he calculated the values of u and n with a time-stepping procedure, 

 first using an approximation to the continuity equation which gives 

 n* , a provisional value of n +1 , by the equation 



n - n 

 r,s+i r 3 s 



At 



+ (ax + n ) 

 r 3 s J 



u - u 

 g+1 ,s r-i,s 



2Ax 



u \ " r+1 > 8 ' Vl ' s + a ] = o 

 ^s L 2Ax J 



(126) 



Then, u is calculated from an approximation to the momentum equation 



u -u u -u +u -u 



r,s+l r,s r+i,s+i r-i ,s+i r+i,s r-i,s 



At 



4Ax 



n , - n + n - n 

 p+ljS+l r-l,s+l r+l,s r-l ,s 



4Ax 



2u 



(ax) ' 



+ 2u - u 



Ax 2 At 



u i i _ u •, , - u , +u 

 2 r+l ,s + i r-l ,s+l r+l ,s r-l ,s 



AxAt 



(127) 



Finally, the continuity equation is used again to give an improved value 

 for ^s+i 



n - ri u -u +u -u 



r,s+l r.s , -. r+i,s + l r-l ,s+l r+l ,s r-l,s 



— — + (ax + n ) 2 — 



At r>s Ax 



u , + u 

 r,s+i v ,s 



2Ax 



(128) 



Street, Chan, and Fromm (1970) and Chan, Street, and Fromm (1970) 

 extended Peregrine's work, using a Marker-and-Cell numerical technique 

 to obtain solutions for waves propagating in one direction. Where values 

 are known at time t, they compute the values of u and w at time 

 t + At using the equations 



At 



U J+l/2,fe = U i+l/2,fe + %x At + Ax" ( Pj,fe " Pj+l,P 



(129) 



At 



V.fc+1/2 = "lk+1/2 + g 2 At + H^j,k ~ P J,k^ 



(130) 



where the coordinate system is shown in Figure 14, p is pressure, g 



61 



