where G(P,Q) is Green's function having a simple pole at the point 

 P(x,y,z) = Q(x',y',z'), P is a point inside the enclosed space defined 

 above, Q is a point on the boundary surface and n is the normal to the 

 boundary surface drawn inward at point Q. This expression is valid pro- 

 vided that the analytic continuation of the velocity potential in the 

 region > z > £ is possible. This is the only assumption for the above 

 formulation. Now we define the Green function in such a way that it 

 satisfies the Laplace equation in the lower half space except a point 

 P = Q, and boundary conditions 



+Y 



,2 '0 3s 



G(P,Q) = 



at z 



(11) 



G(P,Q) = on Z. 



(12) 



It is assumed that the radiation condition 



£im v£ G(P,Q) = or £im JyJ G(P,Q) = 



(13) 



is satisfied too. Since 3cj)/3n = and 3G/3n = 0, the contribution by the 

 integral on I vanishes. If the velocity potential (j) is assumed to fulfill 

 the radiation condition at infinite distance, the integral on Z decays out 

 as the radius of the cylinder tends to infinity. Since £_ is a horizontal 

 plane, the normal n is directed vertically downwards. Then the integral 

 on I can be transformed as 



MM. _ ^ (Q) 3G(P,Q) 

 Q % 



G(P.Q) ^ - *(Q) 3n 



d S, 



- \l [gcp.q) 3 *<*; ; r'- z,) - ♦u'.y..-) ^^ 



dx'dy' 



z' = 



