Understanding and Prediction of Ship Motions 



As before, the quantities 



^kCt) ^ikCx) + 



•-'- CO 



(T) Vik(x, t-T) dr 



can be interpreted in terms of the instantaneous and subsequent response to a 

 sequence of velocity impulses. However, more care is now necessary. If the 

 ship is given a unit velocity impulse in the k-th mode, it will afterwards have a 

 unit displacement in that mode, and the potential for the fluid motion subsequent 

 to the impulse will be 



-Vxj + cp^(x) + YikCx.t) + 02k(^'E) + ^2k<^'S' ^~^) '^^• 



The last two terms clearly represent the disturbance due to the steady dis- 

 placement. If, on the other hand, we think of an impulse of displacement (which 

 is rather difficult to picture), the potential for the later stages of the motion 

 will be: 



-Vxj + q)Jx) + 



3Yil,(x,t) 



2t 



+ Y.kC'^'t) • 



Thus it would not be proper to consider Xjk*^^' t) as the response to a unit im- 

 pulse of displacement. 



So far, we have found only the form of the velocity potential. Before we can 

 write the equations of motion, we must know the pressure distribution on the 

 ship hull and we must then integrate this appropriately to obtain the six compo- 

 nents of force and moment. 



The pressure anywhere in the fluid is given by Bernoulli's Equation: 

 To first order in the small motion variables, this can be approximated by: 



3x, 



(VcpJ- 



Bt 3x, 



• Vcp, 



When we substitute into this equation the expression (11) for the velocity poten- 

 tial, we obtain, after some reduction. 



31 



