SHADOW ZONE AND DIFFRACTION 



67 



If equaUon (113) Lsdivided tlirough by AY/, iiud equa- 

 tion (109) used for c, 



\G dxy IF 





]-' 



(114) 



-Fdy^ Co 



Since the first bracket depends only on x and the 

 second only on y, equation (115) can be satisfied only 

 if each bracket is constant. If we denote the first 

 bracket by — m", the second bracket must be -|-/i^ and 

 we have 



dy' 



+ 



[4^ 



(l + By) ~ ^'jF = (115) 



(116) 



S— • 



The basic problem is to find solutions of equations 

 (115) and (116) which satisfy the boimdary condi- 

 tions. First, we have the boundary conditions for 

 equation (115). In the analysis in Section 2.7.2, these 

 boundary conditions were that the pressure vanished 

 both at the surface and at the bottom. Here, also, 

 the pressure must vanish at the surface. However, 

 the water is so deep that the condition at the bottom 

 disappears. Instead, there is simply the condition 

 that at some distance below the projector no sound 

 is coming upwards ; that is, any sound present at these 

 depths is coming down from shallower depths. Al- 

 though this boundary condition is somewhat compli- 

 cated to formulate exactly, the general result is the 

 same as that foimd in the solution of equation (161) 

 of Chapter 2. In this earlier instance it was foimd 

 that sin 2-ry/Xy, corresponding to F{y) in equation 

 (112), when B is zero, satisfied the two boundary con- 

 ditions only if X„ had one of a number of fixed values. 

 Similarly, the function F{y) can satisfy the two bound- 

 ary conditions only if fi has one of a certain number 

 of values. These values, which are called characteris- 

 tic values of n, may be denoted by jui, A12, Ms, and so on, 

 or more generally by fij, where j can be any integral 

 number. For each of the characteristic values fij, 

 equation (115) has a particular solution Ffy) which 

 satisfies the boundary conditions. 



Once a value of m has been chosen, the solution of 

 equation (116) is very simple. For each value of m, 



G = Afi-'"'^ (117) 



where Aj is an arbitrary constant.* Thus the wave 



" The negative sign must be taken in the exponent so that 

 p, in equation (118) will correspond to a wave moving away 

 from the projector; that is, p, must be a function of lirjl — /jljx, 

 where /i, is positive. 



equation is satisfied by any product of the type 



Pi = Aie'^'^'Fi(y)e-'''f. (118) 



Equation (118) satisfies the boundary conditions at 

 the surface and at great depth since Fj(y) satisfies 

 these conditions. However, the boundary conditions 

 at the vertical plane x = 0, the assumed vertical 

 wall, must also be satisfied. These conditions are that 

 the particle velocity at the sound projector must be 

 Vo cos 2Trft, and that the particle velocity at all other 

 points in the plane x = must be zero. 



To satisfy this boundary condition at the plane 

 X = requires a combination of an infinite number of 

 possible solutions of the form (118). Each Aj must 

 be chosen in such a way that the sum has the re- 

 quired properties. Methods for doing this have been 

 developed, but are beyond the scope of this discus- 

 sion. However, the final result is that the pressure p 

 is the sum of many terms of the type (118) with 

 g2"'/< ^}ig Qjjiy common factor. 



Within the direct sound field a large number of 

 these terms are important, and an exact computation 

 is necessary to find p. In the shadow zone, on the 

 other hand, one term dominates, and the other terms 

 may be neglected. This is because all the m are partly 

 real, partly imaginary, with the result that the abso- 

 lute value of exp (injx) decreases exponentially for 

 sufficiently great values of x. It can be shown that 

 the range at which only one term dominates is ap- 

 proximately the range to the shadow boundary com- 

 puted from the ray theory. This dominant term is the 

 one for which ^ has the smallest imaginary part. 

 Thus, the theory predicts that in the shadow zone the 

 sound intensity falls off exponentially with increasing 

 range, or, in other words, that the predicted trans- 

 mission anomaly in the shadow zone increases line- 

 arly with increasing range. 



Although the exact determination of the different 

 characteristic values m is somewhat involved, it is 

 relatively simple to show how these values depend on 

 the frequency /, the velocity gradient, and the sound 

 velocity Co at the surface. This is useful since it indi- 

 cates how the attenuation into the shadow zone may 

 be expected to vary imder different conditions. In 

 order to investigate this dependence of Hj on the 

 other variables, we rewrite equation (115) in a simph- 

 fied dimensionless form. Let 



and 



4^ 

 cl 



-V? 



y = 



D 



(119) 

 (120) 



