velocities are 0(1). Inserting the expression, Equation (47) in the free 

 surface condition 



H u fe ■* h +w fc) (u2+v2+w2) + ^o w ■ ° (51) 



and collecting terms of 0(1), we obtain on the surface z = £_ 



2 3U 1., T V 1 + 8U lL 2 Sv l v 



x o ^T + u o v o \^T + *T/ + v o ^7 + Y w i 



3w^ 



= " I ( u o h % fy~) (u o +v 5 } " Y C 17 



= Y Q D(x,y) (52) 



The function D(x,y) is expressed by Equation (41). If the double body flow 

 velocity is known, the function D(x,y) is a known function. Then Equation 

 (52) gives the boundary condition at the free surface for the perturbation 

 velocity potential cL defined by 



UTl uyi „ n 



x l = ^T ' V l = ^y" ' W l = ^~ (53) 



as follows. 



- a 2 *-, s 2 *, o ^i 9<j) i 



l o TIT + 2 u o v o ^ + v o 7T" + Y o W = Y o D(x ^ (54) 



8x 3y 



The perturbation velocity potential is harmonic outside the hull surface 

 S and below the curved surface z = X, on which the boundary condition, 

 Equation (54), is satisfied. Since the boundary condition on the curved 



26 



