different orders of magnitude with respect to the slenderness parameter. 

 For example, the potential due to surge or pitch is of order R as R — 0, 

 whereas the potential due to heave is 0(R ). Similar differences will oc- 

 cur in considering the components of each potential which are in phase 

 and out of phase with the respective velocities of the body. In order to 

 circumvent these difficulties without unnecessary higher order perturba- 

 tion analysis, we decompose the velocity potential in the following form: 



$(x, y, z; t) = <t)t (x,y, z; t) + cj)^ (x, y , z; t) + cj) , (x, y, z; t) 



[6] 

 + A [g/w e^^ cos (Kx - cot) + <^j^(x,y, z; t)] 



where <\>t , (^v > ^^'^ ^ i ^^^ linear in the displacements {^,1, ,\\)) and their 

 time derivatives, respectively. The potential A g/w e cos (Kx - cot) 

 represents the incident wave system and the potential Acj)^(x,y, z; t) re- 

 presents the diffracted w^ave potential, corresponding to waves incident 

 on a restrained body. Each potential ^ in Equation [6] must satisfy the 

 free surface boundary condition and the radiation condition; the complete 

 potential $ must satisfy the boundary condition on the body. This condi- 

 tion. Equation [3], is reduced as follows: 



, 9 , v^;^ ^ r . o / -\i ar' ax', ar' ay' ^ ar' az' dR az' 

 {-T— + V$ • V [r' - R(z')l = + i- + 



^ at '^ ' '-■ 9x' at ay' at az' at dz' at 



^ a$ a$ dR . . D / .\ 



+ =0 on r=R(z), 



ar' az' dz' 



or neglecting second-order terms in A, ^ , L, , and ljj, 



||--|f g- [|+ (z -Zg)*^] C°S e+(t-x4;)dR/dz= 



[7] 

 on r = R (z). 



where a dot denotes differentiation with respect to time. Substituting 

 Equation [6] into Equation [7] and separating terms according to their 

 dependence on different displacemients, we obtain the following boundary 

 conditions on the body: 



