a position on the free surface or may move to a position on the body below the 

 surface. In other words, the Intersection point may not move with the fluid. 

 Since the free-surface and hull boundary conditions all involve time 

 derivatives following the fluid motion, they cannot, in general, be directly 

 applied over a time interval to predict quantities at the intersection point. 

 Special methods for handling this point must be developed. However, for 

 heaving motions of an almost wall-sided body, the intersection will be 

 essentially a fluid particle. Therefore for the current study, the body 

 Equations (14) and (15) are applied directly at this point. Inaccuracies in 

 this approach will become apparent in the form of a deviation from zero of the 

 pressure at the intersection point. In fact, such pressure deviations might 

 be used in a method to more accurately follow the intersection point as would 

 be necessary for more complicated body shapes. One such scheme has been 

 tested but has proved to be numerically unstable. 



The pressure at all the grid points along the body including the 

 intersection is calculated from a finite-difference version of Bernoulli's 

 equation: 



p(n+l/2) =/L(n)l2 + L(n)1 2 + L(n+l)1 2 + ["^(n+l)"! 2\/4 



- (g/a2L) [yCn+D + y(n)1 /2 (25) 



The pressure is the pressure at the k-th fluid particle on the body at time 

 t = (n+1/2) At. The force on the body at this time is calculated by 

 numerically integrating the pressure along the body contour. Trapezoidal 

 quadrature is used. 



15 



