Understanding and Prediction of Ship Motions 



1.4 



1.3- 



1.2- 



l.l 



1.0 



.9- 



.8 



.7- 



.6- 



.5 



4 



.3 



.2 



.1 



0. 



4 COMPUTED FROM OBSERVED VECTOR PROCESS 

 o COMPUTED FROM OPEN CIRCLES OF FIGURE 2 



X COMPUTED FROM CONVOLVED SPECTRUM OF 

 FIGURE I (CROSS SPECTRUM NOT SHOWN IN 2) 



"x ° °n 



\ ° 



*x* ♦ 

 ♦ "x ° 



'"x* ♦ 



°oooOO„ -X. ""o 



° O X n 



« X X »X )* X , o 



o " xo" o 



° .xXx? 



""xxx'o 



I I I I I I L 



J I _J_ 



4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 

 h, h» 



Fig. 3 - Coherencies obtained by various procedures 



the cross spectra given in Eqs. (28) and (29) when substituted in the equation for 

 the computation of coherency should yield one. The problem then is why are the 

 computed coherencies so much lower than the theoretical coherencies? 



To find out, the spectrum shown in Fig. 1 was smoothed and points were 

 read from it at five times the spacing of the h values indicated on the horizontal 

 scale of the figure. The smoothed higher resolution curve is shown by the dash- 

 dot curve. Total variance was preserved with reference to the area under the 

 smoothed curve. It was then possible to take Eqs. (28) and (29) and from them 

 compute what the co- spectrum and the quadrature spectrum should have been. 

 These values are shown by the small black dots in Fig. 2. By definition they 

 would yield the coherency of one. 



Now the window through which the co- spectra are viewed is roughly trian- 

 gular in shape, and if, for example, it were peaked at the value h = 9 on one of 

 these figures, it would fall approximately linearly to zero at the values h = 7 

 and h = 11. A linear combination of seventeen of the values given by the black 

 dots thus represents the value given by a diamond. This operation is called a 

 convolution and the results of a seventeen-point centered convolution with a 

 linear growth to the middle value and a linear descent from the middle value is 



91 



