40 



WAVE ACOUSTICS 



tion; they are partly reflected in a direction depend- 

 ing on the direction of the surface and also partly 

 scattered in all directions. Neither does the bottom 

 obey the postulated conditions; it is never infinitely 

 dense; at best it is rocky; at worst it is so muddy 

 that it can hardly be called a boundary. The medium 

 itself, the sea water, is not completely described by 

 its density and its bulk modulus. There are many 

 inhomogeneities in the sea volume, such as bubbles, 

 floating plant and animal life, fish, and others. For 

 all we know, such inhomogeneities may produce a 

 very important part of the observed transmission 

 loss, perhaps as important a part as the variations in 

 soimd velocity. 



The mathematical difficulties should be apparent 

 to anyone who has even glanced at the remainder of 

 the chapter. Even when the boundary conditions can 

 be formulated exactly, and initial conditions are 

 simple, the exact solution of the problem usually can- 

 not be presented. In the general case, it can be proved 

 that a solution exists and is unique, but the solution 

 cannot be written in a formula which would provide 

 a practical basis for intensity calculations. The 

 primary benefit of the rigorous approach is that one 

 can derive certain very useful properties of the sound 

 field, such as the principle of reciprocity and the .de- 

 pendence of intensity on the phase distribution, with- 

 out going into the exact solution itself. 



How, then, are we going to predict the sound field 

 intensity? We certainly cannot go out and measure 

 the intensity in all cases; such measurements are 

 time-consuming, and provide information only about 

 that particular part of the ocean at that particular 

 time. We should have some method for estimating 

 the intensity field, at least qualitatively, so that the 

 observed intensity data can at least be interpreted 

 according to a frame of reference; mere data without 

 some reference to a theoretical scheme are useless. 



In the theory of light, this problem was solved by 

 using the methods of ray optics. The fundamental 

 problems about optical instruments, like those for tele- 

 scopes, can be solved by ray tracing methods without 

 resorting to the exact solution of the wave equation. 

 This ray theory is based on the assumption that light 

 energy is transmitted along curved paths, called rays, 

 which are straight lines in all parts of the medium 

 where the velocity of light is constant, and which 

 curve according to certain rules in parts where the 

 velocity of light is changing. This light-ray theory is 

 valid in all cases where obstacles and openings in the 

 path of the radiation are much greater in size than 

 the wavelength of light. 



In the next chapter, we shall describe the applica- 

 tion of ray methods to underwater sound transmis- 

 sion and shall also examine the validity of this type 

 of approximation. 



