APPENDIX B 

 DERIVATION OF G (y,z) FOR SMALL ARGUMENT 



We expand the exponential function and the exponential integral in Equation (31) 

 for small Kr where y = r sin 6 and z = -r cos 6 



^Kz+iKy = 1 + j^2 + iKy(l+Kz) (B.l) 



^Kz+iK|y| = 1 + Kz + iK|y|(l+Kz) (B.2) 



E^(Kz+iKy) = E^CKre^*^^"®-*) 



= -Y - £n(Kr) -1(71-9) + Kz + iKy (B.3) 



By substitution of Equations (B.l), (B.2), and (B.3) into Equation (31), G is 

 given as 



2D' 



G^^(y,z) = - ^ Re { [ (1+Kz+iKy (1+Kz) ] [-Y-£n(Kr)-i(7T-0)+Kz+iKy] } 

 - i [(l+Kz)+iK|yI (1+Kz)] 



- (1+Kz) [Y+£n(Kr)-Kz+Kye-iTT] (B.4) 



53 



