Bessho 



IV. THE LINEARIZED PROBLEM 



The variational problem for Ps (3.20) Is not satisfactory 

 since there is no reciprocity relation for this form. We must intro- 

 duce the reversed flow potential as was done for Flax's principle. 



Let us consider the integral 



L*(<f>,,*$2) = L*(^2.<i>i) = - i^yy v<^,v^2dT "lyy cia^xdy. (4.1) 



Assuming that <j>i and <t>2 are harnnonic and satisfy the free sur- 

 face condition, we have, by Green's theorena, 



L*{<j>,,'J2) = - i jy Vzz^dS = - i yy ^^iv dS. (4. 



2) 



where S is the surface of a submerged body. This is the recipro- 

 city theorem for a submerged body. [ 8] 



If ^1/ = - 4>v» then 



L*(4>,^) = i ^ y c|.(|)^dS= L(<^,(^), (4.3) 



where 



L is called the modified Lagrangian integral [ 5] , Note that L(<|>,(j>) 

 has a finite value in the linearized case but not in the finite amplitude 

 case. 



If S is the wetted part of a surface -piercing body which is 

 under the waterline before the free surface is disturbed, there is an 

 additional term from the surface integral, [15,16,19,20,21] The 

 reciprocity theorem. In this case. Is 



L*(4»,,'$2) = -i ^L *'^2 ^^ ' ' ^^S ^1^2,. dS 



= iC \C, dy-i Cr ^2*,, dS. (4.5) 



556 



