Understanding and Prediction of Ship Motions 



to the lower half-body. In couplings between heave, pitch, or roll, on the one 

 hand, and svirge, yaw, or sway, on the other hand, it is a hybrid added mass 

 coefficient. The physical situation was described above in the discussion of the 

 meaning of the functions \p^( x) . 



It should be mentioned that b-^ is not a damping coefficient and c^^ is not 

 a buoyancy coefficient. The situation with respect to b^^ will become clearer 

 presently. The statement about c .^ is obvious from Eqs. (A5) and (A6) of Ap- 

 pendix A. 



Relations Between Time- and Frequency-Domain Descriptions 



So long as we had only the ordinary differential equations to describe ship 

 motions, it could never be clear how a ship in a confused sea would be able to 

 respond to each frequency component as if the wave of that frequency existed 

 separately. The experiments described in Chapter II showed that indeed the 

 ship did respond in this way. But certainly we had no basis for expecting, in a 

 random process, that a different differential equation could validly be used for 

 each of the uncountably many frequency components. Such an idea makes non- 

 sense of the whole concept of differential equations of motion. The difficulty, as 

 already pointed out, was that the differential equations were not really differen- 

 tial equations at all, but simply a frequency-domain description. 



Now we have a system of integro- differential equations which purport to 

 describe the ship in a seaway, regardless of the nature of the seaway. This 

 system of equations should, first of all, be capable of representing the ship mo- 

 tions if everything varies sinusoidally. In fact, it is very easy to show that it 

 does. Let us suppose that the exciting forces, whether due to waves [Fj(t)] or 

 other external causes [Gj(t)] , are sinusoidal at frequency w. After a long 

 enough time, it is reasonable to expect that all motions will also be sinusoidal 

 in time, so that we can write 



where a^ and e^ are constants. We substitute this into (13'), noting that the 

 convolution integral can be written 



^k*^^) K.^(t-T) dr = aj^(t-r) K.^(r) dr 



cos (cot + e, ) I K-, (r) sin cot dr 



jk^ 



•' 



- sin(a;t + e|^) I ^jk^^^ ^°^ "^"^ '^^ 



thus obtaining for the left-hand side of (13'): 



35 



