VOLUME REVERBERATION 



257 



depth. Alternatively, Ai could be inferred from the 

 observed transmission loss, if the losses due to at- 

 tenuation and scattering were known. However, it is 

 clear from the discussion in Chapter 5 that these 

 losses are also uncertain. 



Equation (22) is the theoretical expression for the 

 average intensity of received volume reverberation 

 and is the one usually used in the comparison of 

 theory with observation. Also, the computation of the 

 scattering coefficient m from observed reverberation 

 intensities is usually done by the use of this equation. 

 It will be remembered that equation (22) was de- 

 rived on the assumption that the transducer was 

 infinitely far away from the ocean surface or ocean 

 bottom, and therefore that sound traveled from to 

 X on only one path. In actual measiu-ements, the 

 transducer is always near the ocean surface and may 

 be near the bottom as well. The presence of these 

 surfaces increases the number of paths by which 

 scattered energy from any point in the ocean can 

 reach the transducer. Therefore, equation (22) will 

 give erroneously low values for the reverberation 

 intensity if alternative paths from to X, of 

 very nearly equal travel time, exist in the ocean. 

 In the following paragraphs, we shall consider the 

 error in equation (22) caused by the existence of 

 such alternative paths. It should be stressed that we 

 are considering here only the increase of volume rever- 

 beration due to these additional paths. Surface or 

 bottom reverberation will result from scattering when 

 the soimd impinges on one of the bounding surfaces, 

 but we are interested here only in the reverberation 

 resulting from the scattering of sound by the volume 

 elements in the interior of the ocean. 



Possible combinations of alternative paths are 

 pictured in Figure 3. If the ocean surface is calm, 

 the case of Figure 3A, energy will reach the point 

 X from not only along the direct path OBX, but 

 also along the path OAX as a result of specular 

 (mirror-like) reflection from the ocean surface at A. 

 If the ocean surface is rough, however, energy may 

 reach X from along a large, perhaps infinite, num- 

 ber of paths, as indicated in Figure 3B. Because of 

 the principle of reciprocity, the energy returning 

 from X to also travels along these additional paths. 



The existence of these extra paths tends to increase 

 the reverberation intensity received at at time t. 

 To estimate the amount of increase, we note that for 

 every possible path from to Z and back, there will 

 exist an effective scattering volume of the type of 

 S'iS'2 in Figure 2, boimded by two closed surfaces from 



which scattered energy traveling along that path re- 

 turns to at time t. In Figure 3A, illustrating specu- 

 lar reflection, there are four such volumes. One is 



A REFLECTION FROM MIRROR SURFACE 



B REFLECTION FROM ROUGH SURFACE 



C REFLECTION FROM ROUGH SURFACE 

 AND ROUGH SEA BOTTOM 



Figure 3. 

 soatterer. 



Alternative paths from transducer to 



S'iS2 defined in preceding paragraphs, corresponding 

 to the path OBXBO. The others correspond respec- 

 tively to the paths OAXBO, OBXAO, and OAXAO. 

 The volumes corresponding to the paths OAXBO and 

 OBXAO are identical, because the travel time does 

 not depend on the direction of travel along the ray; 

 but the volume corresponding to OAXBO, the vol- 

 ume corresponding to OAXAO, and the volmne S'iS'2 

 corresponding ^to OBXBO are in general all different. 

 For each of these volumes there will be an integral 

 similar to that of equation (13), expressing the con- 

 tribution of the volume to 0(1). Each such integral 

 can be simplified to a surface integral multiplied by 

 Cor/2, as in equation (14). It follows that the average 

 intensity of the volume reverberation should be 

 proportional to the ping length, regardless of whether 



