Bessho 



Alternate representations for these Integrals are 



I* = L*(c},o,'^o) - L*(4>- <l>o» 4»- ?o). (4.12) 



and 



I = Z (Po,Po) - S, (p - po,p - Po), (4.13) 



where the suffix zero stands for the correct solution. These for- 

 mulas show that the variational principle extremizes the Lagrangian 

 of the error and that the stationary values are just given by the 

 Lagrangian. 



The difficulty arises in the case of a surface -piercing body. 

 From (4.12), the functional to be extremized is 



i* = 



L*(+. ♦) + i y (*^o - 't'U dy + i y J (♦ - 4>)x^ dS. (4. 14) 



Taking the variation, we have the boundary conditions equivalent to 

 this variational problem, 



4>x= - gCo» ^x= gCo OJ^ L, (4.15) 



<^i/= - 4>j/ = - Xy on S. (4,16) 



But we have no knowledge of the surface elevation on L, a priori, 

 as this problem may be indeterminant, [ 17,23] We must remember 

 here that the solution is unique only when the detachment points are 

 fixed by the theory of cavitation. [3,14] 



This difficulty may be avoided by introducing a homogeneous 

 solution for the two-dimensional, linearized case (see Appendix C) . 



For the present case, we might proceed as follows: Let us 

 consider the difference between a surface piercing body and the 

 limiting case of a submerged body moving very close to the free 

 surface as in Fig. 3. [ 23] The boundary condition on the water 

 surface above the submerged body must be <}>^ = 0, but since the top 

 is also the free surface, this Is equivalent to 



(|>^= ;x(x,y) =0 on F, (4.17) 



or integrating, we have 



<t>x(x,y,0) = - g;(x,y) = Const = func (y) on T. (4.18) 



558 



