Current Progress in the Slender Body Theory 



Now we view the force in each mode as the sum of contributions from displace- 

 ments in all modes, with transfer functions c. j as in Eq, (3.1), putting for the 

 free surface portions: 



^'r = E^ir^^i- (3.33) 



j=i 



Thus 



(FS) f 



\. ' = J dxS'(s) p.(x,a)) 



(FS) 



^' - J 



dxB(x) pj(x,aj) 



dx x B(x) P:(x,co) 



In terms of the f j of Eq. (3.15), the non-zero c^^^^^ are 



C[[^^ = -p|dx S'(x) (-io; + U ^) fj - pujdx S'(x) i[l^^^ " (x) 



^(FS) 



(3.34) 



-pfdx S'(x) (-ia,+ U^j £3 

 Cir' = -^|dx S'(x) (-ia;+ U ^) f, 

 C3i'^ = -p fdx B(x) (-io; + U ^) fj - pU fdx B(x) fjo^ V''^ ^3.35) 



-> ^ /J (Cont.) 



Cir' = -^Jdx B(x) (-1..+ U^) £3 

 Cir^ == -/'Jdx B(x) (-ia;+ U ^) f^ 



(FS) f / -^ \ f (FS) " 



Cjj = p I dxxBCx) -ioj + U :^ fj + /oU dx xB(x) fjo^ (x) 



151 



