Variational Approaohes to Steady Ship Wave Problems 



APPENDIX C 

 A Variational Principle for the Stream Function 



In the two-dimensional case, we may use a stream function 

 instead of the velocity potential. Let us introduce the stream 

 function as follows: 



<^^(x,z) = i|j2(x,z), <^2(x,z) = - 4ijj(x,z). (C.l) 



Then, the boundary conditions for ijj become 



4;^(x,0) - gi|j(x,0) = and $j(x,0) - g4^(x,0) = 0, (C. 2) 



i|jo(x,z) = - iPq(x,z) = - z on S, (C.3) 



;(x) = - i|;(x,0) and ^(x) = i|j(x,0). (C.4) 



Introducing a modified Lagrangian integral, 



L*(4;,,4;2) = -i J^ V4i,V52dxdy -|J C,?2dx, (C. 5) 



we have, directly, the reciprocity 





y i|;,52^dS= J Vi^dS 



(C.6) 



'S »^s 



In particular , from (C.3), 



= ijj (Vi^o)^dxdy - ij i;^^dx= L(4)o,i|Jo). (C.7) 



The variational prcblem. with the function 



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