1 : 
§i(@) = Dd wR Ar + roe”, (3.3) 
then 
Sj(@) = Sj(@) X gM, (3.4) 
The phase is shifted by wr by this shift in the origin. Accordingly the phase of the 
output should be modified by — row after the cross spectrum analysis. Figure 3.2 shows an 
example of this shift in the analysis of a 5—ft model ship rolling in tank waves?! (beam 
sea, without advance speed). From this figure, we find the amount of shift on cross corre- 
lation should be ro = 9. The results of the analysis are shown in Figs. 3.3 and 3.4. The 
improvement in the coherencies due to the shift is clearly shown in the upper part of Fig. 
3.4. The improvements in phase relation and in gain are also apparent, although these im- 
provements are partly due to the large m(= 60 > 40). 
x10~4 (FEET)? CORRELOGRAM ERUNING 1833 = 
15 . WAVE Qwir) ea 31SEC : 
t tT = 0. 10 
(FEET-DEG) ROLL-WAVE Qpwit) 
0.046 
t 
0.02 
Fig. 3.2. Auto and cross correlations of wave, roll, roll-wave 
for a model ship in tank (run 834). 
(From Yamanouchi.3') 
Of special interest to us is that the improvement in coherencies is also reflected in 
the change of bandwidth of the cross spectrum. Before the shift in origin, the co— and 
quad-—spectra in Fig. 3.3 had a narrower bandwidth than after the shift, showing that the 
blurring effect of the spectral window is smaller for the cross spectra with broader band- 
width than for those with narrower bandwidth. Thus this shift improved the estimate of 
the cross spectra by reducing the leakage of power. Although we refer to improvement, 
we do not have experimental data in regular waves for comparison in this run. However, 
in Figs. 3.5—3.7, which show the same kind of result for run 832 and the data in regular 
waves (Fig. 3.7), we notice the same tendency and conclude that the results obtained by 
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