expansion. Though the free surface condition does not involve the forward 

 velocity, the hull boundary condition does. 



Another choice is the high frequency case. The order of magnitude of 



-1/2 

 the frequency parameter is assumed to be £ . In the first place, we 



assume the Froude number still remains in the order of unity. Because of 



the relation 



K = |2L = _i — __ (i+2f2 cos a- /l+4fi cos a) (135) 



A „ TT 2 / 



2 U cos a 



1/2 

 the ratio of wavelength to the ship's length is of the order of £ , so 



that the wave is not extremely short. Since the wave slope is of the 



1/2 

 order of 6, the wave amplitude is of the order of 6 £ . The velocity of 



the orbital motion of wave is 0(6), so that the order of <})-. is 6 £. In the 



boundary condition on the hull surface for the radiation potential, namely 



-r — = i w(z -xifj) -^r - U ty -~r - U(z -x\J0 -5— r ( -5 — ) (136) 



3n g 3n' dn' g dn' \9z ' 



the first term is of the order of 6, while other terms which are related to 



1/2 

 the forward speed are of the order of 6 £ .On the other hand, the 



1/2 

 lowest order of the free surface condition is 6 and the next is 6 £ . If 



we take the lowest order terms only in both boundary conditions, the so- 



lution is in the case of zero forward speed. We can discuss the effect of 



1/2 

 the forward speed by taking terms of order 6 £ .In this case, the free 



surface condition for the radiation potential becomes 



*\ 







9 *R 









d \ 





+ 



g 





+ 



V 



II 





^ 2 





dz 









3t3x 



dt 

















2 2 



3cj) 9 cj> 3 <J> 3c}> 



+ 2 1)^^-11^-^=0 (137) 



9y 9t3y 3z 2 3t 



Because of the terms except the first and second ones, we cannot solve the 



boundary value problem in two dimensions by ordinary methods. Ogilvie and 



29 

 Tuck employed a successive method by which the solution for the above 



52 



