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



FLUCTUATIONS DUE TO ROUGH- SURFACE SCATTERING 



the perturbation expansion of the pressure and the surface boundary 

 condition. The perturbation parameter, a, is the rms waveheight, 

 which is taken to be small compared to all other lengths in the 

 problem, e.g., the acoustic and surface-wave wavelengths. Implicit 

 in the assumption of small a is also an assumption of small ocean 

 surface slopes. This presents no serious limitation and is generally- 

 valid except in a very rough sea where the waves are breaking and the 

 surface correlation length is the order of a few meters. 



When the expansions are substituted into the equations of Part A 

 and coefficients of like-powers of a are matched, there results the 

 sequence of boundary value problems in Figure 5. The 0(1) problem is 

 the smooth surface problem, while the higher order problems contain 

 the effects of the surface through the boundary conditions. For 

 example, in the 0(a) problem, the surface boundary condition is made 



up of the product of the surface function I, and the z-derivative of 



2 

 the 0(1) pressure. In the o (a ) problem, the complexity of the 



boundary condition increases in a similar fashion and in general 



the surface boundary condition of a higher order problem depends on 



the solution of all the lower order problems. Further, since each 



boundary condition is made up of a sum of products of time-variable 



functions, it is clear that new frequencies will be generated. This 



is, of course, a consequence of the fact that the complete boundary 



value problem is nonlinear in time. But the important point is that 



the bandwidth of the acoustic spectrum increases with the addition of 



higher order corrections. 



A formal procedure can be used in the solution of the sequence 

 of boundary value problems. Since in each case the surface boundary 

 condition has been reduced to the z = plane and since the sound 

 speed depth dependence is the same for each problem, the solutions 



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