where fi 2 ( x i> ^0 is an arbitrary function arising from integration. The 

 momentum equation is 



9^ 2 . dq 1 9q x 9n 2 9n 2 



+ u, + v + + = (91) 



9t 1 3Xj 1 9y 1 dx l 9y x > J 



and the continuity equation is 



9n 2 3(Q 2 ) x 3(Q 2 ) y 



9t x 9x x 9y x 



(92) 



Peregrine (1967) points out that second-order terms will have first- 

 order effects where t-^ is not of small value. He accounts for these 

 effects by incorporating second-order terms into the first-order variables. 



Mei and Le Mehaute (1966) derived a solution for waves propagating 

 in one direction which gives the equations as 



ill i. „ 3r l , m . _-, 3u d 3 9 3 u _ , „ 3u /3\ l9 3d 9 2 u 



9t 



and 



9u 9u 9" 'i ^ * z 



— + u — + — 



at ax a 



where 



and 



-t^d.n)f-|l|^= to ,B|H + (|) d2 ||iLH t93) 



D. (k\ _ $L 9 3 u _ _9_ / 3d\ _3u 

 x \d/ 2 9 2 x9t 9x V 3x/ 9t 



V 3x/ 9x9t 9x ( - y4J 



, _ 9d /9d\ 3 , 9d 9 2 d x d 2 9 3 d ,__, 



B = fl7( d2 |f) C96) 



It can be seen that equation (94) has mixed derivatives, with respect to 

 x and t, where equation (63) has a third-order derivative with respect 

 to x only. Benjamin, Bona, and Mahony (1972) show that the Korteweg- 

 deVries equations with mixed derivatives, such as equation (94), are 

 the preferred form for describing the behavior of long waves. 



Street, Chan, and Fromm (1970) expanded on the work of Peregrine, and 

 for waves propagating in one direction give 



au 3u 3n 1 d 2 a 3 u , d 9d 9 2 u 1 9 2 d 9u fg?1 

 at 8x 3x 3 at3x 2 8x 9t8x 2 3x 2 3t 



and 



f ♦fj[(d + n) u] - (98) 



53 



