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equating now separately the coefficients of /' and f", we get
d*X

dX Adx fidz ydx'
dx

The first, integrated, gives and the second

Hence if we neglect the superfluous constant k */g, the general integral of (4) is, (y A-p^),

therefore, by (c'),

and the actual Telocity of the fluid particles in the direction of the axis of x, is

neglecting quantities which are of the order (al) compared with those retained.

If the initial values of f and w are given, we may then determine f and F\ and we thus see that a single wave, like a pulse of sound, divides into two, propagated in opposite directions. Considering, therefore, only that which proceeds in the direction of x positive, we have

. fl-M _. / f da.

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Suppose now the value of F' (x) = 0, except from x = a to ec = a + a, and Sx to be the corresponding length of the wave, we have

. rdx

and * — f-^= —~ = a very nearly.

Hence the variable length of the wave is

&c = a. V^7 (7).

Lastly, for any particular phase of the wave, we have {dx

t — I -.~ = const:

therefore

is the velocity with which the wave, or more strictly speakingthe particular phase in question,'progresses.

From (5), (6), (7), and (8) we see that if /? represent the variable breadth of the canal and 7 its depth,

f = height of the wave <# j3~^y~*,

u = actual velocity of the fluid particles as /3T^f%.

dx =» length of the wave « 7*,

and = velocity of the wave's motion = **fgy.

ON THE REFLEXION AND REFRACTION

OF SOUND.

From the Transaction* of the Cambridge Philosophical Society, 1838.
[Read Deo. 11, 1837].

ON THE REFLEXION AND REFRACTION OF SOUND.

The object of the communication which I have now the honour of laying before the Society, is to present, in as simple a form as possible, the laws of the reflexion and refraction of sound, and of similar phenomena which take place at the surface of separation of any two fluid media when a disturbance is propagated from one medium to the other. The subject has already been considered by Poisson (Mfm. de I'Acad., &c. Tome x. p. 317, &c.). The method employed by this celebrated analyst is one that he has used on many occasions with great success, and which he has explained very fully in several of his works, and recently in a digression on the Integrals of Partial Differential Equations (Tkforie de la Chaleur, p. 129, &c.). In this way, the question is made to depend on sextuple definite integrals. Afterwards, by supposing the initial disturbance to be confined to a small sphere in one of the fluids, and to be everywhere the same at the same distance from its centre, the formulae are made to depend on double definite integrals; from which are ultimately deduced the laws of the propagation of the motion at great distances from the centre of the sphere originally disturbed.

The chance of error in every very long analytical process, more particularly when it becomes necessary to use Definite Integrals affected with several signs of integration, induced me to think, that by employing a more simple method we should possibly be led to some useful result, which might easily be overlooked in a more complicated investigation. With this impression I endeavoured to ascertain how a plane wave of infinite extent, accompanied by its reflected and refracted waves, would be propagated in any two indefinitely extended media of

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