III-81 



Using the expresssions (III-67) and (III-65) for p. and p , to- 

 gether with (III- 71), it can be shown that for a surface of the form 

 S(x) = h cospx, the boundary condition p +p =0 is also equivalent to 



00 



L '^ ^m-n Jn ^^ V-n> ^ '^'^"^ U^ ^> <"I-81) 



n=-oo 



Then, assuming A^ to be of the form A^ =y^A^^^(kh)^ Heaps (Ref. III-14) 



shows that A is given bv I t| ^1 n| 



n, t ° ■' 



t 

 A,,,= W - ^ V ^''n-/ Vi. t-r (111-82) 



2' [i(t+n) ] I [i(t-n) ] I r=l j 2^ [i(r+j) 1 1 [i(r- j) ] ! 



where j is such that | j| s r, | n-jl s t-r, with r and j both even or odd. 

 Neglecting (kh)* and higher powers, this gives 



A^ = -l + i(kh)^ Y(Yi+Y.i) 



Ai, = i(kh Y) - i(kh)^ Y [| y" + i(YYi+YY.i+ Y±i Yf,) " * ^±, + i Y^i ] 



A±9 = *(kh)^ Y Y±i (III- 83) 



Ai3= -i(kh)«Y[2^ Y^*| Y|,+ iYi, Y±,] 



Heaps (Ref. 111-14) also considered the problem of scattering from a sinusoidal 

 surface when the incident wave is a spherical wave centered a finite distance d 

 below the surface. The scattered wave is represented in the form 



ikr * 

 m=-» 



When powers of (kh)* and greater are neglected, the Bj^ are given by expressions 



similar to (III-83) except that instead of kh, the term h appears, and instead of 



Y , the terms 

 n 



Arthur Sl.lCittlpJmr. 



S-7001-0307 



