It is important to notice that the transmission of energy depends on the fact that the velocity and condensation (which is proportional to the pressure) are in the same phase. The condensed portions of the fluid move in the direction in which the wave is propagated, and the rarefied portions in the opposite direction. It is a consequence of this fact that the work done while the air moves forwards is not undone while the air moves backwards.

We next take the case of waves diverging from a point. The motion to be considered belongs to an important class in which the velocities may be represented as the partial differential coefficients of the same function, called the velocity potential.

Equations (1) are now all contained in the simple equation

If

depends only on the distance r from a fixed point which acts as a source from which the vibrations emanate, we have

This value of is therefore a solution of the differential equation (4). By differentiation

аф

dr

represents the velocity, which it appears does not vary inversely as the distance as might have been expected at first sight. At small distances from the origin the second term is the important one and the velocity varies inversely as the square of r. The origin itself is a singular point at which matter enters and leaves the space. The amount of matter passing per unit time through any sphere having the origin as centre is equal to 42 Ddø/dr which, if r is small, is equal to

2π

λ

4π AD sin vt, and this expression therefore represents the rate at which matter is introduced at the origin. At large distances the first term is the important one. If the difference of phase between any point very near the origin and one at a large distance r away from it, were calculated in the usual way and put equal to 2-r/A, we should commit an error equal to a quarter of a wave-length. This apparent change of phase of a right angle when points near a source and at some distance away from it are considered, has been already referred to several times (e.g. Art. 46). If we were to measure energy simply by the square of the amplitude, equation (4) would lead to the conclusion that the energy does not vary inversely as the square of the distance from the origin as is generally assumed. There is however no reason why it should vary according to the simple law, so long as the energy transmitted follows it. That this is actually the case may be proved as follows. The rate at which energy is transmitted through a sphere of radius r is

= 2πDA® sin (wt — Ir) {l sin (wt – Ir) —–—– cos (ut – Ir)} .

-

·

r

Integrating with respect to the time and leaving out periodic terms,

This expression does not contain r and hence the work transmitted through concentric spheres enclosing the origin is constant. It follows

that the work transmitted in a given time through unit surface varies inversely as the square of the distance. For a fuller treatment of the subject the reader is referred to Lord Rayleigh's treatise on Sound, Vol. II., Arts. 279 and 280.

174. Plane Waves of Distortion in an elastic medium. Let the displacements be parallel to the axis of z and be denoted by , the wave normal being the axis of x. The only force which can do work across the plane ry is the tangential stress which in Art. 129 has been called T, and which according to (7) Art. 131 is equal to nd/da, being zero in the present case. The stress T has been taken to be positive when the portion of matter on the positive side of the plane yz acts on the matter which is on the negative side with a force directed along the positive axis of z Hence for waves travelling in the positive direction, if W be the energy transmitted across unit surface,

and the coefficient of distortion n is replaced by 2/D, we find, as in the case of the sound-wave, leaving out periodic terms,

2

W = DV12 vt

where V, denotes the maximum velocity.

(6),

175. Sphere performing torsional oscillations in an elastic medium. Consider displacements in an elastic medium defined by

so that equations (9), Art. 132, are satisfied if v = n/D. The assumed system of displacements represents therefore a possible wave propagation, the waves being purely distortional.

As does not contain x, y, z explicitly,

It follows that the displacements at any point are at right angles to the radius vector drawn from the origin to that point. As there are no displacements parallel to the axis of a, the displacements are along circles drawn round OX as axis.

Let p be the distance of any point from the axis, so that r2 = x2 + p2. We obtain the amount of the displacement by resolving and in a direction at right angles to p in a plane parallel to the plane of yz. This gives for the displacement:

The angular displacement obtained by dividing the actual displacement by p only depends on r, and is therefore the same at all points of a sphere having the origin as centre. Each such sphere performs torsional oscillations as if it were rigid. any one sphere to be actually rigid and tained by forces applied to this sphere. Our system of equations will then tell us how these oscillations are propagated outwards.

We may therefore imagine the oscillations to be main

In the language of Optics the vibrations at any point are polarized in a plane passing through OX which is the axis of rotation. The angular displacements are

and are nearly equal to the first or second term of this expression respectively, according as r is very small or very large compared with X/2. Comparing large and small values of r, we have here the same change of phase of a right angle which has been noted in Art. 173.

The maximum angular displacement at a distance S from the origin as obtained from (7) is:

and this is the amplitude of oscillation which must be maintained at a sphere of radius S in order to cause an angular amplitude 2A/λr2 at a large distance. If the maintained angular amplitude is B, it follows that for large distances the angular amplitude is

BS2

?2 √1 + X2 (4π3S2) ̄-1

The actual amplitude is obtained on multiplying this expression by r sin 0, where denotes the angle which forms with OX. To calculate the energy communicated by the rigid sphere to the surrounding medium, we make use of the obvious proposition that the energy transmitted through all concentric spheres must be equal and we may therefore simplify the calculation by considering only a sphere of very large radius.

If we write (V, sin 6)/r for the maximum velocity at a large distance, the total energy transmitted through unit surface at any time is by (6)

W = DV,vt sin3 0/r3,

Substituting the value of V1, we find for E, the total energy

The bracket on the right-hand side represents the greatest velocity in the equatorial plane of the rigid sphere.

It should be noticed that the energy transmitted diminishes with increasing wave-length (i.e. increasing period) and this diminution is the more important the smaller the radius of the embedded sphere is compared with the wave-length.

176. Waves diverging from a sphere oscillating in an elastic medium. The problems discussed in this and the preceding article were first solved by W. Voigt*. Kirchhoff † considerably simplified the mathematical analysis and more recently Lord Kelvin ‡

* Crelle's Journal, Vol. LXXXIX. p. 288.

+ Crelle's Journal, Vol. xc. p. 34.

Phil. Mag. Vol. XLVII. p. 480 and XLVIII. pp. 277, 388.