[|x(Hi/D)]^ = (Hi/D)7 - (Hi/D)g, (61) 



where numeral subscripts indicate the points where the ratios are 

 computed. 



For the external mode computations, the continuity and momentum 

 equations are alternately applied at points 2 through 6, while the 

 internal mode computations start at point 4 in the same sequence. 

 The summation of the coupling terms for one complete operation, after 

 eliminating like terms, is 



-eg[U3]3 ^{[^i]2[|x(Hi/D)]E} 



(62) 

 -eg[\te]2 iii^ih^k^^l/^^h + [Ui]i[|x(Hi/D)]p} 



-egL^elg J{[Ui]7[|5(Hi/D)];^}. 



The internal mode computations start at point 4 as discussed earlier. 

 Hence the first and third terms in (62) vanish. The residual terms 

 depend upon the condition at point 7. If point 7 is a solid 

 boundary, then [Ug]^ = I'^^.^-j - and the net energy transferred 

 between modes is zero. The second possibility that point 7 is an 

 open boundary requires that either the depths at point 6 and 7 are 

 set equal or the gradient, C"§x^^l/^^^A ^^ ^®^ ^° zero. In fact the 

 two choices imply one another. It is more rational to set the 

 gradient H^^/D to zero since all these gradients are computed only for 

 the purpose of evaluating the coupling terms. If the interface 

 intersects the bottom at point 7, then the external mode computations 



30 



