where 



U 3*l/Z,k ~ U j-l/2,k + a 2 U J-3/2,fe " Vl/2,^ 

 . (a " I) 2 



CU J-3/2,fe- 2U i-l/2,fe + V/2^ C135) 



At 

 a " U j+l/2,fc Ax 



At 

 1 3+1/2, k Az 



(136) 

 (137) 



Similar expressions can be developed for other convective terms. 



For points near the free surface (Fig. 15), Chan and Street (1970a) 

 give the equation for ?j ^ as 



w 3 \ 



3>k 2(6 6 + 6 6 ) 

 2 k 13 



6 3P1 + 6 1P3 



\pz + S 2ph 



V.{^) V.(^) 



+ R 



J,k 



(138) 



The free-surface position, as given by Chan, Street, and Fromm (1970) 

 at time t + At is 



n J ( *3+l ' n J-l\ 



— = w . - u . I I 



At J J \ 2Ax / 



At 



(139) 



where nj'(t) is the elevation at time t, and Uj and w; the horizontal 

 and vertical velocities, respectively, at the free surface at time t + At. 



For refracting waves propagating in two dimensions in the plane of 

 the water surface, Chen, Divoky, and Hwang (1975) give the equations below 

 using dimensionless expansions similar to those proposed by Peregrine 

 (1967) . A time-staggered scheme is used, with the velocities and wave 

 amplitudes calculated explicitly at alternate time steps of At/2. The 

 amplitudes at t Q + At/2 will be calculated using amplitude at t - At/2 

 and velocities at t • then, the velocities at t Q + At will be calculated 

 using velocities at t Q and amplitudes at t Q + At/2. At time t Q + At/2, 

 the amplitude is 



*3,k - *3,k ~ Mx" [ {< d + n) "V^ - {(d ♦ n) u)._ ik ] 



■w , [l (dt,o7, if>i" ( Cd + n)7 L^-i] C140) 



63 



