LOW FREQUENCY PROPAGATION IN HORIZONTALLY 

 STRATIFIED OCEANS 



Most of the figures discussed before were produced by a computer program 

 designed to model acoustic propagation in a horizontally stratified ocean. For 

 mediums such as this, the acoustic pressure due to a unit point harmonic source 

 situated at (0, 0, Zg) has the integral representation 



-'o 



P (r, z, Zg; o) ) = w / u>X J^ (wXr) G (z, Zs;X, w) dX , 

 where the Green's function G satisfies the depth dependent wave equation 



Id^/dz^ + to2 / c"^(z) - xH I G(z, Zs;X,w) = -25 (z - Zg) 



and suitable boundary conditions. 



The method of solution used here, that is multipath expansion of the integral 

 representation, is quite old, dating back nearly 40 years to Van der Pol and 

 Bremmer. 8 Following Leibiger and Lee, 9 the Green's function is expressed 

 in terms of two linearly independent solutions F+ of the homogeneous depth 

 dependent equation. The solutions F+ are normalized so that their Wronskian 

 equals -2u)i. Upon expanding the denominator of G into a geometric series, 

 the double summation 



oo 4 



P(r, z, Zs;a,) - X) E Pn ^''' ^' ^s^'^) 

 >'=0 n=l 



is obtained. If z < Zg, one sees that 



P^^^ (r, z, Zg; oj) =1 IwJq (wXr) F_(z; X,aj) F+ (Zs;X,w) 



where 7sur and I'bot are boundary reflection coefficients. Other terms of 

 the series are similar, each integral representing a particular ray type. The 

 first four are illustrated in figure 9. It is important to note that, so far, the 

 solution is exact. The validity of the final result depends on the method of 



106 



