^ [r- -R{z')]H(A + v$V)[r' -R(z')]= 

 on r' = R(z') 



(Z) On the free surface, the normal velocity connponent of the free 

 surface must equal the normal velocity component of the fluid particles 

 in this surface, and the pressure must equal atnnospheric pressure. In 

 the linearized theory, these conditions reduce to 



^+g^ = on z.O. [4] 



at2 ^ 9z 



or in the case of a sinusoidal disturbance with frequency oo, 



K$-|- =0 on z = 0, [5] 



oz 



where K = oj /g. 



(3) At infinite distance from the body, the waves generated by the body 

 are outgoing (the radiation condition). 



The free surface condition, Equation [5], and the radiation condition 

 are satisfied by the potential of oscillating singularities beneath the free 

 surface; the boundary condition on the body may be satisfied by a proper 

 distribution of these singiilarities . This distribution may be found from 

 slender-body theory but some care is required in linearizing the present 

 problem. If r' = R(z') is the equation of the body surface over its sub- 

 merged length (-H < z' < 0), we shall assume that R and its first deriva- 

 tive are continuous, that R(-H) = 0, and that the magnitude of the slope 

 |dR/dz'|« 1. The depth H is assumed finite, and it follows that R is 

 small of the same order, as dR/dz'. In the analysis to follow we shall 

 also require that R be small compared to the wavelength of the incident 

 wave system, or that KR « 1. 



We wish to obtain the velocity potential of leading order in the small 

 parameters of slenderness and oscillation amplitudes in order to obtain a 

 consistent set of linearized equations of motion for the body. However, 

 it will turn out that the potentials of different phases of the motion are of 



