Ill- 88 



when the incident wave is given by (III-108). Then, a representation for § is 

 assumed which is of the form 



$ (x, y) = $q(x, y) e^'^^^'^' ^^ (III- 110) 



where u(x, y) goes uniformly to zero as S(x, y) vanishes . After considerable 

 analysis, u is found to be given by 



^ S(x, y) Y (lll-iu) 



1+S(x,y)|](s(x,y)e|^)"g^ 



where 9 = tan (cos"-^ y) and (S(x, y) 6— j signifies that the operator S(x, y)9 



3x 



is to be applied n times. To obtain this result, it is assumed that certain terms 

 can be neglected in solving a non-linear differential equation and that truncation 

 of certain Taylor series expansions is permissible. 



The spectrum (i(X , u) can then be written as 



*(x,m) = 



i_ j j,iMx,y);4<^-)-->']^^^ ^^^_^^^^ 



2 



-00 -CX) 



If the surface S(x, y) can be expressed as a periodic function of x, with period 



— , then u also will be periodic and can be written as 

 P 



00 



^iku(x) ^ ^ g^ ^-inpx (Iii-ii3a) 



n=-oo 



with 



TT 



g 1 C ^iku(x) ^inpx (jjj_^^3^) 



n 2n J 



Arthur a.littleJnf. 



S-7001-0307 



