LAB I ANC A/HARPER: A THEORETICAL APPROACH TO THE PREDICTION OF SIGNAL 



FLUCTUATIONS DUE TO ROUGH-SURFACE SCATTERING 



can be represented by a combination of Fourier transforms for the 

 horizontal spatial coordinates and an expansion in terms of the 

 eigenf unctions (or normal modes) characteristic of the sound-speed 

 depth dependence. We have developed this procedure and as a first 

 application of it we have applied it to a semi-infinite isovelocity 

 ocean. The approach has been utilized by Fessenden (1973) for this 

 geometry, but in a limited way and for different application. We 

 present the application of the theory both for a sinusoidal deter- 

 ministic surface, and for a random surface composed of a sum of 

 sinusoids with random phases and frequencies and possessing a typical 

 Pierson-Neumann frequency spectrum. 



The geometry for the isovelocity case is illustrated in Figure 6. 

 The surface is a moving sinusoid traveling in the positive x direction 

 and the excitation is a point source located on the z axis. This 

 treatment is completely three-dimensional. The procedure for deriving 

 the field expressions at a particular field point, here characterized 

 by the polar coordinates (p, 3, z) is to evaluate the integral repre- 

 sentations by the method of stationary phase. The stationary phase 

 equations were solved numerically. As Figure 6 implies, the field 

 expressions are amenable to ray- theoretic interpretation. There are 

 four contributions to a field point when the analysis is carried to 

 second order consistent with conservation of energy: the direct 

 contribution, the specularly reflected carrier and the up and down 

 Doppler-shifted sidebands. The first order correction yields the 

 two sidebands, while the portion of the second order contribution 

 which is retained corrects for the energy removed from the carrier 

 in establishing the first-order sidebands. The solution of the 

 second-order boundary value problem also yields up and down Doppler 

 sidebands at twice the surface frequency. However, these would re- 

 quire fourth-order corrections for conservation of energy and hence 

 are discarded. 



592 



