y^"^ = ygCek, nAt) 

 Bk ^ '^ 



(20e) 



for k = 1, . . . , M. (The subscript j will be used for a free surface 

 variable and the subscript k for a variable along the body contour.) Thus, the 

 boundary functions are discretized into boundary grid points that are now 

 assumed to move as if they were associated with fluid particles. The 

 evolution Equations (8), (9), and (10) for (j)p(e,t), xp(e,t), yp(e,t) and the 

 evolution Equations (14) and (15) for XB(e5t), yB(e,t) are applicable to these 

 particles and are replaced by finite difference equations based on the Euler- 

 modlfied method. The finite-difference equations are given by 



((,(n+l) = ((,(n) + Atl 

 Fj Fj 



where 



x(n+l) = x(n) + At[4)(n) + (t)('^+l)]/2 

 Fj Fj xj xj 



y(n+l) = y(n) + At[(t)(") + (l)(n+l)]/2 

 Fj Fj yj yj 



,(n) 

 xj 



All 



- (g/La2) 



x(n+l) 

 Bk 



= x(n) + At 

 Bk 



<),(n) + ^(n+1) 

 xk xk 



/2 



y(n+l) 

 Bk 



= y(n) + At 

 Bk 



^(n) + ^(n+1) 

 yk yk 



/2 



|)(m) = ((, (x(m), yC"!), mAt) 



x^ X Gl GH 



(,(m) = (j, (x(m), yCni), mAt) 



yi y GZ Gi 



(21a) 

 (21b) 



(21c) 



(21d) 

 (21e) 



11 



