Murthy 



The expansion in series may safely be expected to be convergent for 

 is of small order I (5 , j3) I and the derivatives of may al- 

 so be assumed to be of the same small order. It is clear from (4-2) 

 that f is also of small order and may therefore be treated as a 

 small constant for the purpose on the expansion. 



Substituting the above series in (4-5), we get 



dt dx dz 3t dx z r zz 



+ 2 v (4> + r* ) .v[V - v$ + r|- (* - V* ) 1 + 0(4> 3 ) = 



z L't x dz t x J 



on z = 0. 

 Now, from (4-2) with w = 



f = — (* - V* ) + (4> 2 ) 

 g t x 7 



so that we have an approximate condition 



<t> - 2V4> + V $ - g(b + 2 V*. V ($ - V* ) 

 tt xt xx eY z * t 2.' 



+ — (<t> - Vd> ) -|- («t> - 2V4> 4 + V 2 4» - g<J> ) + (<f> 3 ) = 



g t X dZ tt Xt XX ° z 



( 4 - 6 ) 



on z = 0. v ' 



IV. 2 Conditions on the Internal Free Surface (IFS) 



If there is a distribution of surface pressure p (x, y;t) on 

 the IFS, we have from Bernoulli's equation (l -8) with u> = 0, 



g 



<J> - V* + „ 

 t x 2 



T (V4>) 2 



p (x, y;t) 



+ -* 



P g 



z = r 



with x, y in the IFS. 



The kinematic condition is the same as that on the EFS and 

 given by (4-1) with o> = 0, 



126 



