a result which might also have been arrived at by direct transformation of the general equation of continuity. To our order of approximation, (11) gives

and eliminating s between this and (10), we find

This is of the same form as (6), so that the solution is

rp = F(r− ct) +f(r + ct)....................(13).

Hence the motion is made up of two systems of spherical waves travelling, one outwards, the other inwards, with velocity c. Considering for a moment the first system alone, we have

which shews that a condensation is propagated outwards with velocity c, but diminishes as it proceeds, its amount varying inversely as the distance from the origin. The velocity due to the same train of waves is

Asr increases the second term becomes less and less important compared with the first, so that ultimately the velocity is propagated according to the same law as the condensation.

168. Let us suppose that the initial distributions of velocity and condensation are given by the formulæ

$ = ¥ (r),

do
dt = x().

.(14),

where x are any arbitrary functions. The value of at any

subsequent time t is then to be found as follows. Comparing (14)

with (13) we find

F (r) +ƒ (r) = r↓ (1o), )

− F' (1') +ƒ' (r) = x (r)

.(15).

Integrating the latter equation, and putting ["rx(r) dr = x,(r), we

obtain

1

− F (r) + ƒ (r) = x, (r) + C,

Χι

where C is an arbitrary constant. This gives, in conjunction with (15),

1

ƒ (r) = { r ↓ (r') + } = x, (~) + §. ................ (17).

с

The complete value (13) of is then found by writing, for r› r — ct in (16), and r + ct in (17), viz.

1

+ 1⁄2 (r + ct) & (r + ct) + 1 = X1 (r + ct).........................(18).

It is obvious from the symmetry of the motion with respect to the origin that

We shall require shortly the value of at the origin. This may be found by dividing both sides of (18) by r, and evaluating the indeterminate form which the right-hand member assumes for r=0; or more simply by differentiating both sides of (18) with respect to r and then making r=0. The result is, if we take account of the relations (19) and (20),

General Equations of Sound Waves.

169. We proceed to the general case of propagation of airWe neglect, as before, the squares of small quantities, so

waves.

that the dynamical equation is

do

c's =

dt

.(10).

Also, writing p=p。(1+s) in the general equation of continuity, (8) of Art. 8, we have, to the same order of approximation

ds d' d' ďo

+ + + = 0..................(22). dt dx dy dz2

The elimination of s between (10) and (22) gives

170. Let us suppose a sphere of radius r described about any point (x, y, z) as centre. Multiplying both sides of (23) by dxdydz, and integrating throughout the volume of the sphere, we find

dr [[[pdxdydz = c° [[[vpdxdydz = off dø
c2 as,

dr

or, writing ds = r'da, so that da denotes an elementary solid angle,

Let us differentiate both sides of this equation with respect to r. The left-hand side gives

The mean value of over a sphere having any point P of the medium as centre is therefore propagated according to the same laws as a symmetrical spherical disturbance of the air. We see at once that the value of at P at the time t depends on

do
dt

the mean initial values of and over a sphere of radius ct described about P as centre, so that the disturbance is propagated in all directions with uniform velocity c. If the disturbance be confined originally to a finite portion of space, the disturbance at any point P external to Σ will begin after a time ", will last for a time", and will then cease altogether;

r

C

с

r1, r, denoting the radii of the spheres described with P as centre,

2

the one just excluding, the other just including Σ.

To express the solution of (23), already virtually obtained, in an analytical form, let the values of and $

аф
dt

when t=0, be

* This result was obtained, in a different manner, by Poisson, J. de l'Ecole Polytechnique, 14me cahier (1807), pp. 334–338. The remark that it leads at once to the complete solution of (23), first given by Poisson, Mém. de l'Acad. des Sciences, t. 3 (1818-19), is due to Liouville, J. de Math. 1856, pp. 1-6. The above references are taken from Liouville's paper.

The equation (24) may be proved also as follows. Suppose an infinite number of systems of rectangular axes arranged uniformly about any point P of the fluid as origin, and let P1, P2, P3, &c. be the velocity-potentials of motions which are the same with respect to these systems as the original motion is with respect to the system x, y, z. If then denote the mean value of the functions P1, P2, P3, &c., ø will be the velocity-potential of a motion symmetrical with respect to the point P, and will therefore satisfy (12). The value of 7 at a distance r from P will evidently be the same thing as the mean value of over a sphere of radius r described about P as centre.

The mean initial values of these quantities over a sphere of radius r described about (x, y, z) as centre are

where l, m, n denote the direction-cosines of any radius of this sphere, and do the corresponding elementary solid angle. Comparing with Art. 168, we see that the value of o at any subsequent time t is

171. We have so far assumed the velocity and the condensation to be so small that their squares and products may be neglected. The results obtained on this supposition are indeed sufficiently accurate for most purposes; but it is worth while to notice briefly the solutions of the exact equations of motion of plane waves which have been obtained, independently and by different methods by Earnshaw and Riemann†.

Riemann's Method.

Riemann starts from the ordinary Eulerian form of the equations of motion and continuity, which may be written