that harmonic waves of these lengths are 
 harmonic components of the waves from the disc, although 
 to produce real harmonic waves feet long would require a 
 movement repeating itself exactly 300 times in each second, 
 and nothing that happens at the source is repeated with this 
 frequency. 
 
 56. Resultant Tones. We may go further than this, and 
 say that there are always, among the imaginary harmonic 
 components of a wave-system, some which could not really 
 be produced by any vibration which the source of sound 
 could execute. The phenomenon of u resultant tones " is one 
 instance of this. As stated in Art. 37, the principle of 
 interference is not exactly true except for infinitely small 
 vibrations. If there are two sources, A and j#, each of which 
 would by itself produce intense harmonic waves, the wave- 
 form of the actual waves can only be approximately made by 
 adding the harmonic wave-forms of the waves which A and 
 B would produce separately. It can, as Fourier's theorem 
 shows, be exactly made by adding harmonic wave-forms, but 
 some others are required besides those of the waves which 
 could really be produced by A and B separately. The most 
 important of these additional harmonic curves are wave-forms 
 of waves which might be produced by two other sources 
 vibrating harmonically with frequencies which are respectively 
 the sum and difference of the frequencies of A and B. The 
 actual waves have therefore four imaginary harmonic com- 
 ponents, and the frequencies with which the waves of these 
 harmonic components would arrive, if they were real, are (if 
 we call the frequencies of A, B, a and b respectively) #, b, 
 a + 5, aI). As stated above, waves may produce a distinct 
 sensation for each of their harmonic components; so that 
 there are two additional sensations when A and B vibrate 
 together, which cannot be produced by either separately. 
 
FOURIER'S THEOREM. 91 
 
 These additional sounds are called respectively the summation 
 and difference tones. 
 
 57. Change of Components. As stated in Art. 36, waves 
 change their form a little as they travel. At some distance 
 from the source their form cannot be made by adding the 
 same harmonic curves which add up to their form near the 
 source. In other words, the harmonic components change as 
 the waves travel ; the components at a distance are not the 
 same as near the source. But if the fundamental is one of 
 the components at one place it always is so at any other 
 place. 
 
 58. Different Wave-systems with same Components. 
 As shown in Figs. 32, 33, the same harmonic curves, in 
 different relative positions, add up to quite different forms. 
 So that two wave-systems, of quite different wave-forms, 
 may have the same harmonic components, in different relative 
 positions. Two such wave-systems set the same bodies in 
 resonant vibration, and therefore cannot be distinguished 
 from each other by merely observing which of a series of 
 resonators they excite. 
 
 In this chapter we have considered chiefly the rise and fall 
 of pressure on a body as waves pass it, because it is to changes 
 of pressure that the resonant vibration of the body is directly 
 due, and the chief use of Fourier's device is to determine 
 what bodies will be set in resonant vibration. Eut the 
 vibration of the particles of the air as the waves pass may 
 also be represented by a curve, and may therefore be con- 
 sidered as the sum of a number of harmonic vibrations, of 
 frequencies which are multiples of the frequencies of the 
 actual vibration. The curve which represents the movement 
 of a particle of air (the displacement curve, Art. 3) is not 
 the same as the curve which represents the rise and fall of 
 pressure in that particle, which is the same as its velocity 
 curve (Art. 15). The two curves, however, can always be 
 made by adding the same harmonic curves in different relative 
 positions. So the harmonic components of the vibration have 
 the same frequencies as the harmonic components of the 
 change of pressure. 
 
92 SOUND. 
 
 The constitution of a non-harmonic vibration means the 
 frequencies and amplitudes of its harmonic components 
 without reference to their relath e phases, and the same term 
 is applied to pressure-changes, wave-systems, &c, So that 
 the vibrations, pressure-changes, or wave-systems represented 
 by the dotted curves in Figs. 33 and 34 are said to have the 
 same constitution, though not the same character. 
 
CHAPTER VII. 
 
 THE EAR AKD HEARING. 
 
 59. Structure of the Ear. The rise and fall of pressure 
 which occur as sound waves arrive at the ear produce, by 
 their effect on its structures, the sensation of sound, and the 
 nature of this sensation depends on the nature of the rise and 
 fall of pressure ; in other words, on the character of the 
 sound waves. We may, therefore, learn much about sound 
 waves from the sensations they produce. But, to understand 
 what physical facts about the waves are indicated by different 
 sensations, we must know something about the structure of 
 the ear. 
 
 This apparatus is a very complex one, but it is constructed 
 on quite simple principles. The external ear, or pinna, has, 
 in ourselves, no connection with hearing, which is not affected 
 by its removal ; but in many animals it is funnel-shaped, and 
 it then serves to increase the intensity of the vibration, in a 
 way explained in the next chapter. A funnel-shaped pinna 
 is also useful for ascertaining the direction from which the 
 waves come. From the pinna a tube, called the meatus, in 
 man about 1 J in. long, leads directly inwards, and is closed 
 at the inner end by a stretched membrane, the tympanum or 
 drum of the ear. This membrane is connected by a chain of 
 fca small bones to another membrane, the fenestra ovalis, 
 which forms part of the wall of the hearing chamber, or 
 cochlea. 
 
 The principle of this hearing chamber may be understood 
 if we imagine a long, low, narrow building of two storeys (a 
 ground floor and one above it), having at one end two doors, 
 one into the ground floor, the other into the upper storey. 
 The floor which separates the storeys is not complete ; it does 
 not extend quite to the further end of the building from the 
 
94 SOUXD. 
 
 doors, and along one side of the building it does not reach, as 
 a solid floor, up to the wall ; along this side, however, but 
 not at the end. it is completed by two sheets of membrane, 
 one completing the ceiling of the lower room, the other the 
 floor of the upper one ; between these membranes there is a 
 space, which is entirely shut off from the rooms, since the 
 membranes meet and join at the end furthest from the doors. 
 The lower and upper rooms are in free communication at the 
 end furthest from the doors, where the floor between them is 
 entirely wanting. The doors are thin and flexible. The 
 whole space inside the building is filled with water. 
 
 In the space between the two membranes there are about 
 3000 rods of different lengths ; these stand on the lower 
 membrane, in two parallel rows running its whole length ; 
 they are fastened at their lower ends to the membrane on 
 which they stand. The rods in each row lean towards the 
 other row, and at their upper ends each touches, and is joined 
 to, its fellow in the other row ; each such pair is free except 
 where the lower ends are attached to the lower membrane. 
 The whole series forms " a sort of gable roof." 
 
 If in such a building the thin material of the door of the 
 upper storey was pushed in, a wave, very similar to an 
 ordinary water wave on a canal, would travel all along the 
 upper room to the further end, depressing all portions of the 
 membranous part of the floor in turn as it passed along ; as 
 the liquid is incompressible, the membranous ceiling of the 
 lower room would be forced down at the same time, and the 
 flexible door of the lower room would be forced out. Every 
 movement of the door of the upper room produces a move- 
 ment of the membranous part of the ceiling of the lower and 
 of the rods supported by it, and if the door is made to vibrate 
 regularly, any rod whose natural vibration period corresponds 
 will be set in resonant vibration. 
 
 In the actual ear, a chamber constructed on a principle 
 similar to the building above described is coiled into a spiral 
 so close that one side of the building, that where the solid 
 floor is complete, becomes a mere central pillar. The cham- 
 ber resembles a snail shell in form ; hence the name cochlea. 
 The upper door is the fenestra ovalis already mentioned ; the 
 other is called the fenestra rotunda. The rods are called the 
 
THE EAB AND HEARING. 95 
 
 fibres or rods of Corti. A. nerve-filament from, the brain is 
 connected to each pair of rods. 
 
 If the cochlea was uncoiled and straightened out, a longi- 
 tudinal section of it would be something like Fig. 35 (#), and 
 a transverse section something like Fig. 35 l>\ ; but these 
 figures merely indicate the general arrangement, not the 
 proportions of the parts. 
 
 "Wlien sound waves reach the tympanum, it vibrates, and 
 its movements cause corresponding, but smaller, movements 
 
 (a) Longitudinal section. (b) Transverse section. 
 Fig. 35. 
 
 /?i", and Scala tympani, the two storeys ; /.o., 
 
 fenestra oralis-, S, stapes, the last of the chain of bones connecting 
 the fenestra ovalis to the tympanum; f.r., fenestra rotunda; SC, 
 Scala cochleae, the space between the membranes ; m.b., membrana 
 "basilaris, to which the rods are attached. 
 
 of the fenestra ovalis, with which it is connected. These 
 movements are transmitted to the liquid which fills the 
 cochlea, and to the rods which are bathed in this liquid. 
 The sensation of sound depends on the vibration of these 
 rods ; usually on their resonant vibration, since ordinary 
 sound waves do not produce sufiiciently great vibration unless 
 the effects of a number of waves are added by resonance. 
 There is no way of ascertaining whether the resonant vibra- 
 tion of each rod produces a separate sensation, because we 
 cannot make one rod vibrate alone. As the rods are of small 
 mass, and move with considerable friction in the liquid, the 
 resonant vibration caused by the movements of the liquid is 
 not much stronger in the rod whose free-vibration period 
 exactly corresponds than in a number of others whose periods 
 are not very different (Art. 48). Harmonic waves, there- 
 fore, set in resonant vibration not one rod, but a group of 
 consecutive rods. The resulting sensation is, however, 
 always a simple one ; it is only when two groups of rods 
 
96 SOUND. 
 
 vibrate, while intermediate ones do not, that we have the 
 sensation of hearing two sounds at once, and this cannot be 
 caused by harmonic waves. 
 
 60. Pitch. The sensation differs according to the fre- 
 quency of the vibration. This difference is called a difference 
 of pitch, and the sensation produced by harmonic waves of 
 greater and smaller frequency are said to be sensations of 
 higher and lower pitch respectively, probably because in pro- 
 ducing, by the voice, waves of great frequency, we have a 
 sensation of using muscles high up in the throat, while the 
 sensation of muscular effort is lower down in the body when 
 waves of smaller frequency are being produced. 
 
 Strictly, the term pitch is applied only to the sensation, 
 not to the vibration which produces it, but the term is often 
 loosely used as equivalent to frequency. 
 
 That pitch depends on frequency of vibration may be 
 shown by fitting two toothed wheels, with different numbers 
 of teeth, on the same uniformly revolving axle (the axle of a 
 heavy top answers well). If a card is held against the 
 wheels in turn, the one with more teeth gives a note of 
 higher pitch. 
 
 Non -harmonic waves produce in the ear the same effect as 
 on any other system of resonators ; that is, they set in reson- 
 ant vibration all the rods whose frequencies of free vibration 
 correspond with the frequencies of any of their harmonic 
 components. The resulting sensation differs very much in 
 different persons, and even in the same person, according 
 to the amount of attention he gives. A person who has been 
 musically trained, and listens attentively, will generally be 
 conscious of hearing, at the same time, the different sounds 
 which real harmonic waves, similar to the harmonic com- 
 ponents, would produce separately. With less training or 
 attention, only a single sensation is perceived, which usually 
 seems to be of rather higher pitch than would be produced by 
 harmonic waves of the same frequency, but is not exactly 
 like the sensation produced by harmonic waves of any fre- 
 quency. There is then a difference between sensations 
 produced by waves of different forms, even if they are of the 
 same frequency, and this is called a difference in the quality 
 
THE EAR AND HEARING. 97 
 
 of the sound. The terms character, timbre, clang -tint, are 
 used by different writers as equivalent to quality. 
 
 61. duality. It is this difference in the quality of the 
 sensations which enables us to distinguish between con- 
 tinuous notes of the same frequency produced by different 
 instruments, a flute and a harmonium, for instance, though we 
 may be quite unable to say which is of higher pitch. But 
 where the sound is not continuous we are aided in our 
 judgment about its origin, by the way in which the sound 
 begins and ends. 
 
 Two wave-systems which are exactly similar in frequency, 
 amplitude, and wave-form must give rise to the same sensa- 
 tion, for there is no respect in which the two can differ. Eut 
 it does not follow that two wave -systems of different wave- 
 form will produce different sensations. 
 
 Quite different curves may be produced (Art. 58) by adding 
 the ordinates of the same harmonic curves in different rela- 
 tive positions, and waves of these different forms set the same 
 resonators in vibration. If the rods of the ear were indepen- 
 dent resonators, it is probable that two wave-systems of different 
 wave-forms would produce exactly the same sensation, if 
 their wave-forms were such as might be formed by adding 
 the same harmonic curves in different relative positions. If 
 this was the case, we could say that the quality of the sensa- 
 tion produced by non-harmonic waves depended on the 
 frequencies and amplitudes, but not on the relative phases, of 
 their harmonic components. 
 
 It was Helmholtz's view that this actually was the case, 
 and he designed the following experiment to prove it. He 
 had 13 tuning-forks, whose frequencies were in the ratios 
 1:2:3:4: &c., mounted each in front of a resonator (Art. 
 54) of corresponding frequency ; the openings of these reso- 
 nators could be closed by sliding doors. All the forks were 
 kept in continuous vibration by the same momentary electric 
 currents passing at regular intervals round electro-magnets 
 placed close to the prongs ; the interval was an exact multiple 
 of the period of the slowest, and therefore of every other, 
 Cork ; and, as the force varied non-harmonically, it produced 
 resonant vibration in them all (Art. 47). The sound of any 
 
 SD. H 
 
98 SOUND. 
 
 fork was hardly audible, except when the door of its resonator 
 was open. Helmholtz first determined, by the method 
 described in Art. 54, the harmonic components of some sound 
 of marked and peculiar quality (say that of a trumpet) which 
 was of the same frequency as the slowest of the forks. He 
 then opened the doors of the resonators opposite the forks 
 whose frequencies corresponded to the harmonic components 
 of the sound of the trumpet, and found that the sound from 
 these resonators vibrating together was similar in quality to 
 that of the trumpet. Xow in this experiment the harmonic 
 waves from the resonators would not, unless by accident, be 
 in the same relative positions as the harmonic components of 
 the waves from the trumpet ; there was, therefore, usually 
 no resemblance between the wave-forms of the two wave- 
 systems. The fact that the sensations were similar seemed 
 to prove that waves of different form produced the same sen- 
 sation if they could be made of the same harmonics in different 
 relative positions. 
 
 This experiment is often described as the synthesis of a 
 given non-harmonic sound. It is, however, only the sensa- 
 tion, not the wave- system, which is reproduced, so that the 
 experiment is purely a physiological (and psychological) one. 
 There is no physical resemblance whatever (except in length) 
 between the original waves and Helmholtz' s copy of them, 
 and if we saw sound waves instead of hearing them, it would 
 be obvious that they were totally unlike. The apparent 
 resemblance is due simply to the very imperfect way in 
 which the ear distinguishes between waves of different form, 
 just as the apparent resemblance between the white lights 
 produced by adding different pairs of complementary colours 
 is due to the imperfect way in which the eye distinguishes 
 between non-harmonic light waves. 
 
 Though Helmholtz failed to detect any difference in the 
 sensations produced by the trumpet and by the forks, it is 
 almost impossible that resonant bodies which, like the rods 
 of Corti, are all connected to the same membrane, should 
 not vibrate differently according to the relative phases of the 
 different harmonic components, and Lord Eayleigh and others 
 have shown by direct experiment that there is sometimes a 
 difference in the sensation produced by waves which have 
 
THE EAR AND HEARING. 99 
 
 the same harmonics in different relative phases. It is there- 
 fore best to state that the quality of the sensation depends 
 on the wave-form of the waves which produce it, but that 
 waves of different wave-form produce nearly the same sensa- 
 tion if there is no difference except in relative position 
 (phase) between the harmonic curves of which the different 
 wave-forms can be built up. 
 
 62. Limits of Audible Sound. Waves arriving with a 
 frequency greater than 38,000 per second produce no sensa- 
 tion of sound. Harmonic waves of smaller frequency than 
 about 33 per second also produce.no sensation,* 4 but non- 
 harmonic waves of Jower frequency than 33, if they have a 
 harmonic component of greater frequency than 33, produce 
 the sensation corresponding to the frequency of that com- 
 ponent. These limits vary considerably in different persons. 
 
 The ear is very sensitive to differences of frequency 
 between sounds of frequencies such as are commonly produced 
 by the voice ; a change of \ per cent, in the frequency of 
 such sounds is easily detected. Outside the limits of the 
 voice, the sensitiveness of the ear to changes of frequency is 
 much smaller. 
 
 63. Musical Sound and Noise. When a series of travel- 
 ling condensations and rarefactions is irregular, and cannot be 
 divided into waves equal in length and similar in wave-form, 
 the sensation produced is that of noise, and has no recognis- 
 able pitch. This may be shown by pressing a card against 
 a revolving wheel notched at the edge with irregular teeth 
 of different sizes. 
 
 64. Loudness. Differences of intensity in the waves reach- 
 ing the ear produce differences of loudness in the sound heard, 
 but no simple relation between the intensity and the loudness 
 can be stated, loudness, indeed, like all sensations, not admit- 
 
 * It is sometimes stated that the waves, when very slow, are 
 heard as separate shocks, but there seems no evidence of this, at 
 any rate in the case of harmonic waves. 
 
100 SOUND. 
 
 ting of quantitative measurement. Of two wave-systems 
 of the same frequency and wave-form, the one of greater 
 intensity sounds the louder, but this is not necessarily the case 
 if the frequencies or wave-forms are different. Waves of very 
 low and very high frequency (near the limits of audible 
 frequency) sound not nearly as loud as waves of the same 
 intensity which are nearer the middle of the audible range. 
 
 65. Discord. When two sources sound together which 
 would, separately, produce harmonic waves of nearly equal 
 length, the actual wave-system consists of alternate groups of 
 more and less intense w.aves (Art. 42). As these reach the 
 ear, the sound heard keeps increasing and diminishing in 
 loudness ; these variations are called beats. These beats may 
 be considered a consequence of the fact, explained above, that 
 there is not much difference between the intensity of the 
 resonant vibration of the rods of Corti caused by waves of 
 exactly their own free -vibration period and that caused by 
 waves of nearly their own period. So that, if two sources of 
 sound have nearly equal vibration frequencies, there are rods 
 which either of the sources would set in resonant vibration, 
 the frequency of this vibration being always that of the 
 source, not the free-vibration frequency of the rod (Art. 48). 
 The movements which these rods would be executing if one 
 of the sources was vibrating alone are sometimes in the same 
 direction, sometimes in the opposite direction, to those which 
 they would be executing if the other source was vibrating 
 alone, and the actual movement of these rods keeps increasing 
 and diminishing. 
 
 From this it will be seen that no beats will be heard if the 
 frequencies of the sources are so different that they do not 
 cause resonant vibration of the same rods. This is found to 
 be the case when the difference of frequencies is more than 
 of the smaller frequency. 
 
 These constant variations in the intensity of the sound, 
 like the variations in the light of a flickering or "bobbing" 
 flame, are very unpleasant within certain limits of frequency. 
 These limits are different for sounds of different pitch. With 
 very low or very high notes, the beats are hardly noticed. 
 For notes of medium pitch (within the range of the voice) 
 
THE EAR AND HEARING . -T'Ol 
 
 they are hardly distinguished if less frequent than 2 per 
 second. From 2 to 10 per second, each beat is heard 
 separately, but the effect is not very unpleasant. Above 10 
 per second the beats are no longer heard separately, but 
 produce the peculiar jarring sensation known as discord. The 
 unpleasantness of this increases with the frequency of the 
 beats up to a certain point, about 30 beats per second for the 
 middle of the voice-range. The effect then becomes less 
 unpleasant, with increasing frequency, and becomes imper- 
 ceptible when the beats are more than 1 to every 4 vibrations, 
 or when they are more than about 80 per second, whichever 
 is first reached. 
 
 As each rod is set in vibration by harmonic components 
 nearly agreeing with itself in frequency, just as if these 
 components were the whole sound, we have the sensation of 
 beats or discord when two non-harmonic wave -systems arrive 
 together, if any two among their harmonic components have 
 frequencies whose differences are within the limits just 
 given. 
 
 In consequence of this, discord between two non-harmonic 
 wave-systems is usually only to be avoided by careful adjust- 
 ment. With harmonic waves this is not the case ; for instance, 
 a tuning-fork giving 300 vibrations per second sounds per- 
 fectly harmonious with another fork giving 400 or any higher 
 number, these frequencies being too different to excite the 
 same rods. Eut if we set two strings vibrating together 
 whose frequencies were 300 and 410, we should probably find 
 the effect unpleasant. The possible harmonics in the waves 
 from the first have frequencies of 600, 900, 1200, 1500, 1800, 
 2100, 2400, &c. ; those in the waves from the second have 
 frequencies of 820, 1230, 1640, 2050, 2460, &c. Among these 
 there are several discordant pairs, e.g., 1200-1230, 2050-2100, 
 2400-2460, and others, if we consider higher harmonics ; 
 and though, when a string is struck, some of the possible 
 harmonics are usually absent, it would be difficult to strike 
 the two strings so as to avoid all the discords. If we try 
 other frequencies for the two strings in the same way, we 
 shall find that there are always discords among the harmonics, 
 unless the ratio of the frequencies is a simple one. If, for 
 instance, the frequencies were 300 and 400, all the harmonics 
 
102 SOUND. 
 
 whose frequencies are not very different agree exactly, so 
 that there is no discord. 
 
 Notes whose frequencies are not in a simple ratio cannot, 
 therefore, he sounded together, if non-harmonic in character, 
 without discordant effect. It is also found that a pleasant 
 effect cannot be produced by sounding notes in succession, 
 unless the frequencies of all these notes are in simple ratios 
 to the frequency of some one note. The reason for this must, 
 of course, be psychological, not physical, since the successive 
 wave-systems do not exist together, and there can be no 
 physical action between them. When a succession of notes 
 all have frequencies in simple ratios to the same note, the 
 same frequencies will often recur among their harmonic com- 
 ponents, and it is probably to a (usually quite unconscious) 
 perception of this recurrence that the pleasurable effect is 
 due. It is not necessary, for this effect, that successive notes 
 should be in such simple ratios to one another that they could 
 be sounded together without discord. 
 
 66. Musical Scales. For musical purposes, then, we re- 
 require to be able to produce a succession of notes whose 
 frequencies fulfil two independent conditions. The frequen- 
 cies must all be in simple ratios to some one frequency, so 
 that the notes can be sounded in succession. Also, it must 
 be possible to form, from the notes, groups in each of which 
 all the frequencies are in simple ratios to each other, so that 
 the notes in any group can be sounded together. A series of 
 notes which fulfils both these conditions is called a musical 
 scale, and the note to whose frequency all the other frequen- 
 cies are in simple ratios is called the key-note of the scale. 
 There are many scales which fulfil the conditions fairly well ; 
 the scale which fulfils them best is called the diatonic scale, 
 or gamut. It consists of a series of notes, whose frequencies 
 are in the ratios of the numbers 
 
 24, 27, 30, 32, 36, 40, 45, 48. 
 
 The last of these is called the octave of the first, being the 
 eighth note from it, and it is also the lowest of another series 
 of notes having the same ratios, and the series is continued 
 both upwards and downwards in the same way. 
 
THE EAK AND HEARING. 103 
 
 The ratio of the frequencies of any two notes is called the 
 interval between them, and the simpler ratios are named from 
 their position in the gamut ; the ratio 2 : 1 is called an 
 octave, 3 : 2 a fifth, '3 : 1 a twelfth, 4 : 3 a fourth, and so 
 on. Thus if the frequencies of two notes are 300 and 200, 
 the first is said to be a fifth above the second, or the second 
 a fifth below the first. 
 
 When one note is an octave above another, all the harmonic 
 components of the first are among the possible harmonic com- 
 ponents of the second; if the two notes contain many 
 harmonics, there is a strong similarity between the sensations 
 produced, so that from the point of view of sensation the first 
 is often considered to be not a different note from, the second, 
 but "the same note an octave higher." This may be shown 
 by fixing on a revolving axle three toothed wheels, having 
 respectively say 60, 120, and 130 teeth. If a card is held 
 in quick succession against the first and second, a resemblance 
 will be noted in the sounds produced which is quite absent 
 if we press the card successively against the first and third, 
 or the second and third, although in the last case the notes 
 are much nearer in pitch than those produced by the first 
 and second. 
 
 For this reason two notes, whose interval is an octave, 
 are called by the same name. The notes of the series given 
 above are called respectively 
 
 Do, Re, Mi, Fa, Sol, La, Si, Do, 
 
 the same names being repeated for the octaves above and 
 below. When it is necessary to distinguish between notes 
 of the same name in different octaves, this is done by dashes 
 above or below the name, thus Mi" means a note two octaves 
 above the note indicated by Mi. 
 
 All the notes of the gamut have frequencies which are in 
 simple ratios to the frequency of any Do or any La of the 
 scale ; either of these may be the key-note. If the Do is 
 taken as the keynote, the ratios of the other notes to it 
 are 
 
 (Do) Re, Mi, Fa, Sol, La, Si, Do, ) 
 
 f"\\ 9 5 4 3 5 15 9 \ far* 
 
 ( L ) T "a 2- a" "8" ^ / <KC ' 
 
 or (1) 1-125 1-25 1-333 1-5 1-667 1-875 2 ) 
 
104 SOUND. 
 
 This is called the major gamut. If the La is taken as the 
 key-note, the ratios of the other notes to it are 
 
 (La) Si, Do, Re, Mi, Fa, Sol, La, ") 
 
 (\\ 3. A 2JL* 3 8. 9. 9 I 
 
 {*; 8 5 20 2 5 5 ^ ) 
 
 This is the minor gamut. The ratios of its notes to the key- 
 note are not so simple as those of the major gamut, and the 
 musical effect is quite different, but, as the physical reasons 
 of this are unknown, it is out of our province. 
 
 The special advantage of the gamut is the large number 
 of harmoni? 1 combinations of three or more notes which are 
 possible in it. Thus, the combinations Do, Mi, Sol ; Fa, La, 
 Do' ; Sol, Si, Re' each consist of notes whose frequencies 
 have the ratios 4:5:6, and such a combination (to which 
 the octaves of any of the notes in it may be added) is quite 
 harmonious. There is no note of the gamut which cannot be 
 combined with two others in this way ; such a combination 
 is called a chord, 
 
 A musical composition consists of notes, or combinations of 
 notes, of certain relative frequencies sounded in succession. 
 The musical effect depends on the relation between the fre- 
 quencies, not on the absolute frequencies themselves, so that 
 the key-note of the scale may have any pitch if the right 
 ratios are kept for the different notes. For example, if we 
 produce in succession, with proper intervals of time, any 
 series of notes whose frequencies are in the ratios of 1, ^f, 
 
 *, , 1, f *, I, f> *> t> 1, H> *, it> f> &c., we get the 
 well-known air " Men pf Harlech." 
 
 All these are gamut ratios, so that if we are provided with 
 the means of producing a series of notes having the gamut 
 ratios (for instance, notes of the frequencies 240, 270, 300, 
 320, 360, 400, 450, with the double and half of each of 
 them, and the doubles and halves of these, and so on), we 
 could produce on this scale any tune or air of Western Europe, 
 
 * f is not a simple ratio, but it is so nearly equal to f that the 
 difference is hardly noticeable. The fourth note of the minor gamut 
 is often considered to be not the Ee of the diatonic series, but a 
 note whose ratio to the key-note is -|. This has a simpler relation 
 to the key-note, but a less simple one to the other notes of the scale. 
 In singing, and on instruments which, like the violin, can produce 
 any note, the one which sounds better is used. 
 
THE EAR AND HEARING. 105 
 
 for all modern Western music is written for the gamut scale. 
 Thus, "Men of Harlech " would be produced on this scale 
 by notes of frequencies 480, 450, 400, 450, 480, 540, 600, 
 &c. It looks at first sight as if a scale of notes of this kind 
 would answer every purpose of music, and that there would 
 be no need to be able to produce any others. There are two 
 reasons why this is not the case. First, the continual re- 
 currence of the same notes, though pleasurable for a time, 
 becomes in the end monotonous, so that it is desirable in a 
 long composition to use several different scales, though the 
 key-notes of these different scales should be such that some 
 notes are common to all the scales used in the same composi- 
 tion. Secondly, the main object of producing a series of 
 notes such as that given above is often to accompany the 
 same notes produced by the voice. Now, the range of the 
 voice is rather limited, and differs in different persons, and 
 many singers would be unable to sing comfortably the higher 
 notes of the " Men of Harlech " series given above. If we 
 have not the means of producing any notes except the single 
 gamut series which we took as an instance, it is not possible 
 to produce a lower series of notes, having the ratios required 
 for "Men of Harlech," except by making each note an octave 
 lower, and this would probably make some of them lower 
 than the singer could manage. It is therefore necessary to 
 have a choice of scales with key-notes of different pitches ; 
 and it is found that an alteration of 7 or 8 per cent, in the 
 frequencies of the scale used is sufficient to make an impor- 
 tant difference in the ease with which a singer can render an 
 air, if the air is one of extended range. In order to suit 
 different singers, then, we require a series of gamut scales, 
 of which each differs from the next by not more than 14 per 
 cent, or so in frequency ; while, to avoid monotony, we re- 
 quire, in the same composition, to be able to change from the 
 scale we are using to another, whose key-note is simply 
 related in frequency to the key-note of the first. Both these 
 requirements are fairly well fulfilled by a series of gamut 
 scales whose key-notes are themselves the successive notes of 
 a gamut. Though any one of these scales has seven notes in 
 each octave, and there are seven scales, the total number of 
 separate notes required is only twenty for each octave, many 
 
106 SOUXD. 
 
 of the notes doing duty in more than one scale. Even this 
 number, however, is unmanageable in a keyed instrument, 
 such as the piano, and in such instruments a tempered scale is 
 adopted.* 
 
 67. Tempered Scales. A tempered scale is one in which 
 the frequencies of the notes are not exactly in simple ratios 
 to the key-note ; by sacrificing this advantage it is possible to 
 make a sufficient variety of scales out of only twelve notes to 
 the octave. The tempered scale most used is called the 
 equally tempered scale. 
 
 The major equally tempered scale consists of a series of 
 notes having to the key-note the ratios 
 
 (1) 2" 2* 2 A 2* 2^ 2 H 2 &C " 
 
 or (1) 1-123 1-260 1-335 1-498 1-682 1-888 2. 
 If we compare these with the gamut ratios 
 
 (1) 1-125 1-25 1-333 1-5 1-667 1-875 2, 
 
 we see that none of them differ by as much as 1 per cent., so 
 that when two notes of the tempered scale which correspond 
 to a perfectly harmonious combination on the gamut are 
 sounded together the beats are not rapid enough to be dis- 
 agreeable. It is this very remarkable approximation of a 
 number of the powers of ^/2 to simple fractions which makes 
 an equally tempered scale possible. 
 
 The equally tempered minor scale consists of notes having 
 to the key-note the ratios 
 
 (1) 2^ 2* 2 tV 2 T * 2* 2* 2, 
 or, (1) 1-123 1-189 1-335 1-498 1-587 1-782 2, 
 while those of the minor gamut are 
 
 (1) 1-125 1-200 1-35 1-5 1-6 1-8 2, 
 the approximation in this case being less perfect. 
 
 * This is the reason usually given for the use of a tempered scale. 
 But there is evidence that tempered scales were used long before 
 key-board instruments, and it is a pure assumption that " natural " 
 scales preceded tempered ones. 
 
THE EAR AND HEARING. 107 
 
 In key -board instruments a series of notes is provided, 
 
 the frequency of each of which is 2^ times that of the note 
 below it ; any one of these notes may be taken as the key- 
 note of either a major or a minor scale. The major scale is 
 formed by taking the 3rd, 5th, 6th, 8th, 10th, 12th, and 
 13th notes above the one chosen as key-note (counting it as 
 1st), the 13th being the octave of the key-note and the lowest 
 note of a similar series. The minor scale is formed by taking 
 the 3rd, 4th, 6th, 8th, 9th, llth, and 13th notes above the 
 key-note, and so on. Practically, key-board instruments 
 are tuned only very roughly to this scale. 
 
 The gamut with simple ratios is sometimes called the 
 11 natural scale," to distinguish it from, the tempered scales. 
 The name is, however, not very correctly applied. It is 
 probable that the use of a scale in which the frequencies are 
 in simple ratios is "natural," i.e., instinctive. But there 
 are many other possible scales of notes in simple ratios besides 
 this particular seven-note scale, and it is probable that it is 
 only custom which makes this scale (which was invented 
 only about three hundred and fifty years ago) seem more 
 natural than other scales which have been used in different 
 countries and times. 
 
 Of these other scales, the most important is the Pytha- 
 gorean, in which the other notes of the octave have to the 
 key-note the ratios 
 
 m 9. 8.JL A 2.7. 2.4.3. 2 
 \*~ J 8 64 3 2 16 128 *" 
 
 This scale, or a near approximation to it, is used in place 
 of the diatonic by most violinists when only a succession of 
 single notes is being produced. 
 
 68. Physiological Difference Tones. The force on the 
 drum of the ear, due to a small displacement, is not exactly 
 proportional to the displacement ; hence (Art. 49) the move- 
 ments of the drum are not exactly the sums of the movements 
 which the harmonic components of a wave-system would 
 produce separately. One result of this is that, if there arrive 
 at the ear waves having harmonic components of frequencies 
 a and J, not only the rods of these frequencies, but also those 
 of frequencies a + b and ab, are set in vibration. The 
 
108 SOUND. 
 
 sensations so produced &TQ physiological resultant tones, and are 
 called the summation and difference tone respectively. They 
 do not correspond to anything in the arriving waves. It 
 has already been pointed out that when two sources vibrate 
 harmonically with frequencies a and #, and disturb the same 
 air, the waves produced have harmonic components of fre- 
 quencies a + 1 and a b. So that there are both an external 
 or physical, and an internal or physiological, reason why 
 sounds corresponding to the frequencies a -f b and a b should 
 be heard when sources of the frequencies a and b vibrate 
 together. Only the part of this sensation which depends on 
 the presence of the corresponding harmonics in the arriving 
 waves can be strengthened by resonators. 
 
 69. Difference Tone not due to Beats. The difference tone 
 corresponds to a frequency ab, and this is also the frequency 
 of the beats which the sources produce. It was long sup- 
 posed that the sound was due to the beats, and that beats, 
 like waves, produced a note when too rapid to be heard 
 separately. This idea, which is now abandoned, seems to 
 have been due to a misunderstanding about the nature of 
 beats. Beats are simply variations in the violence of the 
 vibrations. If you cannot move a thing by shaking it con- 
 tinuously, you certainly cannot move it by ceasing at intervals 
 to shake it ; a fibre of the ear which is of such period that 
 the waves do not set it in vibration will not vibrate simply 
 because the arriving waves are diminished in intensity from 
 time to time. 
 
CHAPTER VIII. 
 
 BEFLECTION OF SOUND. 
 
 70. Reflection with Change of Sign. In Art. 1 1 we 
 described the transmission of a condensed or rarefied condition 
 along a row of elastic bodies such as the carriages of a train. 
 We will now consider what will happen when the pnlse 
 reaches the last carriage, which we will call Z. As in the 
 latter part of Art. 11, we will suppose Z in contact with a 
 fixed obstacle, such as the end of a siding, and that all the 
 carriages are to some degree compressed between the engine 
 and this obstacle. We will also suppose ourselves looking 
 from a position such that the direction from A to Z is from 
 right to left, as in Fig. 16. 
 
 When a condensation, produced in A by a push of the 
 engine, reaches Z, Z moves in its turn to the left, compress- 
 ing the buffers between it and the end of the siding. When 
 Z comes to rest, these are more compressed than those 
 between ]Pand Z, and therefore ^begins to move back again, 
 and stops only when it has transferred its condensed con- 
 dition to Y. Then Y moves in the same direction, and in 
 this way the condensation travels back again to A, each 
 carriage moving in turn a short distance towards the engine. 
 
 In the same way, if a rarefaction travelled from A to Z, Z 
 would move in its turn towards -4, but, as the end of the siding 
 is fixed, this leaves the pressure between Z and the end of 
 the siding less than that between Z and Y, and so Z moves 
 back, Y follows, and a rarefaction travels back to A, each 
 carriage moving a short distance away from the engine. 
 
 In these cases it will be noticed that the condition that 
 travels back is of the same kind as that which travelled to Z 
 from Ay but the direction in which the carriages themselves 
 move is reversed when the wave travels back again. When 
 
110 SOUND. 
 
 this is the case, the wave is said to be reflected ivitJi change 
 of sign. It should, however, be noticed, that as the direction 
 of movement of a wave is deter jiined when the direction of 
 movement of the substance and the kind of pulse (condensa- 
 tion or rarefaction) are given, the wave can only be reflected 
 by reversing the sign of one of these. So that " reflection 
 with change of sign" is really reflection in which it is the 
 velocity and not the condensation of the undulating substance 
 which is reversed in sign. 
 
 Reflection with change of sign takes place not only at a 
 fixed obstacle, but at any point where the movement of a 
 smaller mass is transferred to a larger one. If there is a row 
 of trucks of which, say, those from A to F are empty, and 
 the rest loaded, then, when a condensation produced in A 
 reaches F, a smaller condensation travels on along the rest 
 of the train, while another condensation travels back from F 
 to A. The energy in these two condensations is equal to 
 that in the original condensation. 
 
 In exactly the same way condensations or rarefaction travel- 
 ling through the air (or any substance) are reflected when 
 they reach a substance of greater density. The greater and 
 more sudden the change of density the larger the proportion 
 of the energy which is reflected. 
 
 71. Reflection without Change of Sign. When the move- 
 ment of one portion of an undulating substance is communi- 
 cated to another of smaller mass, a different kind of reflection 
 takes place. Suppose that there is a series of trucks of which 
 those from A to F are loaded, while the rest are unloaded, 
 and that a push is given to A which compresses it. Each 
 truck from A to E moves in turn to the left, and is brought 
 to rest by the increasing pressure between it and the next. 
 When Amoves on in its turn, and the pressure 'between F 
 and G begins to increase, 6r, being lighter, moves on more 
 quickly than the heavier trucks did, so that the pressure 
 between F and G does not increase so fast as that between E 
 and F did ; F therefore moves further than the other loaded 
 trucks before it comes to rest, and G moves the same distance 
 as F. This evidently leaves F in an expanded or rarefied 
 condition, and also leaves a greater space between F and E 
 
REFLECTION OF SOUND. Ill 
 
 than between E and D. .2? therefore begins to move towards 
 jF, expanding its buffers as it does so, and then D towards E, 
 and so on, and in this way a rarefied condition travels back 
 from jPto A, while a condensed condition travels forward from 
 G along the rest of the train. As before, these two contain 
 together the energy of the original condensation. 
 
 In this case it is to be noticed that the reflected pulse is of 
 the opposite kind to the original one, rarefaction instead of 
 condensation, but that the movement of the trucks when the 
 reflected pulse reaches them is in the same direction as it was 
 when the original condensation reached them. The conden- 
 sation is reversed in sign, the velocity is not. This is called 
 reflection without change of sign. A rarefaction arriving at F 
 starts a condensation back in the same way. 
 
 Reflection of this kind happens when a wave which has 
 been travelling in a dense medium reaches a rarer one, for 
 instance, when sound waves produced under water reach the 
 surface, or when waves travelling in air reach the surface of 
 a gas flame. Every condensation that arrives starts a rare- 
 faction back, and vice versa. 
 
 In either kind of reflection, each pulse which arrives at 
 the second medium sends a pulse forward into the second 
 medium as well as one back again through the first, though 
 the forward pulse may be of very small intensity if the 
 difference of density is great. This forward pulse in the 
 second medium is of the same kind as the original pulse in 
 the first medium, neither the kind of pulse nor the direction 
 of movement of the substance being reversed in it. 
 
 72. Reflection in Tubes. As might be expected, the first 
 kind of reflection occurs when sound waves travelling along 
 a tube reach its closed end ; each pulse as it arrives starts one 
 of the same kind back. The second kind of reflection occurs 
 when sound waves passing along a tube reach its open end. 
 
 It is difficult at first to see why this should happen, since the air 
 beyond the end of the tube is not less dense than that in the tube, 
 but the reason is somewhat as follows : In the case described in 
 Art. 71 the reason why a condensation reaching F started a rarefac- 
 tion back again was that when F moved on, the pressure in front 
 did not increase so fast as it had done with the other heavy trucks, 
 
112 
 
 SOUND. 
 
 so that F moved further than they did. The same occurs when a 
 pulse comes to the open end of a tube. Let AB (Fig. 36) be a tube 
 
 
 Fig. 36. 
 
 of 1 sq. cm. area of cross section, and let CD be a thin layer of air in 
 the tuoe. Suppose CD displaced 1 mm. to the right in -3-3^^ sec. 
 At the end of this interval the effect will not have spread beyond 
 1 cm. of the tube on each side of CD, so that an extra hundred cubic 
 millimetres of air have been forced into the cubic centimetre of 
 space immediately to the right of CD, and the pressure there 
 increased accordingly. But if a layer EF at the end of the tube 
 moved a millimetre to the right in -3^00 sec., at the end of this 
 interval the condensation would affect all the air within a radius of 
 1 cm. from the end of the tube, and only the same volume of extra 
 air, 100 cubic mm., has been added to this much larger space. The 
 rise of pressure to the right of EF is therefore not nearly so great 
 as it was in the case of CD. So that when a wave of condensation 
 passes along AB, and each layer of air passes on its energy to the 
 next and comes to rest itself, EF will not have expended all the 
 energy passed on to it by the previous layer when it has moved the 
 same distance as the other layers moved. EF will therefore move 
 further than the other layers did, and so, while it transmits a con- 
 densation to the air in front, it will leave a rarefaction behind it, 
 which will travel back along the tube just as in the case of the 
 railway trucks. In the same way, a rarefaction reaching the end of 
 the tube spreads a rarefaction through the air beyond the end, but 
 starts a condensation back along the tube. 
 
 73. Illustrative Experiments. Take a coil of hard brass 
 
 / III 
 
 I 
 
 Fig. 37. 
 
 wir e about 20 gauge, having (as it usually has when sold) a 
 
KEFLECTION OF SOUND. 
 
 113 
 
 diameter of about 5 inches. Separate the coils so as to form 
 a close spiral, and suspend every fourth or fifth ring by two 
 threads to two parallel wooden rods 8 or 10 feet long, as 
 shown in Fig. 37, arranging the threads so that the turns of 
 wire hang about an inch apart. The longer the threads are, 
 the better ; the distance between the rods may be about 
 equal to the length of the threads. (To make the figure 
 clearer the size and distance of the coils of wire are much 
 exaggerated.) 
 
 If one end of this spiral is fixed to a wall, and condensa- 
 tions and rarefactions sent along it as described in Art. 19, 
 each pulse when it reaches the wall sends a pulse of the 
 same kind back. But when this pulse reaches the free end 
 from which it started, it is reflected again as a pulse of the 
 opposite kind. 
 
 The reflection of sound waves from the surface of a denser 
 medium hardly needs experimental illustration ; echoes are a 
 familiar instance of it. Reflection from a rarer medium may 
 be illustrated by the following experiment : 
 
 Arrange two tubes, each about 3 inches in diameter and 
 3 or 4 feet long, at right angles to each other, as shown in 
 Fig. 38. (Tubes of thick paper are sufficient.) Place a 
 
 B 
 
 Fig. 38. 
 
 watch at A, and your ear at .Z?, with a large book or some 
 newspapers between A and B to prevent sound travelling 
 direct. The watch will be nearly inaudible, but the ticking 
 becomes very distinct if a large flat gas flame is placed at O y 
 with its plane vertical and inclined at 45 to AC and CB. 
 SD. i 
 
114 
 
 SOTTED. 
 
 74. Reflection in Open Air. In the cases of wave-pro- 
 pagation which we have so far considered, the wave-fronts, or 
 continuous surfaces drawn through those points where the air 
 is in the same phase of its vibration, have been either com- 
 plete spheres or have been bounded at the edges by the walls 
 of a tube ; in either case the wave-front can only travel at 
 right angles to itself ; expansion in its own plane is impossi- 
 ble. Eut when wave-fronts, expanding through the air, 
 reach a fixed obstacle, only the pieces of the wave-fronts 
 which are stopped by the obstacle are reflected. We require 
 therefore to know how pieces of wave-front travel through 
 the air when they are not confined at the edges by the walls 
 of a tube. 
 
 Suppose we have a screen or wall EF (Fig. 39) of some 
 material which does not transmit sound, and that in this 
 
 Fig. 39. (b) 
 
 screen there is a hole G. If waves or shells of condensation 
 and rarefaction, such as AB, arrive at this screen from a source 
 to the left of it, a piece of each wave-front passes through 
 the hole, and spreads through the air to the right of the 
 screen. How it spreads depends on the relation between the 
 diameter of the hole and the distance between successive 
 shells of maximum condensation (wave-length). As each 
 shell of condensation arrives at the hole, the air in the hole 
 becomes condensed and moves to the right, exactly as it 
 would do if a solid piston fitting the hole moved to the right, 
 
REFLECTION OF SOUND. 
 
 115 
 
 and, if the distance between a wave-front of maximum con- 
 densation and one of maximum rarefaction is larger than the 
 diameter of the hole, a shell of condensation spreads spheri- 
 cally in every direction through the air to the right of the 
 screen, exactly like the waves produced by the vibrating 
 piston in Eig. 22 ; rarefactions spread similarly in their turn. 
 This is illustrated in Fig. 39 (a) ; the thick circles are wave- 
 fronts of maximum condensation, the thin ones those of 
 maximum rarefaction. In this case the sound is audible at 
 any point. But, if the wave-length is much smaller than the 
 diameter of the hole, the piece of each shell which passes 
 through hardly spreads at all after reaching the other side, 
 but travels at right angles to its own wave-fronts* as shown 
 in Eig. 39 (5). In this case no sound is audible except at 
 points from which straight lines can be drawn to the source 
 of the sound ; we have in fact on the further side of the hole 
 a beam of sound in a definite direction from the source. 
 
 This may be considered as an instance of " interference " (Chap- 
 
 P 
 
 Fig. 40. 
 ter III.). Let H (Fig. 40) be any point from which the source of 
 
 * This would also be the case with the waves produced by a 
 vibrating solid surface if they were short compared to the surface, 
 but practically solid surfaces cannot be made to vibrate fast enough 
 for this, so that the waves spread from them in all directions. 
 
116 SOUND. 
 
 sound would not be visible, and let waves be arriving from the left 
 whose wave-length is much smaller than the diameter of the hole. 
 Then the distances of H" from different parts of the hole differ by 
 more than half a wave-length. We may suppose the hole divided 
 by cross lines into small holes, and we can pair off each of these 
 small holes with another whose distance from JET is half a wave- 
 length greater or less. (It is not self-evident that they can all be 
 paired off in this way, but this is proved in works on Physical 
 Optics, where the subject of transmission of waves is more fully 
 treated.) Let J, K be such a pair. Then, if all the opening except 
 / was blocked up, waves would spread from J in all directions to the 
 right, and a condensation would reach If at the same moment that a 
 rarefaction would reach H if all the opening except K was blocked 
 up, so that, if J and K are open together and the rest of the opening 
 blocked, there is no vibration at H. As this applies to every pair of 
 small holes, there is no vibration at H when the whole opening is 
 free. 
 
 It is evident that this argument does not apply to a point such as 
 P } from which the source would be visible through the opening, for 
 the different parts of the opening are all practically at the same 
 distance from P. So that there are waves from the opening in the 
 direction of P, but not in the direction of H. Also the argument 
 would not apply if the wave-length was greater than twice the 
 diameter of the hole, for then we could not find two parts of the 
 opening whose distances from H differed by half a wave-length. In 
 that case there would be vibration at H, wherever H was taken, 
 that is, the waves would spread in all directions from the opening. 
 There is not a definite line between the two cases ; the smaller the 
 wave-length compared to the opening, the less the pieces of wave- 
 surface which come through the hole spread sideways, and the more 
 nearly the vibration is confined to the regions from which the source 
 would be visible. 
 
 The part played by the screen is simply to limit tlie size of 
 the piece of each, wave-front. It does not matter what limits 
 the size ; any limited piece of a travelling shell of condensa- 
 tion or rarefaction, if small compared with the distance 
 between two successive shells, spreads in every direction in 
 front of it, bnt a piece of a travelling shell which is large 
 compared with the distance between two shells advances at 
 right angles to its own surface, and does not spread. As each 
 piece of shell is accompanied by a nearly constant quantity of 
 energy, the intensity of the sound diminishes very slowly 
 with distance when the pieces of shells do not increase in size. 
 
 The different behaviour of waves of different lengths is 
 easily shown experimentally. In a sheet of roofing felt, 
 
REFLECTION OF SOUND. 117 
 
 say 3 feet square, cut a round hole 6 inches in diameter. It 
 will be found that the sound of a humming top, placed 2 feet 
 from the hole on one side of the felt, is quite audible at any 
 point on the other, and is not perceptibly louder at points 
 from which the top is visible through the hole than at points 
 in quite a different direction from the hole ; the sound is, 
 however, nowhere nearly as loud as if the felt was removed. 
 But if for the top, which gives waves perhaps 4 feet long, we 
 substitute as source of sound a watch, which gives waves 
 from 2 inches to half an inch in length, we shall find that at 
 any point from which the watch is visible through the hole 
 the sound is nearly as loud as if no screen was interposed, 
 while at points from which the watch is not visible it is 
 hardly audible at all. 
 
 There is a similar difference between the behaviour of waves 
 of different lengths when an obstacle is interposed between a 
 source of sound and the ear. If the interval between succes- 
 sive shells of condensation is greater, or even not much 
 smaller, than the diameter of the obstacle, the waves close 
 in round the edge into the space behind the obstacle, and the 
 sound is heard at any point. But waves which are very close 
 compared with the diameter of the obstacle do not close in 
 much, but advance at right angles to their own surfaces, each 
 shell advancing with a hole in it where part has been stopped 
 by the obstacle. The obstacle thus casts a sound shadow 
 whenever the wave-length of the sound is much less than the 
 dimensions of the obstacle. This, like the corresponding 
 case of the opening in a screen, may be shown to be an 
 instance of interference. 
 
 The short waves from a watch are almost entirely cut off 
 from the ear by a quarto magazine interposed, but the longer 
 waves from a small clock are not. This is most strikingly 
 shown by placing the watch between the clock and the ear, 
 so that both are heard at once, but in such a position that the 
 watch sounds much the louder. If a thick quarto magazine 
 is placed in the line from the ear to the two sources of sound, 
 the clock is heard much louder than the watch, even if the 
 watch is audible at all. A large screen (such as a pile of 
 open newspapers) cuts off the sound from both ; a packet of 
 post-cards from neither. 
 
118 
 
 somn). 
 
 When a wave of condensation reaches a plane solid or 
 liquid surface, a wave of condensation begins to travel away 
 from each point of that surface as soon as the original shell of 
 condensation reaches that point. The result is that a reflected 
 shell of condensation is formed, which makes the same angle 
 with the plane surface as the original shell, but slopes in the 
 opposite direction. Some successive stages in the formation 
 of such a reflected shell are shown in Fig. 41, the reflected 
 
 \ F 
 \ " 
 
 4-\ 
 
 y 
 
 ' 
 
 i 4 
 
 Fig. 41. 
 
 part being shown by a broken line. 
 
 If the piece of each shell which is reflected is large com- 
 pared with the distance between two condensed shells, the 
 reflected pieces advance at right angles to their own surfaces 
 without spreading, and the effect of the reflected sound is 
 confined to a reflected beam, which fo^ows the same law as 
 the reflection of light. If the condens'ed waves are farther 
 apart than the diameter of the reflecting surface, the reflected 
 waves spread in every direction from the reflecting surface, 
 and diminish so rapidly in intensity that they cannot be 
 detected at a short distance. The same is true when the 
 reflection is " without change of sign," as from the surface of a 
 gas flame ; hence in the experiment shown in Eig. 38 the source 
 of sound must produce very short waves. 
 
 A reflected sound is called an echo. An echo, to be audible, 
 must be formed by reflection against a surface whose dimen- 
 sions are very large compared with a wave-length of the 
 sound, for otherwise the sound energy intercepted by the 
 surface is dispersed in all directions, and soon becomes 
 inaudible. The sound of a gun requires a cliff or high wall 
 to form a good echo, but a much smaller surface will give a 
 clear echo of a shrill whistle. 
 
 Echoes afford a rough method of finding the velocity of 
 sound. If we shout, and observe the interval before the echo- 
 
REFLECTION OF SOUND. 
 
 119 
 
 is heard, this interval is the time taken by sound to travel 
 twice the distance from us to the cliff or wall. 
 
 When a shell of condensation or rarefaction arrives at a 
 concave solid surface, and is turned back, the reflected shell 
 is concave to the direction in which it is going, if the con- 
 vexity of the original shell was not too great. Some stages 
 of such reflection are shown in. Fig.^ 42, [where the lines 
 
 Fig. 42. 
 
 marked 1, 2, 3, 4, &c., are successive positions of a shell of 
 (say) condensation "which originally started from a source C. 
 As each part of the shell reaches the concave surface, a con- 
 densed condition starts back, so that when one part of the 
 shell of condensation has reached the position 33, other parts 
 have already started back and reached the position 33', so 
 that 3'333' is a continuous shell of condensation, of which the 
 part shown by the complete line is still travelling towards 
 the mirror, while the broken line parts are travelling away. 
 If these reflected portions are large compared with the dis- 
 tance between a condensed and a rarefied shell, they will 
 advance at right angles to their own wave-fronts without 
 spreading, assuming the successive positions 4'444', 555, &c.> 
 
120 
 
 so that the reflected wave may (if the reflecting surface is 
 rightly shaped) converge towards a point F. At this point 
 the intensity of the sound is ver, great, for the diminishing 
 pieces of shells have always the same energy ; the sound is 
 brought to a focus just as light is. After reaching F the 
 wave diverges again, on the other side, assuming successively 
 the positions 11, 12, &c. 
 
 All this is only true if the mirror, and therefore the reflected 
 pieces of the shells, are large compared to a wave-length. If this 
 is not the case, the reflected wave spreads in every direction, 
 instead of converging. In fact, in that case it makes no practical 
 difference what the form of the mirror is. This seems to be lost sight 
 of by some makers of ear trumpets, who attempfc, by parabolic 
 reflectors, to converge the sound waves into the ear. No reflector, 
 whatever its shape, can do this unless it is many times larger than 
 the wave-length of the sound waves, and the waves produced in 
 conversation are not often less than 18 inches long. A funnel- 
 shaped tube concentrates sound to some extent, but it acts not by 
 making the wave-fronts concave, but by communicating the move- 
 ment and energy of each layer of air to a smaller one, so that the 
 amplitude continually increases. 
 
 In Fig. 42, the reflected condensed shell is represented converging 
 till it becomes a mere point. This does not really happen, because 
 before it gets so small it ceases to be large compared with the 
 distance between it and the shells of rarefaction in front and 
 behind, and then it no longer advances only at right angles to itself, 
 but spreads at the edges. Fig. 43 shows, roughly, some successive 
 
 Fig. 43. 
 
 stages in the advance of a concave wave-surface which is no longer 
 large compared to a wave-length. 
 
 If F is the centre of the reflected shells as long as they remain 
 spherical, it is evident that the total distance from C to any point 
 
REFLECTION OF SOUND. 121 
 
 of the mirror, and thence to F, must be constant, for every part of 
 . the same reflected wave-front (999, for instance) started at the 
 same time from 0, has been travelling with the same velocity ever 
 since, and is now at the same distance from F. It can easily be 
 shown that this is equivalent to saying that lines from C and F to 
 any given point of the mirror make equal angles with the reflecting 
 surface; i.e., F and are conjugate foci for light (Text-book of 
 Light, Art. 32), so that the point where the sound is most intense is 
 the point where an image of the source is formed if the mirror is 
 polished. In fact, light waves are reflected just as sound waves 
 Are when the sound waves are short compared to the reflecting 
 surface. 
 
 If the concave surface is a yard across, a watch placed at 
 C is heard as loudly by an ear at F as if the watch was an 
 inch instead of perhaps several feet away, but there is no 
 increase of loudness due to the mirror at a point a few inches 
 from F. If a large tuning-fork is used instead of a watch, 
 the sound will not be perceptibly louder at F than at other 
 points. 
 
 As before, we may consider the intensity of the sound at F, 
 and its absence at points near F, to be due to "interference"; 
 to any point except F there are routes from C (via the 
 reflector) of different lengths, so that condensation would 
 arrive by one route at the same time as rarefaction by 
 another. As long as all routes from C to F vid the reflector 
 are too nearly equal for this to happen, it makes little differ- 
 ence to the intensity at F whether they are exactly equal or 
 not. So that roughnesses or irregularities of the mirror are 
 unimportant if they are much smaller than a wave-length ; 
 the mirror need not be polished, but may be of gutta-percha 
 or sheet lead. 
 
 75. Refraction of Sound. Sound waves may also be con- 
 verged by means of a lens, but the lens must be of gas, not 
 of any denser substance, or nearly all the energy will be 
 reflected at its first surface. If two convex circular sheets 
 of collodion film are cut from a large collodion balloon and 
 attached by their edges to a metal hoop, and the space 
 between them filled with carbon dioxide gas, we have a lens 
 which will answer the purpose. Sound travels more slowly 
 in carbon dioxide than in air, so that when each shell reaches 
 
122 SOTTED. 
 
 the lens it advances more slowly at its centre than at the 
 edges, as shown in Pig. 44, so that when it comes out at the 
 
 Fig. 44. 
 
 other side it is concave instead of convex towards the direc- 
 tion in which it is advancing.- If these concave pieces of 
 shells are large compared with the distance between successive 
 waves, they will converge towards a point F, though, as ex- 
 plained above, they do not really become points. As in the 
 case of the concave mirror, it is only sound whose wave- 
 length is short compared with the diameter of the lens which 
 can thus be converged ; waves of greater length spread in 
 all directions after passing through the lens, as if the lens 
 was a mere hole, and do not converge. 
 
 A similar effect is produced by a concave lens filled with a lighter 
 gas than air; coal-gas is best. Hydrogen does not answer well, the- 
 difference of density between air and hydrogen being so^.great that 
 a large part of the energy is reflected on reaching the lens (Art. 72). 
 
 Prisms of collodion film filled with carbon dioxide have also been 
 made, and give a deviated beam of sound if the wave-length is short 
 compared to the prism, but there is no dispersion of sound of 
 different wave-lengths, as in the case of light, since all sound waves 
 travel with the same velocity. (The spectrum analysis of sound 
 has, however, been performed with a gigantic diffraction grating). 
 
 The statements made above are strictly true only for 
 harmonic waves. If the waves are non-harmonic, we may 
 
REFLECTION OF SOUND. 
 
 123 
 
 conveniently apply Fourier's device, and calculate the result 
 of the reflected or refracted waves as if the " harmonic com- 
 ponents" had a separate physical existence, and were 
 reflected or refracted each according to its wave-length. The 
 different harmonic components are of course quite differently 
 reflected and refracted ; thus a concave mirror converges the 
 shorter harmonic components nearly to one point, while it 
 distributes the longer ones in every direction. The character 
 of non-harmonic waves is therefore often entirely altered by 
 reflection or refraction, or by simply passing an obstacle, 
 having at one place a larger proportion and at another a 
 smaller proportion of harmonic components of very short 
 wave-length. 
 
 A simple experiment shows this very well. Hold a watch at 
 arm's length, and interpose a large sheet of card between it and the 
 ear. The sound is practically cut off. Now try with a post-card 
 instead of the large sheet. The sound is not much fainter when the 
 card is interposed, but it is a much lower note, or rather, as a trained 
 ear will recognise, all the higher components, which give the peculiar 
 sharp click, are cut off, and only the lowest of the notes produced is 
 heard. The longer harmonic components come round the card to 
 the ear; the shorter ones are absent behind the card (Art. 74). 
 
 76. Effect of Wind. When there is a wind blowing, the 
 air close to the ground travels more slowly than that higher 
 up, so that on the side of the source towards which the wind 
 
 J AC 
 
 Fig. 45. 
 blows the waves advance more rapidly above than below* 
 
124 SOUND 
 
 As the wave-fronts, being large compared with, a wave-length, 
 advance at right angles to themselves, an obstacle such as A, 
 Pig. 45, will not cut off the sound coming from a source (7, from 
 an observer at /, for the higher parts of the waves, which 
 have passed over A, soon begin to advance in a slightly 
 downward direction, and so reach the earth at a point 
 beyond. It is for this reason that sound is much better 
 heard on the side of the source towards which the wind 
 blows, especially if there are obstacles between the source 
 and the listener. In the other direction the waves advance 
 less rapidly above than below, and soon leave the earth 
 entirely, as shown in the figure, where the wind is supposed 
 blowing from right to left. (The difference of velocity 
 :above and below is very greatly exaggerated in the figure.) 
 
CHAPTER IX. 
 
 STATIONABY UJSDTJLATIOK 
 
 77. In Chapter IV. we found that, if two sources of 
 sound vibrate at the same time, the air is in the condition of 
 progressive undulation only in the regions in which the 
 waves from the sources, if the sources vibrated one at a time, 
 would travel in the same direction. Of the other vibratory 
 conditions of the air which may exist, the most important is 
 the condition of Stationary Undulation. This is produced 
 in any region of the air if two sources vibrate together which, 
 vibrating separately, would send through the region, in 
 exactly opposite directions, waves of the same length, 
 amplitude, and wave-form. We will first suppose that this 
 wave -form is harmonic. 
 
 Let each of the lines F, Z, 11, 12, &c., in Fig. 46, represent 
 the same region of air, in different conditions. Suppose that 
 there are two similar vibrating springs A and B (not shown), 
 one to the left and the other to the right of the region. Let 
 the top line Y represent, on the plan explained in Art. 15, 
 the condition in which the air in this region would have been, 
 at a given instant T, if A had been vibrating exactly as it 
 is, while B was at rest, and let the second line Z represent 
 the condition in which the same air would have been, at the 
 same instant T, if B had been vibrating exactly as it is, 
 while A was at rest. Y represents a progressive wave- 
 system travelling from left to right, while ^represents an 
 exactly similar wave- system travelling from right to left. 
 The actual condition of the air in the region represented, at 
 the instant T, will be (very nearly) the resultant of the two 
 conditions represented by Y and Z. 
 
126 SOUND. 
 
 N A N A N A N 
 
 Y CCCc-RRRRR-cCCCc-RRR 
 2 CC-RRRRR'. ccCCc-RRRRR 
 
 12 CCC-RRRRR-GCCCC-RRRR 
 
 Uniform 
 density. 
 
 RR-cCCCC-RRRRR-CCCC 
 
 3 
 
 4 D D R . P P ^ r . R D P D R . P. P Stationary, 
 
 e RRR-cCCCc-RRRRR-cCCC 
 
 Uniform 
 density. 
 
 8 CCC-RRRRR-CCCCC-RRRR' 
 
 Fig. 46. 
 
STATIONARY UNDULATION. 127 
 
 This condition is represented in the third line marked 1 1 . 
 The height of the letter C or R at each point, which denotes 
 the degree of condensation or rarefaction there, is the sum of 
 the heights of the letters at the corresponding points of T 
 and Z if they are both C or both 7t, and the difference of 
 their heights if one is C and one R ; and the length of the 
 arrow over each letter, which denotes the velocity with 
 which the air is moving at each point, is the sum or differ- 
 ence of the lengths of the arrows at the corresponding points 
 of ]Tand Z, according as these are in the same or opposite 
 directions. 
 
 On examining line 11, we see that we have a condition of 
 the air quite different from any kind of progressive undula- 
 tion. In every progressive undulation the velocity of the air 
 is greatest where the condensation or rarefaction is most 
 extreme, but here the greatest velocity occurs at the parts 
 which are neither condensed nor rarefied, and the greatest 
 density-differences occur where the air is at rest. So that, 
 though there are regions where the air is condensed and others 
 where it is rarefied, these do not correspond with the " con- 
 densations" and '^rarefactions" of progressive undulation. 
 In progressive undulation a " condensation" means a region 
 where the air is all condensed, and all moving one way. In 
 the condensed region in line 1 1 , half the air is moving one 
 way and half the other. 
 
 Next let us consider what will be the condition of the air 
 in this region an instant later than the moment for which Y 
 and Z are drawn. We can represent the condition in which 
 the air would have been an instant later if A had been 
 vibrating alone by shifting the line Y a little to the right, 
 and the condition in which it would have been if B had been 
 vibrating alone by shifting the line Z the same distance to 
 the left. Suppose each shifted the distance of between two 
 letters, and find the resultant condition again. This time we 
 get line 12. "We notice that, though we have supposed lines 
 Y and Z altered only in position, the resultant condition 
 "has not been displaced either to right or left, but has changed 
 in degree ; the condensations and rarefactions have become 
 everywhere less pronounced, while the velocity of the air has 
 everywhere increased. We see, in fact, that in this mode of 
 
128 SOUND. 
 
 vibration a given condition of the air does not move along as it 
 does in progressive undulation. The greatest degree of rare- 
 faction which existed at the moment shown in line 11 is not 
 to be found in line 12 a little to right or left of its former 
 place ; it no longer exists anywhere. And, though there is 
 nowhere so great a degree of rarefaction as in line 1 1 , the 
 greatest degree of rarefaction which exists anywhere in 
 line 12 is in the same position as the greatest degree of rare- 
 faction in line jf. I \ 
 
 At certain points, those under the *A 9 8 in the top line, the 
 degree of condensation which would be due to one of 
 the progressive wave-systems F, ^is exactly equal to the 
 degree of rarefaction which would be due to the other, so 
 that the actual condition of the air at these points, as shown 
 in line 1 1 , is one of average density. It is easy to see that, 
 if each of the lines !F, Z is shifted onward the same distance, 
 to represent the conditions which would be due to the two 
 wave-systems a moment later, the rarefactions and condensa- 
 tions under the A's are still equal, so that this condition of 
 average density is a permanent one at these points, which are 
 called Antinodes. 
 
 "We see also that there are certain other points, those under 
 the JV's in the top line, where the velocity of the air which 
 would be due to one of the wave-systems Y, Z\$> equal and 
 opposite to the 'velocity which would have been due to the 
 other, so that at these points the air is at rest ; and, as 
 before, we see that this is a permanent condition at these 
 points, which are called Nodes. 
 
 The antinodes are called by some writers " ventral seg- 
 ments."' This name is more properly applied to the whole 
 region between one node and the next, an antinode being the 
 middle point of a ventral segment. 
 
 The existence of points fixed with respect to, the air, at 
 which a definite condition of the air is always to be found, is 
 the most striking peculiarity of this mode of vibration; hence 
 the name Stationary Undulation. 
 
 To get a more exact idea of this condition, we will trace 
 the changes in the region shown in the figure through some 
 further stages, at each stage advancing T and Z one letter as 
 before. The condition of the air at these stages is shown in 
 
STATIONARY UNDULATION. 129 
 
 the lines 1 to 10. These complete a cycle, as the next stage 
 after that shown in line 10 is line 11 again. 
 
 By comparing the successive lines of Fig. 46 we can 
 make out the following important points. 
 
 (1) Nodes occur at the points where the maximum conden- 
 sations of one imaginary progressive wave-system (i.e., the 
 wave-system which one of the sources would produce if it 
 
 . vibrated alone) arrive at the same moments as the maximum 
 condensations of the other. Antinodes occur where a maxi- 
 mum condensation of one imaginary system arrives at the 
 same moment as a maximum rarefaction of the other. 
 
 (2) The distance between two successive nodes, or two 
 successive antinodes, is half the wave-length of the waves 
 which the sources would produce separately. Prom a node 
 to an antinode is one-fourth of this wave-length. 
 
 (3) The period in which the stationary undulation goes 
 through all its changes is the period in which the waves 
 which the sources would produce separately would advance 
 their own wave-length, i.e., it is the vibration period of 
 either source. 
 
 (4) Condensation and rarefaction do not move along as in 
 progressive undulation ; they simply appear and disappear 
 again, to be succeeded by the opposite condition in the same 
 place. 
 
 (5) The nodes are not places of greater average density 
 than the rest of the air, but of greatest variation of density ; 
 each node is a point of maximum and minimum density in 
 turn. The average density is the same at nodes as else- 
 where. 
 
 (6) There is an instant, twice in each complete vibration, 
 when all the air is stationary at the same moment (line 4 or 
 10). This may be called the stationary instant. At a 
 stationary instant every point has the maximum degree of 
 condensation or rarefaction which it ever has, and this is 
 greatest at the nodes and diminishes to zero at the antinodes, 
 alternate nodes being condensed and rarefied^ After the 
 stationary instant all the air, except at the nodes, begins to 
 move from the condensed nodes towards the rarefied ones 
 (lines 5, 11); its velocity at anyone instant is greatest at 
 the antinodes and diminishes to zero at the nodes. The 
 
 SD. x 
 
130 SOUND. 
 
 velocity increases everywhere (line 6 or 12), the velocities at 
 the different points always keeping the same ratio, so that it 
 is always greatest at the antinodep. Meanwhile the conden- 
 sation and rarefaction everywhere diminish, hut fastest at the 
 nodes, so that the degrees of condensation and rarefaction at 
 different points remain in the same ratio. The velocity 
 increases, and the condensation and rarefaction diminish, till 
 we reach an instant (line 7 or 1) when the air has every- 
 where the same density, which is that of the external air. 
 This may he called the moment of uniform density. At this 
 moment the air has at each point the maximum velocity 
 which it ever has at that point. The air continues to move 
 in the same direction, but with diminishing velocity, and the 
 nodes towards which it is moving, which were previously the 
 rarefied ones, now become condensed (line 2 or 8) and those 
 which were previously condensed become rarefied. The 
 velocity everywhere diminishes, and the degree of condensa-- 
 tion and rarefaction everywhere increases (line 3 or 9), till all 
 the air comes to rest at the same moment, and we have a. 
 stationary instant again. Then all the movements begin 
 again, but in the reverse direction, and so on. 
 
 (7) At any given moment all the air between two consecu- 
 tive nodes is moving in the same direction, and all the air 
 between two consecutive antinodes is in the same condition 
 (all rarefied or all condensed). 
 
 (8) Since the velocity of the air at each point varies har- 
 monically, the air at each point moves harmonically, and its 
 amplitude at each point is proportional to the maximum 
 velocity of the air there ; this amplitude is therefore greatest 
 at the antinodes and zero at the nodes. Each particle of air 
 passes its mean position at the moment (line 1 or 7) when it 
 has its maximum velocity. 
 
 78. Energy of Stationary Undulation. The total energy 
 in the stationary undulation is the sum of the energies of the 
 undulations which the sources would produce separately, but 
 it is in a different form. Tor, while in any progressive 
 undulation half the energy is at any one moment kinetic and 
 the other half potential, in the stationary wave-system the 
 whole energy keeps changing from one form, to the other. 
 
STATIONARY UNDULATION. 
 
 131 
 
 When the air is in the condition shown in line 1, Fig. 46, 
 there are no differences of density, and the whole energy is 
 kinetic, depending on the velocity with which the air is 
 moving. At this moment most of the energy is near the 
 antinodes, where the air is moving fastest. When the air 
 is in the stage of stationary undulation shown in line 4, there 
 is no kinetic energy, as all the air is at rest; it is all potential, 
 depending on differences of pressure, and none is at the 
 antinodes. In intermediate stages the energy is partly in 
 one form, partly in the other. 
 
 N 
 
 I 1 
 
 ffijj 
 
 IS 
 
 1 1 
 
 A r 
 
 1 1 1 1 1 
 
 J A N 
 
 ! 1 1 I , 
 
 2 
 
 MM 
 
 i 
 
 Mm 
 
 i i i n 
 
 3 
 
 i i i i 
 
 i 
 
 ii i i i 
 
 I i i 
 
 4 
 
 i i i 
 
 n 
 
 ii i i i 
 
 MM 
 
 5 
 
 i i I 
 
 I 
 
 n i i i 
 
 i i i 
 
 6 
 
 i i" i i 
 
 i 
 
 n i i i 
 
 i i i i i 
 
 7 
 
 i M i 
 
 
 1 1 1 1 1 
 
 i ( i" i' " 
 
 G 
 
 ii 1 1 
 
 
 i i i n 
 
 i i i i i 
 
 fl 
 10 
 
 III 1 
 
 i ! ! I 
 
 
 i 1 1 i i 
 
 MM! 
 
 ,i M i 
 
 iji i i 
 
 ! ! '. '. \ 
 
 11 
 
 II 1 1 
 
 
 1 1 1 n 
 
 III 1 1 
 
 12 
 
 ill i 
 
 
 i i M i 
 
 Illl II 
 
 13 
 
 i i i i i 
 
 i 
 
 i i if ii 
 
 Fig. 47. 
 
 i i i i i i i 
 
 79. Tig. 47 shows in the same form as Fig. 20 the move- 
 ments which take place in a region of stationary undulation. 
 
132 SOUND. 
 
 The vertical lines in each row represent, as before, plane 
 surfaces, seen edgewise, which were equidistant in the 
 undisturbed air, and the air is to "be supposed moving in such 
 a way that the same air always remains between the same 
 two planes. The lines vibrate harmonically about their 
 mean positions; those at the antinodes have the greatest 
 amplitude of vibration, and those at the nodes have zero 
 amplitude. They all cross their mean positions together, 
 but those on opposite sides of the same node are moving in 
 opposite directions. The numbers of different stages corre- 
 spond with those in Fig. 46. It is easy to see, from the 
 successive lines, how the air sways backwards and forwards 
 between fixed planes, the nodes. 
 
 A comparison of Pigs. 20 and 17 shows clearly the funda- 
 mental difference between progressive and stationary undula- 
 tion. In the former all parts of the air move the same 
 distance, but at different times. In the latter they move at 
 the same time, but different distances. 
 
 80. Cheshire's Disc. The movement of which Pig. 47 
 shows successive stages may be shown passing through these 
 stages by a modification of Crova's Disc (Pig. 21), designed 
 by Mr. Cheshire and published in Nature. 
 
 To construct the disc for this purpose, describe a circle of \ inch 
 radius in the middle of a circle of cardboard 8 inches in diameter. 
 Divide the circumference of the small circle into twelve equal parts, 
 numbering the points of division 1 to 12. Draw a diameter from 3 
 to 9, and draw lines at right angles to this diameter from 2 to 4, 
 1 to 5, 12 to 6, 11 to 7, 10 to 8, so dividing the diameter into six 
 parts, not all equal. We will call the points of division of the 
 diameter a, fr, c, &c., so that, including its ends, there will be seven 
 marked points on the diameter, marked respectively 9, a, fc, c, d, e, 3. 
 Taking these points as centres, in this order, describe ink circles, 
 increasing the radius |- inch each time. When you have described 
 the circle with centre 3, go back to e, d, &c., to 9, then 0, &c., back 
 again, making 20 or more circles in all. Mount this disc exactly 
 like the one in Fig. 21, and, on rotating it, the portions of the circles 
 seen through the slit will execute the movements of stationary 
 undulation. 
 
 The movement can also be well shown by the spiral wire 
 of Art. 73, as described in the next chapter. 
 
 81. Stationary Undulation caused by Reflection. It does 
 not often happen that the condition of stationary undulation 
 
STATIONARY UNDULATION. 133 
 
 is due to the simultaneous vibration of two sources. 
 Much more usually it is due to reflection of waves from a 
 single source. In this case the two imaginary component 
 "wave-systems are the waves which the source would produce 
 if no reflection occurred, and the reflected wave which each 
 of these would produce if it arrived by itself at the reflecting 
 surface. The real condition of the air between the source 
 and the reflecting surface (say a tuning fork and a wall) can 
 be found by adding these imaginary conditions, and it is 
 evidently nearly stationary undulation along a line from the 
 fork at right angles to the wall, if the circumstances are such 
 that an incident wave is reflected without much spreading 
 and consequent loss of intensity. This, as we saw in the last 
 chapter, requires that the waves shall be short compared to 
 the reflecting surface ; a shrill whistle, or a squeaker, such 
 as is used in many toys, is therefore better than a tuning 
 fork. If such a squeaker is blown with a steady pressure of 
 air (so as to give always waves of the same length) at a dis- 
 tance of a few feet from the wall of a room, there is stationary 
 undulation between. 
 
 At the wall itself there is a node, for the part of the 
 reflected wave which is just starting back is the reflection 
 of, and similar in condensation to, the part of the incident 
 wave which arrived an infinitely short time ago, and therefore 
 differs infinitely little in condensation from the part of the 
 incident wave which is just arriving ; and a node is a point 
 where the degrees of condensation of tha components are 
 always equal. Other nodes occur every half wave-length of 
 the incident waves from the wall. 
 
 It is only along a line from the source perpendicular to the 
 wall that the direct and reflected waves are in exactly oppo- 
 site directions, so that the condition of stationary undulation 
 strictly exists only along this line, but there will be a condi- 
 tion which is very nearly nodal at all points, not very far 
 from this line, which are at the same distance from the wall 
 as the true nodes, and similarly for the antinodes, so that the 
 nodes and antinodes are surfaces parallel to the wall. Their po- 
 sition depends only on the position of the wall and the length 
 of the waves, not at all on the position of the source, which 
 is not, unless by accident, either at a node or at an antinode. 
 
134 SOUND. 
 
 82. Experiments on Stationary Undulation. The exist- 
 ence of nodes and antinodes may be shown, and their positions 
 found, in two ways. One end of a flexible tube may be 
 inserted into the ear, and the other end moved along tho 
 perpendicular from the source to the wall, keeping the plane 
 of the opening of the tube parallel to this line. A series of 
 points are found where the sound is fainter than at inter- 
 mediate points ; these are the antinodes, for the movement 
 of the air backwards and forwards across the opening, with- 
 out change of density, sends no waves up the tube. When 
 the open end of the tube is at a node, each change in the 
 density of the air sends a corresponding wave up the tube, 
 and loud sound is heard. This is not a very good method. 
 It is clear from Fig. 46 that the absence of change of density 
 at the antinodes depends on the exact equality of the conden- 
 sations of one component wave-system and the rarefactions of 
 the other. Now the rarefactions of the reflected waves are 
 produced by the rarefactions of the incident waves, and do 
 not correspond in wave-form with the condensations of the 
 incident waves unless the incident waves are symmetrical, 
 which never is the case in practice. So that unsymmetrical 
 waves, like those from the squeaker, do not, when reflected 
 " with change of sign," form perfect antinodes at all, 
 though they do form true nodes, or places of no movement. 
 So the antinodes are indicated, not by silence, but only by 
 faintest sound, and the nodes by loudest sound; neither is 
 easy to determine. A sensitive flame, which directly deter- 
 mines the nodes, is much better . 
 
 A sensitive flame is produced by burning, at a burner per- 
 forated with a single pin-hole, coal-gas at a pressure equal 
 to that of 8 or 10 inches of water. It will be found that, as 
 the tap is turned on, the flame, which is like a much elon- 
 gated candle flame, increases in length to about 16 or 
 20 inches, and then suddenly shortens to half that length, 
 flaring at the top and producing a loud noise. If the supply 
 is adjusted so that the flame is just on the point of flaring, it 
 is very sensitive to movement of the air just above the 
 burner, which makes the flame flare as long as the movement 
 lasts, but it is not at all sensitive to changes of pressure in 
 the air apart from movement. Such a flame, held in a region 
 
STATIONAKY UNDULATION. 135 
 
 of stationary undulation, flares everywhere except at the 
 nodes, and thus the nodes, if they are at all perfect, may be 
 very exactly determined. 
 
 This is, of course, a method of finding the length of the 
 waves from the source, which is twice the distance between 
 two consecutive nodes (Art. 77). For very short waves it is 
 one of the best methods. 
 
 This form of sensitive flame requires a gas-bag or gas-holder, 
 since the ordinary gas supply is at too low a pressure, but sensitive 
 flames may be more simply obtained. If gas at the ordinary 
 supply pressure is allowed to issue from a pin-hole burner, and a 
 piece of wire gauze fixed a little distance above the jet, the gas may 
 be lit above the gauze without lighting that between the gauze and 
 the burner, and the flame above the gauze will be blue at the bottom 
 and yellow above. If we increase the distance of the gauze from 
 the burner, the blue part increases and the yellow diminishes, and 
 by trying different distances a position may be found at which the 
 flame is sensitive, the yellow tip entirely disappearing while any 
 vibratory movement of the air, of high frequency, is taking place 
 above the burner. A wide glass tube or lamp-chimney round the 
 flame, standing on the gauze, makes the flame still more sensitive 
 and the effect more visible, as a much longer flame is then obtained, 
 which shortens to less than half its length, and becomes much less 
 luminous, when there is any vibration in the air. The*e flames, like 
 the other, are unaffected at a node, but disturbed at any other point 
 in a stationary undulation. 
 
 Sensitive flames may also be used for showing to an audience the 
 phenomena of reflection and refraction of sound for instance, its 
 concentration at the focus of a concave mirror. 
 
 83. As reflected waves are not quite as intense as the 
 incident ones, stationary undulation produced by reflection is 
 more or less imperfect. It is convenient to consider the 
 incident wave-system as the sum of two systems, one of the 
 same intensity as the reflected waves, the other making up 
 the actual intensity. The first of these, with the reflected 
 system, gives true stationary undulation, so that the actual 
 condition of the air is the sum of a stationary undulation, 
 and a very feeble progressive undulation in the direction of 
 the incident waves. 
 
 Very perfect stationary undulation occurs in organ-pipes, 
 but there it is complicated by resonance. We consider it in 
 the next chapter. 
 
CHAPTER X. 
 
 VIBRATIONS OF AIR IN PIPES. 
 
 84. Wave in a Tube. Take a wide tube, say about 
 3 inches in diameter and 6 inches long, and stand it on a 
 table, so that its lower end is closed. Hold a strip of paper, 
 an inch wide, with its free end over the open mouth of the 
 tube, and tap the upper side of the paper sharply, near the 
 free end, with a penholder, so that the paper moves suddenly 
 towards the tube, but does not reach it. In addition to the 
 noise which would be produced in any case by striking the 
 paper, another sound will be heard which is not produced 
 when the paper is held at a distance from the tube and 
 struck in the same way. This additional sound will he- 
 found to be always the same for the same tube, and quite 
 independent of the size of the paper or the way in which it 
 is held and struck, and most persons will recognise that it is 
 a musical note of definite pitch, which can be matched on the 
 piano or by the voice, while the tap of the penholder on the 
 paper is a mere noise, and has no definite pitch. 
 
 Any large wide-mouthed hollow vessel, a jug for instance, 
 may replace the tube in this experiment, and the sound pro- 
 duced is so characteristic of hollow vessels that any sound 
 which produces the same effect on the ear is commonly 
 termed a " hollow sound." 
 
 The sensation of definite pitch is found, in all cases which 
 can be investigated, to depend on condensations and rare- 
 factions reaching the ear at regular intervals (Art. 63). Now 
 there is nothing regular about the movements of a piece of 
 paper which has been struck ; this is shown by the sound 
 being a mere noise. The tube, then, has in some way the 
 effect of converting the single wave produced by striking the 
 paper into a succession of waves starting at regular intervals. 
 
VIBRATIONS OF AIR IX PIPES. 137 
 
 And it is not the tube itself which does this, but the air con- 
 tained in it, for, if we try the experiment with a metal and a 
 pasteboard tube of the same size and shape, we find no 
 difference in the hollow sounds they give back. 
 
 The way in which this happens is as follows : The con- 
 densation produced by tapping the paper travels down the 
 tube to the table, where (Art. 72) it is reflected, and a con- 
 densation travels up the tube again. When it arrives at the 
 open end, a slight condensation starts off through the outside 
 air, but (Art. 72) by far the greater part of the energy travels 
 down the tube again in the form of a rarefaction. This is 
 reflected, still as a rarefaction, at the closed end, and, when 
 it reaches the mouth again, a slight rarefaction starts off 
 through the air, while a condensation travels back along the 
 tube, and so on. Only a small fraction of the energy leaves 
 the tube each time the wave reaches the mouth, so that the 
 wave may travel hundreds of times up and down the tube, 
 changing its sign each time it reaches the mouth, before it 
 becomes imperceptible. A condensation starts off from the 
 mouth at the end of every fourth single journey of the wave, 
 and a rarefaction half way between each two condensations, 
 and, as these travel away through the air at the same speed 
 as the wave travels in the tube, we have waves, travelling 
 away through the air, whose wave-length is four times the 
 length of the tube. 
 
 A similar action will take place if we tap a strip of paper 
 over the upper end of a tube which is also open at the 
 lower end. In this case the condensation produced by the 
 sudden movement of the paper travels down to the bottom of 
 the tube and there starts a slight condensation off through 
 the outside air, while a rarefaction travels back up the tube. 
 When this reaches the upper end, a rarefaction starts off 
 through the air and a condensation travels down again, and 
 so on. In this case a condensation starts away from the tube 
 through the outside air at the end of every second single 
 journey of the wave in the tube, so that the waves in the air 
 are only twice as long as the tube. An open tube, under 
 these circumstances, gives waves of about the same length as 
 a tube of half its length closed at one end (called a " closed 
 tube "). 
 
138 SOUKD. 
 
 Though a hollow vessel is often said to " resound "to a 
 sudden blow like that of the penholder on the paper, the 
 sound produced is not an instrnce of resonant vibration in 
 the sense in which we have used the term, for the air in the 
 tube was set in vibration by a single violent impulse, not a 
 succession of properly timed small impulses. But, as the air 
 in the tube has, as we have seen, a natural period of vibration, 
 it can be set in resonant forced vibration by impulses of the 
 same period. 
 
 85. Resonant Vibration of Air Columns. Suppose we 
 take a tube open at the top and closed at the bottom, and 
 make a flat spring vibrate over the mouth. Each movement of 
 the spring to wards the mouth of the tube sends a condensation 
 down the tube, and each movement away from the mouth, a 
 rarefaction. When the first condensation returns to the top, 
 it would of itself start a rarefaction down. If at this 
 moment the spring is moving upwards, a rarefaction produced 
 by its movement travels down the tube together with the 
 rarefaction which would in any case be produced by the 
 arrival of the condensation at the top ; we have a rarefaction 
 of nearly double the amplitude of the first wave. When 
 this returns to the top, it would of itself send a condensation 
 down, and, if the spring is at this moment moving downwards, 
 this condensation will be increased by that due to the spring, 
 and so on. Evidently the effect of a large number of 
 impulses can be added up in this way if the period of the 
 spring is exactly such that, each time the wave in the tube 
 returns to the mouth, it finds the spring moving in the 
 opposite direction to that in which it was moving the pre- 
 vious time. 
 
 The condition that the air in a closed tube may be set in 
 resonant vibration by a spring is, therefore, that' the spring 
 must make an odd number of half vibrations in the time that 
 a pulse takes to travel twice the length of the tube. 
 
 In the case of an open tube, a condensation sent down 
 returns as a rarefaction, which would of itself start a con- 
 densation down again. In order that this may be increased 
 by the movement of the spring, the spring must be at that 
 moment moving downwards again, so that it must have 
 
VIBBA.TIONS OF AIR IN PIPES. 139 
 
 -completed a whole number of (double) vibrations while a pulse 
 has travelled twice the length of the tube. 
 
 In trying the experiment it is more convenient to use a 
 tuning fork than a spring, because, unless the spring is very 
 firmly fixed, it shakes its support, and then it does not 
 vibrate very long or regularly. 
 
 A closed tube, then, will be set in resonant vibration by a 
 fork which makes either 1, 3, 5, or any other odd number 
 of half vibrations while a pulse travels twice the length of 
 the tube. In any of these cases each pulse of condensation 
 or rarefaction, on reaching the mouth, gives off a small 
 fraction of its energy in the form of a wave of its own kind, 
 which travels away through the outside air, while the rest of 
 the energy travels back down the tube as a pulse of the 
 opposite kind, increased by the wave which the fork was 
 sending down at the same moment. The wave in the tube 
 keeps on increasing till the energy sent off at each return 
 equals that received from the fork. As the tube sends a 
 pulse off for each one that the fork sends down the tube, the 
 waves start from the tube with the same frequency as from 
 the fork, and are of the same wave-length, but they are of 
 much greater intensity, so that the sound heard, though of 
 the same pitch as that heard when the fork is sounded without 
 the tube, is very much louder. It is not at first clear how 
 this increased loudness can be produced, as of course the tube 
 cannot send out more energy that it receives from the fork, 
 but we shall see presently that, when the air in the tube is in 
 resonant vibration, the air near the mouth moves up and 
 down, keeping time with the fork, and under these circum- 
 stances the fork communicates its energy much more rapidly 
 to the air, as explained in Art. 41. Of course the fork comes 
 to rest much sooner when it makes the tube resound, but 
 while it lasts the sound is much louder. 
 
 86. Condition of the Air in a Resounding Tube. If 
 we produce at the mouth of the tube a single condensation, 
 as in the experiment with the strip of paper, a real wave 
 travels up and down the tube, and would be seen to do so if 
 the air was visible. When a fork vibrates continuously 
 .at the mouth of the tube, producing condensations and 
 
140 SOUND. 
 
 rarefactions alternately, it is convenient to speak as if these 
 actually travelled down the tube, while previous waves, 
 reflected from the lower end, ^vere travelling up again. It 
 has, however, been explained (Art. 37) that this is merely a 
 convenient way of saying that the real condition of the air is 
 one which can be found by adding two such wave-systems, 
 and does not mean that these systems really exist in any 
 physical sense. So that, if we could make the air in a tube, 
 which is resounding to the tuning fork, visible, we should 
 not see waves travelling down and other waves travelling up, 
 or indeed anything moving continuously along the tube 
 either way ; what we should see would be simply the process 
 of stationary undulation described in the last chapter, since 
 this is the condition which we find when we add two 
 imaginary equal wave-systems in opposite directions. 
 
 It is very important to keep in mind that the real physical con- 
 dition of the air in a resounding tube is the condition of stationary 
 undulation, and that the waves travelling up and down are a 
 mathematical fiction, not a physical fact. To describe the condition 
 of the air as one in which waves are travelling along it in opposite 
 directions is as physically incorrect (and as mathematically correct) 
 as to describe the condition of a man who is standing still, by 
 paying that he is walking forwards and walking backwards at the 
 same time. 
 
 All harmonic stationary longitudinal undulation is of the 
 same kind, so that the general description of this process 
 given in Art. 77 applies to the air in any resounding tube if 
 the vibrations are harmonic. To make it a complete account 
 of the movement of the air in a resounding tube it only 
 remains to state where the nodes and antinodes are situated 
 in the tube. This is easily found in any given case. 
 Suppose, for instance, that we hold over the mouth of a tube, 
 closed at the bottom, a fork which makes 2 \ vibrations while 
 a pulse would travel up and down the tube, so 'that a pulse- 
 travels f of the length of the tube while the fork makes a 
 vibration. Pig. 48 shows the position of the fork at the 
 moment when it is producing a maximum of condensation, 
 and the positions which each maximum of condensation and 
 rarefaction previously produced would occupy at that moment 
 if it was solitary. The maximum condensation produced 
 one period ago would be f of the way to the bottom, and 
 
VIBRATIONS OF AIR IN PIPES. 
 
 141 
 
 the one before that would have reached the bottom and come 
 -J of the way np again, while the maximum rarefaction which 
 started 2| periods ago would have just reached the top, so 
 that the maximum condensation due to it would be just ready 
 to start down with that now being produced by the fork. 
 Those pulses which started still earlier would in the same 
 way have positions coincident with later ones, so that the 
 points marked are all the places where there would be 
 
 OPEN TUBE 
 A 
 
 N 
 
 A 
 
 N 
 
 A 
 N 
 
 A 
 f 
 
 RfCf 
 
 CLOSED TUBE 
 
 A 
 
 
 A 
 
 
 A 
 
 
 
 
 
 N 
 
 
 
 N 
 
 
 A 
 
 
 
 A 
 
 
 N 
 
 
 
 
 
 A 
 
 _N. 
 
 a/ 
 
 
 N 
 t 
 
 
 U 
 
 c 
 
 Fig. 48. 
 
 Fig. 49. 
 
 maxima of condensation and rarefaction if each pulse travelled 
 independently. The actual condition of the air can be in- 
 ferred from the positions of these imaginary pulses. JV, N, JV, 
 where an imaginary maximum of condensation travelling 
 down passes another travelling up, are nodes (Art. 77) ; 
 A, A, A, where imaginary pulses of opposite kinds cross, are 
 antinodes. In Fig. 48 the imaginary waves travelling down 
 the tube are shown on the left, and the imaginary reflected 
 waves travelling up again on the right, the directions in 
 which the air would be moving in such waves being shown 
 by short arrows. The actual condition of the air in the tube 
 
142 SOUND. 
 
 (i.e., nodal or antinodal) with the actual direction of move- 
 ment of air is shown by encircled letters in the middle 
 column. The moment chosen is the "uniform density instant." 
 
 In the same way we find that forks which make respec- 
 tively i, f, f vibrations while a pulse travels twice the length 
 of a closed tube throw the air into the conditions of sta- 
 tionary undulation represented by a, #, c of Fig. 49, and forks 
 making |, f, -y- or any higher odd number of half-vibrations 
 in the same time will produce a similar condition with still 
 shorter intervals between the nodes. And by the same 
 method we find that forks which make respectively 1, 2, & 
 vibrations while a pulse travels twice the length of an open 
 tube throw the air into the conditions of stationary undula- 
 tion represented by d, e, /of Fig. 49, and forks making 4, 5, 
 or any higher number of complete vibrations in the same time 
 would produce a similar condition. 
 
 In either open or closed tubes, the number of half-segments 
 into which the tube is divided by nodes and antinodes is 
 equal to the number of half -vibrations made by the spring 
 (and therefore by the air in the tube) while a pulse would 
 travel twice the length of the tube ; the number of half- 
 segments is therefore odd in a closed pipe and even in an open 
 one. A closed end is always a node, and an open end an 
 antinode. The middle point of an open tube must be either 
 a node or an antinode ; in a closed tube it is neither. 
 
 The mouth of the tube being an antinode, the air there 
 simply moves in and out, and may be considered as a piston 
 vibrating at the end of the tube like that in Fig. 22. Con- 
 densation and rarefaction are therefore produced in turn in 
 the air just outside the mouth, and travel away (by the 
 process of progressive undulation) in all directions through 
 the external air. The condensation just outside the mouth 
 is at its maximum when the air just inside the' mouth is 
 moving outwards most rapidly, which is at one of the 
 moments of uniform density of the stationary undulation in 
 the tube ; the maximum degree of rarefaction just outside 
 the tube occurs at the other uniform density instant (Art. 77). 
 It is this latter instant which is represented in Fig. 48, and 
 the letters and arrows round the mouth are intended to 
 indicate that the air there is rarefied and moving inwards 
 
^r j * 1 v _i rv < A , , N L-f I V^ fl I II I 
 
 D> *T OF PHYSICS 
 
 VIBRATIONS OF AIR IN PIPES. 14& 
 
 towards the mouth, while the rarefied condition is travelling 
 away in all directions. It will be seen that this condition of 
 the air outside the mouth might he considered as a continua- 
 tion into the external air, with diminished amplitude, of the 
 imaginary return waves which have been reflected at the 
 bottom. The changes of density produced in the air outside, 
 though it is to them that the sound we hear is due, are very 
 small compared to those which occur in the air in the tube 
 itself. 
 
 As a condensation starts off through the external air each 
 time the air just inside the mouth of the tube moves outwards, 
 the length of the waves in the air outside is the distance a 
 pulse travels in one period of the stationary undulation. 
 And we showed in Art. 77 that the period of a stationary 
 undulation was four times the time required for a pulse to 
 travel a half -segment (from a node to the nearest antinode). 
 So that the waves produced in the external air are always, 
 four times the length of a half-segment of the stationary 
 undulation in the tube. If we examine Pig. 49, keeping this 
 in mind, we can easily make out the following table, in which 
 I is the length of the tube, and t the time required by a pulse 
 to travel this length. 
 
 KIND OF TUBE. 
 
 CLOSED. 
 
 OPEN. 
 
 
 Slowest 
 
 
 
 
 Slowest 
 
 
 
 
 
 or 
 
 
 
 
 or 
 
 
 
 
 Mode of Vibration. 
 
 Funda- 
 
 2nd. 
 
 3rd. 
 
 &c. 
 
 Funda- 
 
 2nd. 
 
 3rd. 
 
 &c. 
 
 
 mental. 
 
 
 
 
 mental. 
 
 
 
 
 Fig. 49. 
 
 a 
 
 b 
 
 c 
 
 
 d 
 
 e 
 
 / 
 
 
 No. o half-seg-\ 
 ments / 
 
 1 
 
 3 
 
 5 
 
 &c. 
 
 2 
 
 4 
 
 6 
 
 &c. 
 
 Length of half-\ 
 segment / 
 
 I 
 
 V 
 
 V 
 
 &c. 
 
 
 
 V 
 
 V 
 
 &c. 
 
 Length of waves\ 
 in external air J 
 
 
 
 # 
 
 V 
 
 &c. 
 
 21 
 
 I 
 
 
 
 &c. 
 
 Period of vibra- / 
 tion \ 
 
 4* 
 
 V 
 
 ^ 
 
 &c. 
 
 2t 
 
 t 
 
 
 
 &c. 
 
 Ratio of frequency") 
 to slowest vibra- V 
 tion J 
 
 1 
 
 3 
 
 5 
 
 fc . 
 
 1 
 
 2 
 
 3 
 
 ,, 
 
144 SOUND. 
 
 Some of the statements just given are only approximately 
 true, for the mouth of a tube is not strictly an antinode. 
 
 The stationary undulation in the tube is not divided 
 from the progressive undulation which starts from it by a 
 definite line ; the reflection of the wave at the open end does 
 not take place exactly at any one point, so that there is an 
 intermediate region, partly in the pipe and partly outside, 
 where the undulation is not exactly either stationary or pro- 
 gressive. There is thus no true antinode at the mouth, nor 
 is the distance from the mouth to the first node quite as 
 great as the distance from a node to an antinode in the tube. 
 The difference is shown, both by experiment and calculation, 
 to be about equal to f of the radius of the tube, so that we 
 must add this quantity to the length of a closed tube, and 
 twice this quantity to that of an open one, to get the length 
 of an exact number of half-segments. Strictly, therefore, 
 these increased lengths should be substituted for I in the table. 
 
 The length of the waves from a closed tube vibrating in 
 its slowest mode is thus 4 (l+a) where a is about f- of the 
 radius of the tube ; from an open tube it is 2 (/-f-2#). The 
 waves from the open tube are therefore not exactly half as 
 long as those from a closed tube of the same length, but 
 rather more ; the note produced by the open tube is rather 
 less than an octave higher than that of the closed one. 
 
 The slowest resonant harmonic vibration possible for a 
 pipe is called its fundamental vibration, and the others are 
 called its harmonics, or overtones. We see from the table 
 that the overtones of a closed pipe have frequencies which 
 are odd multiples of that of its fundamental, while the over- 
 tones of an open pipe have frequencies which include every 
 exact multiple of its fundamental. 
 
 The air in a tube closed at both ends may be sejf, in resonant 
 vibration, as in Kunclt's experiment described in Chapter XII. The 
 modes of vibration possible in this case are like those of an open 
 tube with nodes and antinodes interchanged. As no waves are 
 given off from a tube closed at both ends, the resonance is not 
 audible in this case unless the observer is inside the tube, which 
 may be a long room or passage ; but in smaller tubes the fact of 
 the resonant vibration may be shown in other ways, to be explained 
 in Art. 93. 
 
 So far, we have considered only pipes of uniform bore. Pipes of 
 
VIBKATIONS OF AIE IN PIPES. 145 
 
 varying diameter can also be thrown into resonant vibration ; in 
 this case they divide into segments whose natural vibration periods 
 are equal, but whose lengths, are unequal. A conical pipe, stopped 
 at the apex, may be made to vibrate with 1, 3, 5, &c., half-segments 
 like a cylindrical stopped tube, the apex being of course a node. 
 But, when it vibrates as one half-segment, its period is only half 
 that of a cylindrical stopped tube of the same length, or the same 
 as that of a cylindrical open tube of that length. Also, when it 
 vibrates with 3 half-segments, the first of these (beginning at the 
 apex) is half the length of the tube, and when it vibrates with 
 5 half -segments, the first is -| of the length of the tube, and so on, 
 so that its successive harmonics have to its fundamental the ratios 
 2, 3, &c. Thus both its fundamental and its more rapid modes of 
 vibration correspond in frequency with those of an open cylindrical 
 tube of the same length, though the actual mode of vibration is 
 quite different. 
 
 87. Vibration produced "by Air Blast. The air in a tube 
 may be set in resonant vibration in many other ways. One 
 is to send a blast of air across an open end of the tube. 
 
 It is not very clear how this produces resonant vibration, and 
 various explanations are given. It is sometimes stated that the 
 rushing noise produced by the blast striking the edge of the hole 
 has among its harmonic components a vibration of the frequency to 
 which the tube resounds ; but, if that was the cause, a tube ought 
 also to resound when a rushing sound is produced near its mouth 
 by blowing across the edge of something else, without the blast 
 itself reaching the tube, and this does not occur. The following is 
 perhaps nearer to the true explanation. A very slight difference in 
 the direction of the blast, of air determines whether the air goes 
 into the tube, so producing a condensation, or simply passes across 
 the opening, in which case it exhausts air from the tube, by an 
 action similar to that of spray and scent diffusers. If, as is usually 
 the case, this exhausting action first takes place, the air inside the 
 mouth is rarefied, till the pressure inside the tube becomes so 
 much less than that outside, that the air blast is deflected inwards, 
 so producing condensation; and so on. These conditions travel 
 down the tube, and are reflected, and each condensation, as it reaches 
 the mouth again, deflects the air blast outwards, so that its action 
 increases the rarefaction which would in any case be produced by a 
 condensation reaching the mouth ; and similarly for a rarefaction. 
 The stronger the blast, the more rapidly it exhausts or condenses 
 the air, and the larger the number of rarefactions and condensations 
 which start down the tube before the first returns, after which the 
 action of the blast is simply to increase the waves each time they 
 return to the mouth. 
 
 SD. L 
 
SOUND. 
 
 A blast of proper intensity will throw the air in a tube 
 into any of the modes of stationary undulation into which it 
 might be thrown by a fork ; a very weak blast causing the 
 fundamental vibration, and blasts of increasing strength pro- 
 ducing higher and higher harmonics. It is, however, difficult 
 to produce any but the fundamental mode of vibration in 
 wide tubes. 
 
 The blast of air may be across the end, as in whistling 
 with a key, or across a hole in the side, as in a flute, or 
 directed by means of a passage so that the jet strikes a sharp 
 
 Fig. 60: - 
 
 Fig. 
 
 edge, as in a whistle, or organ pipe with a flute mouthpiece ; 
 a section of the latter is given in Fig. 50. 
 
 Instead of producing any required mode of vibration by 
 adjusting the strength of the blast, we can produce one 
 mode to the exclusion of others by opening additional holes 
 in the side of the tube at points which are antinodes for the 
 particular mode of vibration we require, for no mode of 
 vibration is possible which has not an antinode at every 
 opening. This principle is used in the flute. 
 
 88. Non-harmonic Vibration. The impulses given by 
 the wavering air-blast to the air-column are periodic but not 
 
VIBRATIONS OF AIR IN PIPES. 14? 
 
 harmonic, and may therefore be considered as the sums of 
 harmonic pressure-changes of frequencies which are multiples 
 of the real movement of the blast. (Art. 52.) Each such 
 pressure- change would produce resonant vibration of its own 
 period if the air column had a free-vibration period not very 
 different. The vibration produced is therefore not harmonic, 
 and may have harmonic components of any frequencies 
 (multiples of that of the vibration) which are nearly equal to 
 possible free-vibration frequencies of the air column. Thus 
 the vibration in a closed tube, caused by a blast, may have 
 harmonic components whose frequencies are odd multiples of 
 its own, but not even ones ; that of an open tube may have 
 components which are any exact multiples of its frequency. 
 The harmonic components of the waves sent out are the same 
 as those of the air column itself. 
 
 Any air column, open or closed, if set in resonant vibra- 
 tion by a single tuning fork, vibrates harmonically like the 
 fork, and under these circumstances the resonant sounds from 
 an open and a closed tube are exactly alike. But, when 
 either tube, if not very wide, is set in resonant vibration by 
 a blast of air, it usually vibrates non-harmonically, and in 
 that case the sounds are of different quality, because, as 
 explained above, the closed tube gives waves whose harmonic 
 components are all odd multiples of their fundamental, while 
 the open tube has even multiples as well. Yery wide tubes 
 are not easily set in non-harmonic vibration by a blast of air, 
 so that there is not so much difference in the quality of the 
 sounds from open and from closed pipes when they are wide 
 as when they are narrow. 
 
 There is an important difference between the vibration pro- 
 duced by a fork and that produced by a blast of air. The air 
 column in a tube is of small mass, and there is considerable loss of 
 energy in each vibration owing to the waves sent off; forced 
 resonance, of a period different to its natural vibration, is there- 
 fore easily produced. (Art. 48.) A fork has a natural period not 
 easily altered, and accordingly sets an air column in resonant 
 vibration of this period, even when the natural period of the air 
 column is considerably different. The air blast, on the contrary, 
 has no natural period of vibration, and is controlled by the return 
 of the waves it produced, so that the period of the resonant vibra- 
 tion is exactly that in which a pulse would travel the length of four 
 half -segments of the tube. (Art. 77.) 
 
148 SOUND. 
 
 89. Reed pipes. An air column may also "be thrown into- 
 resonant vibration by means of a reed. The term " reed " is 
 applied sometimes to the whole ';nd sometimes to a part of 
 an arrangement consisting of a strip or tongue, usually of 
 thin metal, fixed at one end to a plate, so that it covers a 
 rectangular opening in the plate, as shown in Fig. 51. The 
 tongue may be either a little larger than the opening, in which 
 case it is a striking reed, or a little smaller, when it is a free 
 reed. In either case the tongue is bent so that its free end 
 stands a little away from the plate, and a blast of air is 
 blown through the opening from the side on which the 
 tongue is fixed. As the air increases in speed, it carries the 
 tongue with it, and so blocks the opening ; when the rush of 
 air stops, the tongue springs back. Thus a succession of 
 puffs of air escape through the opening, producing waves. 
 These may either be allowed to escape into the air, as in the 
 harmonium, in which case the frequency of the puffs is that 
 of the natural vibrations of the tongue ; or they may escape 
 into a pipe whose length is such that it can be set in resonant 
 vibration of the same frequency, as in the clarinet and 
 several kinds of organ pipes. In this case, the tongue being 
 of small mass as well as the air column, each forces the 
 vibrations of the other ; the vibration period is a compromise 
 between that natural to the reed and that natural to the air. 
 
 The tongue of a free reed is not a freely vibrating spring, 
 being affected by the changing pressure of the air blast ; its 
 frequency is not independent of this pressure, as is often 
 stated, but increases with it : this is easily shown on a 
 concertina. The striking tongue, which rebounds from the 
 plate, has its frequency still more increased by an increase of 
 air pressure. The striking form is the one practically used to 
 cause vibration in tubes. 
 
 Tig. 51 shows a reed in conjunction with a conical tube, 
 the box which contains the reed being provided with windows 
 for observing the vibration. The air is blown from below 
 into this box, passes through the reed in the direction away 
 from the reader, and escapes behind the metal plate into the 
 conical tube. The vibration period of the tongue is adjusted 
 by a sliding wire which allows a shorter or longer portion to 
 vibrate. 
 
VIBRATIONS OF AIR IN PIPES. 149 
 
 An air blast against an edge of a hole in a tube sets the air 
 Column in vibration whether the other end of the tube is open or 
 closed ; in either case the hole across which the air is blown is an 
 .antinode, and the air vibrates in and out through it, but the blast sucks 
 out in otie half-vibration the air that it has just blown in in the other ; 
 there is no continuous current of air along the tube, and if smoky 
 air is used for the blast the air in the tube remains clear for a long 
 time. The case is quite different when a tube is made to vibrate 
 by a reed at one end, as in the clarinet. Here the air never passes 
 out of the tube through the reed, so that the tube must be ojgen at 
 the other end ; and as a condensation, returning to the reed end, 
 does not escape there, it is reflected as a condensation, so that the 
 reed end is a node. A reed tube is therefore always a closed tube, 
 and, if cylindrical, can give only the odd harmonics of its funda- 
 mental. As pointed out above, this may be avoided by making the 
 tube conical, as in the French horn, in which, as in many other 
 instruments, the lips of the performer take the place of a mechanical 
 reed. 
 
 90. Effect of Change of Temperature. As the vibration 
 period of an air column depends on the time taken by a 
 pulse to travel its length, the vibration frequency is propor- 
 tional to the velocity of a pulse. Rise of temperature 
 therefore increases the vibration frequency of the air column, 
 the frequency being proportional to the square root of the 
 -absolute temperature (Art. 33). The increase of frequency 
 is really a little less than this, because the pipes lengthen 
 with rise of temperature ; this effect is greater with metal 
 than with wood pipes. In reed pipes the increase of fre- 
 quency with rise of temperature is much less than with flute 
 pipes, because the stiffness of the tongue diminishes as the 
 temperature rises, so that, while the rise of temperature 
 shortens the natural period of the air column, it lengthens 
 that of the reed. Reeds without pipes, as used in the 
 harmonium and concertina, diminish slightly in frequency 
 with rise of temperature. 
 
 91. Vibration of Liquid Columns. Liquid columns may 
 also he set in resonant vibration. If a common tin whistle 
 is immersed entirely in water in a jar, and connected by a 
 tube to a high pressure water supply (such as the ordinary 
 water pipes of a house), the water in the whistle is set in 
 resonant vibration. Not much sound is heard, as sound does 
 
150 SOUND. 
 
 not pass easily from water to air (Art. 71) but the trembling 
 of the jar is easily felt. If the observer puts his ear under 
 the water, the sound is well hea.cl. 
 
 92. Vibrations of Solid Eods. As condensations and 
 rarefactions travel along rods of elastic material exactly as 
 along the air in tubes, and are reflected at free ends of rods 
 exactly as at open ends of tubes, and at fixed or loaded 
 points of rods exactly as at closed ends of tubes, rods can be 
 set in resonant stationary undulation exactly like that of 
 air columns, but different means must be adopted to give the 
 successive impulses. The simplest way is to draw a resined 
 cloth along the rod. 
 
 Resin, like other viscous substances, adheres the more strongly to 
 surfaces over which it moves the more slowly it travels along them. 
 The cloth sticks to the rod, pulling the part with which it is contact 
 along with it, and so producing condensation in front and rarefac- 
 tion behind. These conditions travel to the ends of the rod, or to- 
 any poii.t of it which is fixed or loaded, and are reflected ; and thus 
 travel up and down thft rod, and as they pass any part of the rod 
 that part moves a short distance. As a wave passes the cloth, if it 
 is one in which the particles of the rod move in the same direction 
 as the cloth, the relative velocity of the cloth along the rod is 
 diminished, and the cloth adheres more strongly to the rod, and 
 gives the surface a pull in the direction in which it is already 
 moving, while a contrary action takes place if the wave is one in 
 which the particles of the rod are moving the opposite way to the- 
 cloth. Each wave is therefore increased each time it passes the 
 cloth, and the rod is set in resonant vibration. 
 
 The resonant vibration of a rod fixed or loaded at one end 
 is exactly similar to that of the air in a closed tube, for the 
 waves are reflected in just the same way. The vibration of 
 a rod not firmly fixed anywhere, for instance held in one 
 hand and rubbed with the other, is similar to that of the air 
 in an open tube. In the vibration of a rod clamped in the 
 middle, each half vibrates like any other rod clamped at one 
 end, but the two halves keep time with each other, so that 
 points at equal distances from the middle always move in 
 opposite directions at the same time. A rod fixed at both 
 ends (which may be a stretched wire) vibrates like the air in 
 a tube closed at both ends. If pieces of lead are fixed by 
 
VIBRATIONS OF AIR IN PIPES. 151 
 
 clamps to two points on a stretched wire, and a piece of 
 sandpaper, or resined cloth, drawn along the part of the wire 
 between them, a loud sound is produced, whose pitch 
 depends on the distance between the pieces of lead, but not 
 on the tightness of the wire. This sound is due to the 
 resonant longitudinal vibration of the part of the wire 
 between the lead blocks, which are nodes. The blocks 
 need not be fixed, except to the wire, as the waves are 
 almost totally reflected on arriving at a portion of the wire 
 of so much greater density than the rest (Art. 70). 
 
 Owing to the great number of successive impulses whose 
 energy may be added to cause a resonant vibration, the 
 movement of the air in a resounding tube, or of the material 
 of a rod in resonant vibration, may be very large compared 
 with the ordinary movements of progressive undulation. 
 In an air column the movement of the air may amount 
 to an actual wind, capable of carrying along cork filings 
 or other light powders. Ey drawing, with one hand, 
 a resined cloth along a steel bar, we can make the bar 
 lengthen and shorten to an extent which it would require a 
 direct pull of many tons' weight to effect. Thick glass rods 
 capable of supporting a ton or more, may easily be pulled to 
 pieces in this way. 
 
 93. Experimental Illustrations, The movement of the 
 vibrating air in a tube may be studied in various ways. If 
 the tube, or one side of it, is of glass, we may place it vertically, 
 and let down into it a thin membrane stretched on a horizontal 
 wire ring (as described in Art. 40), with a little sand on the 
 membrane ; the sand dances everywhere except at nodes. 
 Or the tube may be placed horizontally and a light powder 
 (cork dust or lycopodium) scattered in it ; this, being blown 
 about everywhere else, soon collects at the nodes. The same 
 method may be used for a vibrating liquid column, a heavier 
 powder, such as precipitated silica, being used instead of 
 cork dust. 
 
 For demonstrating to a large audience the different modes 
 of stationary undulation of the air in an organ pipe, Konig's 
 manometric capsules (Fig. 52) are very useful. A manometric 
 capsule is a box, shaped like a large pill-box, of which one 
 
152 
 
 SOUND. 
 
 end consists of a stretched membrane. It is fitted into 
 a hole cut in the side of the organ pipe, with the membrane 
 side inwards, in contact with the air in the pipe. Coal-gas 
 
 Fig. 52. 
 
 is conveyed into the capsule by a tube, and out by another 
 to a pin-hole burner, where it burns as a small flame. If 
 any rapid changes take place in the pressure of the air in the 
 part of the organ pipe where the capsule is inserted, the 
 membrane vibrates and the flame flickers, and, though this 
 flickering is too rapid to be easily observed directly, it may 
 be detected by watching the reflection of the flame in a 
 rapidly rotating mirror. It then appears as a band of light 
 toothed along its upper edge, the teeth being the images of 
 the flame at the moments when it is highest. (A similar 
 effect can be produced, without a rotating mirror, by rapidly 
 turning the head from side to side while looking at the flame.) 
 If a number of such capsules are inserted in the side of an 
 organ pipe, the flames of all of them flicker except of those 
 at antinodes, and the positions of these are therefore easily 
 seen. The methods previously given detect the nodes. 
 
 The nodes of a rod in stationary undulation may be shown 
 either by scattering sand on it or by putting a number of 
 
YIBKATIONS OF AIR IN PIPES. 153 
 
 -card or wire rings on the rod and holding it nearly, but not 
 quite, horizontally. When the rod is set in resonant vibration, 
 the rings slip down it till they reach a node. 
 
 Any of the modes of resonant vibration of an air column may be 
 imitated by means of the spiral wire of Art. 73. A short heavy 
 pendulum, of adjustable length, must be suspended vertically over 
 one end of the spiral, so that the bob of the pendulum hangs between 
 the horizontal rods and on a level with them. The pair of strings 
 supporting that end of the spiral are then to be disconnected from 
 the rods and tied to the pendulum bob, so that the end coil of the 
 spiral now hangs from the bob, while the others hang from the rods. 
 The other end of the spiral must be fixed if the vibration in a closed 
 tube is to be represented ; free for an open tube. If the pendulum, 
 which represents the tuning fork used with an air column, is set 
 swinging in the direction of the length of the spiral, only an ir- 
 regular movement of the coils results until the period of the 
 pendulum is adjusted so that it fulfils the condition necessary in 
 order that periodic impulses may cause resonant vibration of a rod 
 or fluid column ; the condition explained in Art. 85. The spiral 
 then begins to vibrate quite regularly, with definite nodes where 
 the coils do not move, and forms a good illustration of the process 
 of longitudinal stationary undulation. With a little practice, the 
 hand, moved regularly and rapidly backwards and forwards, may 
 replace the pendulum bob, and gives better results. 
 
 The resonant vibration of gas columns and rods affords a 
 means of determining the velocity of sound in the respective 
 gases and solids ; these methods are explained in Chapter XII. 
 
 EXAMPLES II. 
 
 In the following examples the velocity of sound in air is to be 
 taken as 33,240 + 60 cm. per second, t being the centigrade tempera- 
 ture. Unless otherwise stated, the temperature is to be taken as C. 
 
 ELEMENTARY. 
 
 1. Give the lengths of the three shortest closed tubes, and of the 
 three shortest open tubes, which would resound to a tuning fork 
 making 200 vibrations per second. 
 
 , 2. A whistle making 2000 vibrations per second is placed a metre 
 from a wall. At what points between the whistle and the wall are 
 antinodes to be found ? 
 
154 SOUND. 
 
 3. A whistle is sounded near a wall, and the nearest node to the 
 wall is 3 cm. from it. Determine the frequency of the whistle. 
 
 4. Give in each case the four Jowest vibration frequencies 
 possible in a tube 3'324 metres long (a) when open at both ends, 
 (ft) when open at one end, (c) when closed at both ends. [Neglect 
 the correction for radius.] 
 
 5. If a tube makes 340 vibrations per second when the temperature 
 is 16 C., what is its frequency, in the same mode of vibration, when 
 the temperature is 51 C. ? [Neglect expansion of tube.] 
 
 6. The air in a closed tube 34 cm. long is vibrating with two- 
 nodes and two antinodes, and its temperature is 51 C. What is- 
 the wave-length of the waves produced in the air outside the tube, 
 if the temperature of that air is 16 C. ? 
 
 7. A closed tube 15 cm. long resounds, when full of oxygen, to a 
 given fork. Give the length of a closed tube, full of hydrogen, 
 which will resound to the same fork. 
 
 8. If the velocity of sound in hydrogen is 126,000 cm. per second, 
 and in air 33,300 cm. per second, what is the length of the waves 
 which will be produced in the surrounding air by blowing an open 
 organ pipe, a metre long, with hydrogen, the pipe being also full of 
 hydrogen. 
 
 ADVANCED. 
 
 1. A vertical tube 1 metre long and 4 cm. in diameter is gradually 
 filled with water, while a tuning fork, making 500 vibrations per 
 second, is held over the upper end. At what positions of the water 
 surface will the tube resound (taking the correction for diameter of 
 the pipe into account) ? 
 
 2. What must be the diameter of a closed tube 2 ft. long in order 
 that it may resound to the lowest note given by an open tube 4ft 
 long and 8 ins. in diameter ? 
 
 3. Where must a conical tube, closed at the apex, be cut in two in 
 order that each part may resound to the same note ? What is the 
 ratio between the frequency of this note and the note to which the 
 
 whole tube would resound ? 
 
 ^ 
 
 4. A wooden rod 1 yd. in length floats (when compelled to float 
 vertically) with 2 ins. of its length out of water, and if rubbed 
 longitudinally, without being firmly fixed anywhere, the lowest note 
 it can be made to give has a frequency of 256. Find Young's 
 Modulus for the wood in poundals per square foot. 
 
 5. A brass wire, 2 metres long and 1 sq. mm. in sectional area 
 weighs 16 gm., and when it is hung up by one end, and 20kilog. 
 are suspended from the other, it elongates by 6 mm. What note 
 will it give when rubbed in the direction of its length ? 
 
CHAPTER XL 
 
 TRANSVERSE UKDULATION. 
 
 94. Transverse Wave in a Cord. Take a long rope AB 
 (Fig. 53) and fasten one end B firmly to a wall at a point 
 about six feet from the ground. Hold the other end A in 
 your hand, about four feet from the ground, and stretch the 
 
 Fig. 53. 
 
 rope till the part nearest to your hand is about horizontal. 
 It will not be quite straight ; but we neglect that at present. 
 Raise your hand suddenly a few inches, so that A is at A. 
 The immediate result of this is that a short portion A'C, 
 close to your hand, is in an altered condition, not rarefied or 
 condensed, but sloping instead of horizontal, the rope having 
 the form A CB. The point C is now acted on by two forces, 
 due to the stretched condition of the string, along CA and 
 CB, and the resultant of these is upwards. C therefore 
 begins to move up, and it stops only when it has moved as 
 far as A moved, so that a portion of the rope A'C' is hori- 
 zontal again, while the sloping condition exists in another 
 portion C'D, the rope having now the form A C'DB. D then 
 begins to move up in the same way, and so the sloping con- 
 dition which was first produced in A C travels all along the 
 rope. 
 
156 SOUND. 
 
 Now this continuous movement of a condition of slope 
 along the rope has been effected by each portion of the rope 
 moving in turn a short distance in a direction at right angles 
 to the rope : A to A 1 , C to C', and so on. Further, each 
 portion moved only while it formed part of the sloping 
 section ; as soon as the sloping condition had passed it, it was 
 at rest again. 
 
 The whole process is in many respects very similar to the 
 transmission of a pulse of condensation or rarefaction, as 
 described in Art. 11. In both cases a condition of altered 
 relative position of the particles travels continuously, while 
 each particle in turn moves a short distance and then stops. 
 In both cases the altered relative position exists where the 
 particles are moving, and when the particles come to rest 
 they are in their original position relative to their neighbours, 
 though not in space. In both cases the velocities with which 
 the particles at different points are moving are proportional to 
 the difference between their actual and their ordinary relative 
 position. The chief differences are ( 1 ) that in the experiment 
 we have just described the " altered relative position " of 
 the particles is altered relative direction, not altered relative 
 distance; slope, not condensation or rarefaction; (2) the 
 short movement which each particle executes in turn is at 
 right angles to the direction in which the condition travels, 
 not in that line, as in Art. 11. For this reason this kind of 
 motion is called transverse progressive undulation. 
 
 95. Velocity of Transverse Waves. It can be shown 
 (see Appendix D) that the slope produced by the movement 
 of A (i.e., the angle between the changed direction and the 
 original one) is proportional (as long as it is small) to the 
 velocity with which A was displaced, but that the velocity 
 with which the sloping condition travels along the rope does 
 not depend on how A moved, but only on the mass of each 
 unit length of the string and on the force with which the 
 string is stretched. Even without investigating the exact 
 relation, it is evident that, the more tightly the string is 
 stretched, the greater the resultant force on (7, and therefore 
 the quicker C will move up to (7', and the sooner D will 
 begin to move, and so on. So that the velocity of a trans- 
 
TRANSVEKSE UNDULATION. 157 
 
 verse wave depends on the force with which the string is 
 stretched. The velocity of a longitudinal wave, on the other 
 hand, is the same whatever the tightness of the string, a& 
 explained in Art. 27. Even if a string or wire is stretched 
 to the point of breaking, the velocity of a transverse wave 
 along it is always much less than that of a longitudinal one. 
 A fuller investigation shows (see Appendix D) that the 
 velocity with which a sloping condition travels along a rope is- 
 
 /force with which the rope is stretched 
 mass of unit length of the rope 
 
 the stretching force being measured in dynamical units, as 
 explained in Art. 27. 
 
 96. Reflection of Transverse Waves, When the sloping 
 condition arrives at the fixed point B, it is reflected, and 
 travels back again to A. The direction of the slope is the 
 same in the reflected as in the original wave, but the move- 
 ment of each particle of the rope while it forms part of the 
 reflected wave is in the opposite direction to its motion while it 
 formed part of the original wave. This corresponds to the re- 
 flection of a condensation or rarefaction at the closed end of a 
 tube, and is reflection with change of sign. 
 
 If such a transverse wave arrives at a free end of a rope, it 
 is also reflected, but in this case the slope is reversed, and 
 the motion of the particles is not. This corresponds to the 
 reflection of condensations and rarefactions at the open end of 
 a tube. The wave produced by cracking a whip is reflected 
 in this way. 
 
 Every movement which we cause A to execute at right 
 angles to the length of the rope is repeated in turn by each 
 particle ; later and later the further from A. So that, until 
 the waves reach .Z?, the past displacements of A are the 
 present displacements of the successive points of the rope. 
 The rope is in fact a displacement curve representing the 
 history of the movements of A. 
 
 If we move A up and down harmonically, the rope itself 
 is thus thrown into harmonic waves, which travel along it 
 away from A. The continuous line in Fig. 54 shows the 
 form of the rope at an instant when A has been vibrating 
 harmonically for some time ; the arrows show the relative 
 
158 SOUND. 
 
 velocities of the material of the rope at different points. 
 These velocities are proportional at each point to the slope of 
 the rope there, and are in opposite directions at points where 
 
 Fig. 54. 
 
 the rope slopes oppositely. An instant later every point 
 of the rope has moved a short distance in the direction of the 
 arrow attached to it, and the form of the rope is the dotted 
 cnrve. Thus, while each particle of the rope has moved up 
 or down, the form of the rope has moved to the left. 
 
 If the stretching force is different in different parts of the string, 
 the velocities with which the waves travel along different parts are 
 proportional to the square roots of the stretching forces, as shown 
 above. The number of waves which pass any point in a second is of 
 course equal to the number that start in a second, and is the same 
 in all parts of the string. The waves are therefore longer where 
 they travel quicker, the length of a wave varying as its velocity 
 varies as it goes along, as in longitudinal waves. 
 
 97. Transverse Stationary Undulation. Next suppose 
 we move both A and B up and down harmonically with the 
 same frequency. A series of harmonic waves, of equal wave- 
 length, will start from each end towards the other. After 
 these wave-systems have met in the middle of the rope, the 
 principle of superposition shows that the actual displacement 
 at any given moment, of every part of the rope, can be found 
 by adding the displacements which would be due at that 
 moment to the wave-systems separately. 
 
 Let -X", Fig. 55, be the form which the rope would have, 
 at a given moment T, if only the waves from A travelled 
 along it, and let Y be the form which it would have at the 
 same moment if only the waves from B travelled along it. 
 The actual form of the rope at the moment T is that found 
 by adding the ordinates of X and Y\ it is shown in line 8. 
 The forms which would be due an instant later to the waves 
 
THAISTSVEKSE UNDULATION. 
 
 159 
 
 from A and from B respectively can be found by shifting X 
 to the left and Y an equal distance to the right. If we 
 suppose each advanced -J- of a wave-length, and add the 
 
 Fig. 55. 
 
 ordinates again, we get line 1. By advancing X and Y 
 again each of a wave-length, we get line 2, and so on. 
 (Only 5 out of the 8 such stages of a complete cycle are 
 shown, the remaining 3 being simply 1, 2, and 3 inverted.) 
 We see that the actual movement of the cord is one in which 
 its form is always a harmonic curve, and that, twice in each 
 complete cycle, there is a moment when every part of the 
 cord has, simultaneously, its maximum displacement, and is 
 therefore at rest (stationary instant) while twice in each cycle 
 the curve becomes a straight line. We see also that there 
 are certain fixed points nodes through which the cord always 
 passes, so that the displacement at these points is always 
 zero, but that it is at these points that the greatest changes 
 of slope occur. Also that between these there are other 
 points antinodes where the greatest displacements occur, 
 but where the cord is always parallel to its original position, 
 so that there are no changes of slope. The movement of the 
 cord is thus one of transverse stationary undulation. We see 
 also that the distance from one node to the next is half a 
 
160 SOUND. 
 
 wave-length of either of the wave-systems which would he- 
 caused hy the movement of A or B alone, and that the 
 stationary undulation goes through the complete cycle of 
 movements in the time in which either of the progressive- 
 undulations, due to A or JS alone, would advance one wave- 
 length, or twice the distance "between two nodes. 
 
 Thus, if I is the distance between two nodes and Fthe 
 
 21 
 velocity of a transverse travelling wave, is the period in 
 
 which the stationary undulation goes through its changes, 
 
 fr 
 and is its frequency or the number of times it does this- 
 
 2il 
 
 in a second. As 
 
 / stretching force 
 
 mass per unit length 
 
 this frequency = - / stretching force m 
 21 V mass per unit length 
 
 If, when the cord is in this condition of stationary undula- 
 tion, we fix any two nodesj say C and D, the movement of 
 the cord is of course unaffected, since C and D were stationary 
 already. The condition of stationary undulation will therefore 
 continue till the energy of the string has been partly Com- 
 municated to the air in sound waves, and partly converted 
 into heat in the string itself. When two points on a string 
 are fixed, a solitary travelling wave in the part between 
 would run backwards and forwards between them, being 
 reflected each time it reached either, and the stationary 
 undulation of a cord fixed at two points may be conveniently 
 considered as the resultant of two fictitious wave-systems 
 travelling in opposite directions and continually reflected in 
 the same way. 
 
 We saw above that, if we had any stretched string, we 
 could, by sending harmonic waves of length 21 along it from 
 both ends, throw it into a condition of stationary undulation 
 with nodes I apart, and that if the mass per unit length of 
 this string is m, and the force with which it is stretched/, 
 
 the frequency of its stationary undulation is . / 
 
 2t/ y 
 
 L. Also 
 m 
 
TK AN S VERSE UNDULATION. 
 
 161 
 
 that, if we fix any two nodes xl apart (x being any whole 
 number), the part of the string between them will continue 
 to vibrate with the same frequency in x loops separated by 
 nodes. At each stationary instant such a string is a motion- 
 less harmonic curve having x single bends. So that, if we 
 took a similar string, similarly stretched, fixed it in the form 
 of a harmonic curve having x single bends of length /, and 
 then let it go everywhere at once except the two ends 
 (that this is impracticable does not matter for our present 
 purpose), it also would vibrate in the same way and with the 
 same frequency. (This can also be proved directly, without 
 considering the movement as the resultant of two fictitious 
 travelling waves.) The more bends we made, the shorter I 
 would be, and the greater the frequency, the frequency being 
 proportional to the number of bends. If, for instance, we 
 could bend a string into the form of the continuous harmonic 
 curve jP, Eig. 56, and let go, it would vibrate from that 
 position to that of the dotted one and back, with a frequency 
 
 /Z_, 
 Y wi> 
 
 Fig. 56. 
 I being in this case the whole length of the string. 
 
 We will call this frequency n. If we could bend the same 
 string into the form of the continuous line G and let go, it 
 would vibrate in two loops, with a frequency 2n, since I is 
 now only half the length of the string. Similarly for three 
 
 SD. 
 
 M 
 
162 
 
 SOUND. 
 
 or more bends. In any case, if the string starts from a har- 
 monic form, every point of it vibrates harmonically with the 
 same frequency. 
 
 98. Quality of Sound from String. Next suppose that 
 we could bend the string into the form H, whose ordinates 
 are the sums of the ordinates of F and G. When we let it 
 go, the movement of each point of the string will be the sum 
 (in the sense explained in Art. 7.) of the movement which it 
 would have executed if it had been bent in the form F and 
 that which it would have executed if it had been bent into 
 the form G. In other words, the string vibrates in two 
 loops from side to side of an imaginary line which vibrates- 
 like F, in the same way as the string G vibrates in two loops 
 
 Fig. 57. 
 
 from side to side of a straight line. Pig. 57 shows some- 
 stages in this movement, the dotted line being the imaginary 
 line which moves as the string F did. The movements of 
 different points of the string are not similar, nor, usually, 
 harmonic, but the motion of each point is the sum of two 
 harmonic vibrations of frequencies n and 2n. 
 
 The same would be true if we began by bending the string 
 into a form which was the sum of any number of harmonic 
 curves which, like F and G, have an exact number of single 
 bends in the length of the string. Now, when we pull a 
 point of the string to one side, the form the string assumes- 
 is that of two straight lines meeting at an obtuse angle at 
 that point. This is not a form which looks likely to be the 
 sum of a number of harmonic curves, but Fourier's Theorem, 
 or an easy extension of it, shows that any of the angular 
 forms which can be produced by pulling a point of the 
 string to one side, is a form which might be made by adding 
 the ordinates of harmonic curves of which the length of the 
 string contains an exact number of single bends. It shows- 
 further that, of the infinite number of such curves, all are- 
 
TRANSVERSE UNDULATION. 163 
 
 required except those which cut their axis line at the point 
 at which the string was pulled aside. Those which cut the 
 axis very near this point are required only of small ampli- 
 tude. 
 
 Suppose, for instance, we pull the string CD (Fig. 58) to 
 one side at a point E, \ of the way from C to D. The form 
 CED is one which might be produced by adding the ordin- 
 ates of harmonic curves which have 1, 2, 3, 4, 6, 7, 8, 9, 11, 
 &c., half- waves in the length CD , those which have any 
 
 Fig. 58. 
 
 multiple of 5 half -waves being omitted, because they would 
 cross the axis under the point E. So that, if n is the frequency 
 with which the string would vibrate if bent into the form of 
 a single harmonic half -wave, the motion of each point of the 
 string, when it has been pulled into the form CED and 
 released, is the sum of harmonic movements whose frequencies 
 are n, 2n, 3%, 4n, 6%, &c. Vibrations of these frequencies 
 are the harmonic components of the real movement of the 
 string. 
 
 Similarly, if we pull a string aside at a point ~ of its length 
 
 
 
 from one end, ^- being a fraction in its lowest terms, any 
 
 harmonic curve which has xb single bends in the length of the 
 stiing (x being any whole number) cuts the axis at the point 
 where the string was pulled aside. The harmonic com- 
 ponents of the vibration produced when the string is let 
 go are vibrations whose frequencies are all the multiples of 
 n which are not multiples of bn, n having the same meaning 
 as above. 
 
 If we strike a string, as in the piano, the harmonic com- 
 ponents of the vibration produced have the same frequencies 
 as if we had pulled it to one side at the same point, but 
 
164 SOUND. 
 
 their relative amplitudes are different. A hard hammer, 
 which touches only a very short piece of the string, and 
 remains in contact with it a very short time, produces a 
 vibration whose high-frequency harmonics are intense, and 
 conversely. 
 
 !N"ext, suppose we touch the vibrating string at some other 
 point with a light object such as a feather or camel' s-hair 
 brush. This stops all modes of vibration of the string 
 except those which have a node at the point touched, and 
 the subsequent motion of the string is the sum of those 
 harmonic components of its previous movement, which have a 
 node at this point. If, for instance, after plucking the 
 string CD at \ of its length from one end we touch it at a 
 point i of its length from one end, all harmonic vibrations 
 become impossible except in 3, 6, 9, 12, 15, &c., loops. 
 Of these its motion included all except that in 15 loops, so 
 that the subsequent motion is the sum of the harmonic 
 vibrations which it could execute in 3, 6, 9, 12, 18, &c., 
 loops, all multiples of 3 which are not multiples of 5 being 
 included. The harmonic components of this vibration have 
 of course frequencies 3n, 6%, 9n, 12n, I8n, &c. ; n having the 
 same meaning as before. 
 
 Similarly, if a string which has been plucked or struck at 
 a point of its length from one end is then touched at 
 
 a point 4 fr om one en ^ ( T~ an ^ 4 being fractions in 
 d o d 
 
 their lowest terms), the harmonic components of the subse- 
 quent vibration have frequencies which include all the 
 multiples of n which are multiples of dn but are not multiples 
 of In. 
 
 The ends of a stretched string must be fastened to some 
 solid body, for instance, to pegs or screws in a board, and 
 when the string vibrates its pull on the pegs varies and the 
 board vibrates also, and so produces air waves. The movements 
 of the air so produced correspond nearly to those of the board, 
 and the movement of the board, though it does not correspond 
 closely to that of the string, is the sum of the movements which 
 the harmonic components of the vibration of the string would 
 
TRANSVERSE UNDULATION. 
 
 165 
 
 produce separately, so that the movement of the board has a 
 harmonic component corresponding in frequency to each 
 harmonic component of the movement of the string. (The 
 relative amplitudes of the components may be very different 
 for the board and string, since some components may cause 
 resonant vibration of the board, and not others. There may 
 also be components of the vibration of the board which are 
 not components of that of the string. ) The string itself also 
 produces waves in the air, but for the reason explained in 
 Art. 22 they are insignificant in intensity compared with 
 those from the larger moving surface of the board, and a 
 vibrating string would hardly be audible if it could be 
 fastened to absolutely immovable points. 
 
 99. Resonant Vibrations of Strings. The harmonic com- 
 ponents of the air waves, and therefore the quality of the 
 sensation they produce, thus depend on the point at which 
 the string is pulled aside or struck, but we cannot make the 
 string vibrate harmonically or produce harmonic air waves 
 by pulling or striking it. But it may be set in resonant 
 vibration by impulses agreeing in frequency with any of its 
 modes of vibration; if these impulses are harmonic in character, 
 the resonant vibration will be harmonic. For instance, if we 
 attach a thread to one prong of a tuning fork, as shown in 
 Fig. 59, it will be found that when one of certain definite 
 
 Fig. 59. 
 
 loads is placed in the scale-pan, and the fork excited by 
 drawing a violin bow across one prong, the thread is thrown 
 into strong stationary undulation, the number of loops depend- 
 ing on the weight in the pan . This occurs only if the weight is 
 such that one of the modes of free stationary undulation of 
 of the thread is nearly of the same frequency as the fork ; the 
 weights which are required to make the same string vibrate 
 
166 SOUND. 
 
 in 1, 2, 3, &c., loops are inversely proportional to the 
 squares of these numbers. This is "Melde's experiment. " 
 
 The fork may also be placed so that it vibrates in the 
 direction of the length of the thread, instead of at right angles 
 to that direction, so that the fork pulls the thread instead of 
 shaking it up and down. In this case there is resonant 
 vibration when the weight is such that one of the modes of 
 vibration of the thread has half the frequency of the fork, 
 for the thread is straight when the prong is at one end of its 
 swing, and loosest when the prong is at the other, so that a 
 quarter vibration of the thread takes place in half a vibration 
 of the fork. 
 
 Merely bringing the stem of a vibrating fork into contact 
 with the board on which a string is stretched will set the 
 string in resonant vibration if the frequency of one of its 
 modes agrees nearly with that of the fork. This may be 
 used to find the frequency of a fork, if the string is stretched 
 by means of a weight so that the stretching force is known. 
 This weight, or the length of the string, is altered till the 
 fork throws the string into resonant vibration, which is 
 detected by placing a little folded piece of paper astride the 
 string ; the paper is thrown off when resonance occurs. The 
 mode of vibration is determined by placing a large number of 
 such " riders " at different points along the string ; those 
 which are at or near nodes are not thrown off. The mass per 
 unit length of the string is ascertained by weighing and 
 measuring it or a similar piece. The frequency of the string 
 in the mode in which the foiik sets it vibrating can be calcu- 
 lated by the formula given above, and this is the frequency 
 of the fork. This method is not very accurate, partly because 
 the rigidity of the string makes its frequency rather greater 
 than that given by the formula, which is strictly accurate 
 only for a perfectly flexible string, and partly because, owing 
 to the small mass of the string, it is not easy to determine 
 exactly when the resonance is at its maximum, and the free- 
 vibration frequency of the string therefore equal to that of 
 the fork. The latter cause of error may be avoided by ad- 
 justing the string until the note produced by twanging it or 
 drawing a violin bow across it is the same as that of the fork, 
 as shown by the absence of beats, or, as very slow beats are 
 
TRANSVERSE UNDULATION. 167 
 
 not easily heard, till the beats have a frequency which is 
 easily counted. The frequency of the heats must of course he 
 #dded to or subtracted from the calculated frequency of the 
 string. The action of the how, which is resined, is the same 
 as that of the resined cloth in Art. 92 ; though it produces 
 resonant vibration, its impulses depend on the vibration 
 already existing, and the vibration it produces has the 
 frequency of the free vibration of the string. 
 
 This experiment is most easily carried out on a sonometer, 
 Tvhich is an instrument for experimenting on the vibrations of 
 strings. Its usual form is shown in Pig. 60. A A is a long 
 
 Fig. 60. 
 
 TDOX with holes in its sides. Two wires are stretched over 
 the two fixed bridges B\ B, the upright faces of which are a 
 metre apart. One of the wires passes over a pulley and is 
 stretched by weights ; the other is stretched by a wrest pin, 
 D, which is turned by a key. A third bridge C is movable, 
 and a little higher than the others, so that it presses the string 
 and reflects transverse waves. 
 
 A simpler form, with one string, is called a monochord. 
 
 The formula n = 7 \l is most easily proved experiment- 
 ally by means of the same apparatus, the string being 
 stretched by a weight, Z, /, m can be observed, and the 
 calculated value of n compared with that determined by 
 direct measurement. For this, if we wish to avoid all 
 assumptions, we can use the photographic method, or the 
 method of tracing on a revolving drum, or the vibration 
 microscope. If we assume that the pitch of the note pro- 
 duced corresponds to the frequency of the vibrations of the 
 string, the frequency may also be measured by the siren. 
 
168 SOUND. 
 
 These methods are described in the next chapter. The 
 calculated and observed frequencies agree very closely. 
 
 From the formula n = 4 12. several facts at once follow 
 
 2/V m 
 
 which are sometimes called the "laws" of the transverse- 
 vibration of strings. The most important are : 
 
 (1) If the stretching force and mass of unit length remain 
 constant, the frequency varies inversely as the length. 
 
 (2) If the length and mass per unit length remain constant, 
 the frequency varies directly as the square root of the 
 stretching force. 
 
 (3) If the length and stretching force remain constant, the 
 frequency varies inversely as the square root of the mass of 
 unit length. 
 
 The last of these may be put in a different form for round 
 strings or wires, viz. : 
 
 (30) If the length, stretching force, and material of several 
 strings are the same, their frequencies are inversely as their 
 radii (or diameters). 
 
 These laws, being merely proportional, are true whatever 
 units are used to measure the different quantities. In this, 
 as already pointed out, they differ from the absolute formula 
 
 n = . I!., which cannot be deduced from these laws, as it 
 21 \ m 
 
 contains more than they do. 
 
 The "laws" are of course completely proved by the 
 experiments which prove the absolute formula, but they may 
 also be illustrated by less elaborate experiments which do not 
 involve the measurement of an absolute frequency. These 
 experiments are often said to "prove" the laws, but the 
 proof is far from conclusive, and involves several assump- 
 tions. 
 
 "We shall evidently require some means of determining 
 whether the changes of frequency produced by changing the 
 length, stretching force, or thickness of the string correspond 
 with the changes predicted by the laws given above. As 
 explained in Art. 65, most persons easily recognise whether 
 the interval between two notes is exactly an octave, and it 
 was shown by an experiment that when it is an octave, the 
 
TEANSVEKSE UNDULATION. 169 
 
 frequency of one vibration is twice that of the other. We 
 shall assume, then, that the perception of this relation 
 between the sounds is sufficient proof that the frequencies of 
 the strings are as 2 : 1. With this assumption, the following 
 experiments may be considered to illustrate the laws, though 
 it would be easy to devise quite different laws with which 
 the experiments are equally consistent. 
 
 Law 1. Adjust the tension of the string which is stretched 
 by the wrest-pin till it gives the same note as the other string 
 when made to vibrate, which is best done by drawing a 
 resined violin bow across it. Insert the movable bridge C 
 under the string F, but not under G, and adjust the distance 
 CB till the portion of the string F between B and C gives, 
 when plucked or bowed, a note an octave higher than that 
 given by G. It will then be found that the length BC is 
 almost exactly half the length BB'. 
 
 Law 2. Next remove the bridge C and increase the weight 
 E till F gives a note an octave higher than 6r, and 
 therefore an octave higher than F did originally. It 
 will then be found that the total weight at E has been 
 increased very nearly to four times its original amount, 
 allowing for the weight of the rod in each case. 
 
 Law 3. Next substitute for F a brass wire of 20 gauge, 
 and load E till this wire gives the same note as G. Then 
 replace it by a brass wire of 25 gauge, and load this till 
 it gives a note an octave higher than G. It will be found 
 that the weight required is about the same as that used for 
 the 20-gauge wire. Ey weighing equal lengths of the two 
 wires, or by measuring with a screw-gauge, it can be shown 
 that the mass of unit length of 20-gauge wire is about 4 times 
 as great as of 25 -gauge wire, or that the diameter of the 
 former is twice that of the latter. 
 
 The quantity J stretching force whicll appears ^ 
 
 V mass per unit length 
 
 the expressions for velocity of a transverse wave and fre- 
 quency of a transverse vibration may be put in a different 
 
170 SOUND. 
 
 form. The ratio stretching force ^ fte 
 area ot cross section 
 
 string. The ratio mass per unit length . g ^ ^ rf ^ 
 
 area 01 cross section 
 
 material. The ratio stretching force . g therefore e L 
 mass per unit length 
 
 to, though not the same as ~ ; the latter may therefore 
 
 density 
 
 be substituted for the former. The mass of unit length of a 
 string is sometimes called its linear density. 
 
 100, Transverse Vibration of Rods and Plates. A wave of 
 transverse displacement may be sent along a rod of any elastic 
 material in the same way as along a stretched cord, but the rod need 
 not be stretched, and it is the elasticity of the material, not tension, 
 which restores the successive portions of the rod to their original 
 relative positions. The velocity of such a transverse wave, if har- 
 monic, is independent of the thickness b of the rod in the direction 
 perpendicular to the displacement ; proportional directly to the 
 thickness t in the direction of the displacement, and to the square 
 root of Young's Modulus Y for the substance ; proportional inversely 
 to the square root of the density d and to the length of the wave A. A 
 non-harmonic wave has no definite velocity, but the movement is 
 the sum of the movements which would be due to the harmonic 
 components, each travelling with the velocity proper to its wave- 
 length. This dependence of velocity on wave-length makes the laws 
 of transverse vibration of elastic rods much more complex than 
 those of strings. In elastic rods stationary transverse undulation 
 is produced, as in strings, by the interference of wave systems, which 
 are travelling in opposite directions, and which have been reflected 
 from the ends of the rod. The modes of stationary transverse 
 undulation possible in a bar depend on what points, if any, are fixed, 
 and whether the bar at these points is fixed both in direction and in 
 position, as when a rod is held in a vice, or fixed only in. position, as 
 when it merely rests on fixed supports, as in the toy called the har- 
 monicon. In any of these cases there may be nodes and antinodes, 
 ^ind an end which is fixed in position but not in direction is always 
 a node. An end fixed in direction is not a true node, since it is not 
 
 * " Tension " is still sometimes used as equivalent to " stretching 
 force," but it is better to avoid this and use the term only for the 
 stress, or ratio of stretching force to area of cross section. Compare 
 footnote to Art. 27. 
 
TRANSVEBSE UNDULATION. 
 
 171 
 
 -a point of maximum change of slope (Art. 97) ; nor is a free end a 
 true antinode, since it is not a place where the rod moves parallel to 
 itself. The point of no transverse motion nearest to a free end is 
 also not a true node, since the change of slope there is not as great 
 as at the free end itself. All the true nodes are equidistant ; the 
 distance from an end fixed in direction to the nearest node, or from 
 a free end to the nearest true node, is lj times the distance between 
 two true nodes ; the distance from a free end to the nearest point 
 of no transverse motion is a little less than a third of the distance 
 between two true nodes. From these rules the different possible 
 modes of harmonic vibration in any given case are easily determined ; 
 the simpler oney, for a bar entirely free (or supported at two nodes) 
 are shown in Fig. 61, for a bar fixed in direction at one end in Fig. 62. 
 
 :sr 
 
 Fig. 61. 
 
 Fig. 62. 
 
 'The frequency of the vibration, being the reciprocal of the time 
 required for two travelling waves to pass each other whose length A. 
 is twice the distance D between two true nodes, is independent of 6, 
 proportional to t and ^Y, and inversely to Vd and D 2 . From this 
 it follows that, ( 1 ) if two rods vibrate in the same mode, their frequencies 
 are proportional directly to t and v/F, and inversely to Vd and to 
 the squares of their lengths; (2) if the same rod vibrates in two 
 different modes, the frequencies are inversely proportional to the 
 squares of the lengths of pieces which vibrate in the same way, for 
 instance, the pieces A, B or the pieces 0, D, in Fig. 62. The frequency 
 of each mode of vibration, relatively to the other modes of the same 
 figure, is shown by the number to the right of the diagram. 
 
 If a rod is bent into a U-shape, and macle to vibrate with an even 
 number of nodes, two of these nodes are very close together at the 
 bend, and at the antinode between them there is not only no change 
 of direction, but very little transverse motion, as shown in Fig. 63. 
 If the vibrating rod is held by this point, it shakes the holder 
 very slightly, and so loses very little energy except by communi- 
 cating it directly to the air ; it thus vibrates a very long time. The 
 
172 SOTOD. 
 
 tuning-fork (Fig. 31) is an application of this principle. It may 
 
 be made to vibrate by striking one prong on an inelastic substance 
 
 such as lead, or by drawing a viol'n bow across one 
 
 prong, or (if the prongs are nearer together at the 
 
 points than at the bend) by drawing between the points 
 
 of the prongs a rod of wood just too large to pass 
 
 without bending them. The relative frequencies of 
 
 the different modes of vibration do not differ much 
 
 from those of a straight bar fixed at one end (Fig. 62). 
 
 The slowest mode (that shown in Fig. 63) lasts much 
 
 longer than the others, so that the vibrations of a 
 
 tuning-fork which has been vibrating for some time 
 
 are always in this mode, and are (practically) perfectly 
 
 harmonic. 
 
 It will be noticed that the frequencies of the Fig. 63. 
 different modes of free harmonic vibration of a bar 
 or tuning-fork are not exact multiples of the slowest, as they are 
 in the case of perfectly flexible strings and (nearly) in perfectly 
 cylindrical or conical tubes. The following laws apply to most cases 
 where the more rapid modes are not exact multiples of the slowest, 
 e.g., to uniform rods, tuning-forks, plates, bells, columns of air of 
 other forms than cylindrical or conical, such as resonators (Art. 106). 
 
 (1) The free vibration of such a body, if not harmonic, is the sum 
 of two or more of its harmonic modes occurring together. . The 
 slowest of these is called the fundamental, and the others partials^ 
 upper partials , or overtones. These terms can be applied to any 
 components of higher frequency than the fundamental, whether 
 exact multiples of the fundamental or not. The term harmonics is- 
 restricted to exact multiples. 
 
 (2) When such a body is set in forced vibration by a periodic 
 force, all the components of its motion are exact multiples of the 
 frequency of the force, so that all the upper partials are harmonics, 
 of the fundamental. The components which have any considerable 
 amplitude correspond to components of the periodic force which 
 nearly agree in frequency with possible free vibrations of the body. 
 
 It must be remembered that, even when a bar or fork vibrates 
 quite harmonically, the air-waves from it are not exactly harmonic 
 (Art. 22), and therefore they have harmonic components which are 
 exact multiples of the frequency of the bar. So that, when a bar 
 is in free non-harmonic vibration, the waves from it have two 
 distinct sets of harmonic components : one set which correspond in 
 frequencies to different modes in which the bar can vibrate, but 
 are not exact multiples of the fundamental frequency ; the other- 
 set having frequencies which are exact multiples of the fundamental- 
 frequency, but do not correspond to anything in the movement of 
 the bar. The former set are called the non-harmonic overtones, the 
 latter the harmonic overtones. The latter are etill present when 
 the motion of a fork has become quite harmonic. 
 
T1UNSVERSE UNDULATION. 173 
 
 If a tuning-fork vibrates in front of a spherical resonator whose 
 slowest vibration frequency is the fundamental frequency of the 
 fork, none of the overtones of the fork correspond with free- 
 vibration frequencies of the resonator, and therefore the forced 
 vibration of the resonator has no intense overtones, but is nearly 
 harmonic. The waves so produced are the most nearly harmonic 
 waves which can be made. 
 
 Plates of glass, metal, or any elastic solid, clamped at one point 
 and free everywhere else, can be made to vibrate transversely, as 
 already explained in Art. 40. In the stationary undulation pro- 
 duced, there are certain lines on the surface which have no move- 
 ment in space, but where the surface slopes first one way, then the 
 other. These lines are called nodal lines, and the parts of the 
 surface separated by a nodal line are moving, at any one moment, 
 in opposite directions. The antinodes, or places where the surface 
 moves parallel to itself, are also lines. The nodal lines can be found 
 by scattering sand on the surface and drawing a violin bow across 
 the edge of the plate, which for this purpose must be fixed horizon- 
 tally ; the sand collects along the nodal lines, as it is thrown into 
 the air from every other point. The figures so produced are called 
 Chladni's figures. In the case of a circular disc fixed at its centre 
 the nodal lines are radii, and divide the disc into an even number of 
 equal sectors. There are four such sectors if the edge of the disc is 
 touched only by the bow, but if, while the plate is bowed, we touch 
 with our fingers two points on the edge whose distance apart is 
 contained an even number of times in the circumference, there will 
 be that number of radial nodal lines, and two of them will run from 
 the centre to the points touched by the fingers. There may also be 
 circular nodal lines concentric with the plate. 
 
 The frequencies of the same plate vibrating with different 
 numbers of radial nodes are proportional to the squares of these 
 numbers. The frequencies of plates of the same material vibrating 
 with the same number of radial nodes are inversely as the squares of 
 the radii and directly as the thicknesses of the plates. 
 
 Circular stretched membranes vibrate in modes which are very 
 similar, though the vibrations depend on the force with which they 
 are stretched, not on elasticity, and the relative frequencies are not 
 those of plates. 
 
 Plates of other forms can also be made to vibrate, their nodal 
 lines being often very complex. In the case of square and rect- 
 angular plates they have a general resemblance to Lissajous' figures. 
 The following device, due to Wheatstone, enables us in many cases 
 to predict their forms : A rectangular plate D, Fig. 64, might be 
 supposed divided into a series of equal rods by parallel cuts in either 
 of two directions. We will call these the A and B series respectively. 
 Among the modes of transverse vibration possible for these imagin- 
 ary rods there is sure to be one mode for those of the A series, and 
 one for those of the B series, of about equal frequency. Suppose 
 
174 
 
 SOUND. 
 
 the rods made to vibrate in these modes, but with exactly equal 
 frequencies and amplitudes, all the rods of the same series executing 
 simultaneously the same movement, and the rods of both series- 
 reaching their stationary instants simultaneously. In the figure,. 
 
 I 
 
 JU 
 
 Fig. 64. 
 
 the shaded parts of the rods are parts which might be moving all ir* 
 one direction at the same time ; while the unshaded parts were 
 moving in the other direction. The actual movement of any part 
 of the plate is the sum of the movements of the corresponding 
 portions of the two fictitious rod-systems vibrating as above, and 
 the nodal line of the actual vibration is one like that in C, at every 
 point of which the movements of the fictitious rods would either 
 both be zero or equal and opposite. 
 
 A bell, which is simply a concave circular plate, bears to a flat 
 plate the same relation as a tuning-fork to a straight bar, the 
 advantage gained by the bent form being the same as explained 
 above in the case of the fork. The modes of vibration of a bell are 
 the same as those of a circular plate, the edge being divided by 
 nodal points into any even number, not less than 4, of oppositely 
 moving segments. This may be tested by holding a suspended pith 
 
 Fig. 65. 
 
 ball against different parts of the edge, or by inverting the bell, 
 filling it with water, and bowing the edge ; ripples will be seen to 
 proceed from all points except the nodes. 
 
 The curved form makes a difference which in the case of a bell is 
 
TRANSVERSE UNDULATION'. 175 
 
 of some importance. The edge of a bell vibrating in four segments 
 keeps changing its form from that of the dotted line in Fig. 65 to 
 that of the broken one and back again, and its edge always passes 
 through the points ABFG, which are therefore considered nodes. 
 These nodes, however, are not points where the substance of the 
 edge is not displaced, as they are in a flat plate. The arc ACB 
 is longer than the arc ADB, so that particles of the rim which were 
 at A, B before the vibration began are at A', B' when the rim has 
 the dotted form and at A", B" when it has the broken one. (The 
 distances A A', &c., are exaggerated in the figure.) There is therefore 
 a tangential vibration of the material of the bell at A, B, F, G, as well 
 as a radial vibration at D, H, J,K, and a vibration making an oblique 
 angle with the edge at intermediate points. As one of these move- 
 ments cannot take place without the other (unless the edge alters 
 in length), the whole movement is produced if we either make D 
 vibrate radially or make A vibrate tangentially, so that striking the 
 edge of the bell at D, or drawing a violin bow across that part of the 
 edge, produces exactly the same vibration as applying a resined 
 finger at A and drawing it along the edge. So that, when we run a 
 wet finger round the edge of a tumbler, it vibrates just as if we struck 
 it at a point 45 from the finger, and as the finger moves round, the 
 mode of vibration moves round also, the finger being always at a 
 point of purely tangential vibration, or node. 
 
 If one end of a rod is twisted, the next portion twists, and a wave 
 of twist or torsion travels along the rod. The velocity of such 
 torsional waves depends on the rigidity of the substance, and their 
 chief interest is as a means of determining this. They are reflected 
 and form nodes and antinodes, exactly like longitudinal vibrations 
 of rods. 
 
 101. Law of Linear Dimensions. A very important law, 
 which applies to every kind of vibration of solids or fluids, is 
 due to Bernouilli and is called the Law of Linear Dimensions. 
 It states that bodies of geometrically similar form, and the 
 same material, differing only in dimensions, when they vibrate 
 in the same manner, have periods proportional to their linear 
 dimensions. 
 
 EXAMPLES III. 
 
 ELEMENTARY. 
 
 1. If four strings of the same length and material, but of diameters 
 in the ratios 1:2:3:4, are all stretched to half their breaking 
 stress, compare their vibration frequencies. 
 
176 SOUND. 
 
 2. A string is stretched by the weight of 60 Ibs. and it is found 
 that a hump (or small transverse wave) produced by striking it 
 travels along it at 30 feet per secor i. By what weight must it be 
 stretched to make the hump travel 90 feet per second ? 
 
 3. A string stretched by the weight of 10 Ibs. gives a certain note. 
 By what weight must a second string of tho same material, but of 
 twice the length and twice the diameter, be stretched to make it 
 give a note an octave higher than that of the first string ? 
 
 4. Four violin strings, all of the same length and material, but 
 of diameters in the ratios 4:3:2:1 are to be stretched so that 
 each gives a note a fifth above the preceding. Compare the forces 
 necessary. (For meaning of " a fifth above " see Art. 66.) 
 
 ADVANCED. 
 
 5. A string whose mass is 500 grams hangs vertically, and a 
 mass of 400 grams is fastened to its lower end. A transverse wave 
 an inch in length is produced by shaking the lower end, and travels 
 to the top of the string. What is the length of the wave at the 
 middle and at the top ? 
 
 6. A rope weighing half a pound to the foot is stretched hori- 
 zontally by a force equal to the weight of 100 Ibs. If the rope is 
 shaken at one end, with what velocity do the waves run along it ,? 
 
 7. A wire a metre long, weighing 2\ grams, is stretched on a 
 sonometer by a weight of 18 kilograms, and when its length is 
 adjusted to 80 centimetres and a vibrating fork applied, the string 
 throws off a rider placed at any point except two, besides the ends. 
 Find the frequency of the fork. 
 
 8. A bar 2 feet long and -^ inch square, fixed at one end in a 
 vice, makes 20 vibrations per second. How many vibrations will a 
 bar of the same material, 1 foot long, \ inch wide, and \ inch thick, 
 make per second if made to vibrate transversely in the direction of 
 its thickness ? 
 
 9. In performing Melde's experiment, it was found that the 
 string vibrated in 5 loops when 10 grams was placed in the scale- 
 pan. What mass must be placed in the scale-pan to make 
 the string vibrate in 7 loops ? [Neglect the weight of the scale- 
 pan.] 
 
CHAPTER XII. 
 
 ACOUSTIC MEASTJEEMEISTTS. 
 
 In this chapter we shall describe some of the most im- 
 portant experimental methods of investigating vibration and 
 undulation. 
 
 102. Frequency. The frequency of the vibrations of a 
 point of a solid body may be determined by several methods. 
 The best is certainly the photographic. The surface of the 
 body round the point whose vibrations are to be counted is 
 smoked, and a small bright mark made at the point by 
 scratching with a diamond or affixing a very minute drop of 
 mercury with a little grease. This point is strongly illumin- 
 ated by a beam of light, which is interrupted once in each 
 second for a very short time by a swinging pendulum. A 
 mirror is made to revolve on an axis parallel to the vibrations 
 of the point, and an image of the image of the point in this 
 mirror is formed by a lens on a photographic plate. The 
 trace left on the plate, when " developed," is a displacement 
 curve of the movement of the point, and is a wavy line inter- 
 rupted by very short spaces corresponding to the momentary 
 eclipses of the light at the end of each second. The number 
 of vibrations per second can be counted on the plate. 
 
 Graphic methods, in which the vibrating body is made to 
 trace its movements, are less accurate, since the necessary 
 friction, though it may be made very small, always affects the 
 frequency (and still more the character) of the vibration. 
 The Vibroscope, shown in Fig. 66, is a good example of this 
 method. The vibrating body is fixed in such a position that 
 the point whose movements are to be counted moves near to 
 the surface of the cylinder, and parallel to its axis. The 
 
 SD. N 
 
178 
 
 SOUND. 
 
 cylinder is covered with smoked paper, and a fine wire, attached 
 with wax to the vibrating body, just touches the paper. As 
 the handle is turned, and the body vibrates, the wire traces 
 a wavy line on the paper ; this line forms a screw thread 
 
 Fig. 66. 
 
 round the cylinder, which is made to advance in the direction 
 of its length by a screw thread cut on the axle. When this 
 apparatus is used for measuring frequencies, the vibrating 
 body is insulated and connected to one end of the secondary 
 of an induction coil, while the other end is connected to the 
 revolving cylinder. The pendulum of a (seconds) clock is 
 arranged to make contact, as it passes its lowest, point, with 
 a drop of mercury, and so to close and break, once in each 
 second, a circuit which includes pendulum, mercury drop, 
 the primary of the induction coil, and a battery. The spark 
 which passes between the wire and the smoked paper, each 
 time this circuit is broken, knocks off a little spot of the soot 
 at the point in contact with the wire at the moment ; the 
 number of double bends between two successive spots on the 
 wavy line traced by the wire gives the frequency. 
 
ACOUSTIC MEASUREMENTS. 
 
 179 
 
 Both these methods give the displacement curve of the movement 
 of a point, so that they may be used to determine the character as 
 well as the frequency. A fairly accurate measurement of the 
 frequency of a tuning fork may be made by means of the simple 
 
 Fig. 67. 
 
 apparatus shown in Fig. 67. The tuning fork A is fixed in a nearly 
 vertical position, and a bristle or pointed piece of thin sheet metal B 
 is fastened with wax to one of its prongs. The prongs are pressed 
 together till they are close enough to enter a square notch in the 
 metal plate C, which holds them in this compressed position. A 
 smoked plate of glass D is suspended from two pins E, F by a silk 
 thread G hanging over them in two loops, one of which is cemented 
 to the upper edge of the glass plate, while the other is caught under 
 one of the points of the metal plate C. When C is withdrawn, 
 which should be done quite horizontally, the plate D falls freely 
 (the friction of the silk thread on the pins being negligible), and at 
 the same instant the prongs, suddenly allowed to fly apart, begin to 
 vibrate. The point B thus traces on the falling plate a wavy line 
 similar to H, the bends of which at the lower end are too close to 
 be distinguished, because traced when the plate was falling very 
 slowly, but widen out as we go up. In any part of the trace where 
 the bends are quite clear and distinct we mark three points, X, Y, Z, 
 
180 SOUND. 
 
 such that there is some exact whole number a of double bends 
 between X and Y, and the same number between Y and Z. Let the 
 distance XY be d^ and YZ, d. 2 . Then the average velocities of the 
 
 plate while the parts XY, YZ passed the tracing point, were -^ and 
 
 j & 
 
 2S2 respectively (n being the frequency of the fork), and these were 
 
 a 
 the actual velocities at the mid-instants of these intervals. So the 
 
 plate gained a velocity " - l -l while the pointer traced a waves, 
 a 
 
 in seconds, so its acceleration was 2 ~ . This = g,* which 
 n or 
 
 is 981 in England if centimetres and seconds are the units. Hence 
 
 It is difficult to get on a freely falling plate a continuous trace 
 representing more than sec., and this limits the accuracy with 
 which n can be found. A better method is to fix the glass plate 
 to the bob of a heavy seconds pendulum, so that the face of the 
 glass is vertical and parallel to the plane of swing. The fork is 
 held so that the bristle vibrates vertically and touches the plate. 
 The trace on the plate is a wavy line, the bends of which are of 
 unequal lengths, as the plate moves with different velocities in 
 different parts of the swing. In any part of this trace where the 
 waves are not too crowded to be distinct, mark off any three con- 
 secutive lengths containing each the same number a of waves. 
 Let d ]9 d. 2 , cZ 3 be these lengths. The longer the portion of the trace 
 occupied by them, the more accurate the result. Let 6 be the angle 
 
 whose cosine is * - - , and T the time of a double swing of the 
 
 2(^2 
 
 pendulum. Then the frequency of the fork can be shown to 
 
 , 360 x a 
 
 be -^' 
 
 In all these graphic methods the attachment of a tracing 
 point, by increasing the mass, diminishes the frequency of 
 vibration, and this must be allowed for. It is tyest done by 
 comparing the frequency of the vibrating body, before and 
 after the attachment of the tracing point, with the nearly 
 equal frequency of a fork which remains in the same condi- 
 
 * This experiment is also used as a method for finding g, the 
 frequency of the fork being assumed known, but, as g is a quantity 
 which can easily be measured many times more accurately than the 
 frequency of the fork can possibly be known, this cannot be con- 
 sidered as more than a very rough method for finding g. 
 
ACOUSTIC MEASUREMENTS. 181 
 
 tion in both, experiments ; the change clue to the mass of the 
 tracing point is thus determined. 
 
 There are several good methods of making this comparison 
 between the frequencies of nearly equal forks. One is simply 
 to count the beats heard when the two are sounded together ; 
 the nnmber of beats per second is the difference of the 
 frequencies. If two forks, A and B, are very nearly equal in 
 frequency, the beats cannot be heard ; in this case take a 
 third fork, (7, of about the same frequency, and load its 
 prongs with wax till it gives about four beats per second when 
 sounded with A. Then determine by counting beats how 
 much faster A and B are, respectively, than C- the 
 difference between these two differences is of course the 
 difference between A and B. 
 
 Where a third fork is not used, it will be necessary to 
 determine not only the difference between A and B, but 
 which of them has the higher frequency. Most persons can 
 decide this by ear ; if there is any difficulty, the prongs of 
 one fork must be loaded with a little wax, and the beats 
 counted. If they are now faster than before, the fork that 
 has been loaded was the slower, and vice versa. 
 
 The device of a third fork may be made to give very 
 accurate results by employing Lissajous' figures instead of 
 beats to determine the difference of frequency. In this 
 method, if the nearly equal frequencies of A and B are to be 
 compared, a third fork, C (Fig. 68), has a convex lens fixed 
 to one prong, and is adjusted by sliding movable masses 
 along the prongs till its frequency is a little less than that of 
 A and B. It is fixed so that its prongs, and their plane of 
 motion, are horizontal. One of the forks to be compared, 
 say A, has a small bright dot scratched with a diamond on 
 the smoked end of one prong m, and is fixed vertically, with 
 the dot immediately under the lens of (7, and so that the 
 movements of the dot are at right angles to those of the lens. 
 Trie dot is illuminated by a condensing lens c. If A and C 
 are both made to vibrate, the virtual image of the dot seen 
 from above through the lens has a movement which is the 
 sum of the movements of the dot and lens ; it describes, 
 according to the relative phases of these movements, one of 
 the forms of the top line of Fig. 9, the curve described 
 
182 SOUND. 
 
 (which, by persistence of vision, appears as a continuous 
 bright line) assumes in turn all the forms of the series, 
 going through the whole cycle in the period in which 
 
 Fig. 68. 
 
 A gains one complete vibration on C. As the moments at 
 which the curve becomes a straight line can be observed very 
 exactly, the difference of frequency between A and (7, which 
 is the number of complete cycles per second, can be found 
 within -^Q of a vibration per second. Similarly, JS can be 
 compared with (7, and the difference between A and B 
 deduced from the two results. 
 
 The superiority in exactness of this method to that of 
 audible beats results from the great accuracy witlj which wa 
 can ascertain when one fork has gained an exact whole 
 number of vibrations on the other, and the fact that the 
 vibrations can be observed for a much longer time than they 
 are audible. 
 
 A still better arrangement is to place the lens /at such a 
 distance above m that a real image of m is formed, and to 
 examine this real image through a fixed eyepiece ^, as shown 
 in the figure. 
 
ACOUSTIC MEASUREMENTS. 183 
 
 The character, as well as the frequency, of the non -harmonic 
 vibration of a point may be determined by the use of a lens-fork. 
 If a globule of mercury is attached with grease to a string, and 
 observed through the vibrating lens after the string has been 
 plucked or struck, a curve is visible if there is a simple ratio between 
 the two periods, but the curve is not one of Lissajous' figures, since 
 one of the movements, that of the string, is not harmonic. It is an 
 easy matter to determine what movement of the string, combined with 
 the harmonic movement of the fork, would give the observed curve. 
 
 By means of this vibration microscope we can adjust a fork very 
 exactly to a frequency which has any simple ratio to that of C. 
 The fork can be made quicker by filing it near the points, or slower 
 by filing it at the bend, and is to be adjusted till a Lissajous' figure 
 corresponding to the required ratio is seen, and changes only very 
 slowly. Two forks can be adjusted to have a given ratio to each 
 other by adjusting them, in turn, to any convenient ratios to the 
 lens-fork which have the required ratio to each other. 
 
 The vibratory movements of the air caused by a vibrating 
 body at any point which is always at the same distance from 
 the body must agree in frequency with those of the body 
 itself ; the exact repetition of the same causes must produce 
 the same effects. Frequency is indeed the only particular in 
 which vibrations of the air necessarily agree with those 
 of the body causing them. Any of the methods for deter- 
 mining the frequency of the movements of the air can there- 
 fore be used to determine the frequency, though not the 
 character, of the vibrations of a solid body. 
 
 To determine the frequency of the movements of the air 
 at a point, an apparatus called a Phonautograph (Pig. 69) 
 may be used. A large funnel, usually made parabolic in ' 
 form, is closed at the smaller end by a stretched membrane ; 
 below the centre of this membrane, at right angles to its surface, 
 is fixed a short bristle. The funnel is fixed so that the mem- 
 brane is parallel and very close to the surface of a smoked 
 cylinder exactly like that of the Vibroscope. A spring, which 
 touches a point of the membrane exactly above the centre, 
 insures the membrane always vibrating with a vertical 
 nodal line through the centre ; the part of the membrane 
 which carries the style is therefore never displaced in the 
 direction of the axis of the funnel, but merely alters its 
 direction, vibrating on a fixed vertical axis. The point of the 
 bristle therefore moves parallel to the axis of the cylinder, 
 
SOUND. 
 
 and always remains just in contact with the smoked surface, 
 and its movements correspond in frequency, though not in 
 any other respect, with those of the air. 
 
 Seconds can be marked on the trace left by the bristle by 
 the ^electrical method described above for the vibroscope, or a 
 tuning-fork of known frequency can be made to trace a wavy 
 line on the cylinder by the side of that traced by the bristle, 
 and the frequency of the bristle determined by comparing its 
 movements with those of the fork. 
 
 Fig. 69. 
 
 The frequency of air waves can also be found by means of 
 a counting siren. Any instrument in which a sound is 
 produced by puffs of air escaping at regular intervals through 
 holes or notches in a rotating plate or cylinder, is called a 
 siren. The name was given because most forms of such 
 an instrument may be blown with water in water instead of 
 with air in air ; this is however not peculiar to the siren, but 
 is equally true of nearly all wind instruments (Art. 91), and 
 Homer's Sirens differed from the mechanical ones in this 
 respect. A common but not very good form of counting 
 siren is shown in Fig. 70. The lid of the box C and the 
 revolving plate D contain the same number of holes, and a 
 pun issues simultaneously from all the holes each time those 
 in the plate coincide with those in the lid, so that there are 
 
ACOUSTIC MEASUREMENTS. 
 
 185 
 
 as many successive puffs in each rotation of the disc as there 
 are holes in the circle. The rotations of the disc are 
 counted by wheels, similar to the counting apparatus of a 
 gas meter, which are driven by a screw on the axle of 
 the disc, and can be made to begin counting by pressing a 
 knob, and stopped by releasing it. Air is blown into the 
 box A, and the velocity of the disc increased till the pitch of 
 the note heard is nearly that of the sound whose frequency 
 is to be determined. There is nothing harmonic about the 
 
 Fig. 70. 
 
 process by which the waves from a siren are produced, and 
 the waves, being not at all harmonic in form (Art. 55), have 
 many harmonic components, and can excite in the ear or 
 other resonators vibrations of higher frequencies than that of 
 the actual puffs, and it is only after much practice that it is 
 possible to be certain whether it is the frequency of the 
 puffs, or one of its multiples, which is equal to the frequency 
 of the waves from the other source. When we are sure that 
 it is the lowest in pitch of the sensations due to the siren 
 (which is often not the loudest), which corresponds to the 
 
186 SOUND. 
 
 lowest of the sensations due to the other source, the velocity 
 of rotation should be slightly diminished till beats are heard 
 at the rate of about four per second. The counting mechanism 
 is then set in action for a definite time, say thirty seconds, 
 while the observer keeps count of the total number of 
 beats heard during this interval, and regulates the rotation 
 so that the beats do not become too rapid to count or too 
 slow to distinguish. The total number of rotations recorded, 
 multiplied by the number of holes in the disc, gives the 
 number of waves from the siren during the observed interval 
 of time, and this number, plus the number of beats counted, 
 is the number of waves which left the other source during 
 the same interval. 
 
 In the siren shown in the figure, the disc is made to rotate 
 by the air issuing from the holes, which are drilled obliquely 
 in both lid aud disc, but slope opposite ways, as shown in 
 section at E. This is a very bad arrangement, for the speed 
 can only be increased by blowing harder, which also increases 
 the loudness of the sound, and beats are not very distinctly 
 heard except between two sounds of nearly equal loudness. 
 It is much better to drive the disc independently by an 
 electromotor or other easily controlled mechanism. 
 
 It must not be forgotten that beats are also heard if any of 
 the harmonic components of the sound from the siren nearly 
 agree in frequency with any of the harmonic components of 
 the sound from the other source, so that, if the waves from 
 the other source are also very far from harmonic, the effect 
 on the ear is that of several series of beats, of different 
 frequencies, happening at the same time. This renders the 
 counting of the beats of the fundamental vibration very difficult. 
 
 103. Konig's Wave Siren. By substituting for a disc with 
 holes a disc with its edge cut in harmonic waves, the siren may be 
 made to give more nearly harmonic sounds. In this case, the air is 
 blown from a single radial slit, which is entirely or only partly 
 covered by the disc, according to the position of the latter. The 
 amount of air issuing from the slit varies nearly harmonically if the 
 teeth of the disc are of harmonic form, so that the waves as they 
 start are approximately harmonic. They become less so, however, 
 as they proceed, so that the difficulty of beats between the harmonic 
 components is not entirely overcome. 
 
 By cutting the edge of such a disc into waves or shallow teeth of 
 
ACOUSTIC MEASUREMENTS. 187 
 
 different forms, waves of different wave-form can be produced, and 
 M. Konig showed that discs whose edges are cut into different 
 curves which have the same harmonic components (Art. 58) may 
 produce sounds of different quality, and he considered that this 
 disproved Helmholtz's view (Art. 61) that the quality of the sound 
 due to wave systems having different forms but the same com- 
 ponents is the same. M. Konig's experiment, however, in no way 
 disproves Helmholtz's view unless we assume that the harmonic 
 components of the waves produced are the same as the harmonic 
 components of the form of the teeth of the disc, and there is no 
 reason to believe that this is the case. There are, however, more 
 satisfactory arguments against Helmholtz's view. 
 
 These are the most important methods of measuring 
 frequency directly. If the velocity of sound in a medium is 
 known, and the length of the waves is measured, the frequency 
 can be at once determined, since 
 
 frequency x wave-length = velocity. 
 
 104. Wave-length. No method is in use for directly 
 measuring the wave-length of a travelling undulation ; it 
 would, perhaps, be difficult to devise one. A method based 
 on the interference of two systems travelling in the same 
 direction was given in Art. 40. More usually the travelling 
 undulation is converted, by reflection, into a stationary un- 
 dulation, and the wave-length of the travelling undulation 
 determined by finding the distance between the nodes and 
 antinodes of the stationary one. 
 
 One method of doing this has been given in Art. 82 ; it is 
 the best when the waves are very short. For longer waves, 
 such as those from a tuning-fork, a long glass tube about 
 2 inches wide, and closed at one end, may be fixed vertically, 
 and water poured in until the air column above the water is 
 of such a length that it resounds when the vibrating fork is 
 held over the mouth of the tube. If the tube is long enough, 
 a number of lengths can be found which resound to the same 
 fork. As explained in Art. 85, a closed tube resounds when- 
 ever the distance from the point of reflection, just outside 
 the open end, to the closed end, is an odd number of quarter 
 wave-lengths of the waves arriving from outside. The distance 
 between any two consecutive levels of the surface of the 
 water which leave air columns of lengths which will resound 
 
188 SOUND. 
 
 is half a wave-length, of the waves from the fork, or the 
 distance from the highest of such water-levels to the point of 
 reflection (which is about *8 of the radius of the tube above 
 the level of the opening) is a quarter of the wave-length. 
 It is from the last fact that the wave-length is most easily 
 found ; the resonance of the longer columns being less distinct. 
 
 105. Velocity. If the frequency of the fork is already 
 known, this experiment gives the velocity of sound, which is 
 wave-length x frequency. By surrounding the tube with a 
 larger one, and filling the space between with water of 
 different temperatures, we can determine the change of 
 velocity due to a given change of temperature in the air 
 of the inner tube. In this experiment, mercury should be 
 used to adjust the length of the air column in the inner tube 
 instead of water, as, if water was used, the air would be mixed 
 with a large proportion of water vapour at high temperatures. 
 
 The tube can be filled with other gases instead of air, and 
 the velocities determined. For gases lighter than air, the 
 open end of the tube must be turned downwards, and the 
 length of the column adjusted by means of a sliding piston 
 instead of liquid. 
 
 Owing to the small mass of the air column, its vibrations 
 are easily controlled, so that it resounds nearly as loudly when 
 it is a centimetre too long or too short as when it is of a 
 length which would vibrate freely with the same frequency 
 as the waves arriving from outside. Any single determina- 
 tion of the wave-length by this method is therefore very 
 uncertain, though the average of a large number of indepen- 
 dent determinations is fairly reliable. A much better way is 
 to use a closed organ-pipe whose length can be adjusted by a 
 sliding piston. The tube is kept sounding by an^ air-blast, 
 and the piston adjusted till the note is the same as that of 
 the fork, as shown by the absence of beats, In this case the 
 air column is vibrating with its free -vibration frequency. 
 But an organ-pipe does not work well unless its mouth is 
 much smaller than the cross section of the tube, and the 
 narrowness of the opening increases the time required for the 
 air to move in and out, so that an air column much shorter 
 than a quarter of the wave-length of the fork vibrates with 
 
ACOUSTIC MEASUREMENTS. 189 
 
 the same frequency as the fork. In an organ-pipe, there- 
 fore, the distance from the mouth to the nearest node differs 
 from a quarter of a wave-length by a considerable, and 
 rather uncertain, amount, so that we cannot determine the 
 wave-length of a note from a knowledge of any one length 
 of the column which will give the note. Two observations 
 are necessary. First blow gently, so that the air column 
 vibrates in its fundamental mode (Fig. 49, a\ and observe 
 what length gives the same note as the fork. Then blow 
 harder so as to make the column ribrate in its second mode 
 (Pig. 49, #), and increase the length of the column till it again 
 gives the note of the fork. In each experiment there is a 
 node at the piston, and in the second experiment there is 
 also a node at the point where the piston was in the first, 
 since the frequency is the same in each case. So the differ- 
 ence between the lengths of the column in the two experi- 
 ments is half a wave-length of the waves from the fork. 
 
 A whistle fitted by means of a cork into one end of a long 
 glass tube 2 inches in diameter answers well for this ex- 
 periment ; it can be blown by foot-bellows. 
 
 The difficulty about the exact position of the point of 
 reflection is avoided by another method, due to Kundt. 
 Though a wave-length is actually measured, it is not really 
 a method of determining the wave-length of a given wave- 
 system, but of comparing the lengths of waves of the same 
 frequency in different substances, and so the velocities of 
 sound in those substances. One form of the apparatus is 
 shown in Pig. 71. A wide glass tube BB', about 4 feet long 
 and 2 inches in diameter, is closed at one end by a tight-fitting 
 
 Fig. 71. 
 
 cork JT. This cork grasps tightly the middle of a metal or 
 glass rod AA\ on the inner end of which is fixed a disc a of 
 wood or card fitting the tube very closely without being tight. 
 A cork V sliding in the other end of the tube serves to adjust 
 the length of the air space between itself and the card disc. In 
 this space is scattered lycopodium powder or fine cork dust. 
 
190 SOUND. 
 
 A cloth, dusted with resin, or (for a glass rod) moistened 
 with alcohol, is drawn along the outer half of the rod, and 
 when this squeaks the rod is in longitudinal stationary 
 undulation, the middle being a node and the ends antinodes. 
 If the movable cork is placed in different positions in the 
 tube, it will be found that when it is at certain points, 
 the powder collects in equidistant ridges, at right angles to the 
 length of the tube, when the rod squeaks ; if lycopodium 
 powder is used, it may form rings right round the tube. The 
 ridges or rings always divide the space from the movable 
 cork to the card disc into an exact number of equal parts. 
 The ridges are nodes in the stationary undulation which is 
 produced in the air in this space. In may seem as if the disc 
 should not be a node, as it moves ; but, owing to the addition 
 of a large number of impulses, the maximum amplitude of 
 the vibrating air becomes many times greater than that of 
 the disc, which is therefore practically at a node. The 
 maximum resonant vibration, and the most clearly defined 
 ridges, are produced when the length of the air column from 
 the cork to the disc is such that one of its modes of free 
 stationary undulation has exactly the frequency of the 
 vibrations of the rod, and when this is the case the distance 
 between two ridges is exactly half the length of the waves 
 which would be produced if the disc was vibrating in free 
 air ; but, owing to the small mass of the air in the tube 
 compared with that of the rod, the air is set in resonant 
 vibration nearly as strong, and leaves ridges nearly as distinct, 
 when the distance from cork to disc is considerably different, 
 so that there is no certainty that the distance between two 
 ridges is more than an approximation to half a wave-length of 
 the waves which would be produced in free air. By repeat- 
 ing the experiment many times, adjusting the cork each time 
 till the ridges seem clearest, and taking the average, very 
 reliable results can be obtained. 
 
 The length of the rod (of glass, suppose) being half the wave- 
 length in glass, and the distance between two ridges being 
 half the wave-length in air, of waves of the same frequency 
 (that of the vibration produced by rubbing the rod), it follows 
 that the ratio of the velocity of sound in air to that in glass 
 is the ratio of the distance between two ridges to the length 
 
ACOUSTIC MEASUREMENTS. 191 
 
 of the glass rod, so that either of these velocities can be 
 determined if the other is known. Or if the frequency of 
 the rod is determined by the vibroscope or siren, or by calcu- 
 lation,* the absolute velocity of sound in air can be found. 
 By filling the tube with different gases in turn, the relative 
 velocities of sound in them can easily be found ; for they are 
 proportional to the spaces between consecutive ridges ; the 
 inlet tube m is for this purpose. Liquids may be used 
 instead of gases ; in this case a heavier powder, such as 
 finely divided silica, must be used. As the walls of the tube 
 are not perfectly rigid, the velocity of sound along the liquid 
 in a tube is not nearly so great as in a large volume of the 
 same liquid (Art. 31), so the results are not of great value. 
 
 Eods of different materials may be tried in place of the 
 glass one, the tube being always filled with air. The velocities 
 of sound in the different rods are proportional to the numbers 
 obtained by dividing the length of each rod by the length of 
 the segments into which the ridges divide the air space when 
 that rod is sounding ; for these numbers are the ratios of the 
 velocities of sound in the rods to velocity in air. 
 
 For any of these purposes, except the comparison of different 
 rods, the rod and disc may be dispensed with, and the wide tube 
 itself made to vibrate longitudinally. This is done by holding it by 
 the middle, and drawing a cloth, moistened with alcohol, along 
 one half. The tube is closed at both ends by corks, one at least of 
 which must be adjustable in position as in the last experiment. 
 When the distance between the corks is exactly or nearly an exact 
 number of half wave-lengths in air of the frequency of the vibrations 
 of the glass, the air is thrown into resonant stationary undulation, 
 the corks being nodes (very nearly, like the disc in the other form). 
 
 106. Wave-form. No accurate method seems to be known 
 of determining the wave-form of an undulation. The 
 phonautograph described above gives traces which are 
 different for waves of different character, and these traces are 
 sometimes loosely spoken of as the wave-forms, but neither 
 this method nor any method which depends on the yielding of 
 
 * If Young's Modulus for the glass, and its density, are known, 
 the velocity of a travelling wave in it can be found from Art. 27, 
 and the period of the stationary undulation of the rod, when vibrating 
 with one node as in this experiment, is the time that a travelling 
 wave would require to travel twice the length of the rod. 
 
192 SOUND. 
 
 a membrane can give the wave-form as defined in Art. 15. If 
 a membrane could be massless and perfectly loose, it would 
 move exactly as the air moved, aitd might be made to trace a 
 displacement curve of the movement of the air. If a mem- 
 brane could be stretched infinitely tightly, its displacement at 
 each instant would be proportional to the pressure on it at 
 that instant, and it might be made to trace a true pressure- 
 curve or wave-form, though the amplitude of this would be 
 indefinitely small. The movement of any real membrane 
 is a compromise between them, modified considerably by the 
 mass of the membrane and its liability to be thrown into 
 resonant vibration if any of its own very numerous modes of 
 free vibration agree in frequency with any of the harmonic 
 components of the waves reaching it. 
 
 The wave-forms of undulations then cannot at present be 
 determined with any accuracy, and are, in fact, unknown, 
 except in the regions close to large vibrating surfaces, where 
 they cannot differ much from the velocity curves of the 
 surfaces themselves. We can, however, determine, by the 
 method explained in Art. 54, the frequencies of the harmonic 
 waves whose wave-forms would add up to that of the actual 
 waves. That is not at all the same as knowing the actual 
 wave -form, because we cannot find either the relative 
 amplitudes or the relative phases of these harmonic waves at 
 all accurately, and harmonic waves of the same frequencies, 
 but of different amplitudes and relative phases, may add up 
 to quite different wave-forms. Still, for many purposes, to 
 know the frequencies of the harmonic waves whose wave- 
 forms would add up to the actual wave-form, is as useful as 
 to know the wave-form itself, and this can be ascertained by 
 means of resonators. A resonator is a globular or cylindrical 
 box with a wide hole at one end, and a small one v fitted with 
 a tube, at the other. Its frequency of free vibration is found 
 by blowing across the wide hole, and determining the frequency 
 of the sound produced, by the siren, phonautograph, or other 
 method. This frequency can be altered by slightly altering 
 the size of the larger hole, for a tube whose diameter is 
 diminished at an open end has a much longer vibration period 
 than an ordinary open tube. In the case of cylindrical 
 resonators, which are made in two pieces to slide one inside 
 
ACOUSTIC MEASUREMENTS. 193 
 
 the other like the joints of a telescope, the frequency can 
 also be adjusted by altering the length. A set of such 
 resonators is prepared whose free-vibration frequencies are 
 exact multiples of the frequency of the waves to be analysed, 
 and, if the tubes from the smaller openings of these resonators 
 are placed in turn in the ear, we hear a loud v sound from 
 those whose free-vibration frequencies correspond to those of 
 the harmonic components of the arriving waves. 
 
 As the components can only be detected by this method 
 one at a time, it is only suitable for the analysis of sounds 
 which can be produced continuously. For those of short 
 duration we must be able to determine which resonators are 
 vibrating at a given moment. For this purpose the smaller 
 opening of each resonator is fitted with a manometric capsule, 
 and the flames from all the .capsules are arranged close one 
 above another in front of a rotating mirror, as in Pig. &$r 
 The appearance of the flames in the mirror shows which 
 resonators are vibrating. In this way the analysis can be 
 shown to a large audience. 
 
 107. Phonograph. Edison's Phonograph is similar to the 
 phonautograph in principle, but a cylinder covered with wax 
 is substituted for the smoked one, and a very narrow chisel 
 for the tracing point. The centre of the membrane is in this 
 case made an antinode, and the point of the chisel therefore 
 moves perpendicularly to the surface of the cylinder, not 
 parallel to it, and so digs a trench of varying depth as the 
 membrane vibrates. A section of this trench, parallel to its 
 length and perpendicular to the surface of the cylinder, shows 
 the form of the bottom, which depends on the quality of the 
 sound, but is not the " wave-form " in any accurate sense of 
 the term. It is, however, (like the trace of the phonauto- 
 graph) a curve whose harmonic components are nearly the 
 same as those of the waves. If the chisel is now replaced by 
 a blunt point, and the cylinder, after being replaced in its 
 original position, turned again in the same direction as at 
 first the blunt point presses on the undulating bottom of the 
 trench already cut by the chisel, and the membrane to which 
 the point is connected repeats the movements by which the 
 trench was originally cut. It thus produces waves in 
 
 SD. o 
 
1 94 SOUXD. 
 
 the air similar to those which, hy their arrival, caused 
 the original movements of the membrane, though they are 
 reversed, the condensations of the original waves being 
 represented by rarefactions in those afterwards produced by 
 the membrane. Though in this and other respects there is con- 
 siderable difference between the wave-forms of the original 
 ound and the, sound reproduced by the membrane, the two have 
 nearly the same harmonic components, and therefore seem to 
 the ear so nearly of the same quality that, if the original 
 sound was that of the speaking voice, the different vowels 
 (which are only special qualities) are easily distinguished in 
 the reproduction. 
 
 It is worth noticing that it is the imperfection of the ear 
 as a detector of differences in air waves that makes the 
 phonograph and telephone possible. The waves they give 
 out are in form often so different from the original waves 
 that, if the ear had as great a power of distinguishing between 
 different wave-forms as the eye has of distinguishing different 
 outlines, it would be impossible to recognise any resemblance 
 to the original sound. + 
 
 108. Amplitude. The amplitude of the vibrations of any 
 point of a solid body can of course be found from its trace on 
 the vibroscope cylinder. jS^o method is known of measuring 
 the amplitude of the vibrations of gases ; such estimates as 
 those in Art. 24 are arrived at by assuming that all the 
 energy observed to be expended in keeping the source in 
 vibration is converted into sound, and determining the ampli- 
 tude of vibration necessary at any given distance from the 
 source in order that the air at that distance may transmit 
 energy at the same rate as it leaves the source. The true 
 amplitude, of course, cannot be greater than this, but it is on 
 theoretical grounds that it is believed to be not much less. 
 
APPENDIX. 
 
 A. (ART. 5). 
 
 Let B (Fig 1 . 72) be a point revolving uniformly round a circle of 
 radius a in a period t, and A a point moving up and down a diameter 
 CD so as to be always on the same level as B, so that A moves harmoni- 
 cally. Let B l be the position of B when OB has moved through an 
 
 angle 8 from the position at right angles to CD. At this moment the 
 displacement OA l of A from its mean position is a sin 8. The 
 
 velocity of B is ^ in the direction B E, and the component of this 
 velocity in the direction parallel to 00 is BF, or BE cos 0, or 
 - sin (0 + 90). This is the velocity of A when it is at A v Now 
 
 B + 90 is the angle through which OB will have rotated from OX a 
 quarter of a period later. So the velocity of a harmonically vibrating 
 
 point at any instant is times the displacement which it will have 
 
 a quarter of a period later. 
 
 Now the velocity of A is the rate of change of the displacement 
 of A, so that, if we take the displacement of A to represent any 
 quantity which varies harmonically, the velocity of A at the same 
 instant represents the rate at which that quantity is changing. Hence 
 
196 SOUND. 
 
 the rate of change of any quantity that varies harmonically, and has 
 a mean value = 0, is times the value that the quantity itself will 
 
 have a quarter of a period later. Aiid the acceleration of A is the 
 rate of change of the velocity of A . The acceleration of A when 
 
 at A l is therefore times the velocity it will have a quarter of a 
 
 period later, or - ^ sin (0 + 180), or ^ sin 0. It is therefore pro- 
 portional to the displacement of A, but in the opposite direction. 
 The ratio of the acceleration to the displacement is ^-, and is 
 
 therefore independent of the amplitude. If, therefore, the force on a 
 body, due to displacement, is such as to produce an acceleration which 
 is always k times the displacement, the body will vibrate harmonically 
 
 in a period t such that k = -^-, and this period is independent of the 
 
 amplitude. 
 
 B. (AET. 23). 
 
 The term intensity of an undulation is, used by some writers to mean 
 the quantity of energy which flows per second through a square centi- 
 metre parallel to the wave -fronts, and by others to mean the quantity 
 of energy in a cubic centimetre of space at a given moment. We 
 have adopted the former definition, which has the high authority of 
 Lord Rayleigh ; but the other seems to be coming more into favour. 
 It makes some difference to the proofs which definition is adopted ; if 
 the first, it follows easily that the intensity is inversely as the square 
 of the distance, but it is difficult to give a satisfactory elementary 
 proof that the intensity is proportional to the square of the amplitude. 
 From the second definition this latter proposition follows at once, but 
 it is then difficult to prove the former. In many books the definition 
 is given so loosely that it may be taken to have either meaning, and 
 then both propositions deduced from it. 
 
 If the definition we have adopted is used, the intensities of the 
 transmitted and the reflected wave, when waves reach the surface 
 between two media, are together equal to the intensity of the incident 
 waves, but this is not true on the second definition. 
 
 ^ 
 
 C. (AET. 37). 
 
 The name interference is applied by some writers to all cases of 
 superposition, and limited by others either (a) to those cases in which 
 the actual distribution of energy is very different from that of the 
 imaginary systems (see footnote to Art. 42) , or (b) to those cases in 
 which the vibration due to the sources together is less intense than 
 that which would be due to one of them alone. "We adopt (a) as being 
 usual in England, though (b) has the high authority of Lord Kayleigh. 
 
APPENDIX. 197 
 
 D. (ARTS. 14, 27, 95). 
 
 VELOCITY OF A LONGITUDINAL WAVE ALONG A ROD OB FLUID COLUMN 
 
 (ART. 27). 
 
 Let a force of / dynes be applied to one end A of a very long rod AB, 
 and let the force be directed towards B. The first centimetre of the 
 rod shortens till it exerts a force of / dynes on the next centimetre, 
 and then this shortens till it exerts the same force on the next, and 
 so on. As the process is exactly the same for each successive centimetre, 
 the compressed condition extends, with uniform velocity, along the 
 rod. Let this velocity be V, and let the mass of each centimetre be m, 
 and let the amount by which each (original) centimetre is diminished 
 in length, when there is a force / at each end of it, be I. As each 
 centimetre becomes compressed in its turn, all the centimetres between 
 it and A, which are compressed already, advance towards through 
 a distance /. As V fresh centimetres become compressed in each 
 second, VI is the velocity with which the portion already compressed 
 moves towards B. In each second V fresh centimetres are added to 
 this portion, and each of these, when compressed, has a momentum Vim. 
 So Y'-lm is the increase of momentum of this portion per second. 
 But rate of change of momentum is equal to the force producing it. 
 
 Hence /= VHm or V= \//. 
 V m 
 
 If now the force at A is diminished to f } dynes, the first" centimetre 
 has a force of /i dynes at one end and / dynes at the other, and it 
 therefore moves more slowly than the next centimetre, which has a force 
 /"at each end, until it has expanded to such an extent that it only exerts 
 a force of / t dynes on the second, after which the velocity of the first 
 centimetre becomes uniform, and that of the second diminishes, and 
 so this diminished velocity and condensation extend along the rod 
 with a uniform velocity V\. Let l } be the amount by which each 
 original centimetre elongates as it changes from the more to the less 
 compressed condition. Then the distance from A to any given particle 
 of the part of the rod which is in the more compressed condition 
 increases in each second by V-J^ So the difference of the velocities of 
 the material of the rod in the less and more compressed portions is 7Vi 
 and 1\ centimetres, of mass m each, pass from one condition to the 
 other in each second. So that the momentum of the portion of the 
 rod between A and a given particle in the more condensed portion 
 diminishes in each second by Vfl-pn. That portion of the rod has a 
 force of / at one end and of f\ at the other, and, therefore, its momentum 
 changes in each second by ff\. Hence 
 
 r,V,m =/-/ or T, = 
 But, within the limits within which Hooke's law is practically true, 
 
 - ; hence V l = V. 
 
 
198 SOUND. 
 
 So that every variation in the force applied at A produces a cor- 
 responding change in the density of the material close to A, and this 
 
 change extends along the rod with th<"- velocity \J~ . 
 
 V Itn 
 
 RELATION BETWEEN CONDENSATION AND VELOCITY (ART. 14). 
 
 Density of compressed portion of rod 1 171 
 
 ^ TT"S a ^ * J = , 7 = 1 + I nearly. 
 
 Density of uncompressed portion ot rod 1 I 
 
 Change of density 
 Original density 
 Velocity of movement of compressed material _ F7 __ , 
 Velocity of advance of compressed condition V 
 Change of density _ velocity of substance 
 
 Original density velocity of wave 
 
 As the denominators are constants, 
 velocity of undulating substance 
 
 oc difference between its actual and average density. 
 
 VELOCITY OF A TRANSVERSE WAVE ALONG A STRING (ART. 95). 
 
 Let ABj Fig. 73, be a very long string, stretched horizontally by a 
 force of * dynes at each end, so that a small portion A is in equilibrium 
 
 G, 
 D a 
 
 ixr 
 
 D A s "C H s ' J P B 
 
 Fig. 73. 
 
 between two forces of * dynes, one AC along AS, and the other AD in 
 the opposite direction. Apply at A another force A F of /dynes in 
 an upward direction, and let the resultant of AF and AD be a force r 
 in the direction AG. A will then move in the direction AF, and it 
 can never be in equilibrium until the part of the string close to A is 
 in the direction opposite to AG. When a portion of the string has 
 thus assumed the position A , H, the forces on a small portion at H are 
 a force r in the direction HA lt and a force s in the direction HB. 
 
APPENDIX. 199 
 
 These are equal and parallel to the forces r, s on A when the force 
 was first applied ; II therefore begins to move upwards exactly as A 
 did, and another portion HJ becomes equally sloping, and so on. As 
 the forces on A } would accelerate it if A^TI was more horizontal 
 than A\G\, and retard A l if A^H was less horizontal than A^G^ A 
 moves in such a way that ATL\& always opposite to AG, and similarly 
 for each succeeding portion. Thus, as point after point of the string 
 begins to move sideways, A continues to move in the direction AA l in 
 such a way that the part of the string between A and the point of the 
 string which is just beginning to move is parallel to the line GA. 
 Let A^ be the position A has reached when the sloping condition has 
 extended to P. AA. 2 P is really isosceles, but in practice AA.^ is very 
 short compared with the other sides, and we may consider AA% as 
 
 A A AT? f 
 
 perpendicular to AP. Hence =* = ^- 2 = J-. If V is the velocity 
 
 ./i-t A^AJ^ s 
 
 Vf 
 with which the sloping condition extends along the string, '- is the 
 
 velocity with which the part of the string which is already sloping 
 moves upwards. If m is the mass of one centimetre of the string, 
 
 - is the momentum of each centimetre of the sloping portion, and, 
 
 5 
 
 as V fresh centimetres are added to this portion per second, 3 is 
 the increase of momentum per second. This is equal to the force 
 
 which produces it. Hence / = Z^ or V = v / . 
 
 s V m 
 
 Just as in the corresponding proof for a longitudinal wave, every 
 variation of / produces a corresponding change of slope in the part of 
 the string close to A , which slope is always proportional to /, and every 
 change of slope produced in the part close to A extends along the 
 
 string with the velocity \ / . 
 y m 
 
 The reader who has not studied dynamics may be surprised that a 
 constant force applied to A gives A instantly a velocity which does 
 not afterwards increase, not an increasing velocity. It must be re- 
 membered that the effect of a force is to cause a momentum which 
 increases as long as the force lasts. If the mass in movement is 
 constant, its velocity must increase, but if, as in the cases considered 
 above, the mass in movement keeps increasing as long as the force 
 lasts, the velocity of the part in motion may be constant. 
 
ANSWERS. 
 
 EXAMPLES I. (page 60). 
 
 1. -25f. 2. 899 C. 3. 34,533 cm. per sec. 
 
 4. (i.) 1139-6; (ii.) 1139-6. 
 
 5. (a) infinite; (b) 100; (c) 400; (d) 0. 6. 44,294. 
 7. 130,550. 8. 1,408,741. 
 
 EXAMPLES II. (pages 153, 154). 
 
 Elementary. 
 1.41-55; 124-65; 20775: 83*1; 166'2 ; 249'3. 
 
 2. 4'155 and every odd multiple of 4'155 from wall. 3. 5540. 
 
 4. (a) 50, 100, 150, 200; (b) 25, 75, 125, 375; (c) 50, 100, 150, 200. 
 
 5. 360. 6. 42-815. 7. 60. 8. 52f cm. 
 
 Advanced. 
 1.15-02; 48-26; 81'5 cms. from top. 2. 8 inches. 
 
 3. At the middle ; 2 : 1. 4. 139,260,000 poundals per sq. ft. 
 
 5. 714-8. 
 
 EXAMPLES III. (pages 175, 176). 
 1. All equal. 2. 540 pounds' weight. 3. 640 pounds' weight. 
 
 4. 1024 : 1296 ! 1296 : 729. 5. vV inch 5 H inch - 
 
 6. 80 feet per sec. 7. 498'3. 8. 100. 9. 5'1 prams. 
 
INDEX. 
 
 A. PAGE 
 
 Acoustic pendulum 1 
 
 Adiabatic elasticity 49 
 
 Air blast 145 
 
 Amplitude 9 
 
 , measurement of 194 
 of forced vibra- 
 tions 78 
 
 Antinodes 128 
 
 B. 
 
 Beam of sound 115 
 
 Beats 73, 100, 108- 
 
 Bell 174 
 
 Blackburn's pendulum 17 
 
 C. 
 
 Change of sign 110 
 
 of components 91 
 
 Character 97 
 
 Cheshire's disc 132 
 
 Chladni's figures 173 
 
 Chord 104 
 
 Clang-tint 97 
 
 . Clarinet 148 
 
 Closed tube 137 
 
 Cochlea 93 
 
 Components 81 
 
 Concertina 149 
 
 Condensation 27 
 
 Conical pipes 144 
 
 Constitution 92 
 
 Corti, rods of 95 
 
 Crova's disc 95 
 
 SB. 
 
 Diatonic scale 102 
 
 Difference tone 108 
 
 Discord 100 
 
 Displacement curve 4 
 
 Distance, effect of 4O 
 
 Doppler' s principle 45 
 
 Drum of the ear 93 
 
 Ductus cochlearis 95 
 
 Dynamical units 46 
 
 E. 
 
 Ear, structure of 93 
 
 trumpets 74 
 
 Echo 118 
 
 Elasticity . 48 
 
 Energy 35,71, 130 
 
 Equally-tempered scale ... 106 
 
 Ether 3 
 
 F. 
 
 Fenestra ovalis 93 
 
 rotunda 94 
 
 Fifth 103 
 
 Flute 146 
 
 mouthpiece 146 
 
 Forced vibration 75 
 
 Fourier's theorem 82 
 
 Free reed 148 
 
 vibration 75 
 
 French horn 149 
 
 Frequency 3 
 
 , measurement of 177 
 
 Fundamental .. 85 
 
202 
 
 INDEX. 
 
 G. - PAGE 
 
 Gamut 102 
 
 Gases, velocity in 53 
 
 Gas flame, reflection from.., 113 
 
 H. 
 Harmonic analysis . . 88 
 
 components 
 
 curve 
 
 variation 
 vibration 
 
 84 
 
 ... 7 
 
 waves 30 
 
 Harmonicon 149 
 
 Harmonics 85, 172 
 
 Harmonograph 17 
 
 Heat, production of 41 
 
 Helmholtz 97 
 
 Hollow sound 136 
 
 Hopkins' s tube 69 
 
 I. 
 
 Intensity 196 
 
 Interference 196 
 
 Intervals 103 
 
 Isochronous vibration ... 9, 196 
 Isothermal elasticity 49 
 
 K. 
 
 Kaleidophone 14 
 
 Key-note 102 
 
 Kinetic energy 36, 131 
 
 Konig's capsules 151 
 
 ,, wave-siren 186 
 
 Kundt's tube 189 
 
 L. 
 
 Laplace's formula 50 
 
 Law of inverse squares 40 
 
 of linear dimensions ... 175 
 
 Laws of plates 173 
 
 of rods 170 
 
 of strings 168 
 
 Limits of hearing 99 
 
 Linear density 170 
 
 Liquids, waves in 45, 149 
 
 Lissajous' figures 13 
 
 Loudness 99 
 
 M. PAGE 
 
 Major gamut 104 
 
 Melde' s experiment 166 
 
 Membrana basilaris 95 
 
 Minor gamut 104 
 
 Molecules 55 
 
 Monochord 167 
 
 Musical scales 102 
 
 sound... 99 
 
 N. 
 
 Natural scale 107 
 
 Newton's formula 50 
 
 Nodes 128 
 
 Noise... 99 
 
 O. 
 
 Octave 102 
 
 Open tube 137 
 
 Ordinate 5 
 
 Organ pipe 146 
 
 Overtones 144,172 
 
 P. 
 
 Partials 172 
 
 Pendular vibration 7 
 
 Pendulum, acoustic 1 
 
 , Blackburn's 17 
 
 Period 3 
 
 of forced vibration . 78 
 
 Phase 10 
 
 Phonautograph 183 
 
 Phonograph 193 
 
 Physiological resultant 
 
 tones 107 
 
 Pitch 96 
 
 Plates, vibration of 173 
 
 Potential energy 36 
 
 Progressive undulation 26 
 
 Pulse 26 
 
 Pythagorean Scale 107 
 
 Q. 
 
 Quality 97 
 
 of sound from string 162 
 
INDEX. 
 
 203 
 
 R. PAGE 
 
 Rarefaction 25, 27 
 
 Rectilinear vibration 3 
 
 Reeds 148 
 
 Reflection of sound 118 
 
 with change of 
 
 sign 110 
 
 ,, without change of 
 
 sign Ill 
 
 of transverse 
 
 waves 157 
 
 Refraction of sound 121 
 
 Resonance 76 
 
 Resonators 88, 192 
 
 Resultant tones 90 
 
 Rigidity 175 
 
 Rods, vibration of 150, 170 
 
 S. 
 Scala tympani 95 
 
 vestibuli 95 
 
 Scales 102 
 
 Sensitive flames 134 
 
 Sensitiveness of ear 99 
 
 Shadows, sound 117 
 
 Sine curve 8 
 
 Siren 184 
 
 Konig's 186 
 
 Solids, waves in 45 
 
 Sonometer 167 
 
 Sound, cause of 1 
 
 Specific heats 51 
 
 Spiral wire 33, 112, 153 
 
 Stationary undulation 128 
 
 Stress 47 
 
 Striking reed 148 
 
 Summation tones 91, 108 
 
 Superposition, principle of 
 
 62, 196 
 
 Synthesis of sound 98 
 
 T. 
 
 Temperature '. 53, 149 
 
 Tempered scales 106 
 
 Timbre 97 
 
 Tisley's Harmonograph 17 
 
 Torsional vibration 175 
 
 PAGE 
 
 Transverse undulation... 155, 159 
 waves, velocity 
 
 of 156 
 
 Tubes, reflection in Ill 
 
 ,, , sound in 41 
 
 , stationary undulation 
 
 in 140 
 
 Tuning-fork 172 
 
 Tympanum 93 
 
 Undulation 
 
 U. 
 
 , longitudinal 
 ,, , progressive 
 ,, , stationary .. 
 
 ,, , torsional 
 
 ,, , transverse .. 
 Upper partials 
 
 V. 
 
 Velocity curve 
 
 Velocity of air ... &7, 42, 56, 
 
 of sound 46, 
 
 of transverse wave 
 , measurement of . . . 
 
 Ventral segment r 
 
 Vibration 
 
 forced 
 
 free 
 
 harmonic 
 
 longitudinal 
 
 torsional 
 
 transverse 
 
 Vibration microscope 
 
 Vibroscope 
 
 Volume elasticity 
 
 W. 
 
 Wave 30 
 
 Wave-form 28 
 
 Wave-front 39,114 
 
 Wave-length 29 
 
 Waves, harmonic 30 
 
 Wheat stone's Kaleidophone 14 
 Wind, effect of 123 
 
 27 
 
 26 
 26 
 128 
 175 
 155 
 172 
 
 6 
 
 198 
 197 
 198 
 188 
 128 
 3 
 
 75 
 
 75 
 7 
 
 26 
 175 
 155 
 181 
 177 
 
 48 
 
 Y. 
 
 Young's Modulus 
 
 47, 48 
 

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