The value of G* for k = reduces to the two-dimensional source potential. Equation 

 (30), as 



OC 



G*(y,z;0) = - |:^ J 



z|£| -iy£ 

 d£ (40) 



'-0O |£l - K 



An approximation similar to Equations (33) and (34) can be derived for the transform 

 G*(y,z;k). An asymptotic expansion of Equation (39) for Kr << 1 is derived by 

 Ursell in the special case U = 0. 



For the case U > 0, the asymptotic expansion is given by (see Appendix C) 



G*(k,K) = G^^ - ^ f*(k,K,K) (41) 



where G„ is Equation (33) and 



V |k|/ 



f*(k,K,K) = ln(^j^^] -iTT + ttG^* (42) 



SOLUTION OF THE POTENTIAL FUNCTION WITH THE 

 SLENDER-BODY ASSUMPTION 



Under the slender-body assumption the length of the ship is far larger than the 



beam or the draft. We separate the fluid domain into two regions: the outer region 



where (y,z) is of the order of the length, and the inner region where (y,z) is of the 



order of the beam or the draft. In the outer region, the three-dimensional Laplace 



equation is solved with the free-surface condition. Equation (27), and the radiation 



condition. The inner solution is governed by the two-dimensional Laplace equation 



with the free-surface condition. Equation (28) , and the body condition, Equation 



(19), on the ship hull. Then, the inner and outer solutions are matched in the 



overlap domain to determine the slender-body solution. 



10 



