Van Uatev and Beal 



Is for water of uniform depth, but for practical purposes as long as 

 the minimum depth (at i = I) is greater than half the length of the 

 longest waves with significant energy results will be quite satis- 



factory. Accordingly the depth used in Eq. (4) is taken as the depth 

 at i = I. Equation (4) may be rewritten in indexical notation as 



^'Jk S...0-.. 

 ijk ij 



1/2 

 r^di 



L Vd.U 



H^wO (23) 



'3'"|J 



^'ijk = B'ijh cos (/CjjSj'ji^ - a)jt|jn) (24) 



where , 



Bjjjj = envelope elevation function in dimensional system, (ft) 



Tijj^ = wave elevation in dimensional system, (ft) 



rQ,do = radius and depth of cavity in dimensional system, (ft) 



^iik ~ pa-th length to i station along k^*^ ray. Also indexed 

 j since path varies with frequency 



Vji = group velocity 



- ^^ ■ h^-^T k ' TUS^) ,25, 



J = third-order Bessel function of the first kind 



1/2 



/ dv' \ _ / , / tanh g-jj X / sinh Z<x\\ - Zq-jj cosh Zq-jj + (Tij 1 \ 



(26) 



The wave number, K\\ , obtained from the equation 



Wj = g/Cjj tanh /Cjjhj (27) 



and may be approximated in closed form by 



2 / 2 V 1/2 



K, =^(coth^) . (28) 



264 



