260 



THEORY OF REVERBERATION INTENSITY 



to the volume elements at Y without hitting the 

 surface. 



To evaluate the expression (28), we must first 

 specify the volume element dV. It is not convenient 

 to define the element dV in the same way as was done 

 for volume reverberation. Instead, we set up a 

 cylindrical coordinate system {p,4>,z) whose axis OP 

 is the vertical line through the transducer, as in 

 Figure 5; and we define dF as the infinitesimal vol- 

 ume lying between p and p + dp, </> and <p + d(t>, 

 z and z + dz. Then the value of rfF is 



dV = p d(iidzdp. 



We integrate over the intersection of the volume 

 S'iS'2 with the surface scattering layer, which is as- 

 sumed to have depth d. This volume is an annulus 

 (ring-shaped figure) determined by z varying from 

 zero to d, 4> varying from to 2ir, and p varying from 

 S'l to (S2. Then equation (28) becomes 



G(t) = -^— \ dz\ d4,\ pmh?hh'dp, (29) 

 47r Jo Jo Jsi' 



where p is integrated from S'l to S'^. In the ocean, it 

 can be assumed that on the average sound rays are 

 bent only in a vertical direction. Then, the distance 

 in the p direction from S[ to S'2 is independent of the 

 polar angle <i>, but may depend on the depth z. 



In order to put equation (29) in a form suitable for 

 calculations, we shall have to make several additional 

 simplifying assumptions. First, we assume that the 

 only factor in the expression (29) which depends on 

 the depth coordinate z is the scattering coefficient m, 

 and that m depends only on z. Then equation (29) 

 becomes 



\ mdz]\ d4>\ ph^hh'dp. (30) 



4x \Jo /Jo Jsi' 



Git) 



This assumption is readily defended. Since the depth 

 of the surface scattering layer is usually small com- 

 pared to the horizontal range from the transducer, 

 there is little difference in initial ray direction be- 

 tween the ray which reaches a point Y' in the ocean 

 surface and the ray which reaches the point Y" a 

 depth d below Y' (Figure 5). Therefore, the product 

 66' is practically independent of z in our volume of 

 integration. Again, since the depth of the surface 

 scattering layer is usually small, the horizontal dis- 

 tance traversed by a ray in its passage from S'l to S'2 

 changes but little from the top to the bottom of the 

 layer. For the same reason it can be assumed that 

 there is usually little difference in transmission loss 

 among the various paths from the transducer to 



points in the volume of integration.* There is little 

 reason for the scattering coefficient to vary with any- 

 thing but depth, as long as the grazing angles of the 

 rays on the surface do not vary appreciably over the 

 volume of integration; this will be the case if the ping 

 length is sufficiently short compared to the range of 

 the reverberation. 

 We may rewrite equation (30) as 



where 



- FF' c r 



G(t) = m' d<t> ph'bb'dp, (31) 



iw Jo JSi' 



m' = f mdz. 



It should be remarked that the disappearance of d in 

 equation (31) is of Httle consequence. There is ordi- 

 narily no way to accurately estimate d in any par- 

 ticular case; it is just the depth down to which scat- 

 terers which depend on sea state appear in significant 

 quantity. Without committing ourselves as to the 

 exact size of d, we may give the factor m' real 

 physical meaning by redefining it as 



-' = r 



mdz 



(32) 



where m is the backward scattering coefficient of the 

 scatterers causing surface reverberation, is dependent 

 on sea state, and is negligible below some unspecified 

 depth. It seems likely that the depth at which m be- 

 comes negligible is usually small enough so that the 

 lack of dependence on z of h^bb' can be assumed. If 

 not, or if for any other reason the assumptions used 

 to derive equation (31) from equation (28) are not 

 satisfied, the first integral in equation (30) cannot be 

 regarded as a separate factor, and the concept of an 

 overall surface scattering coefficient m' has no mean- 

 ing. One situation in which equation (31) does not 

 apply, while equation (28) does, is for surface rever- 

 beration in the presence of sharp negative tempera- 

 ture gradients near the range where the limiting ray 

 leaves the surface. This situation is pictured in 

 Figure 6. Strictly speaking, in this case the surface S'2 

 does not intersect the ocean surface at all; but S'2 may 

 be drawn to intersect the surface, as in Figure 6, with 

 the understanding that the transmission loss is in- 

 finite to the shaded volume. Under these circum- 



' It must be noted that this assumption ignores the possi- 

 bility of the image effect described in Section 5.2.1. The effect 

 of image interference on the behavior of surface reverbera- 

 tion is briefly discussed in Section 14.2.1. 



