III-104 



2 . Stochastic Surfaces 



The foregoing developments have all assumed that the sea surface 

 possesses a definite functional description z = S(x, y) . This description was 

 generally taken to be one-dimensional and periodic in nature; that is, the sur- 

 face was generated by a family of lines parallel to one of the coordinate axes . 

 In such a situation, the scattered wave resulting from a plane incident wave 

 could be expressed as the sum of plane waves . To obtain the magnitude of the 

 scattered plane waves, we studied the surface z = S(x) = h cos px. 



In reality, the sea surface is extremely complex and varying in time . 

 Generally it is not locally periodic nor one dimensional, although it may be ap- 

 proximately so when considered from a macroscopic viewpoint . As mentioned 

 earlier, there may also be inhomogeneities near the surface . Accordingly, any 

 results based on a model such as S(x) = h cos px can only be suggestive of the 

 true scattering behavior . 



Marsh, et a]_. (Ref. III-27) have approached the problem of scattering 

 from the surface by using a spectral description of the sea. In particular, they 

 use the Neumann-Pierson spectrum A^(uu), where A^(uo)daj gives the contribution 

 to the mean square wave height due to surface waves having frequencies between 

 'V and ijirt- dm. The surface is furthermiore assumed to be isotropic, i.e., the 

 correlation of the surface height at two points depends only on the distance be- 

 tween them . This means that the surface waves have (on the average) no well 

 defined direction. Using such a description of the surface. Marsh obtains an 

 approximate expression for the intensity scattered in the specular direction. 

 Since his treatment is a little obscure, and contains somie numerical' errors, 

 we shall present it in some detail . 



As in the other treatments, the incident wave is a plane wave, of the 

 form 



ik(ax+8x+vz) (III-122) 



p.^^(x,y,z) = e 



and the scattered wave is represented by 



p (x,y,z)= V I e^^(^"^^y+^"> l(X,u)dXdu (III-123) 



sc 



n-" 



where \, u, v are direction cosines satisfying (III-48). 



;artbur H.ltittlcllnc. 



S-7001-0307 



