[From the Philosophical Magazine, Vol. xxxiv. p. 52, (January, 1849.)]

ON SOME POINTS IN THE RECEIVED THEORY OF SOUND*.

I PROCEED now to notice the apparent contradiction at which Professor Challis has arrived by considering spherical waves, a contradiction which it is the chief object of this communication to consider. The only reason why I took no notice of it in a former communication was, that it was expressed with such brevity by Professor Challis (Vol. XXXII. p. 497), that I did not perceive how the conclusion that the condensation varies inversely as the square of the distance was arrived at. On mentioning this circumstance to Professor Challis, he kindly explained to me his reasoning, which he has since stated in detail (Vol. XXXIII. p. 463) †.

* The beginning and end of this Paper are omitted, as being merely controversial, and of ephemeral interest.

The objection is put in two slightly different forms in the two Papers. The substance of it may be placed before the reader in a few words.

Conceive a wave of sound of small disturbance to be travelling outwards from a centre, the disturbance being alike in all directions round the centre. Then according to the received theory the condensation is expressed by equation (1), where r is the distance from the centre, and s the condensation. It follows from this equation that any phase of the wave is carried outwards with the velocity of propagation a, and that the condensation varies inversely as the distance from the centre. But if we consider the shell of infinitesimal thickness a comprised between spherical surfaces of radii r and r+a corresponding to given phases, so that these surfaces travel outwards with the velocity a, the excess of matter in the shell over the quantity corresponding to the undisturbed density will vary as the condensation multiplied by the volume, and therefore as r2s; and as the constancy of mass requires that this excess should be constant, s must vary inversely as r2 not r.

Or instead of considering only an infinitesimal shell, consider the whole of an outward travelling wave, and for simplicity's sake suppose it to have travelled so far that its thickness is small compared with its mean radius r or at, t being

The whole force of the reasoning rests on the tacit supposition that when a wave is propagated from the centre outwards, any arbitrary portion of the wave, bounded by spherical surfaces concentric with the bounding surfaces of the wave, may be isolated, the rest of the wave being replaced by quiescent fluid; and that being so isolated, it will continue to be propagated outwards as before, all the fluid except the successive portions which form the wave in its successive positions being at rest. At first sight it might seem as if this assumption were merely an application of the principle of the coexistence of small motions, but it is in reality extremely different. The equations are competent to decide whether the isolation be possible or not. The subject may be considered in different ways; they will all be found to lead to the same result.

1. We may evidently without absurdity conceive an outward travelling wave to exist already, without entering into the question. of its original generation; and we may suppose the condensation. to be given arbitrarily throughout this wave. By an outward travelling wave, I mean one for which the quantity usually denoted by contains a function of rat, unaccompanied by a function of rat, in which case the expressions for v and s will likewise contain functions of r — at only. Let

We are at liberty to suppose f (z) = 0, except from z=b to z = c, where b and c are supposed positive; and we may take ƒ" (2) to denote any arbitrary function for which the portion from z =

b

the time of travelling from the origin to the distance r. Then assuming the expression (1), and putting the factor r outside the sign of integration, as we are at liberty to do in consequence of the supposition made above as to the distance the wave has travelled, we have for the quantity of matter existing at any time in the wave beyond what would occupy the same space in the quiescent state of the fluid,

very nearly, or 4p4t, putting 4 for the value of the integral

the inner to the outer boundary of the wave. Hence the matter increases in

quantity with the time.

to z = c has been isolated, the rest having been suppressed. Equa

✈ (r) being an arbitrary function of r, to determine which we must substitute the value of 4 given by (2) in the equation which has

This equation gives ↓ (r) = C + D/r, C and D being arbitrary constants, whence

y = do _f' (r — at) _ f (r — at) _ D

dr

r

2.2

..(4).

Now the function f(z) is merely defined as an integral of ƒ'(z) dz, and we may suppose the integral so chosen as to vanish when z=b, and therefore when z has any smaller value. Consequently we get from (4), for every point within the sphere which forms the inner boundary of the wave of condensation,

Again, if we put ƒ(c) = A, so that ƒ (z) = A when z>c, we have for any point outside the wave of condensation,

The velocities expressed by (5) and (6) are evidently such as could take place in an incompressible fluid. Now Professor Challis's reasoning requires that the fluid be at rest beyond the limits of the wave of condensation, since otherwise the conclusion cannot be drawn that the matter increases with the time. Consequently we must have D = 0, A = 0; but if A = 0 the reasoning at p. 463 evidently falls to the ground.

2. We may if we please consider an outward travelling wave which arose from a disturbance originally confined to a sphere of radius e. At p. 463 Professor Challis has referred to Poisson's expressions relating to this case. It should be observed that Poisson's expressions at page 706 of the Traité de Mécanique (second edition) do not apply to the whole wave from rat — €

1

to r = at +e, but only to the portion from rat — e to r = at; the expressions which apply to the remainder are those given near the bottom of page 705. We may of course represent the condensation s by a single function 1/ar. x (r−at), where

x(-2)=f' (2), x (2) = F′′ (2),

z being positive; and we shall have

A

= [° ̧x (e) dz = ƒ (e) − ƒ (0) + F (e) — F (0).

=

- E

Now Poisson has proved, and moreover expressly stated at page 706, that the functions F, f vanish at the limits of the wave; so that ƒ (e) = 0, F(e)=0. Also Poisson's equations (6) give in the limiting case for which z = 0, ƒ (0) + F(0) = 0, so that A=0 as before.

3. We may evidently without absurdity conceive the velocity and condensation to be both given arbitrarily for the instant at which we begin to consider the motion; but then we must take the complete integral of (3), and determine the two arbitrary functions which it contains. We are at liberty, for example, to suppose the condensation and velocity when t=0 given by the equations

from rb to r = c, and to suppose them equal to zero for all other values of r; but we are not therefore at liberty to suppress the second arbitrary function in the integral of (3). The problem is only a particular case of that considered by Poisson, and the arbitrary functions are determined by his equations (6) and (8), where, however, it must be observed, that the arbitrary functions which Poisson denotes by ƒ, F must not be confounded with the given function here denoted by f, which latter will appear at the right-hand side of equations (8). The solution presents no difficulty in principle, but it is tedious from the great number of cases to be considered, since the form of one of the functions which enter into the result changes whenever the value of r+ at or of rat passes through either b or c, or when that of r at passes through zero. It would be found that unless ƒ (b) = 0, a backward wave sets out from the inner surface of the spherical shell contain

ing the disturbed portion of the fluid; and unless f(c) = 0, a similar wave starts from the outer surface. Hence, whenever the disturbance can be propagated in the positive direction only, we must have A, or ƒ (c) —ƒ(b), equal to zero. When a backward wave is formed, it first approaches the centre, which in due time it reaches, and then begins to diverge outwards, so that after the time c/a there is nothing left but an outward travelling wave, of breadth 2c, in which the fluid is partly rarefied and partly condensed, in such a manner that frs dr taken throughout the wave, or A, is equal to zero.

It appears, then, that for any outward travelling wave, or for any portion of such a wave which can be isolated, the quantity A is necessarily equal to zero. Consequently the conclusion arrived at, that the mean condensation in such a wave or portion of a wave varies ultimately inversely as the distance from the centre, proves not to be true. It is true, as commonly stated, that the condensation at corresponding points in such a wave in its successive positions varies ultimately inversely as the distance from the centre; it is likewise true, as Professor Challis has argued, that the mean condensation in any portion of the wave which may be isolated varies ultimately inversely as the square of the distance; but these conclusions do not in the slightest degree militate against each other.

=

If we suppose b to increase indefinitely, the condensation or rarefaction in the wave which travels towards the centre will be a small quantity, of the order b1, compared with that in the shell. In the limiting case, in which b∞, the condensation or rarefaction in the backward travelling wave vanishes. If in the equations of paragraph 3 we write b+x for r, bo (x) for ƒ' (r), and then suppose b to become infinite, we shall get as σ (x), v=σ (x). Consequently a plane wave in which the relation v as is satisfied will be propagated in the positive direction only, no matter whether fo (x) dx taken from the beginning to the end of the wave be or be not equal to zero; and therefore any arbitrary portion of such a wave may be conceived to be isolated, and being isolated, will continue to travel in the positive direction only, without sending back any wave which will be propagated in the negative direction. This result follows at once from the equations which apply directly to plane waves; I mean, of course, the approxi