REFRACTION OF SOUND. 405 



reflected wave is here shown to be such that, at the moment tlie re- 

 fracted wave disappears, the intensity of the reflected becomes exactly 

 equal to that of the incident one. If we moreover suppose the vibra- 

 tions 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 here given for all media, elastic or 

 non-elastic. When, however, both the media are elastic, it is reuiarkable 

 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 quan- 

 tity 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 (yss), and let the axis of % be parallel to the line of inter- 

 section 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 

 Ai are the densities of the two media under the constant pressure P and 

 s, A'l the condensations, we must have 



|A (1 + .y) = density in the upper medium, 

 IA,(1 +*,) = density in the lower medium. 



|P(1 + As) = pressure in the upper medium, 

 1^(1 + A^s^) = pressure in the lower medium. 



Also, as usual, let (p be such a function of x, y, x, that the resolved 

 parts of the velocity of any fluid particle parallel to the axes, may be 

 represented by 



dcp d<p d(p 



dx ' dy ' dx ' 



In the particular case, here considered, (p will be independent of z, and 

 the general equations of motion in the upper fluid will be 



3 F 2 



