SINUSOIDAL MOTION 



If we consider a sinusoidal ship oscillation such as h = f e for S of 



t s 



Equation (1), we substitute 



4> 1 = ^(x,z) e ia)t (21) 



into Equation (12), integrate with respect to t, and obtain 



' J l=JH V ^l dZdX -IiJ *l d * 

 D l 



+J f*j dx-J (Vl^^d. (22) 



S S T 



s J 



This is exactly the same Lagrangian that Bai and Yeung used. Working from Equation 

 (18), we can apply eigen solutions to the whole field by using 



S 



where 



s 



, , /2 itot 

 h / /h +1 = F e 

 t x 



9 

 A similar functional was used by Sao et al. to solve the problem of a heaving 



oscillation of a dock. 



ITERATIVE SCHEME 

 For problems with the linear free surface condition, we can completely localize 

 the numerical scheme in D with £ t < t for any X with the help of the convolution 



form. However, for nonlinear problems, the finite-element technique has to rely on 



2 

 an iteration. We may thus use an iterative free surface condition on z = 



in 



