in which : 

 C 



Van Oovtmevssen 



J (kr) Y (ka) - J (ka) Y (kr) 

 n n, r n, r n 



J (ka) + i Y (ka) 



n, r n, r 



€ = 1 for n = 



n 



c = 2 for n i 



*-n 



For the case that the cylinder does not extend to the bottom, 

 Garret f 5 J has derived an analytical solution, using variational 

 principles. 



II. 3 Numerical solutions 



For the body of arbitrary shape, the velocity potential can be 

 found from numerical methods. At the Netherlands Ship Model Basin 

 a computer program has been developed for the numerical calculation 

 of the velocity potential, using a source distribution over a surface 

 inside the body. According to Lamb £6^ the potential function can 

 be found from : 



4> s (x) =jj q (a) % (x,a) dA 



(ID 



in which : 



3" (x, a) = the Green's function for a source, singular in ^ 

 a_ = vector which describes the surface A, on which the 



sources are located, 

 q (a) = the unknown source strength. 



The Green's function represents the contribution to the velo- 

 city potential in x due to a unit wave source located in a. A Green's 

 function which satisfies the boundary conditions in the free surface, 

 at the bottom and the radiation condition, has been given by John [ 7J 



: 2 -V 2 

 k~d -v 2 d +\> 



tf (x,a) = 2tf 2 k " I cosh k (c+d) coshli(z+d)[ Y q (kr.) - iJ Q (kr.)j 



(12) 



...+z 



in which : 



~ 4 (k n 2 + \> 2 ) 



r . cos k (z+d) cos k n (c+d) K^^r ) 



n=i dk + d\) -v) 

 n 



962 



