256 



THEORY OF REVERBERATION INTENSITY 



(?(<) = 



CoT F-FVm dS 

 2 4x dQ 



Jm 



4>Wie,,f>)da. (17) 



These latter assumptions are not always realistic. 

 The assumption that transmission loss in the ocean 

 depends only on the range and oceanographic condi- 

 tions, and not at all on the initial ray direction, is 

 probably a poor one for volume reverberation in a 

 nondirectional transducer, because transmission loss 

 in a vertical direction may differ appreciably from 

 transmission loss in a horizontal direction. Even in 

 a highly directional transducer of the sort used by 

 the Navy for echo ranging at 24 kc, this assumption 

 may be in error, since at long range rays leaving the 

 projector only a few degrees apart may travel along 

 widely separated paths; such a divergence occurs, for 

 example, when the refraction theory predicts a split- 

 ting of the beam. Moreover, when spht beams occur, 

 TO will not be independent of (6,0) if the scattering 

 strength of the ocean is not independent of depth. 

 For, if the overall scattering strength of the sea is 

 not the same at all depths, then a pair of rays which 

 become widely separated by the prevaihng refraction 

 may reach portions of the ocean with different scat- 

 tering strengths; in such a case m, evidently will de- 

 pend on {d,cj)). Of course m may always be regarded 

 as an average over the entire volume of the ping, and 

 thereby may be removed from under the integral 

 sign in equation (16). But if the scattering strength 

 and transmission loss in the ocean really vary with 

 angle within the main transducer beam, this type of 

 averaging procediu-e makes the value of to depend on 

 the directivity pattern of the gear; in this event re- 

 moving TO from under the integral sign has little 

 significance. 



In equation (17) the dependence of reverberation 

 on the directivity pattern of the gear is contained 

 wholly in the integral, which can be evaluated from 

 the known directivity patterns of the transducer as a 

 projector and a receiver. If the transmission loss 

 obeys the inverse square law, that is, if the losses due 

 to refraction, absorption, and scattering are neg- 

 lected, then 



r'' dQ 

 and equation (17) becomes 



G(t) = v^-2 fHe,4>Wi6,<t>)dn, (18) 



where r, the range of the reverberation, is equal to 

 Cot/2. In this ideal case, then, the average intensity 



of the received volume reverberation varies inversely 

 as the square of the time following midsignal. 



In general, however, the ocean is far from ideal and 

 this simple law would not apply. To compare the 

 observed time variation of the received reverberation 

 in the general case with that predicted by the ideal 

 formula (18), the general formula (17) is written as 



or, in decibels 



101ogG(<) = 10 log (^yj + 10 log [F-F') 



+ 10 log TO - 20 log r -f J, -h 20 log (r^h) 



-lOlogr^^, (20) 



where 



J. = 10 log — fbid,<t,)b'ie,<l>)dQ. (21) 



4:ir J 



The transmission anomaly A is defined (see Section 

 3.4.1) by 



A = H -20log r. 



By comparing with equation (8), it is evident that 



A = -101og(?-2/i). 



By substituting this expression for A into equa- 

 tion (20) 



10 log G(i) = 10 log (^^) + 10 log (F ■ F') 



+ IGlogTO - 20 log r + J^-2A+ Ai (22) 



where — 10 log r'^{dQ/dS), the transmission anomaly 

 which would result if the normal inverse square 

 divergence were disturbed only by the effect of re- 

 fraction, has been replaced by Ai. It is apparent 

 from the preceding discussion that the quantities 

 A, Ai, and to in equation (22) must be interpreted 

 as averages over that portion of the effective scat- 

 tering volimie which Ues within the main transducer 

 beam. 



The quantity Ai, which depends on refraction 

 alone, cannot be measured directly. In principle, Ai 

 could be computed from the known temperature 

 structure of the sea, according to the methods out- 

 lined in Section 3.4. However, this computation is 

 difficult, frequently inaccurate, and often totally im- 

 practical because the observed bathythermograph 

 [BT] pattern may not extend to a sufficiently great 



