this gives, by (7), in the case of a divergent wave-system,

.(11)

if ro2 vanishes at the inner and outer boundaries of the system*.

(12)

286. The determination of the functions ƒ and F in (6), in terms of the initial conditions, for an unlimited space, can be effected as follows.

Let us suppose that the distributions of velocity and condensation at time t0 are determined by the formulae

where, x are arbitrary functions. Comparing with (6), we have

Again, the condition that there is no creation or annihilation of fluid at the origin gives

The formulae (14) and (15) determine the functions ƒ and F for positive values of z; and (16) then determines ƒ for negative values of z†.

As a very simple example we may suppose that the air is initially at rest, and that the initial disturbance consists of a uniform condensation 8, extending through a sphere of radius a. We have then (2) = 0, whilst x (z) = c2s, or 0 according as za. At a distance r(> a) from the origin, the motion will not begin until t = (r− a)/c, and will cease when t = (ra)/c. For intermediate instants we shall have

The disturbance is now confined to a spherical shell of thickness 2a; and the condensation s is positive through the outer half, and negative through the inner half, of the thickness.

We shall require, shortly, an expression for the value of at the origin, for all values of t, in terms of the initial circumstances. We have, by (6) and (16),

287. We proceed to the general case of propagation of expansion-waves. We neglect, as before, small quantities of the second order, so that the dynamical equation is, as in Art. 285,

Also, writing p = Po (1+s) in the general equation of continuity, Art. 7 (5), we have, with the same approximation,

Since this equation is linear, it will be satisfied by the arithmetic mean of any number of separate solutions 41, 42, 43, .... As in Art. 38, let us imagine an infinite number of systems of rectangular axes to be arranged

...

...

uniformly about any point P as origin, and let P1, P2, P3, be the velocitypotentials of motions which are the same with respect to these systems as the original motion is with respect to the system x, y, z. In this case the arithmetic mean (4, say) of the functions P1, P2, P3, will be the velocitypotential of a motion symmetrical with respect to the point P, and will therefore come under the investigation of Art. 286, provided r denote the distance of any point from P. In other words, if o be a function of r and t, defined by the equation

where is any solution of (4), and do is the solid angle subtended at P by an element of the surface of a sphere of radius r having this point as centre, then

The mean value of over a sphere having any point P of the medium as centre is therefore subject to the same laws as the velocity-potential of a symmetrical spherical disturbance. We see at once that the value of at P at the time t depends on the means of the values which & and dat originally had at points of a sphere of radius ct described about P as centre, so that the disturbance is propagated in all directions with uniform velocity c. Thus if the original disturbance extend only through a finite portion of space, the disturbance at any point P external to Σ will begin after a time r/c, will last for a time (rar)/c, and will then cease altogether; 71, 72 denoting the radii of two spheres described with P as centre, the one just excluding, the other just including Σ.

To express the solution of (4), already virtually obtained, in an analytical form, let the values of 4 and a/at, when t = 0, be

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

*This result was obtained, in a different manner, by Poisson, "Mémoire sur la théorie du son," Journ. de l'École Polytechn. t. vii. pp. 334-338 (1807). The remark that it leads at once to the complete solution of (4) is due to Liouville, Journ. de Math. t. i. p. 1 (1856).

where l, m, n denote the direction-cosines of any radius of this sphere, and So the corresponding elementary solid angle. If we put

Hence, comparing with Art. 286 (21), we see that the value of 4 at the point (x, y, z), at any subsequent time t, is

1 д Απ οι

ff (x + et sin o cos w, y + ct sin 0 sin w, z + ct cos 0) sin 0 dł dw

ป

ffx (x + ct sin & cos w, y + ct sin 0 sin w, z + ct cos 0) sin 0 do dw,

x

which is the form given by Poisson*

(9)

288. The expression for the kinetic energy of the fluid contained in any given region is

where stands for dp/dt. By Green's Theorem (Art. 43), this may be put in the form

We have seen (Art. 280) that, subject to a certain condition, W represents

the intrinsic energy.

The complete interpretation of (3) may be left to the reader. In various important cases, e.g. when the boundary is fixed (a/an = 0), or free ($ = 0), the surface-integral vanishes, and we have

"Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques," Mém. de l'Acad. des Sciences, t. iii. p. 121 (1819).

For other proofs see Kirchhoff, Mechanik, c. xxiii., and Rayleigh, Theory of Sound, Art. 273.

This leads to a proof of the determinateness of the motion consequent on a given initial distribution of velocity and condensation. For if 1, 2 were two distinct forms of the velocity-potential satisfying the prescribed initial conditions, then, in the motion for which = 1 − 2, T + W would be constantly null, since it vanishes initially. Since every element of T and W is essentially positive, this requires that the derivatives of 4 with respect to x, y, z, t should all vanish; i.e. 1 and 2 can only differ by an absolute constant*. The argument applies, of course, to all cases where we can predicate the vanishing of the surface-integral in (3).

Simple-Harmonic Vibrations.

289. In the case of simple-harmonic motion, the time-factor being eit, the equation (4) of Art. 287 takes the form

It appears on comparison with Art. 280 that 2/k is the wave-length of plane waves of the assumed period (2π/σ).

In the case of symmetry with respect to the origin, we have by Art. 285 (5), or by transformation of (1),

If the motion is finite at the origin we must have B = 0, and (4) reduces to

It may be noticed that this solution may be obtained by superposition of systems of plane waves, the directions of propagation being distributed. uniformly. Thus, for a system of plane-waves whose direction of propagation makes an angle 0 with a given radius vector r, we have

and the mean-value of this for all directions through the origin is

.(6)

(7)

* Kirchhoff, Mechanik, c. xxiii.

The time-factor is omitted here and elsewhere for shortness.