Current Progress in the Slender Body Theory 



6 



.(FS) 



'[7 = E'i^i. (3-14) 



j = l 



with fj = fj(x,aj) as the transfer function between motion in the jth mode and 

 the free surface interaction potential at station x . A similar relationship would 

 hold for the wall interaction f ( ^^^^^ ^ which is not of interest to us in the pres- 

 ent work. 



The free surface interaction transfer functions f j are obtained in Ap- 

 pendices I and II by the use of the hull boundary condition, the result being again 

 in the form of Eq. (1.2), i.e.. 



(3.15) 



where 



(3.16) 



•' - 00 



Qi(x) = -(-^'^ + " ^) ^'(""^ 



QaCx) = 



Q3(x) = -(-i^+ u£) BCx) 



Q4(x) = 



QgCx) = + f- iw + U ^J xB(x) 



QfiCx) = 



are flux transfer functions, and K^ ^ ^ is an absolute kernel independent of hull 

 geometry and defined by 



^(i)(''^ " -^ dke"''"'/3(k) coth /3(k) , (3.17) 



where 



, ^ , (-i« - ikU)2 

 cosh /3(k) = -!^ 



I 



m (for the case co real, see Eq. (A3. 6)). 



The kernel K^ i)(x) depends on the frequency oj and speed u as parameters 

 as well as the variable x, and will sometimes be written K^ i)(x, w,U) to emphasize 



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