Computation of the Motion of Long Water Waves 



^i^j =^i.li ^St<-u^- v^).,^. (38) 



A^k =^iM ^St< -u|^- v|X>,^,^ (39) 



where Uj^^j and Vjj^^ contribute to UjAj and Vji+^ , respectively, 

 through the convection process and ( ) represents the as yet un- 

 specified finite difference approxinnation of the enclosed terms. 



Before Eqs. (37) can be employed to compute the new velocities, 

 the p field must be obtained. Consider the finite -difference conti- 

 nuity equation [ see Eq. (35)] 



n*l n+l n+l n+l 



,.,,:Mi,p^.:vr_i±L_.o (.0, 



Substituting Eqs. (37) into Eq. (40) and requiring d|^*' = leads to 

 the pressure equation 



= 1/Pi^li +Pi-li 4- Pij^l •^Pij-l +R. \ (41) 



Here 



'-'^h*^) <*^> 



* * 



_ 1 r ^i4i -^4i , ^ii-i -^iJ-a" 1 



^ij ' 6t L 6x 6y J 



(43) 



Near the free surface "irregular stars" (Fig. 8) must be used to de- 

 rive an appropriate pressure equation so that, in the discrete sense, 

 the free surface condition p = Pg is applied to the exact location. 

 Let T||, T)2, "Hj, T|^ be the lengths of the four legs of the irregular 

 star (Fig. 8) and p, , P2» p,, p^ be the value of p at the ends of these 

 legs. Then, it can be shown the irregular-star pressure equation is 



Equation (44) reduces to Eq. (41) when Eq. (44) is applied to an 

 interior cell. 



173 



