HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



The variable displacement of the surface in Figure 11, at the 

 point of entry, produces a variable phase delay 6 (Equation 13). 

 To the simplest approximation it is actually a difference of the 

 slant path in the water associated with the entry angle and the 

 slant path in the sediment associated with the refracted angle. 

 This makes the effect much smaller than in scattering, say, from a 

 free surface or from a reflecting bottom. That is, only the differ- 

 ence in the acoustic delays in the two media accumulates, so that a 

 large surface displacement actually produces a relatively small 

 change in the phase. Hence, Equation (13) is the variable phase 

 to be inserted across the area of integration, being the random 

 displacement of the surface. Again assuming that the normal gradient 

 of the field in the bottom is the vertical gradient, there are two 



final approximations: first, that the angle 6 in the Green's 



e 



function is the same as the refracted angle 9 of the wave entering 



the bottom; and second, that the wavenumber k for the refracted 



wave and k for the Green's function are the same. With this 

 o 



approximation the field <I>(P) is expressed in Equation (14) as the 



integral over the insonified region of the refracted wave incident 



on the bottom times the Green's function integrated over the insoni- 

 fied area. 



Hence, the wave impinging on the bottom is refracted in the 

 bottom yielding (}) . There is a phase variation with x across the area 

 of insonif ication, but the Green's function to the linear approxima- 

 tion used here has exactly the same phase variation because of the 

 agreement of phase at the boundary. That is, the X variation of phase 

 in the one function is exactly canceled by the variation in the 

 Green's function, leaving only the variable phase delay associated 

 with roughness. 



290 



