The three-dimensional Green function which satisfies Equation (27) is given by 



9 

 Wehausen and Laitone as 



G(x,y,z) = 



i7/-/ 



gu e exp (-ixu cos 9-iyu sin 9) 



2 

 gu - (a>fUu cos 9) 



d9 



(35) 



Equation (35) is the potential function of a unit source which is located at the 

 origin and moves with constant velocity U. Equation (35) is multiplied by a constant 

 value, -1/4tt, for convenience. With the change of variables, u cos 9 = k, u sin 9 = £ 

 and udud9 = dkd£, G(x,y,z) can be rewritten as 



00 a 



;(x,y,z) = -^ j e"^^ ^^ ^ } 



_£e_ 



2 7 

 z(k +£^) 



1/2 



-iy£ 



1/2 



d£ 



-«> g(kV£^) - (a>fkU)^ 



(36) 



If we define the Fourier transform 



f*(k) 



OO 



-I 



f(x) e dx 



(37) 



and the inverse Fourier transform 



^ 2^ J 



f(x) = ^ I f*(k) e ^^ dk 



(38) 



then the Fourier transform of the three-dimensional Green function is given by 



o 



GMy,z;k) = -|^ J 



2 2 ^/2 



z(k +£ ) -iy£ 

 I e 



1/2 



d£ 



(39) 



-_oo g(k2+£2) ' - (a>fkU)2 



