The integral containing ^ in Equation (87) is the Froude-Krylov force. If we 

 substitute ^ for (()„ in Equation (82), we have 



-Jj.. ,,^ds = -{fu^n. dS 



(88) 



Here, we have neglected the higher-order terms. With Equation (88), the Froude- 

 Krylov force is given by 



f . = -p \^^ n. (^ia>.U 1^) ^^ dS 



-itot 



= -ipoj 



[[ n. ^ 

 o JJ J o 



dS e 



-i6Jt 



(89) 



where 'P is given in Equation (71) . The heave excitation force is given by 



fg = 2pgA 



W 



K z 



1-e 



cos (K x-OJt) cos K y dxdy 



(90) 



and the pitch moment is 



f^ = 2p, 



■A\ 



K z 



-1+e 



X cos (K^x-cot) cos K y dxdy 



K ^^ K z 

 2pgA-^ II e ° (K^z-1) 



sin (K x-OJt) cos K y dxdy 



(91) 



where K = K cos 3, K = K sin 3, and the integral limits z^ and z are given in 

 Figure 2. 



22 



