Lee 



The value for the function W. which has all the required properties 

 can be shown to be 



Ae*^'y 

 W, =^^ cos K'x. (C-4) 



It tcikes little more effort to solve for Wig. We find it by using a 

 transform. Let 



00 



* 

 W 



00 



*3 = \ e-""'w,3(x,y) dx. 



The Laplace equation requires that 



P w. + W,, =0 

 ^ Is Isyy 



or 



W*(p;y) = c(p)e""\ (C-S) 



If this is substituted into the Fourier transform of Eq. (C-3),the left- 

 hand side yields 



W*y - KW* = (IpI - K)c(p) 



and the right-hand side yields 

 >oo r»oo 



pOO poo 



a\ e'"*" sinK'lxl dx = 2A \ e-'"* sin K'x dx 

 J-oo "^0 



1 Jo p2-K'2 



(C-6) 



where the apparent improper integral above is interpreted as a 

 generalized function.^ Thus we find that 



HI T,x / X -2AK' - 2AK' 



(IpI -K)c(p)=p--^ or c(P)-(p2.K'2){|p| -K) 



' Another way of interpreting this is that of Lighthill [ 1967] who let 

 a,'=c., + j€. e^O so that e^*''"'' = e^*'°"'' e'''* where K;=(a)|-eVg 

 approaches K' when fx (= e2tOo/g) —' 0. 



948 



