HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



He obtains the curve shown in Figure 3 for the dimensionless displace- 

 ment, u, of the wave that is refracted in the bottom as a function of 

 the dimensionless grazing angle, 3. A is the actual displacement, and 

 a the grazing angle. For the model from Hamilton's paper, numerical 

 values are shown for the grazing angle in degrees and the horizontal 

 displacement in meters on the beam between entry and reemergence. 



Brekhovskikh shows that the wave theory and the ray theory give 

 good agreement beyond 3=1. The subsequent discussion will be 

 restricted to grazing angles for which ray theory can be employed in 

 the bottom with some safety. 



When there is attenuation in the bottom (Figure 4) , there will be 

 losses on the refracted path and presumably the subsequent reflec- 

 tions will be of minor importance. In Morris's paper, the plane wave 

 reflection coefficient is used and the interference between the returned 

 paths after successive bounces is extremely sensitive to the grazing 

 angle. Hence, the resulting bottom-loss curves have a strong ripple 

 associated with the interference. If the interference is removed by 

 separating Paths 1 and 2, either in space or in time, (or if there is 

 an intromission condition with a very small reflection coefficient 

 for Path 1) , then Path 2 should dominate the field. 



The theory in which the velocity is strictly a linear function 



2 



of depth (rather than n linear as above) has been developed exten- 

 sively in a paper by Pekeris (1945) who solved the Green's function 

 for a point source in a linear gradient medium, and in a later paper 

 by D. H. Wood (1969). The Green's function is given by Equation (4) 

 in Figure 5 for a source at the origin of the coordinate system where 

 z is the depth and r is the distance from the source. 



279 



