Thus, as in the previous section, we can easily derive the corresponding linear free 

 surface boundary value problem using Equation (5) with the exact body boundary con- 

 dition in Equation (11). 



USE OF CONVOLUTION 

 Equations (6) and (12) are Lagrangians in a time-dependent, two-dimensional 

 space with nonlinear and linear free surface conditions, respectively. They could 

 be localized in space but not in time. Namely, we had to specify the conditions on 

 S at t = and t = t with a sufficiently large T. On the other hand, we did not 



require the initial condition <J> = at t = 0. In addition, such time T when t)> 1 = 



3 

 on S may be too large for practical use. For linear free surface boundary condi- 

 tions, we can treat our variational problem in the same way as those who have treated 

 vai 



by 



variation principles for linear initial value problems ' using convolutions defined 



> 1 * <f> 2 = 4> 1 (x,z,t) 4> 2 (x,z;i-t) 



dt 



1 + —3 

 i 8z 



We change Equation (12) to 



+ 



S S 



If we use the identity relation 



J 



1 y 2 v 2 1 



(14) 



J =11 ^ V *i * ^i dzdx ~Ji ] *1 * *ltt dx 



J" \ **i^-J (v i* 2 ) * *2n dz (15) 



