INTERFERENCE OF SOUND-WAVES.

891

in opposite directions. Let a disk of card-board of the same size be divided into the same number of sectors, and let alternate sectors be cut away, leaving only enough near the centre to hold the remaining sectors together. If the card be now held just over the vibrating disk, in such a manner that the sectors of the one are exactly over those of the other, a great increase of loudness will be observed, consequent on the suppression of the sound from alternate sectors; but if the card-board disk be turned through the width of half a sector, the effect no longer occurs. If the card is made to rotate rapidly in a continuous manner, the alterations of loudness will form a series of beats.

It is for a similar reason that, when a large bell is vibrating, a person in its centre hears the sound as only moderately loud, while within a short distance of some portions of the edge the loudness is intolerable.

887. Interference of Direct and Reflected Waves.1 Nodes and Antinodes. Interference may also occur between undulations travelling in opposite directions; for example, between a direct and a reflected system. When waves proceeding along a tube meet a rigid obstacle, forming a cross section of the tube, they are reflected directly back again, the motion of any particle close to the obstacle being compounded of that due to the direct wave, and an equal and opposite motion due to the reflected wave. The reflected waves are in fact the images (with reference to the obstacle regarded as a plane mirror) of the waves which would exist in the prolongation of the tube if the obstacle were withdrawn. At the distance of half a wave-length from the obstacle the motions due to the direct and reflected waves will accordingly be equal and opposite, so that the particles situated at this distance will be permanently at rest; and the same is true at the distance of any number of half wave-lengths from the obstacle. The air in the tube will thus be divided into a number of vibrating segments separated by nodal planes or cross sections of no vibration arranged at distances of half a wave-length apart. One of these nodes is at the obstacle itself. At the centres of the vibrating segments-that is to say, at the distance of a quarter wave-length plus any number of half wave-lengths from the obstacle or from any node -the velocities due to the direct and reflected waves will be equal and in the same direction, and the amplitude of vibration will accordingly be double of that due to the direct wave alone. These 1 See note C, page 895.

are the sections of greatest disturbance as regards change of place. We shall call them antinodes. On the other hand, it is to be remembered that motion with the direct wave is motion against the reflected waves, and vice versa, so that (§ 869) at points where the velocities due to both have the same absolute direction they correspond to condensation in the case of one of these undulations, and to rarefaction in the case of the other. Accordingly, these sections of maximum movement are the places of no change of density; and on the other hand, the nodes are the places where the changes of density are greatest. If the reflected undulation is feebler than the direct one, as will be the case, for example, if the obstacle is only imperfectly rigid, the destruction of motion at the nodes and of change of density at the antinodes will not be complete; the former will merely be places of minimum motion, and the latter of minimum change of density.

Direct experiments in verification of these principles, a wall being the reflecting body, were conducted by Savart, and also by Seebeck, the latter of whom employed a testing apparatus called the acoustic pendulum. It consists essentially of a small membrane stretched in a frame, from the top of which hangs a very light pendulum, with its bob resting against the centre of the membrane. In the middle portions of the vibrating segments the membrane, moving with the air on its two faces, throws back the pendulum, while it remains nearly free from vibration at the nodes.

Regnault made extensive use of the acoustic pendulum in his experiments on the velocity of sound. The pendulum, when thrown back by the membrane, completed an electric circuit, and thus effected a record of the instant when the sound arrived.

888. Beats Produced by Interference. When two notes which are not quite in unison are sounded together, a peculiar palpitating effect is produced; we hear a series of bursts of sound, with intervals of comparative silence between them. The bursts of sound are called beats, and the notes are said to beat together. If we have the power of tuning one of the notes, we shall find that as they are brought more nearly into unison, the beats become slower, and that, as the departure from unison is increased, the beats become more rapid, till they degenerate first into a rattle, and then into a discord. The effect is most striking with deep notes.

These beats are completely explained by the principle of interferAs the wave-lengths of the two notes are slightly different,

ence.

BEATS.

893 while the velocity of propagation is the same, the two systems of waves will, in some portions of their course, agree in phase, and thus strengthen each other; while in other parts they will be opposite in phase, and will thus destroy each other. Let one of the notes, for example, have 100 vibrations per second, and the other 101. Then, if we start from an instant when the maxima of condensation froin the two sources reach the ear together, the next such conjunction will occur exactly a second later. During the interval the maxima of one system have been gradually falling behind those of the other, till, at the end of the second, the loss has amounted to one wavelength. At the middle of the second it will have amounted to half a wave-length, and the two sounds will destroy each other. We shall thus have one beat and one extinction in each second, as a consequence of the fact that the higher note has made one vibration more than the lower. In general, the frequency of beats is the difference of the frequencies of vibration of the beating notes.

NOTE A. § 869.

That the particles which are moving forward are in a state of compression, may be shown in the following way:-Consider an imaginary cross section travelling forward through the tube with the same velocity as the undulation. Call this velocity v, and the velocity of any particle of air u. Also let the density of any particle be denoted by p. Then u and p remain constant for the imaginary moving section, and the mass of air which it traverses in its motion per unit time is (v – u) p. As there is no permanent transfer of air in either direction through the tube, the mass thus traversed must be the same as if the air were at rest at its natural density. Hence the value of (vu) p is the same for all cross sections; whence it follows, that where u is greatest p must be greatest, and where u is negative p is less than the natural density.

whence

u ρ
v

Po
P

that is to say,

If po denote the natural density, we have (v-u) p=v po, the ratio of the velocity of a particle to the velocity of the undulation is equal to the condensation existing at the particle. If u is negative-that is to say, if the velocity be retrograde --its ratio to v is a measure of the rarefaction.

From this principle we may easily derive a formula for the velocity of sound, bearing in mind that u is always very small in comparison with v.

For, consider a thin lamina of air whose thickness is dx, and let du, dp, and dp be the excesses of the velocity, density, and pressure on the second side of the lamina above those on the first at the same moment. The above equation, (v-u) p=v po, gives (v-u) dp-pdu=0, whence or, since u may be neglected in comparison with v,

=

δυ v-u
δρ P

δυ v δρ

δη

The time which the moving section occupies in traversing the lamina is and in this

v

time the velocity of the lamina changes by the amount du, since the velocity on the

1

second side of the lamina is u+du at the beginning and u at the end of the time. The force producing this change of velocity (if the section of the tube be unity) is - dp, or -1.418p, and must be equal to the quotient of change of momentum by time, that is

This investigation is due to Professor Rankine, Phil. Trans. 1869.

NOTE B. § 874.

The following is the usual investigation of the velocity of transmission of sound through a uniform tube filled with air, friction being neglected: Let x denote the original distance of a particle of air from the section of the tube at which the sound originates, and x+y its distance at time t, so that y is the displacement of the particle from the position of equilibrium. Then a particle which was originally at distance x + dx will at time t be at the distance x+x+y+dy; and the thickness of the intervening lamina, which was originally dx, is now 8x+8y. Its compression is therefore

dy

δι

dy

or ultimately -d and if P denote

the original pressure, the increase of pressure is -1.41 P

d

dx

dx

y

(1·41 Pd) 8x, or 1·41 Pd 8.

dx

dx;

behind a lamina 8x above the pressure in front is and if D denote the original density of the air, the acceleration of the lamina will be the

The term F (x-vt) represents a wave, of the form y=F (x), travelling forwards with velocity v; for it has the same value for t1+ôt and x1+v. dt as for t1 and x1. The term f(x+vt) represents a wave, of the form y=f (x), travelling backwards with the same velocity.

In order to adapt this investigation, as well as that given in Note A, to the propagation of longitudinal vibrations through any elastic material, whether solid, liquid, or gaseous, we have merely to introduce E in the place of 1.41 P, E denoting the coefficient of elasticity dy of the substance, as defined by the condition that a compression is produced by a force

dx

The following is the regular mathematical investigation of the interference of direct and reflected waves of the simplest type, in a uniform tube.

STATIONARY UNDULATION.

895

Using x, y, and t in the same sense as in Note B, and measuring x from the reflecting surface to meet the incident waves, we have, for the incident waves,

a denoting the amplitude, and λ the wave-length. For the reflected waves, we have

since this equation represents waves equal and opposite to the former, and satisfies the condition that at the reflecting surface (where x is zero) the total disturbance y1+Y2 is zero. Putting y for y1+y2, we have, by adding the above equations and employing a wellknown formula of trigonometry,

x The factor sin 2π vanishes at the points for which x is either zero or a multiple of λ

›, and attains its greatest values (in arithmetical sense) at those for which is λ, or λ plus a multiple ofλ. On the other hand, the factor cos 2 vanishes at the latter

84 points, and attains its greatest values at the former. The points for which sin 2π እ vanishes are the nodes, since at these points y is constantly zero; and the points for which cos 2π vanishes are the antinodes, since at these the extension or compression is λ constantly zero.

The motion represented by equation (3) is the simplest type of stationary undulation.

NOTE D. § 888.

The following is the mathematical investigation of beats for two systems of waves of equal amplitude but slightly different wave-length and period, travelling with the same velocity.

By hypothesis m1 is m2 very small compared with m1+m2; hence the factor cos 1⁄2 (m1 — m2) 0 remains nearly constant for an increment of which causes (m1+m2) to increase by 2π. The expression therefore represents a series of waves having a wave-length intermediate between A and λ (since (m+m2) is intermediate between m1 and m2), and having an amplitude 2a cos (m1 — m2) 0 which gradually varies between the limits zero and 2a.