Variational Approaches to Steady Ship Wave Problems 



where we have used (A. 1) and integrated (A. 7) by parts. We have 

 also assumed that the potential and surface elevation are continuous 

 over S and F, including L. 



Making use of asymptotic characters (A, 5) and (A, 6) of S, 

 we obtain the asymptotic expansions of ^ as follows: 



*<P>x-^ri 2??A^T^B, (A. 9) 



Zirr 



where r = PO and 



^"jj *'''^^'*"gj *'''^^' (A. 10) 



B=yj [<j>^- 4kJ dS -ij [<t,-x4>Jdy. (A. 11) 



The expression for A may also be written as 



A =yy 4>^dS -ijj* 4>xxdxdy =yj <f>^dS +yy 4>2dxdy, (A.IO') 



by using (A. 3). Thus A is the total outward flux from the water 

 domain. This must be zero; otherwise, we would have a large source 

 of the resistance other than from the wave and splash. 



We also have the kinematic condition on the surface of the ship, 

 <j>y= - Xy on S, (A, 12) 



Therefore, 



yj <f>^dS=-^y x^dS = 0, (A. 13) 



where S is the wetted surface of the ship below the undisturbed 

 water surface. From (A, 10) and (A, 13), we have 



1\ <}>,dy = - \ ; dy = 0. (A. 14) 



But this condition is not adequate in practical cases. One way to 

 avoid this difficulty nnay be to take the real wetted surface as S, 

 On the other hand, for the consistency of the theory, it nnay be 



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