Ill- 89 



Substituting (III- 113) into (III-112) gives 



^(\, ^) = - 2n6(u) 



^ B_^6(x-a-ffi) (III-114) 



which, when substituted into (III-107), gives 



Zik{cL^x+ Y z) 

 B^e (III- 11 5a) 



n=-oo 



where 



np 

 n k 



'. = ('-^)* 



(III- 11 5b) 



with the usual conventions (III-48) on the sign of y • This is exactly the type of 

 expansion obtained in the other approaches dealing with periodic surfaces . The 

 coefficients are given as the Fourier coefficients of the function e^'^^'-'^', as in 

 (III-113). 



Parker's development requires that S(x, y)^ 0; thus, a sinusoidal one 

 dimensional surface may be given by 



S(x) = h(l-cospx) (III-116) 



Then, it turns out that 



= c +2_] c sin(npx+b ) (III-117a) 



where 



ku 



o 



n=l 



2kh Ya 



c = • (III-117b) 



n 



i+{phf e^ r 



Arthur B.lUtlcJnir. 



S-7001-0307 



