As explained in the preceding section, the present theory is a conse- 

 quence of the inclusion of the second term in the slender body expansion. 

 Nevertheless it is by no means a second order theory, but still a first 

 order theory. The formal perturbation procedure, taking successive 

 approximations, starting from the lowest order term and proceeding to the 

 second order approximation, never leads to the same result as the above. 

 This fact is a peculiar feature of the asymptotic expansion of the singular 

 solution such as the present problem. 



WAVE PRESSURE ON SLENDER SHIPS 



Boundary Value Problem for the 

 Diffraction Potential 



Although the strip theory is employed in the usual practice of pre- 

 dicting wave exciting forces on ships, the diffraction problem does not 

 admit the use of the strip theory in longitudinal waves. Although the 

 strip theory is an acceptable approximation in high frequencies for the 

 radiation problem, the short wavelength associated with the high frequency 

 invalidates the condition of slow variation along the ship's axis. In the 

 case of long waves, on the other hand, the frequency becomes low and the 

 three-dimensional effect comes in as in the radiation problem. 



Now we consider first the case of a ship with forward speed in long 

 waves, so that we assume U and (0 are both of order of unity. The boundary 

 value problem for the diffraction potential has been given previously, but 

 its solution needs some contrivance. The boundary value problem in the 

 near field is the two-dimensional Laplace equation 



3 2 * D 3 2 * D 



D + — r^ = (188) 



2 2 

 9y 3z 



with the hull surface boundary condition of Equation (104) 



3^ " " 3^ < 189) 



69 



