h 



Salve sen 



- h 1 



lim r = lim T- (Ut7 - U/Se'^"' cos vx + Uh) 



h^co h 



1 1^/32 



= lim U-^^U U. (A13) 



Hence, condition 2 is also satisfied, and a Stokes wave is therefore a possi- 

 ble wave behind a two-dimensional body. 



Having concluded this we may now obtain the wave resistance for any two- 

 dimensional body in terms of the Stokes wave far behind the body. The "exact" 

 formula for the wave resistance, as derived in Appendix B, is 



where x^ denotes any vertical plane behind the body. Evaluating this expres- 

 sion far downstream, using the velocity potential, Eq. (Al) and the wave eleva- 

 tion, Eq. (A4), it is seen that the wave resistance correct to the third order in /3 

 is 



R = jPg/32 + o(/?'»). (A15) 



Let us now compare the wave resistance, Eq. (A15), obtained by the second- 

 order Stokes wave to the resistance obtained by the linear theory. The linear- 

 ized velocity potential and wave elevation are 



(D = Ux - Vae^y sin i/x (A16a) 



and 



Tj = a cos vx , (A16b) 



where a is the linearized wave amplitude. The well-known wave resistance for 

 the linear theory is therefore 



R=\pga'. (A17) 



It is important here to distinguish between the linearized wave amplitude a 

 and the second-order Stokes wave amplitude /5, If these amplitudes are mis- 

 takenly set equal, we have by Eqs. (A15) and (A17) the incorrect conclusion that 

 the linear theory and the second-order Stokes wave both yield the same wave 

 resistance. 



In fact we have that 



/S = a + 0(a2) , (A18) 



626 



