Understanaing and Prediction of Ship Motions 



Cummins proposed a solution in the following form: 



6 6 ^t 



<D(x,t) = ^ (i^(t) V^i^Cx) + Yl ^k(^' t-^) ^\,(r)dT, 



k = 1 k = 1 ''- CO 



(7) 



where "/'^(x) satisfies: 



a) 0k " 



b) 



30W 



on Xj = , 

 n-ik- k=l,2,3, 



Q'ik-a'^^' k = 4,5,6, onS^, 



(8) 



with Sq the mean position of the hull, and where x^^ix, t) satisfies: 



Bt^ Bx, 



b) V^ = , 



dn 



c) 



By,. 



Bt 



B0k 

 Bx, 



on X, = , 



on S„ , 



on X3 = 0, for t = , 

 for all X \\hen t = , 



(9) 



There is no particular difficulty in showing that (7), along with conditions 

 (8) and (9), does satisfy (6a) and (6b). The verification will not be carried out 

 here. In any case, Cummins did not suggest how to find any of the twelve func- 

 tions, i//k and y^ , and it is simply assumed that they can be found and that they 

 will satisfy the other conditions of the problem, namely, (6c) and (6d). It is 

 much more interesting to investigate the meaning of the different parts of the 

 solution. 



The functions 4']^(x) are the velocity potentials for separate, much simpler 

 problems. These functions are originally defined only in the fluid region, that 

 is, outside the body and in X3 < 0. But condition (8a) implies that iPy. is anti- 

 symmetric with respect to the X3 = plane, and so we can interpret it physi- 

 cally in a much larger region. For example, consider the case of the heaving 

 ship. That is, let a3(t) be the only non-zero motion variable. We can think of 

 the body being extended by having its reflection in the free surface added to it, 

 the whole space outside of the body now being filled with fluid. If the extended 

 body now moves as a unit, the velocity potential for the hydrodynamic problem 

 will be just i3(t) i/'3(x), with <^^(x) satisfying (8a) and (8b). This is a classical 

 Neumann problem, and the same picture fits the pitch and roll modes. 



For the other three degrees of freedom, the physical problem to which 

 \(x) pertains is not so clear. For k = 1 (surge), for example, the body must 



27 



