Newman and Tuck 



€53^^ = pJdxxB(x) ^-ioj + U ^j £3 



(3.35) 



I dx xB(x) (-io) + U ^] 



C^^^^ = P I dx xB(x) ( -io; + U ^1 f^ . 



Finally, using the convolution integral representations (3.9) and (3.15) for f ^^^ 

 and f j respectively, we have 



cir^ = pjj dxd^ S'(x) S'(i) [K(x-i,co,\]) - K(x-^,0,U)] 



cir^ = pj fdxd^ S'(x) B(^) K(x-^,aj,U) 



Cir^ = -pj|dxd^ S'(x) ^B(a K(x-^,a;,U) 



CsT^ = pJjdxd^B(x) S'ca [K(x-^,a;,U) - K(x-^,0,U)] 



€33^^ = p JJdxd^ B(x) B(a K(x-^,aj,U) (3.36) 



C^r^ = -pjjdxd^ B(x) ^B(a K(x-^,aj,U) 



Cgi^^ = -p J Jdxd^xBCx) S'C^) [K(x-^,aj,U) - K(x-^,0,U)] 



CsT^ = -pJJdxd^xB(x) BCa K(x-^,a;,U) 



^sT^ = p Jl dxd^xB(x) ^B(^) K(x-^,a;,U) , 



where 



K(x,aj,U) = ^io; + U^V K^j^(x,a;,U) . (3.37) 



This K is the kernel for all heave and pitch motions, but for surge induced mo- 

 tions the kernel is 



'k - lim k\ , 



aj-> / 



the additional correction being only of importance for non-zero forward speed 

 and arising from the correction (3.27) to the pressure field due to displacement 

 of the ship in the steady flow field. But we can easily see that without this cor- 

 rection the results for (say) C^^P^ would be nonsensical, for as aj->o, c^^j^^ must 

 represent the restoring force in surge (i.e., change in wave resistance) due to a 



152 



