LABIANCA/HARPER: A THEORETICAL APPROACH TO THE PREDICTION OF SIGNAL 



FLUCTUATIONS DUE TO ROUGH-SURFACE SCATTERING 



The reversal of the sidebands is illustrated in Figure 17 where the 

 receiver is now above the source. Here the direction of the three 

 power curves is that of the surface wave (g = 0°) and the up Doppler- 

 shifted sideband is seen to die out more rapidly than the down 

 Doppler-shifted one. 



The planes of symmetry which I have just illustrated can be 

 predicted mathematically, and indeed we have done this in the two 

 cases of 3 = 90° and z = z'. The B = 90° symmetry is perhaps intui- 

 tively acceptable, but there is no a priori reason to expect the 

 z = z' symmetry to exist. In fact, although we can describe all of 

 the geometric consequences of the latter symmetry, there does not 

 appear to be an obvious physical explanation for it. And as far as 

 the reversal in the amplitudes of the sidebands about this plane is 

 concerned, we have not been able to assign a simple physical explana- 

 tion to it either. Perhaps the complicated geometrical structure 

 precludes this. 



The generalization of the isovelocity half space problem to the 



case of a random surface is now presented. Figure 18 shows the time 



series for the surface simulation, as well as the resulting pressure. 



Following Shinozuka and Jan (1972) , we simulate the surface as a sum 



of sinusoids. Here the phases y . are random and uniformly distributed 



between and 27t, and the frequencies Q. have a small random part about 



a mean value. The means are equally spaced over some band and the 



random part prevents the process from being periodic. The amplitudes 



h are deterministic, being determined from the height of the power 



J 

 spectrum at the mean value off].. The power spectrum can be a measured 



3 

 spectrum or a typical Pierson-Neumann distribution. In any case, 



the simulated process is ergodic with respect to its mean and auto- 

 correlation. For large samples (i.e., large M) , the process becomes 

 Gaussian. 



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