Understanding and Prediction of Ship Motions 



3{f} = Fourier transform of f(t) 

 ■^"* f(t) dt . 



If f(t) = for t < 0, then 



3{f} = 5^{i} - i^Ji} 

 where 



3^{f} = Fourier cosine transform = f(t) cos cot dt , 









Sgif} = Fourier sine transform = f(t) sin cot dt . 



•^ 



The fimctions K.. (t) have this property, and so we can rewrite the transform of 

 (13'): 



6 



+ i[a.b., +a.3,{K.,}]| 3{a,} = 3{F. .G-}. (15) 



Tf we multiply this equation by e^'^* and, in (4), let 



x^Ct) = e'"*3{ai^} , 

 fj(t) + g/t) = e^-*3{Fj+G3} , 



then this equation is clearly equivalent to (4). In words, this means that taking 

 the Fourier transforms of the equations of motions (that is, of the true equations 

 in the time domain) is equivalent to breaking the forcing function into its fre- 

 quency components and determining the response to each of these components. 

 Such a result can hardly be considered as surprising, in view of what is common 

 knowledge in control theory about the relationship between time- and frequency- 

 domain descriptions of a linear system, but it was, until recently, a missing link 

 in our arguments about ship motions in non- sinusoidal waves. 



It was assumed above that the disturbances were transients such that, for 

 all k, a^(t) ^ as t -» CO, so that all transforms existed in the conventional 

 sense. The character of a ship is such that this assumption may well not be 

 warranted, and for generality a modification of the results is necessary. In Ap- 

 pendix B, it is shown that if the ship system is stable, that is, if all a^(t) re- 

 main bounded for all time, then Eq. (15) is still valid even if the transform of 



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