Linear equations : 



3u Sn rcT\ 



— + g — - = (57) 



3t 3x 



^ + d — = (58) 



3t 3x 



Finite- amplitude equations 



3u 3u 3n _ rc m 



— +u — + g — - = (59) 



3t 3x 3x 



_3n + 3[(d + nj u] = Q 



3t 3x 



(60) 



Boussinesq equations : 



3u 3u 3n 1 ,2 3 u . ,,., 



— +u — + g — = — d t (61) 



3t 3x 3x 3 9x 2 



lH + 3[(d + n) u] = Q (62) 



at 3x > 



In addition, for waves traveling in only one direction, the Boussinesq 

 equations may be reduced to the Korteweg-deVries equations which are 

 then written as 



2 M + 2(gd) 1/2 ^ + 3u ^ + i d 2 (gd) 1/2 ^ = (63) 



, = /dV- u + ui___3xi_ (64) 



W 4g 6(gd) 1/2 



When considering the means of describing the propagation of long- 

 period waves, the parameter, U, should be evaluated where U is 

 defined as 



-I® 



(65) 



and sometimes referred to as the Stokes or Ursell parameter. The 



importance of this parameter was first noted by Stokes (1847) when he 



stated that the parameter must be small if his equations were to remain 

 valid for long waves. 



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