Submerged Two-Dimensional Bodies 



and the other, which has a phase shift t and an amplitude « of second order in 

 a, has the form 



7]2(x) = S COS y(x-T) + ... , (30) 



where both Eqs. (29) and (30) are correct to the second order in a. 



From the theory of Stokes waves (Appendix A) it follows that the velocity 

 potential for these two Stokes waves can be written as 



and 



0j(x,y) = - alle'^'^" ^ sin vx , 

 ^jCx.y) = - SUe^^^~ ^ sin v(x-r) 



(31) 



(32) 



Assuming now that the potential and the wave elevation far downstream can 

 be written as the sum of the two Stokes waves above, we have that 



ue'^^y-b) 



and 



a sin vx + S sin v(x-t) 



Tj - Tjy-k- Tj^ - a COS VX+ ~ va^ cos 2vx + 8 cos v{x- t^ . 



(33) 



(34) 



Note that the potential, Eq. (33), and the wave elevation, Eq. (34), are valid far 

 downstream, and that both are correct to the second order in a. 



The "exact" formula for the wave resistance derived by Havelock (8) and 

 by Wehausen and Laitone (6c) using energy considerations and also derived in 

 Appendix B by momentum considerations, is 



^M 



b + 7,(x ) 



R = -^ 



^x'c^o'y) + <^y'(''o.y) 



1 , 



(35) 



where c/>(x,y) is the "exact" velocity potential, t7(x) is the "exact" wave eleva- 

 tion, and Xq denotes any vertical plane behind the body. 



Evaluating this expression far downstream using the velocity potential (Eq. 

 (33)) and the wave profile (Eq. (34)) it can be shown that the wave resistance 

 correct to the third order is 



R = — pg [a^ + 2aS cos vt] . 



(36) 



Since by Eq. (34) the trough-to-crest wave height correct to order a^ is 



H = T7(0) - 7](7t) = 2a + 2S cos vt , (37) 



the resistance R can also be expressed in terms of the wave height 



607 



