where n(a,t) = (n^jny) is the unit normal directed into the fluid given by 

 (n^.ny) = OBy/8a, -9B^/9a)/ [OB^/9a)2 + OBy/8a)2]l/2 



The required boundary condition for ij) at (Bx(a,t), By(a,t)) on the body 

 contour is thus 



d(|) 



— = (|)xnx + <t)vny = nx3Bx/9t + nySBy/St (6) 



9n ■" ■' ■' 



The free-surface coordinates xp(e,t) and yp(e,t) have been 

 parameterized in terms of e. The parameterization e is chosen such that for 

 fixed e the functions xp(e,t) and ypCejt) describe the path of a fluid 

 particle. In other words, e is a Lagrangian variable. The velocity potential 

 on the free surface is also parameterized in terms of the Lagrangian variable 

 e by (f) = (t)p(e,t). An equation to be satisfied by (})p(e,t) can be obtained 

 from Bernoulli's equation, which can be expressed as 



p + 9(|)/9t + ((t)2 + <j,2)/2 + (g/a2L)y = (7) 



X y 



where p is pressure, g is the acceleration of gravity, a is the frequency of 

 the forced harmonic heaving, and L is the characteristic length. Bernoulli's 

 equation is valid on the free surface and throughout the fluid region. At the 

 free surface, the pressure is assumed to be zero and a particular case of 

 Bernoulli's equation, the dynamic free-surface boundary condition, results: 



dlj)^ 04" 9 9 



— ^ (e,t) = — (xp(e,t),yp(e,t),t) = (4.^ + <t>^)/2 - (g/a^L) yp(e,t) (8) 

 dt Dt '^ * X y " 



Here D/Dt = ^■^'b/'d-x. + iJiyS/Sy + 9/9t is the derivative following the motion 

 of a fluid particle. The kinematic free-surface boundary condition, which 



