CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 



57 



VELOCITY 



PROJECTOR DEPTH 

 



I ^. 



HYDROPHONE DEPTH 



\ \ 



Figure 19. Ray path in succession of linear gradients. 



since 6^ — 6^ has the value 2{a/co)h2 by equation (81). 

 Similarly, we have 



Jo e^ eg Jo (^ 



dy' 



2ia/co)y'y 



_ A, (do - dh) 

 ~ ^ {a/co)do9h 



_ ^ J 1 hi 



^ 6o6h 2(^0 -\- dk) 



Substituting these expressions into equation (66), we 

 find for the transmission anomaly 



1 h 



A =10 log 





+ 



do9k 2(^0 -\- 6h) 



)] 



= ''''&i' + tm^))\' 



(82) 



since from equation (37), 



Oo 2(^0 + &h) 



If the gradient is sharp, and if the range is con- 

 siderable, the angle 6q will generally be small com- 

 pared with the angle at the hydrophone, dh. Also, if 

 the hydrophone is not too far down, we may assume 

 that the fraction h^/hi is not too large. In that case, 

 the second term in the parentheses in both equations 

 (80) and (82) is small compared with unity and may 

 be omitted as negligible. In either case we have, then, 

 as a rough estimate of layer effect, the simple rela- 

 tionship 



10 logl 



(83) 



Combination of Linear Gradients 



In this subsection we shall derive a formula for the 

 intensity along a ray which has passed through a suc- 

 cession of layers in each of which the sound velocity 

 changes linearly with depth in the layer. This con- 

 dition is of considerable practical importance since 

 most velocity-depth curves can be replaced in good 

 approximation by a number of linear gradients. 



The assumed velocity-depth pattern is shown in 

 Figure 19. There are n + 1 layers, labeled 0, 1, 2, 3, 

 • • ■ n, in which the velocity gradients are ao, ai, - • • a„ 

 respectively; the term Ui represents the velocity 

 change in the I'th layer in velocity units per foot of 

 depth increase in the layer labeled i, where i takes 

 any integral value from 1 to n. The velocity at pro- 

 jector depth is Co," at the top of layer 1, Ci; at the top of 

 layer 2, c^; and so on, as indicated in Figure 19. The 

 ray direction is ^0 at the projector, 61 at both the 

 bottom of layer and the top of layer 1, and so on; 

 and, finally, 5„ at the bottom of the (n + 1) layer, 

 which is assumed to be the depth of the receiving 

 hydrophone. The total horizontal range covered by 

 the ray is R; the component of horizontal range 

 covered in each layer is designated by J?o, ^i, • • • Rn, as 

 indicated. 



We shall compute the intensity at the range R by 

 means of the formula (71), which is generally ap- 

 plicable. To use this formula, we must first derive 

 an explicit expression for dR/ddo in terms of param- 



