Although we have discussed the two-dimensional time-dependent problem, Equations 

 (19) through (24) can be extended to the three-dimensional time-dependent problem. 

 That is 



= JjJ I V *1* V *1 dxdydz-^ J] 



'1 y ltt 



dxdy 



- JJ $ 1 * f. dxdy + Ih * (f) dxdy 



( z =o)nD 1 



S J 



and so on. 



CONCLUDING REMARKS 

 We have formed functionals with both linear and nonlinear free surface boundary 

 conditions. For the former but not the latter case, we could apply a convolution 

 integral. However, the body boundary condition is satisfied exactly in both cases. 

 In many cases, the flow field near an arbitrary body is of interest, and eigen solu- 

 tions with linear free surface conditions are known. Even in any large unsteady 

 motion such as ship slamming, the free surface condition for a short period in the 

 beginning may be linear, then the convolution may be applied in the early time 

 period. Especially in the slamming problem, the peak pressure is known to be reached 

 early in the beginning and estimation of the early pressure distribution on the 

 slamming body is required. With this functional, we can find the solution for an 

 arbitrary body numerically by such methods as the finite element technique or singu- 

 larity method. Thus, a wide application of such functionals can be expected. 



12 



