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which is the precise angle at which light polarized perpendicular to the plane of reflexion is wholly transmitted.
But it is not only at this particular angle that the reflexion of sound agrees in intensity with light polarized perpendicular to the plane of reflexion. For the same holds true for every angle of incidence. In fact, since
which is the same ratio as that given for light polarized perpendicular to the plane of incidence. . (Vide Airy's Tracts, p. 356)*.
What precedes is applicable to all waves of which the front is plane. In what follows we shall consider more particularly the case in which the vibrations follow the law of the cycloidal pendulum, and therefore in the upper medium we shall have,
$ = a sin (ax + by + ct) + /8sin (— ax + by (8).
Also, in the lower one,
<f>t = a, sin («,# + by + et): and as this is only a particular case of the more general one, hefore considered, the equation (7) will give
If <y, >7, or the velocity of transmission of a wave, be greater in the lower than in the upper medium, we may by decreasing a render at imaginary. This last result merely indicates that the form of our integral must be changed, and that as far as
« [Airy on Tkt Vndidatnry Theory of Optict, p. 1X1, Art. 128.]
This result is general for all fluids, but if we would apply it to those only which are usually called elastic, we have, because in this case <y'A — yfA„
a/A aV tan « = =* . oA, ay
and therefore, by substitution,
tan e ~' ay* = /* V/*» tan* 0 - sec1 0,
because — , and - = tan 0. 7 a
As e = — e,, we see from equation (9), that 26 is the change of phase which takes place in the reflected wave; and this is precisely the same value as that which belongs to light polarized perpendicularly to the plane of incidence; (Vide Airy'B Tracts, p. S62*.) We thus see, that not only the intensity of the reflected wave, but the change of phase also, when reflexion takes place at the surface of separation of two elastic media, is precisely the same as for light thus polariied.
* Airy, «5t tu/p. p. 114, Art. 133.
As o = /S, we see that when there is no transmitted wave the intensity of the reflected wave is precisely equal to that of the incident one. This is what might be expected: it is, however, noticed here because a most illustrious analyst has obtained a different result. (Poisson, MJmoires de VAcademie des Sciences, Tome X.) The result which this celebrated mathematician arrives at is, That at the moment the transmitted wave ceases to c exist, the intensity of the reflected becomes precisely equal to that of the incident wave. On increasing the angle of incidence this intensity again diminishes, until it vanish at a certain angle. On still farther increasing this angle the intensity continues to increase, and again becomes equal to that of the incident wave, when the angle of incidence becomes a right angle.
It may not be altogether uninteresting to examine the nature of the disturbance excited in that medium which has ceased to transmit a wave in the regular way. For this purpose, we will resume the expression,
<!>, = Be*'*sin f = Be*''sin (% + ct);
or if we substitute for B, its value given by the last of the equations (10); and for a', its value from (11); this expression, in the case of oidinary elastic fluids where 7*a = T*, A,, will reduce to
^, = 2a/is cos e. e~ * * * sin (by+ cl), \ being the length of the incident wave measured perpendicular to its own front, and 6 the angle of incidence. We thus see with what rapidity in the case of light, the disturbance diminishes as the depth x below the surface of separation of the two media increases; and also that the rate of diminution becomes less as 6 approaches the critical angle, and entirely ceases when 6 is exactly equal to this angle, and the transmission of a wave in the ordinary way becomes possible.