The first approximation for the free surface elevation is given by 

 Equation (31) or 



^o = " lk (1 - u2 o- v2 o } n (48) 



z=0 



Then the kinematical condition of the free surface, Equation (8), gives 



H Q 9C 3w Q 



w = u ^ — + v„ 3 Z, 3 — at z=0 (49) 



1 dx dy dz 



This relation provides the boundary condition at the free surface for the 



first approximation of the perturbation velocity potential. However, the 



right-hand side is determined by the double body flow which does not 



present a wavelike motion, so that the boundary value problem gives the 



solution which is not wavelike. This result contradicts the actual 



phenomena. In order to avoid this contradiction, one has to revise the 



basic assumption. Here we employ the hypothesis which was proposed by 



Ogilvie. The basic assumption is that the perturbation velocity is 



wavelike and the wavelength is proportional to the Froude number squared. 



This means that the differentiation results in the change of order of 



magnitude by the order of wave number. Secondly, it is assumed that the 



wavelike nature appears in the first approximation of the perturbation 



2 

 velocities, i.e., u, , v, , w.. which are 0(Fn ), but not in the first 



approximation of the free surface elevation £_. The second approximation 



for the surface elevation is given by 



h = " T n ^oWi* n (50) 



z=0 



4 2 



and is 0(Fn ). Since u.. , v, , w.. are 0(Fn ) and their differentiation 



_2 

 reduces the order by Fn , the derivatives of the perturbation 



25 



