Variational Approaches to Steady Ship Wave Problems 



When 4*1 - 'I'* T2~ T ^^^ ^v- - ^v> L becomes 



I-*(<|>,^) =i J ^^x^dS + i^J ^x^dS, (4.6) 



where n is the inner normal to the waterline curve L in the hori- 

 zontal plane. Thus, the first term in the right-hand side of (4.6) is 

 the correction for the change of the wetted surface S. [16] This is 

 justified, on the one hand, by the dynamical meaning of the Lagrangian 

 and, on the other hand, by the linearization procedure of the pre- 

 ceding section. 



For the case of a pressure distribution over the water surface, 

 we may integrate (4. 5) by parts and make use of the formulas in 

 Appendix A. This results in the expression 



= ill ^P' ^ p^^'^^2 ^" ^^ = ill ^^2 "■ pgy^. ^- ^y- 



(4.7) 

 Thus, the reciprocity becomes [8] 



^*tP. 'P^' = h^L P,?ad- dy = ^|C U,^ dy (4.8) 



where 



X*(p,,52) = L*(^^,'ij - f yy ;,?2 dx dy. (4,9) 



Making use of these reciprocities, we may easily show the 

 equivalence of the boundary value problemi to the variational problem 

 for the functional I , where 



S 

 for a submerged body, and 



I* = iyy [<l4v- (^-^W dS, 



(4.10) 



I* = 



s 



for a pressure distribution. [ 24,26] 



557 



i^yy [pl- (p-p)Co] dxdy, (4.11) 



