so that the three-dimensional part vanishes when the frequency becomes 

 infinite, and the fluid motion becomes purely two-dimensional for which 

 the strip theory holds exactly. 



Now let us consider the boundary value problem. When a ship is in 

 heaving and pitching oscillations, each transverse section has a vertical 

 velocity V(x) e , where V(x) is related to the mode of the oscillation. 

 When the slender body approximation is employed, the boundary condition 

 satisfied by the velocity potential $ e at the surface of the body is 



to 7 " " V(x) 3n"' 



(163) 



If we introduce the expression for $, given by Equation (161), the boundary 

 condition can be written in the form 



,(2D) 



3n' 



(x)- -| K I a^(x')N(K|x-x' |)sgn(x-x , )dx' 

 -I 



dz 

 3n' 



(164) 



Here we introduce the solution of the two-dimensional problem of a heaving 

 cylinder for which the boundary condition at the body surface is 



,(2D) 



3n f 



dz 

 9n' 



(165) 



With this solution, we put the coefficient of the source term as 



a " A o 



(166) 



Then the coefficient a n for the boundary condition of Equation (164) 

 satisfies the equation 



a Q (x) = 



(x)- - K aJ(x')N(K|x-x' | )sgn(x-x' )dx' 



-I 



A Q (x) 



(167) 



61 



