HORIZONTAL RAY THEORY FOR NEARLY HORIZONTALLY 



STRATIFIED OCEANS 



At long ranges or in shallow water, the effects of horizontal variations in 

 the sound-speed or bottom characteristics are often large and not readily modeled 

 by any of the techniques discussed previously. A nearly horizontally stratified 

 ocean is one in which the horizontal variation is slow. 



This rather vague notion is quantized by introducing a small parameter e 

 and assuming that the properties of the medium depend -^n the horizontal coor- 

 dinates X, Y only through the combinations 15 



x= eX, y= eY . 



This being so, let us seek solutions of the reduced wave equation in the form 



oo 



P (X, y, z; e)« exp \ e(x, y)/ie[ Z) A^ (x, y, z) (ie)" . 

 Each A^, in turn, is assumed to have the form 



00 



A^ (X, y, z) - L a[ \x, y) ^y. (x, y, z) , 

 iC — u 



where the V'k a-i'e orthonormal eigenfunctions of the depth dependent wave equa- 

 tion 



— ^ + K (X, y, z)^^= X^ .(.^ 

 oz 



subject to the appropriate boundary conditions. 



Upon substituting our ansatz into the reduced wave equation and comparing 

 similar powers of ie, one finds that the phase function, e , satisfies the hori- 

 zontally dependent eikonal equation 



£) • (0 



2 2 



= >^„ (X, y) , 

 y/ ^ 



where Xn is one of the eigenvalues, x,, computed above. 



This equation, like the ordinary eikonal equation, may be solved by using 

 ray tracing techniques. Note, however, that all depth dependence is missing. 

 The pressure depends on depth only through the vertical eigenfunctions. It may 

 also be shown that the leading amplitude, a(^), satisfies the conservation law 



114 



