states that no fluid particle on the free surface can leave the free surface, 

 is expressed by the two equations 



dXp Dx 



—^ (e,t) = — (xp(e,t), yp(e,t), t) = (|) (xp(e,t), yp(e,t), t) (9) 



dt Dt * " ^ " ^ 



dyp. Dy 



— £■ (e,t) = — (xp(e,t), yp(e,t), t) = cj) (xp(e,t), yp(e,t), t) (10) 



dt Dt '^ " ^ " " 



At t = 0, the velocity potential on the free surface is given by 



(t)F(e,t=0) = (11) 



and the free surface is such that 



yF(e,t=0) = (12) 



The parameter e and the function XF(e,t) can be arranged so that 



XF(e,t=0) = e (13) 



It is convenient to parameterize the fluid at the body contour by a 

 Lagrangian variable e so that x = XB(e,t) and y = yB(e,t) along this 

 boundary. This is possible since fluid particles along the solid boundary can 

 never leave that boundary, except possibly to become free-surface particles at 

 the intersection of the body and the free surface. Thus 



dx^ Dx 



— 2. (e,t) = — (xp(e,t), y„(e,t), t) = (j) (xg(e,t), yB(e,t), t) (14) 



dt Dt "^ ^ ^ ^ '^ 



— S-(e,t) = — (xg(e,t), yB(e,t), t) = <}, (xg(e,t), yB(e,t), t) (15) 



