66 



RAY ACOUSTICS 



predicts a discontinuous drop of intensity from a 

 finite value on one side to a zero value on the other. 

 It was shown in Section 3.6 that the ray theory can- 

 not be trusted whenever it predicts such a rapid 

 change of intensity in a distance of only a few wave- 

 lengths. Thus, it is necessary to use the wave equa- 

 tion directly to compute the intensity of sound which 

 penetrates the so-called shadow zone. 



The simplest case of a shadow zone is that pro- 

 duced by a screen in front of a light source. As shown 

 in Figure 26, the ray theory predicts that no light 



SOURCE 





/ / 



. / / / 



\\ 



'' SHADOW ZONE \^, 



\ 



/ / 



/ / 



/ / 

 / / 





\ \ 

 \ \ 



FiGUKE 26. Optical shadow zone produced by screen. 



can reach the shadow zone behind the screen. When 

 the rays carrying the energy are curved, as in Figiu-e 

 24, it is the surface of the ocean that intercepts the 

 curved rays and "casts a shadow." In either case, 

 however, some energy actually appears inside the 

 predicted shadow zone, and the wave is said to be 

 "diffracted." 



The computation of diffracted sound in the shadow 

 zone is a rather complicated problem in the general 

 case. To indicate the type of analysis required, and to 

 show the general nature of the results, a simplified 

 problem will be considered here. As shown in Figure 

 27, a sound projector is assmned to be placed against 

 a vertical wall, which extends down to great depths. 

 The introduction of the wall simplifies the problem 

 without changing the final results essentially. The 

 water is assumed to be so deep that bottom-reflected 

 sound may be neglected. The projector face is as- 

 sumed to be so wide that the horizontal spreading of 

 the sound beam may be neglected; thus, only the 

 two-dimensional problems need be considered. The 

 sound velocity c is assumed to vary according to the 

 law 



c- = 



cl 



1+By 



(109) 



where B is a constant, and y represents depth below 

 the surface. Since B is in practice very small, this 

 gradient is indistinguisliable from a linear gradient 

 at depths of interest. The exact velocity gradient at 

 the depth y is given by 



dc cl B 



dy^ ~2c{l+Byr' 



Thus, at the surface, where y = 0, the velocity 

 gradient —5 is given by 



-6 = ^=-^^ ■ (111) 



2 dy y^o 



The gradient (109) is chosen instead of a simple 

 linear gradient not for physical reasons, but because 

 it simplifies the following computations. 



(110) 



\ PROJECTOR 



VELOCITY-^ 



Figure 27. Sound shadow cast by sea surface. 



To solve the wave equation imder these conditions, 

 it is necessary to use the method of normal modes 

 developed in Chapter 2. In particular, we must find a 

 solution to the wave equation (27) in Chapter 2 which 

 satisfies the boundary conditions we shall impose. As 

 in Section 2.7.2, we look for a solution which is the 

 product of three functions, one dependent only on 

 the time t, another dependent on the depth y, and the 

 third, a function of the horizontal distance x. The 

 coordinate z need not be considered in the two- 

 dimensional case under discussion. 



Following the analysis of Section 2.7.2, we there- 

 fore write 



p(x,y,z,t)=e'''f'F(y)Gix). (112) 



By substitution of equation (112) into the wave equa- 

 tion (27) of Chapter 2, and by dividing through by 

 C^, it is found that F and G satisfy an equation of the 

 form 



G--, + F-- + -^FG = 0. 

 dy^ dx^ & 



(113) 



