Submerged Two-Dimensional Bodies 



and in general the complete second-order problem must be solved in order to 

 obtain p> correct to the second order. 



In other words, the Stokes wave theory gives us the form of the potential 

 (Eq. (Al)) and the shape of the wave (Eq. (A4)) correct to the second order; how- 

 ever, the magnitude of the wave elevation cannot be obtained without solving the 

 complete second-order problem. 



A very useful result can be obtained from the theory of Stokes waves. By 

 the wave profile, Eq. (A4), it follows that 



^ 2 

 where H is the wave height. Therefore the wave resistance, Eq. (A15), becomes 



R=|pj4T^0(H^). (A19) 



4 "^ V2 



Applying this result to experimental work, we have that Eq. (A19) gives the wave 

 resistance from measured waves correct to the third order in wave height. 



Appendix B 



DERIVATION OF THE "EXACT" WAVE RESISTANCE FORMULA 

 AND A NOTE ON A VARIABLE RESISTANCE PARADOX 



Consider a two-dimensional body in a uniform stream of velocity u, and let 

 us apply the momentum theorem to the fluid region bounded by the plane x = a 

 far ahead, another plane x= c behind the body, the bottom y = -h, the surface of 

 the body, and the free surface. The horizontal force on the body is then given by 



R= r (P + pOx^) dy - r (p + P<1>^^) dy , (Bl) 



•' x= a x= c 



where $ = Ux + 0. Introducing the Bernoulli equation, Eq. (3), and the continuity 

 condition 



r $x dy = I I'x dy , (B2) 



we have that 



R = pg J -y dy + pg I y dy + ^ r {^^ - <^^) dy " ^ f (^y'" "^x") ^V 



(B3) 



627 



