264 



THEORY OF REVERBERATION INTENSITY 



Rit) = 10 log ( Y j + 10 log {^) - 30 log r 



+ JM - 2A'. (51) 



In other words, if 4' is used instead of A in equation 

 (45), then the proper surface reverberation index to 

 use is Js{0) rather than Jg{6). The same rule applies, 

 of course, in the evaluation of equation (39) or equa- 

 tion (43) 



12.4 



BOTTOM REVERBERATION 



Bottom reverberation is defined as reverberation 

 arising from sound scattered back to the tranducer 

 by scattering centers in the ocean bottom. These 

 scattering centers are thought to usually lie in a very 

 thin layer on the bottom. Thus the formulas for the 

 dependence on range of bottom reverberation will be 

 similar to those formulas for surface reverberation, 

 and can be derived from them by simple changes of 

 notation. We have then, from formulas (39) through 

 (45), 



10 log Git) = 10 log ^y j + 10 log (F-F') 



+ 10 log ^^)- 30 log r (52) 



+ Jeie) - 2A, 



1 r'' 



with Jeie) = 10 log — I h(e,4>)h'{e,4>)d<i> (53) 

 27r Jo 



R'{t) = 101ogG(<) - lOlog (F-F') 



= 10 log yjj + 10 log (^y) - 30 log r 



+ Jeie) - 2A. (54) 



To 



Rit) = 10Iog(5(<) - 10 log iF-F') + 10 log 



T 



= 101og(^°) + 101og(^)-301ogr 



+ Jsie) - 2A. (55) 



In these formulas w" is the backward scattering 

 coefficient of the bottom per unit area of the bottom. 

 The coordinate system in equation (53) is similar to 

 that in Figure 5; however, the transducer is usually 

 directed downward instead of upward as in Figure 5. 

 Equation (41) may be used to evaluate Jaid) in 

 equation (53); and as before JaiO) should be re- 

 placed by JsiO) if the transmission anomaly as 

 usually measured is used instead of the actual trans- 



mission anomaly along the ray path. The quantity 

 m" is in general a function of the range since it 

 probably depends on the angle of incidence of the ray 

 on the bottom. 



It should be noted that the similarity of the 

 formulas for bottom and surface reverberation does 

 not imply that bottom and surface reverberation 

 arise from similar mechanisms. The bottom scatter- 

 ing originates in irregularities in bottom contour; 

 these irregularities may vary from the fine separa- 

 tions between grains of sand to such macroscopic 

 irregularities as large rocks and underwater cliffs and 

 valleys. 



12.5 EXPIJCIT AND TACIT ASSUMPTIONS 



The preceding derivations of the theoretical for- 

 mulas for volume, surface, and bottom reverberation 

 were based on many assumptions, not all of which 

 were stated explicitly. In this section we shall discuss 

 briefly the significance and probable validity of these 

 assumptions. Because of present uncertainties re- 

 garding scattering sources and the infinite complexity 

 of the ocean, this discussion is partly qualitative and 

 not clear-cut. 



It was pointed out in Section 12.1 that though 

 equation (1) governs the propagation of reverbera- 

 tion in the ocean, a complete solution of equation (1) 

 for the reverberation received under given conditions 

 is not obtainable. However, certain general properties 

 of solutions of equation (1) are known, and will be of 

 use here. 



The strength of an acoustic disturbance can be ex- 

 pressed either in terms of pressure amplitude or sound 

 intensity. In practical appUcations, the sound in- 

 tensity is a more convenient quantity; while in 

 theoretical discussions based on the wave equation 

 (1) the acoustic pressure is more convenient. In equa- 

 tion (1), the sound intensity does not appear ex- 

 plicitly; in fact, it is impossible to derive a simple 

 differential equation, whose dependent variable is 

 the sound intensity, which like equation (1) ex- 

 presses the fact that the disturbance is a wave 

 traveling through the ocean with the velocity c. In 

 discussing the implications of equation (1), we shall, 

 therefore, be directly concerned with the soimd pres- 

 sure p. To tie in the discussion with the preceding 

 sections of the chapter, we must relate the sound in- 

 tensity to the sound pressure and also to the voltage 

 generated across the terminals of the receiving circuit. 



To simpUfy the discussion, we shall assume, to be- 



