Current Progress in the Slender Body Theory 



p = icopcpe , 



and thus the six forces and moments are 



- i(Dt f f 

 F^ = - icope I 0COS (n,x^)dS, 



(2.9) 



(2.10) 



where cos (n,x.) denotes the direction cosine for i = 1, 2, 3 and the generalized 

 direction cosine 



X..2 cos (n,x..j) - Xj.j cos (n.x-.j) 



for i = 4, 5, 6. Substituting (2.7) and (2.8) in (2.10) it follows that 



( WALL) ( FK) 1 - icJt r f 



Fi = F\ + F\ ' ^ ^icop e J J dS cos (n,x.) 



Q(^) Gi(x, 0,0:^,0,0) d^, (2.11) 



-'L 



where F^'''^^ denotes the "Froude-Krylov" exciting force from the undisturbed 

 incident wave potential 0j. The last term in (2.11) contains all of the free sur- 

 face effects due to the presence of the body. This can be reduced further by 

 noting that 



Gi(x,0,0;^,0,0) = -ttK {Ho(K|x-^|) + Yo(K|x-^1) - 2iJo(K|x - ^| )} , (2.12) 



where Hq , Yg , and Jg are the Struve function, Bessel function of the second 

 kind, and Bessel function of the first kind, respectively. 



One important consequence of (2.11) is that for transverse oscillations 

 (sway, roll, and yaw). 



(WALL) (FK) 



F: = F + F 



(i= 2, 4, 6) . 



(2.13) 



Of course higher order terms including free surface effects could be re- 

 tained. In particular the damping coefficients for sway, roll, and yaw can be 

 found fairly easily from the energy flux at infinity (Newman, 1963), in the form 



27T 



<iJ/oK' 



d(9sin2 6' 



I 



iKx CO s 



S(x) + m .(x)/p 



S(x) Zo(x) + J2 B^(x) - m^^Cx)/^ 



X S(x) + m,2(x)//9 



Vdx 



(2.14) 



where mjjCx) and m2 4(x) are the two dimensional added mass coefficients of the 

 section for the sway force due to sway and the sway force due to roll. 



141 



