= - £ c\_Lnlr> ^ 



9r 



= i cos e + 



(^^-57) t»i 



— -i- = 4.(2 - ZG)cose + o(r— -i: + o(R2y [9] 



or \ dz / 



or dz \ dz / 



^^A K7 r / ? dR\ 1 



= - Gje^^ cos e sin wt - ( KR cos'^ 9 + Icosoot 



ar [ \ dz/ J 



(94>A \ -, 



R^— j+ 0(r2) 



[11] 

 = CO e^^ [- cos e sin ojt + [ i KR + + i KR cos ZeJ cos wt j 



/ 94)Av 

 + 0[R 1+ OIR"^) 



To satisfy the above boundary conditions, we ennploy slender-body 

 theory.^ For example, the potential satisfying Equation [8] is an axial 

 line of horizontal dipoles, of moment density j | [R(z)] per unit length. 

 Thus in an infinite fluid, 



4>£ = ieJ° [R(z')]^|- [r2 + (z-z'/]'' dz; [ 



12] 



To satisfy the free surface and radiation conditions, we substitute 

 for the source potential [ r'^ + (z - z ^ )'^ ] ^ , the potential of an oscillating 

 source under a free surface.^ With this substitution we obtain, in place 

 of Equation [12]: 



/" - 



+ j:"^^e^(-+-l)j^(kr)dk)dzi [13] 



+ TTcKI r° [R(zi)]2 e^<^+^l)^[j (Kr)] dz^ 



J_H ax u 



