Newman and Tuck 



Now as a consequence of slenderness, 3(ji( i)/Bn as given by (A2.3) with the 

 g of (A2.5), is the normal velocity across the cross-sectional curve in planes 

 normal to the x axis. Hence, the net flux across this curve may be calculated 

 in the form 



= dx f] Q.L., (A2.7) 



j = i 



where 



Q.dx = |g3 



dS 



is the flux transfer function for the jth mode, the integration proceeding around 

 the equilibrium hull cross section curve beneath the plane z = o, with dS obtained 

 from (A 1.9). Thus 



-ico + U :^ ) S'(x) , 

 dx/ 



where S(x) - - /zdy is the area of cross section below the equilibrium free 

 surface z = 0. 



r f / ( 2D) ( 2 



.Jdz - J ^dz 0(0)^^ - dy 0(0 



(2D) 



But since ^^q^ satisfies the 2D Laplace equation 



(2D) (2D) 



• r 1 r/j a(2D) ,(2D) 



■iojLzj z=o + J M^ ^(0)^^ + dy 



(0). 



= -ICO 



[Z],=n + 



z = 



h°>.]. = o' ^^^'^^ 



where [ ] ^^^ indicates that the difference between the values of the enclosed 

 qviantities at the two points of intersection of the cross section with the free 

 surface is to be taken. But 4>[o°^ is by definition the 2D double body potential, 

 i.e., 0^^°^ = on z = 0, so that 



(0) 



158 



