Computation of the Motion of Long Water Waves 



p 1 1 



Ujj = Ujj - 2 a^{ui^,j - Uj.ij) - 2 P{uij^, - Ujj.,) 



"2 R(^i + li - ^i-lj) (25) 



"2 R(^ijH- ^ij-|) (26) 



PI 1 



^ij = "nij - -J ^(iH*lj - "Hi-lj) - 1 P('nij+| - "Hij-i) 



- 2(V+R)(uj„j - Uj.,^) - ^(v+R)(v,.j,, - Vjj.,) (27) 



The remaining Eqs. (19-21) of the corrector step were treated in a 

 similar manner; where P values appeared, the values from Eqs. 

 (25-27) were introduced. To test the resulting linearized form of 

 Eqs. (19-21) we introduced the Fourier component solution (or error) 



_ —>}: i(er, x + o-jY) ict 



W = W e . e 



where we seek to determine if <o values, either real or complex, 

 exist such that W is a solution of the difference equation and where 



= constant 



for representative wave lengths \\ and \2 ^^ ^^^ ^" ^^^ y-directions 

 respectively. Now, let fx = e"^^* so 



— ^ — ^* n i(o",x+CT--y) . ^^ 



W = W fi e (28) 



at any point in time and space. Thus, we insert Eq. (28) in the 

 linearized u, v, ri equations and obtain 



167 



