252 



THEORY OF REVERBERATION INTENSITY 



will be seen later that the function g{t) is zero every- 

 where in the ocean except inside a thin, roughly 

 spherical shell; this shell has the projector as its 

 center, an average radius depending on the value of t, 

 and a thickness depending on the length of the 

 emitted signal. 



With these assumptions, it is possible to derive 

 theoretical formulas for the reverberation intensity 

 as a function of gear parameters and oceanographic 

 conditions. This chapter will be concerned only with 

 the average reverberation intensity to be expected in 

 a series of pings. No attempt will be made in this 

 volume to predict the level of the reverberation from 

 one specified ping. The observed levels of the rever- 

 beration from pings only a few seconds apart often 

 differ by many decibels. Discussion of the average 

 value of this fluctuation will be deferred until 

 Chapter 16. 



Although the above assumptions can be defended, 

 they are by no means obvious and require elabora- 

 tion. In particular, it is necessary to specify carefully 

 the meaning of the terms "backward scattering 

 coefficient" and "average reverberation intensity," 

 which are introduced in assumptions 4 and 5. The 

 average reverberation intensity is defined as the 

 average from ping to ping. That is, if we measure 

 the reverberation intensity on a succession of n pings 

 with each measurement performed at a definite time t 

 after midsignal, then the average reverberation in- 

 tensity at time t is 





(3) 



where Gi{t) is the reverberation intensity measured 

 on the z'th ping; the symbol S means summation over 

 all the pings. 



The number of pings averaged must be large 

 enough to smooth out the effects of fluctuation, yet 

 not so large that such external factors as wave height, 

 water depth, and amount of suspended matter in the 

 ocean can change materially during the series of 

 pings. In practice, the number of pings averaged has 

 usually been between 5 and 12, with not more than 

 about 60 seconds between the first ping and the last. 

 Some discussion of the validity of this averaging 

 procedure is given in Chapter 16. 



Also, we must specify more exactly the meaning 

 of the backward scattering coefficient m. If we con- 

 sider a volume V made up of many small volume 

 elements dV, then, strictly speaking, dV can scatter 

 sound only if sound energy reaches it and if it con- 



tains some scattering substance. Thus, if dV lies en- 

 tirely within some rigid scatterer, such as a bit of 

 metallic dust, practically no sound reaches dV be- 

 cause almost all the sound impinging on the scatterer 

 is scattered at the surface of the scatterer. Another 

 difficulty is that there is no way to predict the loca- 

 tions of the scatterers on any one ping. For these 

 reasons it is impossible to predict how much scat- 

 tering from a specified voliune element dV will occur 

 on any one ping. We can, however, speak of the 

 average scattering power of the ocean in the neigh- 

 borhood of dV. The backward scattering coefficient 

 m for a volume V, in the neighborhood of and in- 

 cluding dV, is defined as follows. Let V be insonified 

 by a plane wave of unit intensity n times in succes- 

 sion. Let 6, be the energy scattered per second per 

 unit solid angle in the backward direction, during 

 the ith trial, by the volume V. Then m for V is 

 defined by 



m 



4:TrJ n i 



(4) 



The factor 47r is introduced so that in cases where 

 the scattering is the same in all directions, the average 

 amoimt of energy scattered per second in all direc- 

 tions will be just mV. With the definition of m given 

 by equation (4) that the average energy scattered by 

 dV per second per imit incident intensity per unit 

 solid angle in the background direction is just 

 (m/4x)dF, it also follows that (m/4ir)dF is just the 

 intensity of the scattered sound from dF at imit 

 distance from dV when the incident sound has unit 

 intensity. 



Evidently the volume F in equation (4) cannot be 

 chosen arbitrarily if the definition of m is to have any 

 significance. F must be chosen small enough that m 

 can vary with position in the ocean and can thereby 

 indicate the variation with position of the average 

 number and strength of the scatterers. However, 

 since it is desired that m not vary discontinuously 

 from point to point, V must not be chosen too small. 

 Because so little is known about the scatterers re- 

 sponsible for reverberation, it is difficult to formulate 

 the conditions on F any more precisely than this. 

 Some further discussion of the significance of as- 

 sumption 4, as well as the other previous assump- 

 tions, is given in Section 12.5. That section is not a 

 complete treatment of the problems involved, but 

 may assist the reader to understand the physical 

 ideas underlying the derivation of the theoretical 

 formulas for reverberation. 



