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which the surface of separation in a state of equilibrium should also be in a plane of infinite extent.

The suppositions just made simplify the question extremely. They may also be considered as rigorously satisfied when light is reflected. In which case the unit of space properly belonging

to the problem is a quantity of the same order as \= —|— inch,

and the unit of time that which would be employed by light itself in passing over this small space. Very often too, when sound is reflected, these suppositions will lead to sensibly correct results. On this last account, the- problem has here been considered generally for all fluids whether elastic or non-elastic in the usual acceptation of these terms; more especially, as thus its solution is not rendered more complicated. One result of oar analysis is so simple that I may perhaps be allowed to mention it here. It is this: If A be the ratio of the density of the reflecting medium to the density of the other, and B the ratio of the cotangent of the angle of refraction to the cotangent of the angle of incidence, then for all fluids

the intensity of the reflected vibration _A—B
the intensity of the incident vibration A +B'

If now we apply this to the reflexion of sound at the surface of still water, we have A > 800, and the maximum value of B<%. Hence the intensity of the reflected wave will in every case be sensibly equal to that' of the incident one. This is what we should naturally have anticipated. It is however noticed here because M. Poissbn has inadvertently been led to a result entirely different

When the velocity of transmission of a wave in the second medium, is greater than that in the first, we may, by sufficiently increasing the angle of incidence in the first medium, cause the refracted wave in the second to disappear. In this case the change in the intensity of the reflected wave is here shown to be such that, at the moment the refracted wave disappears, the intensity of the reflected becomes exactly equal to that of the incident one. If we moreover suppose the vibrations of the incident wave to follow a law similar to that of the cycloidal pendulum, as is usual in the Theory of Light, it is proved that on farther increasing the angle of incidence, the intensity of the reflected wave remains unaltered whilst the phase of the vibration gradually changes. The laws of the change of intensity, and of the subsequent alteration of phase, are given here for all media, elastic or non-elastic. When, however, both the media are elastic, it is remarkable that these laws are precisely the same as those for light polarized in a plane perpendicular to the plane of incidence. Moreover, the disturbance excited in the second medium, when, in the case of total reflexion, it ceases to transmit a wave in the regular way, is represented by a quantity of which one factor is a negative exponential. This factor, for light, decreases with very great rapidity, and thus the disturbance is not propagated to a sensible depth in the second medium.

Let the plane surface of separation of the two media be taken as that of (yz), and let the axis of e be parallel to the line of intersection of the plane front of the wave with (yz), the axis of x being supposed vertical for instance, and directed downwards; then, if A and A, are the densities of the two media under the constant pressure Pand «, sx the condensations, we must have

|A (1 + s) density in the upper medium,
(A,(l + «,) = density in the lower medium.

(P (14- As) = pressure in the upper medium,
(P (1 + Axs^j = pressure in the lower medium.

Also, as usual, let <f> be such a function of x, y, z, that the resolved parts of the velocity of any fluid particle parallel to the axes, may be represented by

d<p d<j> d<f>
dx* dy' dz'

In the particular case, here considered, <f, will be independent of z, and the general equations of motion in the upper fluid will be

ds , d'd>

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dy where

, PA,

The above are the known general equations of fluid motion, which must be satisfied for all the internal points of both fluids; but at the surface of separation, the velocities of the particles perpendicular to this surface and the pressure there must be the same for both fluids. Hence we have the particular conditions

d<t>_dfr 1

dx~ dx L (where x = 0),
As = A js, j

neglecting such quantities as are very small compared with those retained, or by eliminating s and we get

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The general equations (1) and (2), joined to the particular conditions (A) which belong to the surface of separation (yi), only, are sufficient for completely determining the motion of our two fluids, when the velocities and condensations are independent of the co-ordinate 2, whatever the initial disturbance may be. We shall not here attempt to give their complete solution, which would be complicated, but merely consider the propagation of a plane wave of indefinite extent, which is accompanied by its reflected and refracted wave.

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Since the disturbance of all the particles, in any front of the incident plane wave, is the same at the same instant, we shall have for the incident wave

<]>=f(ax + by + ct),

retaining b and e unaltered, we may give to the fronts of the reflected and refracted waves, any position by making for them

<f> = F (a'x + by + ci),

Hence, we have in the upper medium,

$ =f{ax + by + ct) + F (dx + bv + ct) (4),

and in the lower one

+ + (5)

These, substituted in the general equations (1) and (2), give ^==^(a'+br)y

<? V) (6).

c* = 7Vai,+ J') )

Hence, a' = + a, where the lower signs must evidently be taken to represent the reflected wave. This value proves, that the angle of incidence is equal to that of reflexion. In like manner, the value of a,, will give the known relation of sines for the incident and refracted wave, as will be seen afterwards.

Having satisfied the general equations (1) and (2), it only remains to satisfy the conditions (A), due to the surface of separation of the two media. But these by substitution give

of {by + ct)-aF Q»y + ct) - af, {by + ct),

A {f{by + ct) + F' {by+ct)} - (by + ct),

because a = — a, and x = 0.

Hence by writing, to abridge, the characteristics only of the functions

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or if we introduce 6, 0,, the angle of incidence and refraction since

cot 8 = *,

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A^ oote,

and therefore ^ = r —-y,

y A~cotF'

» '+'

A cot 6

which exhibits under a very simple form, the ratio between the intensities of the disturbances, in the incident and reflected wave.

JBut the equations (6) give

and hence

sin 0 sand' the ordinary law of sines.

The reflected wave will vanish when
A, cot ff
A cot 0;

which with the above gives

Hence the reflected wave may be made to vanish if <y* — 7,' and (vA)s — (7,A,) have different signs.

For the ordinary elastic fluids, at least if we neglect the change of temperature due to the condensation, A is independent of the nature of the gas, and therefore.

A - A, or 7SA = y'At.

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