Ill- 87 



The different approaches thus far described can be summarized as 

 follows. The Rayleigh-La Casce and Tamarkin-Heaps approaches basically 

 start with the scattered wave Pg^ expressed as a sum of plane damped and un- 

 damped waves. The surface is assumed to be of the form S(x) = h cos px. Using 

 the condition that the total field must be zero at the boundary and employing the 

 expansion (III-71), an infinite set of simultaneous equations is obtained for the 

 coefficients A^. Explicit solutions for the first few coefficients are calculated 

 for small values of (kh) . 



Brekhovskikh and Lysanov initially express Pg^ in terms of a Helmholtz 

 integral. Then, assuming periodicity of the surface, they also obtain an expression 

 giving pg^ as the sum of plane waves . The coefficients A^^ are obtained as explicit 

 functions, i.e., they are not expressed as functions of each other. The Lysanov 

 formulation gives the coefficients as an infinite series involving Bessel functions, 

 while the Brekhovskikh approach expresses the coefficients in terms of a single 

 Bessel function. 



Parker (Ref . III-33) has approached the surface scattering problem by 

 using the spectral representation of the scattered wave given in (III-49) . Without 

 the time dependence, this is 



00 03 



Psc= { f*(^-u)e^^<''^+^y + ^^>dXdM (III-107) 



n 



where X^ -l-^^ + n^ = l and v ^ If 1 - X^ - ^^ s and 1 v> otherwise, of. (III-48). 

 As in the other treatments, tHe incident wave is taken to be a plane wave of the 

 form 



^ gik(ax + Yz) (III-108) 



inc 



Then, the boundary condition that p^nc + Psc = ^^ the surface becomes an integral 

 equation for 6(\ , u), as given in (III-50) . 



In his development, Parker works primarily with $(x, y), the Fourier 

 transform of 0(X, ^) . With z - S(x, y), he first shows that 



lim $(x, y) = $^ (x, y) - - e^^"" (III- 109) 



S-0 



Arthur m.Hittle Jnt. 



S-7001-0307 



