Understanding and Prediction of Ship Motions 



Since we set X3 = o in the line integral, the vector x now lies in the Xj - Xj 

 plane, as does n also. Therefore the correction term is non-zero only for j = 3. 

 That is, it contributes only to the yawing moment. For motions limited to the 

 longitudinal plane, the correction term is clearly zero. 



To conclude this appendix, we write down the complete expressions for 

 force and moment components in the steady coordinate system, using the poten- 

 tial function in essentially the form proposed by Cummins, Eq. (11): 



^j^^) == Xjo - E /^jk ^k(t) - E bj, i^Ct) - ^ c^^ a,(t) 



k=l k= 1 k= 1 



6 t 6 t 



"E f <^k(^) Ljk(t-T)dT- 2 r a^(r) M.^(t-r)dT 



k=l-a) k=l*^-co 



where 



ho-- P 



1 



B^ij(x) 



3n 



gx, + V 



Bx, 



dS , 



(Al) 



(A2) 



A^jk = P 



1 



s^ Bn 



30ij(x) 



s 



i//i^(x) dS , 



^'/'ik(x) 

 ^2kC^) - V -^ + Vcp„ • W'luC?) 



dS , 



^jk = P 



I 



3>Aij(x) 



Bn 





-Vh^(x).V3^ 



dS + pY 



{ 



B>Aj.(x) 



Bn 



B^Tlia-LV-)) 



d^ 



(A3) 



(A4) 



(A5) 



^jk = ^jk + 















+x 



-X. 



+X2 -Xj 



+X3 -X, 



o < 



-X, +x. 



+X, -X 



-X, 



3„ "2 



O ( 



+x. 



+x. 



+X, -X, 







+x 



(A6) 



73 



