III-83 



;^ljslll-£il _ i r r i yx-Xi)+yy-yi)+k^(z-S(xi))| dk^dk. 



l£-rxl ^ ^ J J ^ ^ -I ~f^ 



^ (III-89) 



in the Helmholtz formula 



Psc<^' y- ^> - 4;^ J \TF V | r-rj y - Psc Fn ^ [ r-r,| Jr ^'"''^^ 

 S - - // 



and substitute (III- 87) and (III- 88) to yield: 



S -00 -00 



(III-91) 



i(ka -k^+np)Xg-k vi + (kY - k^) S(xi) 



•e ^ dk dk dxi dy- 



X y -^^ 



where (a', P', y') are the direction cosines of r' = Xi . If the function S(x) 



describing the surface is periodic, with period — , we can set 



P 



k 

 2k 



r (a-a')a+(v-v-)c 1 ^^^V-k )S(x,) ^ y-^ inpx, 



z -■ n=-a> 



Substituting this in (III-91) gives 



i(k x+k y+k z) (III-93) 



i;2n( 



00 00 



i(k 



x y z 



P3,(x.y,z) = -— 2^ j J j A_^a 



n=-oo S -0° -00 



i(ka-k^+np)xi -k yi 



. e ^ dxi dyi dk dk 



x y 



arthut Sl.IlittI«?Jn«r. 

 s-7001-0307 



