the refracted ray into the Stoneley wave. This effect has been previously described by 

 Hawker, Focke and Anderson (1977). 



There is almost no loss when Cj^ = Cg (i.e. y^ ~ 54°) and when cj^ = Cp (i.e. ,7b * "^4°) 

 (see Figure 25). Between these two angles the loss increases because of shear wave genera- 

 tion in the basement. Otherwise the bottom loss is mostly due to absorption of energy 

 from the refracted ray. This loss increases, in general, with frequency. This effect is 

 apparent in Figure 25 at frequencies above 50 Hz. There is also an interference effect that 

 causes the bottom loss surface to be wavy. This is caused by the coherent addition of the 

 refracted and reflected rays. 



It is clear that bottom interaction is somewhat complicated at low frequencies. 

 However, by considering the fundamental physical processes that are responsible for the 

 bottom interaction the total sound propagation field can be determined. 



EQUIVALENT SEDIMENT LAYERS FOR USE WITH THE P.E. MODEL 



In previous sections it is shown how the bottom reflection coefficient R is incorpo- 

 rated in the propagation models. In the normal mode formalism, the use of R results in an 

 exact solution. In the case of ray theory, a shght error is introduced by the use of a plane 

 wave coefficient when a spherical coefficient is called for. However, the error is neghgible 

 at frequencies above ~ 100 Hz. When there are appreciable horizontal changes in the sound 

 speed or bathymetry we must use either perturbation solutions of wave theory or the P.E. 

 (Parabolic Equation) model. At the present time almost all efforts in model development 

 center around the P.E. model. This is due probably to the simplicity of the P.E. method 

 ana the early development work by the Acoustic Environmental Support Detachment 

 (AESD). 



As do most acoustic propagation algorithms, the P.E. model starts with the reduced 

 wave equation 



V^ )// + K^ ,// = , 



where K has been defined before and \p is the potential function. The first step in the 

 solution is to assume a product form for \p in which one term contains the range variation 

 that would occur if there were no horizontal changes and the other term represents the 

 depth (z) dependence of \p plus a small range dependence due mostly to horizontal changes. 

 That is, 



V/(r,z)-U(r,z)Hol (k^r) , 



where H^ (k^r) is a Hankel function of the first type and order zero and k^ is a separation 

 constant. Substitution of the above form of \p into the wave equation results in the follow- 

 ing second order partial differential equation. 



3% .^, 9U^d2u^/„2 u2\TT-n 

 9r2 ° 9r az2 ^ ° ^ 



57 



