VOLUME REVERBERATION 



255 



With customary receivers of ordinary dimensions, 

 the scattered sound returning from the ranges of 

 interest (say greater than 50 yd) is for all practical 

 purposes a plane wave at 0. Thus, using the results 

 (10) and (11), we have 



Watts output from dV = 



m 



—¥F ■ F'b{d,(l>)b'(e,<t>)dSds. 

 4ir 



(12) 



In equation (12), h' has been set equal to h. This as- 

 sumption that the transmission loss is the same on 

 the outgoing and returning journeys will be justified 

 on the basis of the laws of acoustics in Section 12.5.5. 

 In addition, b{d,<j)) and b'{B,4>) are very nearly equal 

 for most transducers. 



Using assumption 5 of Section 12.1, we next ob- 

 tain an expression for the average reverberation in- 

 tensity Git) at time t. Integrating equation (12) over 

 the volume between the surfaces S'l and S2 of Figure 2 

 gives 



G(t) 



F-F' rr 



4x J J 



mh^b{e,4,)b'{e,<t>)dSds. (13) 



In equation (13), the dependence on range is con- 

 tained principally in dS, h, and m; these quantities 

 also depend on the direction ^,0 at which the ray 

 which reaches a particular volume element leaves the 

 projector. However, equation (13) can be simplified 

 as follows. To a good approximation, the extension 

 of any ray between the surfaces S[ and S'2 can be con- 

 sidered equal to cot/2, where co, the average sound 

 velocity along the ray, is always only slightly dif- 

 ferent from the sound velocity at 0. Equation (13) can 

 therefore be rewritten as 



- Cot F-F' C 



Git) = ^^;^jm¥b{fi,4>)b'{e,4,)dS, (14) 



where the integral is to be evaluated on some average 

 surface perpendicular to all the rays. It can usually 

 be assumed that this representative surface is the 

 surface reached at time t/2 by the sound emitted at 

 midsignal; in Figure 2 this surface is labeled S^. 



This assumption for the surface of integration in 

 equation (14) will not be valid if the average value of 

 mWb{d,4>)b'{0,(j))dS along any ray does not occur 

 near 183. For example, this assumption fails when the 

 ping length is not small compared to the range of the 

 reverberation. By using the sirnplifjdng assumption 

 that the sound intensity decays inversely as the 

 square of the distance, it is easy to show directly 

 from equation (13) that at close range (< not much 



greater than t/2), the reverberation intensity may 

 not be regarded as proportional to Cot/2; rather it is 

 proportional to the factor 



"2" 



1 



cg<^- 



c?r^ 



(15) 



Another situation for which the average value of 

 mh^b'dS may not occur near S3 occurs when the 

 rays are curving very sharply. For most oceano- 

 graphic conditions, this error introduced by ray 

 bending is not appreciable. However, when the layer 

 effect discussed in Section 5.3 is present, the error 

 might be significant. In that oceanographic situa- 

 tion, the ping travels out of an isothermal layer into 

 an underlying region of sharp temperature gradient; 

 and the sound scattered from parts of S[S2 below the 

 isothermal layer has a higher transmission loss to 

 the transducer than sound scattered from above the 

 layer. 



Although equation (14) cannot be used as it stands 

 for the calculation of volume reverberation levels, it 

 nevertheless is significant. It implies that irrespective 

 of the directivity pattern of the transducer, and of the 

 oceanographic conditions, the average intensity of 

 the received volume reverberation should be propor- 

 tional to the ping length. This important conclusion 

 is based, of course, on the various assumptions made 

 previously. 



In equation (14), write dS = {dS/dQ)d^, where d^ 

 is the element of sohd angle in the direction (&,</>). 

 Then equation (14) can be further simpUfied if it is 

 assumed that the transmission loss in the ocean de- 

 pends only on the distance traversed by the ray en- 

 tering or leaving the transducer, and not at all on 

 the direction of the ray. Then h and dS/dQ are inde- 

 pendent of {9,<t>), and equation (14) can be written as 



G{t) = T^^^ fmb{d,<t>)b'(e,<t>)dQ. (16) 

 2 iir dQJ 



The term dS/dQ is placed in front of the integral sign 

 in equation (16) because it is a measure of the trans- 

 mission loss due to refraction; dS/dQ is, in fact, just 

 the reciprocal of the intensity diminution due to 

 normal inverse square divergence plus refraction, ac- 

 cording to Chapter 3. 



Finally, if it is assumed that scattering in the ocean 

 is independent of the initial ray direction {d,<t>), the 

 backward scattering coefficient m can also be re- 

 moved from under the integral sign. This yields as 

 our end result for the average reverberatioii intensity 



