484 



VIGOURETJX AND HERSEY 



[chap. 12 



Since the relation between oj and 6 is not simi3le, calculation of U from (23), 

 (22) and (18) is complicated. Pekeris (1948) shows that U is equal to ci at the 

 frequency corresponding to the critical angle ; the phase velocity, V, is also C\ 

 for that angle. As 6 increases, U decreases more rapidly than V , goes through c, 

 and reaches a minimum, after which it tends towards c sin d ; at grazing in- 

 cidence, i.e. at infinite frequency, V and U are both equal to c. 



O. Velocity Gradients 



Since salinity and temperature of the sea can vary with depth as well as 

 locality, the velocity of propagation of sound varies with depth, as can be seen 

 from (8), not only because of the change of pressure but also because of changes 

 in salinity and temperature. The component of gradient of velocity in a direc- 

 tion perpendicular to the ray causes the ray to bend away from the region of 

 high velocity towards that of low velocity ; thus, as a rule, sound rays in the sea 

 are not straight but curved. As, however, the gradients are small, the curva- 

 tures are small, and in shallow water it may not be necessary to take them into 



VELOCITY 



RANGE 



SURFACE 



I 

 I- 



Q. ^ 

 UJt 

 D 



Fig. 2. Sound rays from wide-angle projector when velocity decreases with depth 



account ; but in the deep ocean, where long ranges are reached before a ray 

 arrives at the sea-bed, the angle of incidence can differ considerably from that 

 which the ray makes with the vertical at the source of sound, and this effect 

 must be allowed for when calculating the intensity as a function of range or 

 depth. In these calculations it is in general permissible to assume that over 

 the ranges of interest the velocity is everywhere the same at the same depth, so 

 that the variation with depth is the same everywhere. In such circumstances 

 repeated application of Snell's law (14) to slices bounded by horizontal planes 

 shows that if c is the velocity and 6 the angle which a ray makes with the 

 vertical, and Co, ^o the values of these quantities at some starting point, e.g. 

 at the source of sound, 



c cosec 6 = cq cosec ^o- (24) 



It is thus possible, by proceeding in steps, to trace a ray leaving the source 

 at any angle with the vertical. 



It is instructive to consider in the first instance two cases of propagation 

 affected by velocity. The first is the case when velocity decreases with depth 

 part of the way and afterwards remains constant right to the bottom. In this 

 case (Fig. 2) the rays are eventually bent downwards and, apart from some 



