EXPLICIT AND TACIT ASSUMPTIONS 



269 



place in the ocean. Thus, the vaUdity of this assump- 

 tion, like that of all the other assumptions we have 

 made, depends on the properties of the scatterers in 

 the ocean. 



In considering the validity of this assumption, we 

 may confine our attention to multiple scattering in 

 the body of the ocean, since there is little likelihood 

 of direct multiple scattering from one surface scat- 

 terer or bottom scatterer to another. Experiments on 

 volume reverberation ' have shown that at short 

 ranges (up to a few hundred yards) multiple scat- 

 tering in the body of the ocean probably can be neg- 

 lected. If scattering in the body of the ocean is 

 isotropic, that is, if the backward volume-scattering 

 coefficient m is really the average amount of energy 

 scattered in all directions per imit intensity per unit 

 volume, then it can be concluded from the magnitude 

 of m that multiple scattering is certainly negligible 

 at all ranges of interest in echo ranging. 



However, it is possible that forward scattering in 

 the ocean is appreciably greater than backward 

 scattering. It has been suggested that the high at- 

 tenuation of sound in the ocean at supersonic fre- 

 quencies results from forward scattering of sound by 

 the temperature microstructure in the ocean. On 

 present evidence, it seems unlikely that forward 

 scattering alone can account for attenuation in the 

 ocean,'' but if appreciable wide-angle scattering does 

 occur, then at long ranges the neglect of multiple 

 scattering in the theoretical reverberation formulas 

 is not justified. The predicted dependence of volume 

 reverberation on range [equation (24)] would be 

 changed if volume reverberation contained much 

 multiply scattered sound. The evidence discussed in 

 Chapter 14 suggests that multiple scattering in the 

 ocean probably can be neglected at ranges of opera- 

 tional interest in echo ranging. However, more evi- 

 dence is needed before any definite conclusions can 

 be reached. 



12.5.5 Fermat's Principle and the 

 Principle of Reciprocity 



It was important to show that multiple scattering 

 can be neglected in the computation of reverberation 

 intensity since that assumption enabled us to deline- 

 ate the volume S1S2 in Figure 1 within which ap- 

 preciable scattering is taking place. The determina- 

 tion of volume S'iS'2 (Figure 2) was then based on an 

 application of Fermat's principle. Fermat's principle 



*> This point is discussed in Section 5.4.1. 



is a theorem about the properties of equation (1); 

 it states that when a sound travels between two given 

 points, it always follows a path such that its travel 

 time is a maximum or a minimum." This maximum 

 or minimum value is the same no matter which of the 

 two points is the starting point and which the finish- 

 ing point. Thus, provided the refraction conditions 

 and boundary conditions are not changing with time, 

 the ray paths and travel times from the transducer 

 out to a scatterer, and from the scatterer back to the 

 transducer, are exactly the same. However, refraction 

 and boimdary conditions in the ocean are not con- 

 stant with time. The existence of thermal fluctua- 

 tions, and the fact that BT patterns vary from one 

 hour to the next, show that refraction conditions 

 change with time; and surface waves are an example 

 of changing boundary conditions. 



Complete elucidation of the effect of these chang- 

 ing refraction and boundary conditions would be 

 highly complicated, and, as usual, lack of information 

 about the scatterers would make it difficult to be 

 precise. However, it seems justifiable to assume that 

 short-term fluctuations in thermal microstructure, 

 or such variations in boundary conditions as waves 

 or random movements of the scatterers, do not 

 modify the average equality of the travel times and 

 ray paths. Moreover, the long-term variations evi- 

 dent on the BT trace are too slow to affect the aver- 

 age equality in a series of pings lasting about a 

 minute. Thus, it appears that the use of Fermat's 

 principle was justifiable in Section 12.2, in the 

 delineation of the effective scattering volume S'iS'2. 



Another theorem about the properties of equation 

 (1) is the principle of reciprocity. In the ocean, trans- 

 mission loss is thought to be a combination of ordi- 

 nary inverse square spreading, refraction, absorp- 

 tion, and scattering. According to the principle of rec- 

 iprocity,' if the refraction and boundary conditions 

 are not changing with time, then that part of the 

 transmission loss which is due to inverse square 

 spreading and refraction will be the same for trans- 

 mission from a nondirectional projector at to the 

 point X.as for transmission from a nondirectional 

 projector at X to 0. Of course, the source at (the 

 echo-ranging projector) is not nondirectional; and 

 the source at X (the scatterer) may not be nondirec- 

 tional, since scattering is not necessarily the same in 

 all directions. For directional sound sources, the 

 reciprocity theorem requires modification, because of 

 the possibiUty of reflections and multiple paths be- 

 tween and X. However, along any definite ray path 



