SURFACE REVERBERATION 



261 



stances, the transmission loss varies rapidly with 

 depth in some portion of the volume of integration, 

 and the assumptions used to derive equation (31) 

 from equation (28) are not satisfied. 



We next make assumptions which enable us to 

 integrate out the variable p in equation (31). The 

 integral in equation (31) is taken over an annulus 

 whose horizontal cross section is the ring-shaped area 

 between the circles of radii PU and FF in Figure 5. 



Figure 6. Region of surface scattering in presence of 

 sharp downward refraction. 



If the ping length is assumed to be sufficiently short 

 compared to the range, then /i^bb' varies but Httle in 

 the distance from U to V, and equation (31) be- 

 comes 



_ p.p' r^" r^ 



Git) = m' h?hh'd4, 1 pdp (33) 



Air Jo JSi 



p . p' r^^ /•*' 



= m'K' I bie,(t>W(e,<t>)d4> I pdp. (34) 



47r Jo JSi 



The step from equation (33) to equation (34) is 

 justified only if the transmission loss h is inde- 

 pendent of the polar angle (j). With rays bending only 

 in the vertical direction, there is ordinarily no reason 

 why the average transmission loss should depend on 

 this variable. In equation (34), 6 is some average 

 angle of elevation of rays which strike the surface 

 between the two circles of radii PC/ and PF in Figure 

 5. If the ping length is sufficiently short compared 

 to the range, this average value of 6 may be assumed 

 to be the angle of elevation of the ray which leaves 

 the projector at midsignal and hits the surface after 

 a travel time t/2. In other words, this average value 

 of 6 is the angle of elevation of that ray which passes 

 through the curve of intersection of the ocean sur- 

 face and the surface Si defined in Figure 2. 

 Now, by simple calculus, 



L "^^ = (v,. - ('2I.- 



(PU + c/F)2 (puy 



in Figure 5 



(--^0 = 



= UVi PU + 



P{UV) 



(35) 



where, if the ping length is sufficiently short com- 

 pared to the range, p, the mean value of p in the 

 annulus, may be assumed equal to the value of p 

 where Ss intersects the ocean surface. UV is the dis- 

 tance on the surface from S'l to S2. 



Figure 7. Expanded drawing of ray between projector 

 and ocean surface. 



Next, we evaluate p(UV). Referring to Figure 7, 

 if PW = p, if rfs is the increment of arc along the ray 

 from to IF in the time interval dt, and if dp is the 

 corresponding increment of horizontal range, then 



dp = ds cos a 

 and 



/.(/2 /'t/2 



p = PW = I ds(t) cos a(t) =1 c cos adt, 

 Jo Jo 



since cdt is always equal to ds. Also, since the bending 

 is in the vertical direction only, we have by Snell's 

 law 



cos a cos a' 



where c' is the velocity of sound at the surface, and 



a' is the angle of elevation of the ray OW at W. It 



follows that 



rtl2^2 



p = I — , cos a'dt 



JO c 

 t cos a' 

 2^^' 



(36) 



where c is some average sound velocity. To calculate 



