Therefore, the inner solution for the diffraction potential in the case of 

 co = 0(1) is expressed by the form 



<|, = (K (2D) "U ^g-O + ^l+iiLz) ^ , 



where <J) is the solution of the two-dimensional problem with the above 



2 

 mentioned boundary conditions. The factor CO z/g in the third term on the 



1/2 

 right-hand side can be added only when U = 0(e ) and should be omitted 



-1/2 

 in the case of U = 0(1). Another case to be considered is CO = 0(e ). 



This is the short wave case, but some complication appears if we apply the 



slender body theory. The basic idea of the slender body is that the field 



equation in the near field can be reduced to the two-dimensional Laplace 



equation. However, the short wavelength hinders the above possibility. In 



-1/2 

 the case of CO = 0(e ) with U = 0(1), the ratio of the wavelength to the 



1/2 

 ship's length is A/£ = 0(e ) and the variation of the flow field along 



the x-axis is related to the wavelength. If the order of the diffraction 



potential is 6 £, the order of magnitude of each term in the Laplace 



equation in the near field is 



^D ^\ 3 \ 



2 2 2 

 8x dy 3z 



(6) (6e _1 ) (Se -1 ) 



Therefore, omitting the term of higher order than e, we get a two- 

 dimensional Laplace equation. However, the omitted term in the long wave 



2 

 case has been of higher order, £ . Therefore, the validity of the two- 

 dimensional equation becomes much weaker in comparison with the long wave 

 case. This may damage the accuracy appreciably. The free surface 

 condition in this case is 



3 V 3 *° + , „ ^ + , „ 3 *° ^ ^O 3 *D + . „ , 3 V 



^— + g ^ + 2 U . ,. + 2 U -r -r— 5 r— -5— + g U X, x~ = 



„ 2 b 3z dtdx dy dtdy _ 2 dt ° w „ 2 

 dt dz dz 



at z = (201) 



72 



