Variational Approaches to Steady Ship Wave Problems 



P=T-V-\V ct){x,. + c|>») dS. (3.7) 



Assume that <j) is harmonic and, for simplicity, assume that the 

 integral over an inspection surface at infinity vanishes. Making use 

 of Green's theorem we have 



P = - yy^ [ <|)x + i<V4>)^] dT - ij J C,^ dx dy 



= --\\\ pdT+ Const, (3.8) 



'D 



where 



Const = |ll H^dxdy-|\l z^ dx dy, (3.9) 



and H is the depth to the bottom. 



Taking the variation, we have 



6P = i C \ p6v dS - \ \ 6.|>(4>^ + x^) dS. (3. 10) 



H J Jp J Jp^5 



Therefore, when 



p=0 on F and 4>j,+x^=0 on S and F, (3.11) 



P is stationary. This result was first given by J. C. Luke [12,13] , 

 who pointed out that the volume Integral of the pressure is equivalent 

 to the Lagrangian, 



Furthermore, we may write (3.8) as 



P = M - H, (3.12) 



where 



H = T + V (3.13) 



and 



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