Coupled Ship Motions 

 4'-^(x,y,z,t) = S(t) 0i(x,y,z) + H( t ) 0,i(x,y,z,t) (2.16) 



where S(t) is the Dirac delta function and H(t) the Heaviside unit function. 

 Note that &(t) has the dimension 1/T. 



Denote by Pj(x, y, z, t) the linearized pressure arising from the unit dis- 

 placement, the pressure being evaluated on the displaced surface of the body but 

 expressed in terms of coordinates on the undisplaced surface s^. Then from 

 the linearized form of Bernoulli's principle, we have 



p.(x,y, z, t) 



3t 



+ V^(x,y, z) -vj. - ^ a.(x,y,z,t) • V (v^\ x, y, z)) 



Cx,y,z) on S . (2.17) 



In Eq. (2.17), the last term on the right corrects for the fact that coordi- 

 nates on the undisplaces surface are used in expressing the pressure on the 

 displaced surface. In that term s. has the forms of Eq. (2.3) or (2.4) with 

 x.(t) = H(t) , the Heaviside unit function. 



Let fij(t) be the hydrodynamic force or moment on the body in the jth 

 mode arising from the unit displacement applied at t = o to the body in the ith 

 mode. Then 



fij(t) = - f fpiCx.y, z, t) n-Mj dS i = 1,. ..6 



fij(t) = - J J p-(x,y,z,t) r X n •/!._ 



J = 1, 2, 3 



dS i = 1, ...6 



J = 4, 5, 6 



(2.18) 



(2.19) 



In Laplace's equation and in the Eqs. (2.6) through (2.9) satisfied by i//. , the 

 coefficients of i/^i are independent of time. From this it follows that if the unit 

 displacement is applied at t = r instead of t = , the resulting force or mo- 

 ment will be fij(t - r). Moreover, all the equations are linear. Thus we may 

 invoke Duhamel's principle and write that f^^Ct) , the force or moment in the jth 

 mode corresponding to the velocity x.(t) in the ith mode, is given by 



t 



f/(t) = I fij(t-T) i.(r)dT 



In particular, when x.(t) = HCt)e"^'^S 



(2.20) 



'^' f fij(T')ei"^'dT'. (2.21) 



•^0 



411 



