Bessho 



^"JJJ ^'''^'^' ^^'^^^ 



M is the total momentum of the system in the x-direction and becomes 

 equal to twice T, 



M = 2T = - \ \ 4>XydS=\\ 4><f);,dS, (3.15) 



when ^ satisfies the boundary condition. 

 Hence , 6P = means that 



6H = 6M. (3.16) 



That is , when the variation of the total energy equals that of the total 

 momentum in the x-direction, the potential satisfies the boundary 

 conditions (3»11). 



For purposes of application, it may be convenient to write P 

 as 



P=-yy 4>(x^ + i4>»,) dS-fjl ^^dxdy. (3.17) 



This principle is applied to a regular, two-dimensional wave- 

 train in Appendix B. In general, there is some difficulty in the appli- 

 cation of this theory since the integrals P and L may not be finite. 

 This is because the kinetic energy exceeds the potential energy for a 

 finite amplitude wave. [1,2,4] 



One way to bypass this difficulty may be to assume a flow 

 model like the Riabouchinsky model [ 3] in cavitation theory (see 

 Fig. 2); however, this may be impossible in the three-dimensional 

 case. Another way may be to introduce Rayleigh's friction coef- 

 ficient so that the waves far downstream will die out. In any case, 

 there are still some problems which make us hesitant to begin the 

 actual numerical computations. 



Finally, let us consider the linearization of the free surface 

 condition. Neglecting higher order terms in the integral over the 

 water's surface and assuming that 



g;(x,y) = -c|>,(x,y,0), (3.18) 



we have 



554 



