Current Progress in the Slender Body Theory 



and thus, from (A2.8), q^ = 0. 

 Similarly 



Q3 = i^[y],=o - 



(2D) 

 ( ) „ 



But now if the waterline of the ship is described by 



y - ± - B(x) , 



( 2D; 



then [y] ^^0 = B(x) . Also it is clear that the boundary condition (A2.6) for 

 reduces to 



(2D) 1 



\, -- ± 2UB'(x) 



at the free surface z = 0; i.e., 



,(2D) 



UB'(x) 



Thus 



Q3 = + iwB(x) - UB'(x) 



■io) + U :r^) B(x) 



Now 



Q4 = iojjydy + i^ Jzdz - Jdz 0^0°^ " Jdy '^[oT 



, 



,(2D) 



provided the ship (and hence the steady flow 0^^^ ) has transverse symmetry, 

 which is the case of interest. 



Finally, by similar reasoning to that for Q^, Q3, we have that 



159 



