Variational Approaches to Steady Ship Wave Problems 



E*= D - I. (1.7) 



Therefore, Flax's principle produces an approximate solution which 

 makes the error integral (1.6) stationary. [23] 



This method suggests powerful means for obtaining approxi- 

 mate solutions , but unfortunately it has been applied only to thin 

 hydroplanes and wings. [ 7] 



n. GAUSS' VARIATIONAL PRINCIPLE 



In this section, we assume there is no free surface. Then 

 the velocity potential has the following representations for the source- 

 sink and doublet distributions: 



*i'P) = ?il£ ^ri^ <^^<°>' i = ci.2.--- <2-') 



and 



<i>i(P) = ^yy M-i(Q)-|7 7|^7QydS(Q), i = 0,1,2,... (2.2) 



Here, quantities with the suffix zero stand for the correct solutions 

 while those with other suffices are not necessarily correct. For these 

 potentials we have the following reciprocity relations: 



S "^ ■■'s 



and 



yy ct),o-2ds=yy or,4,2ds, (2.3) 



yy H2^l/S = y[ J^,ct^2,dS, (2.4) 



U 4,,cf>2^dS=yj <f>2*,^dS. (2.5) 



Gauss's variational principle for the Dirichlet problem states 

 that if we consider the functional 



G=iyy ((j>- 2f)o-dS, (2.6) 



where 



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