Understanding and Prediction of Ship Motions 



The solution, (7), of the free surface problem is of just the form commonly 

 used in control theory. The motion of the body is considered to be made up of a 

 sequence of impulsive motions; for each impulse there is an immediate fluid re- 

 sponse (due to the incompressibility of the fluid) and an extended response, the 

 latter lasting much longer than the impulse itself. The quantities a.^(t) are the 

 inputs and quantities x^^(x., t) are the impulse response functions for the veloc- 

 ity potential. 



If the ship has forward speed, the situation is more complicated in practice, 

 but in principle the approach is the same. Let us again use two coordinate sys- 

 tems, one moving steadily with velocity V, where |v| equals the mean speed of 

 the ship, the other system being fixed to the ship. We can then define the vector 

 displacement of a hull point by the same expressions as in (5). 



Let the potential be represented generally by: 



(I)(x,t) = -Vxj + cp^(x) + cPiCx,t) , 



where [-Vx^ + cpp(x)] is the potential for steady flow past the ship fixed in its 

 undisturbed position, that is, cp^Cx) satisfies: 



V 



3x, 



Bx, 



on S, 



where 



v^(x) = V[-Vxj + cp^Cx): 



Again S^ is the surface of the undisturbed hull. Then the free surface condition 

 on (Pj(x,t) is readily found: 



32 



r2 



7 - 2V ^-^ + V2 '- + g ^ =0 on x = . (10a) 



3^2 BtBxj 3^^2 3x3 ^ 



The body boundary condition on cp/x, t) is not so readily determined, for it 

 may be shown that the body condition must be satisfied on the exact, instanta- 

 neous surface* of the hull. However, Timman and Newman (1962) have proved 

 that a consistent first order theory results if the following condition is used: 



f3a(x, t) 1 



n • VcPi = n • I + Vx [a(x,t) x v^(x)] . (10b) 



If we tried to apply the time -dependent boundary condition directly on the mean 

 surface of the hull, we would have only the first term in the braces. The second 



*This has not been done properly by such eminent authors as Havelock and 

 Haskind, and it has led to some long-standing wrong ideas. For a thorough 

 discussion, see Timman and Newman (196Z). 



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