SECT. 4] 



SOUND IN THE SEA 



481 



forces. This property is commonly found, but not everywhere. Where the sea- 

 bed consists of sohd rock a more general treatment is required (e.g. see Ewing, 

 Jardetzky and Press, 1957). 



If p, c and pi, ci denote the density and velocity of propagation in the sea 

 and in the ground below (Fig. I), for the present assumed homogeneous and of 



Fig. 1. Reflection of sound at boundary between elastic media. 



infinite extent, a wave, p, of phase a>t + kz cos d — ky sin d incident at an angle 6 

 to the normal is in general split into two waves. One, pr, of phase cot — kz cos 6 — 

 ky sin Q, is reflected at an angle 6 on the other side of the normal ; the other, pt, 

 of phase a>^-|- A:i2; cos ^i — ^'ly sin ^i, enters -the ground at an angle di, where 

 (Rayleigh, 1878), 



(14) 



(15) 



:i6) 



As in general Ci is greater than c, the transmitted wave is deflected slightly 

 away from the normal until a critical value, sin-i (c/ci), of 6 is reached which 

 makes sin di unity, after which there is total reflection. In these circumstances 

 (15) takes the form 



Pr 

 V 



cos ^ + 7 1 



pC •^\ C2 



plCl 



pc 



COS 6 



Jci^ sin^ d 



:i7) 



Thus Pr has the same amplitude as p but its phase is 



cot — kz cos d — ky sin 6 + 2x, 

 where 



[(ci/c)2sin2^-l]V'2 



tan Y = 



(piCi/pc) cos 6 



(18) 



