where 



W = UCZq) VCZq) - U'CZq) VCzq) . 



The zero depth, source depth, receiver depth and bottom depth are 0, z^, z and z^, respec- 

 tively. The zero depth may be set at the air-water interface or at some other convenient 

 point. It represents the depth above which no sound is refracted or reflected to the 

 receiver. The horizontal wave number is k, r is the horizontal distance between the source 

 and receiver, Jq is the Bessel function of the first kind of order zero, U is a solution of the 

 z-separated part of the wave equation, i.e., U" = (k^ - cj2/v2(z))U, that satisfies the 

 boundary condition at z = 0, and V is a solution of the z-separated part of the wave equa- 

 tion, i.e., V" = (k^ - aj2/v2(z))V, that satisfies the boundary condition at z = z^. For- 

 mally, our treatment will be restricted to (z < z < z^); however, a similar development 

 for (0 < z < Zq) is easily derived. 



It is easy to show that dW/dz is zero so that W is independent of depth. Also, we 

 are free to specify the value of U and V at a selected depth which we, for convenience, 

 indicate by a bar and replace U and V by U and V where Uiz]^) - VCz^) = 1. It follows 

 then that in the limit Zp ^ zj,, W = Wzj^ = [i Cb(l-R) - (1+R) U'Cz^)] /(1+R). Therefore, 

 i// can be expressed as 



■=/ 



(l+R)U(Zo)V(z)Jo(kr)kdk 

 ^ (iC^ - U'Cz^)) - R(iCb + U'(Zb)) ■ 



For the general sound speed profile it does not appear feasible to separate the direct sound 

 paths from the bottom reflected paths. However, if the water has a constant sound speed 

 then U(z) = exp [iCCz^-z)] and V(z) = [exp - iCCz^-z) + R exp iCCz^-z)] /(I+R). In this 

 case it follows that 



/oo oo 



,„ ie(z-zj /* , iC(2zu-z-z^) 



(i/C)e o J^Ckr) kdk + / (i/e)Re ^ ° J^Ckr) 



kdk 



^D ^R 



If the bottom reflected field i//j^ can be measured directly then we have 



oo 



/iC(2zu-z-Zn) 

 (i/C)Re b °^j^(kr)kdk 



•^ o 



and R can be determined experimentally by use of the Hankel Transform 



^=/ \i'RJo(kr)rdr/[i/C)expiC(2zb-z-Zo)] . 

 *'o 



This is not a practical procedure, however, because quadrature sampling would be required 

 to determine the real and imaginary parts of i//j^. That is Real {^j-p) = Pr cos0 and 



47 



