220 DYNAMICAL THEOKY OF SOUND 



front. Let P l be any point on Si, and P' the corresponding 

 point on S f , so that P-f is the path of a particle of the 

 medium in the time Bt. On the principles of optics, the new 

 position $ 2 of the wave-front is obtained as the envelope of 

 a system of spheres of radius c$t, described with the various 

 points P' of S' as centres. If P 2 be that point on the 

 envelope which corresponds to P', P^P Z will be an element of 

 a ray, and P'P Z an element of the wave-normal. Also since 

 Pf U8t, where U is the velocity of the medium, the " ray- 

 velocity" (PiPJfo) is the resultant of the wave- velocity and 

 the velocity of the medium. 



In the present question the velocity U is horizontal, and 

 a function of the altitude (y) only. If i/r, < denote the 

 inclinations to the horizontal of the ray and the wave-normal, 

 respectively, we have 



n V9 T)~ c s Q r> () 



or <f> = T|T H simjr, (2) 



c 



if U/c be small, as will usually be the case. 



To ascertain the law governing 

 the change of direction of the ray, 

 consider first the case of refraction at 

 the common horizontal boundary of 

 two uniform currents U, U'. If <, <' 

 be the inclinations of the wave-normal 

 on the two sides of the plane of 

 discontinuity, we have 



c sec < 4- U = c sec </>' + U', (3) 



each side expressing the horizontal velocity of the trace of 

 the wave-front on the plane in question. Since a continuous 

 variation of U can be approximated to by a series of small 

 discontinuities, we infer that (3) will still hold if <, U and 

 <f>', U' refer to any two positions on the same ray. This gives 

 the altered law of refraction. Lord Rayleigh points out that 

 since sec $ <%. 1, <f>' will become imaginary if 



(U'- Z7)/c> sec </> - 1 (4) 



