If U and U) are of the order of unity, the lowest order term is 0(6) and the 



next term is 0(Se) . Although these are linear with respect to <j> , if c}> 



and <j) are assumed to be known functions, the cross products between 

 w 



derivatives of (f> 1 and (f> make the boundary value problem intractable. If 

 only the lowest order is taken, the free surface condition for the radiation 

 potential becomes 



R = at z = (113) 



3z 



Thus, the double body condition holds again. The boundary condition for the 

 diffraction potential becomes, on the other hand, 



3 *D d \ 



3z 



where C is the wave profile of the incident wave, since we have the 

 w 



relation 



3T + U ax" + * ? w = ° < 115 > 



Now let us construct the boundary value problem. The field equation 



is the Laplace equation, but the governing equation in the near field is 



reduced to the two-dimensional form in the y-z plane because of the singular 



perturbation. Therefore, the boundary value problem is to find a plane 



harmonic function with the boundary condition at the hull surface, where the 



normal velocity is prescribed, and with the condition Sty /dz = at z = 



R 



for the radiation potential. The condition of infinity if- left unspecified, 

 yielding an indef initeness to the solution. Now we assume the solution of 

 the two-dimensional problem of the Laplace equation 



2 2 

 3 _& 3 (j> _ 



9y 



2 + — f = (116) 



45 



