and (3.2a and b) , it then may be seen that the Green function G(C,x) also satisfies 

 the following equations 



V^G = 6(x-?)6(y-n)5(2-C) in z < I (3.3a) 



for C < 

 [8 -(f+iF8 +ie)^]G = on z = I (3.3b) 



2 

 V G = in z < I (3.4a) 



for ? = 

 [8 -(f+iF9 +i£)^]G = -6(x-C)6(y-n) on z = ) (3.4b) 



where V is the differential operator (9 ,9 ,9 ). Equations (3.3a and b) and (3.4a 



X y z ^ 



and b) will be used in the next section for obtaining integral identities satisfied 

 by the velocity potential. 



A well known expression for the Green function, in terms of a double integral, 

 can be obtained by using a double Fourier transformation of Equations (3.1a and b) 

 with respect to the horizontal coordinates E, and r\. This Fourier representation of 

 the Green function is given by 



-y -^ 2 2 2-1/2 2 2 2 -1/2 



4uG(C,x) = -[(x-0^+(y-n)^+(z-C)^] "-'^ + [(x-?)^+(y-n) +(z+C) ] ' 



OO CO 



_l r , r exp[(z+s)(y^vy/^i{(x-Oy+(y-n)v}] 

 ^ -^ . J (y'+v2)l/2 _ (f-Fy+ie)2 



The "Cartesian Fourier integral representation" (3.5) can also be expressed in 

 the form of a "polar Fourier representation" by performing the change of variables 

 y = A. cos9 and v = X sin6, which express the Cartesian Fourier variable y and v in 

 terms of the polar variables A and 0. These equivalent double-integral representa- 

 tions were first obtained by Haskind and Brard, and later by Hanaoka, 



19 20 21 22 



Stretenski, Eggers, Havelock, and Wehausen, and are therefore well known. 



10 



