Chen, Divoky, and Hwang (1975), using a stability criterion obtained 

 by Benjamin, Bona, and Mahony (1972), use a higher order solution for 

 the amplitude when 



d < 



(20 At) 1/3 



where the variables are expressed in dimensionless form. The solution 

 then becomes 



v.*- ,, w.;lta[{ Cd * ,,) "U.*-l td * ,0H U*] 

 [l w> "'- T L*»i i { c4 .* ,0 *}rf.*-ii 



At (higher order derivative terms) (145) 



At 

 2 Ay 



The higher order derivatives are approximated by central difference 

 equations as follows 



3 3 u _ 1 



3x3 = 2 (Ax) 3 [u j+2,k ~ 2U J+I,k + 2U J-I,k " U J-2,^ 



— -— — [v., " 2v-.\ + 2v. . -v. 7 J 

 3y 3 2 (Ay) 3> k+2 J' k+l 3 > k ~ l 3> k - 2 



(146) 



(147) 



3 2 d 



- — [d. , - d. , -d. , + d. . J (148) 

 xAy L .7+l.fe+l .7-l»fe+l J+i,fc-l j-l,fe-l J J 



3x3y 4AxAy j 



etc. Computed surface elevations were smoothed when one of the following 

 conditions was satisfied: 



(a) A crest or trough has wave amplitude less than 25 percent 

 of the maximum wave amplitude at that instant; 



(b) the local velocity component (u) or (v) has a different 

 sign from the average value of the surrounding four points; 



(c) at a matching point where equations change from linear to 

 higher order equations . 



Smoothing is accomplished by the average 



65 



