nonlinear free surface conditions. For the linear free surface problem, a func- 

 tional for the variational principle is obtained with a convolution rather than a 

 general integral. The convolution cannot be applied to a nonlinear free surface 

 problem; the condition is required for large values of time. A nonlinear solution 

 derived using an iteration scheme having the linear convolution form is also dis- 

 cussed. The time integration can be eliminated if the motion is sinusoidal. 



This formulation can be applied to problems of water entry and body slamming, 

 as well as to arbitrary body motion. 



NONLINEAR PROBLEM 

 Since problems dealt with here can be generalized easily to three-dimensions, 

 for simplicity we first consider a two-dimensional problem in the rectangular Car- 

 tesian (x,z) coordinate* plane. When a body whose surface is represented by 



S^ [z=h(x,t)] (1) 



enters the water surface S (z = 0, t < 0; z = h(x,t), t > 0) at time t = 0, or when 



r — 



a semisubmerged or fully submerged body starts to move at time t = and either exits 



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 the water or stops moving at t = t , then the boundary conditions for a velocity 



potential cf> are as follows: 



) = for t < everywhere 



i (V*) 2 - d> t + gh = on S p (2) 



h - V<j>V (h-z) = 



' on S p and S (t) (3) 



Here, S (t) is the submerged body surface varying with time t, and n is the normal 

 direction into the fluid. We consider potentials <f> 1 in the domain D and (J>„ in the 



^Definition of notations are given on page iv. 



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