Newman and Tuck 



Also since in terms of this hull equation the magnitude of the element of surface 

 area is 



^ dS = dxdy [l + Z^2 + Zy2j'^^ 



we have for the outward vector element of surface area 



ndS = (i Z^+ j Zy-k) dxdy . 



This may be written in a manner not dependent on the choice z = Z(x, y) of hull 

 equation, viz. 



n dS = dx 



i ^ (zdy) + j dz - k dy 



(A1.9) 



where dy and dz = Zydy denote components of arc length along the cross sec- 

 tion cvirve. The area of a vertical strip from the free surface to the cross sec- 

 tion curve is -zdy so that the x component of ndS represents the decrease in the 

 area of this strip in passing from station x to station x + dx. The integral iden- 

 tities (A1.3) to (A1.8) may also be proved directly using the expression (A1.9) 

 for ndS, but appear to be more easily derivable from Gauss's theorem, as indi- 

 cated above. 



APPENDIX II 



EVALUATION OF FLUX TRANSFER FUNCTIONS Q: 

 AT FINITE SPEED 



The hull boundary condition for <?i^ j ^ , on an arbitrary body with unit normal 

 n at a point where the hull displacement is 



a = [i ^j(t) + j ^2(t) + k ^3(t)] + [i ^^(t) + j CgCt) + k^eCt)] X r , (A2.1) 



can be written 



— ^ = n . [-ioja + Vx (ax V0^g^)j (A2.2) 



(Timman and Newman, 1962). The second term inside the square brackets 

 gives the induced normal velocity due to non-uniformity of the steady flow cp^^^ 

 in which the body oscillates. On separating out the contributions due to each 

 mode ^j(t), j = 1, ... 6, we can write (A2.2) as 



■^ = Z ^i^-) ^i • (A2.3) 



156 



