III-74 



We could alternatively have written (III-56) as: 



p (x,z,t) = p (x+n§, z, t+kan^/tJj) 

 sc sc 



(III- 57) 



Thus, the scattered wave is the same at Pq and Pj, . Substituting (III-53) and 

 (III-55)in (III- 56), we find: 



, . iakn§ / , ff \ 



p (x,z)e = p (x+ns,z) 



sc sc 



(III- 58) 



It follows that p (x,z)e ^ must be a periodic function with period ? ; there- 



fore, it can be expanded in a Fourier series: 



p (x,z)e 

 sc 



-ik 



00 



ax V^ . . , /2TTinx\ 



(III- 59) 



However, p (x,z) must satisfy the Helmholtz equation 



3~2"P + ^—3- P + k^ p = 

 Sx *^sc at ^sc ^sc 



(III- 60) 



and this permits us to solve for the functions Ajj(z). In fact, each of the terms, 

 A (z) [ — = — I e^ ^, must satisfy (III- 60) separately, since the functions 



[ — = — jeikax are linearly independent. Consequently, A (z) satisfies: 



A (z) - k= 



/ ^2nn \ = 



dz" n 

 Let us introduce the sequence of direction cosines: 



A (z) = 

 n 



(III- 61) 



n §k n n 



(III- 62) 



Arthur Jl.littlcJnt. 



S-7001-0307 



