street, Chan and Fromm 



For waves of the KdV type that we wish to study, U;^ = 0(1), 

 If 





€ = i1o/do (2) j 



and 



(T = do/Lo « 1 (3) 



for long waves, then from Eq. (1), U^^ = 6/o-^= 1 and € = cr^. This 

 is the relation on which Peregrine [ 1967] based his expansion-in-a- 

 parameter analysis. He pointed out also that d(x,y) = 0(1) and the 

 derivatives of d equal 0((r) are necessary restrictions; otherwise, 

 the variations in the depth of water are shorter than the incident 

 waves and tend to generate shorter waves, thus upsetting the scheme 

 of the approximations. 



Under the above conditions Peregrine [ 1967] obtained the 

 momentum equations 



U^ + UU, + VUy + Tl^ = ^ d[ (du),^ + (dv),y]^ - -^ d\ U„ + V,y]^ 



(4) 



0<x<L, , 0<y<L2, 0<t 



1 12 



v^ + uVjj + Wy + T|y = -^ d[ (du)^y + (dv)yy]^ " "g: ^ ^ ^ « ■*" ^yY^t 



(5) 



0<x<L|, 0<y<L2, 0<t 



and the continuity equation 



Ti^ + [ (d + ri)u], + [ (d + Ti)v]y = 



(6) 

 0<x<L|, 0<y<L2, 0<t 



These equations and the appropriate auxiliary conditions described 

 below are solved numerically by a straightforward finite difference 

 scheme that is presented in Section 3.2, 



The auxiliary conditions are the boundary and initial condi- 

 tions appropriate to Eqs. (4-6) for motion in a vertical- walled tank 

 (Fig. 1), For solid, vertical walls the velocity perpendicular to the 

 wall is zero at the wall and non-breaking waves reflect perfectly 

 from the wall. If n is the normal coordinate and U and V are 



154 



