Using the dynamic boundary condition on the free surface, one finds: 



n(x,t) = - i {|i + i V(t>.V<j)} on y = n. (E-10) 



g dt Z 



Substituting the expansions into this equation yields: 

 en^^^ (x,t) H- ^2^(2) ^ 3 _ i ^|i 1 ^^.^^} 



- ^|-{|^^V(^-v*} -^OCn^). (E-ll) 



g -^y 3t 2 y^Q 



Substituting for <j), the right-hand side becomes: 



1,3*^1) 2 3*^2) ,2 (1)2 a^Cl) 2 



2 (1) -2,(1) _ 



_^JL_{9i__} . oieh, ony = 0. 

 g 3y3t <- ^ > / 



First Order e: 



,(l)(,,t) _lil^!lXx.O^. (E-12) 



^ g 3t 



2 

 Second Order e : 



n^^^rx tl - i(n^^^ iiill^ ii^} _ JL(f9i^/ ^ f^t^l^. 



n (x,t) - - g in g^g^ + -j^-i 2g u— 93^) + C-^^3 i 



on y = 

 or 



n (,x,tj - " g ^- g at 3y3t 3t ^" 2g ^ •■ 3x '' 



3<f>(l) 2 

 + (^V~^ ^ on y = 0. (E-13) 



Using the first-order relationship above in the second-order boun- 

 dary condition on the free surface (E-9) , one finds : 



(E-14) 



,2,(2) ' 3^(2) _ 



. 1 3,C1) 3 32,(1) 3^(1) 

 g 3t 3y ^ 3^2 S 9y 





" 3x 3x3t 



3y 3y3t 



150 



