Substituing the expansion for <t> into the free-surface boundary con- 

 dition: 



e ^y' ^ - . e ^^ . ge ^^ . g. -^ 

 3^(1) , 3^(1) . 2, 8*^2) ^ (2) 





(^) 2 36^2) 



+ e 



9y' ■■ 3t 3t ■• 

 r^ 3 -^ 3 -.1 ;3<i)'^^^ ^ 34) '-^-' t. 2;3(}.'^^^ -> li^-*-iT 



t^ 37" J 37V'^ 8ir^ '-^T^^ ' ^~^' "~^^^^' 



t^^ 3r- ^ " 37- J^ ' ' ^ 33r ^ " -17" ^^^J °^^ ) = '^ ^^-^^ 



on y = n . 



Now use the Taylor expansion for ())(x,ri,t) and neglect terms of or- 

 der e-^ in the boundary condition: 



, 3^^^^^(x,0,t) (1), ,, ^V\^ 2,V'^ 

 e{ — ^ ' -' ■ + en (x,t) — ^— J "^ ^ — 2~ 



3t 3y3t 3t 



+ ge{^=^ — + en^ (x,t) — ^-^5-} + ge ^ 



3y - -'- 3^2 - - 3y 



„ ,.(1) „2,(1) ,,(1) „2.(1) , 

 "■ 3x 3t3x 3y 3t3y 



Grouping terms by order: 



First Order e : 



^ . °v = on y = 0. (.E-oJ 



3t2 ' ^y 



2 

 Second Order e : 



,V'^ 3,t2) 3 2 (1) (1) CD 32 (1) 

 ---- — ^ ^ y -- + ri*- -" { — - + g — !— } + 2 -— — 



^2 ^ 3y 3y ^^2 ^ 3y 3x 3x3t 



dt dt 



.214^^=0 ony = 0. (E-9) 



3y 3y3t 



149 



