Salve sen 



And as a -»-oo it follows that the "exact" wave resistance formula under the as- 

 sumption of ideal fluid, irrotational flow, and negligible surface tension is 



41 



T)( C ) 



^x'(C'y) + ^y'c^-y) 



dy + 4g^'(0- (B4) 



We note that the plane x = c may be taken at any distance behind the body and 

 that V is the wave elevation measured relative to the undisturbed free surface 

 far ahead of the body. 



This expression is "exact" and therefore when applied to a linear problem 

 will yield the wave resistance correct to the second order and when applied to a 

 potential correct to the second order will give the wave resistance correct to 

 the third order. 



Sharma* shows that if the continuity condition, Eq. (B2), is not applied, 

 however, the wave resistance becomes 



R - y r (-0x' + -^y') dy + |- gT]^ + pV ^ U dy - pU J (U - J dy . (B5) 



x = c x = a x=c 



He then introduces the linearized velocity potential and wave profile and gets 

 the "variable resistance paradox" 



R - — a^ + pga^ sin vx . (B6) 



This paradox shows the kind of erroneous conclusions which follow from a 

 first-order theory if the equations are applied directly without a specific knowl- 

 edge of the order of magnitude of each term involved. Equation (B6) was ob- 

 tained using a linear theory; hence continuity is satisfied only to the first order. 

 Therefore any second-order terms resulting from the continuity condition have 

 no meaning and should have been disregarded by Sharma. 



''S. D. Sharma, "A Comparison of the Calculated and Measured Free-Wave Spec- 

 trum of an Inued in Steady Motion," in "International Senninar on Theoretical 

 Wave-Resistance," Vol. 1, University of Michigan, 1963, p. 255. 



628 



