SHIP MOTIONS 



175 



: 4 e 8 ic iz 14 



ojg, I/sec 



Fig. 19 Co- and quadrature spectra, wave and pitch, modified to 

 reference phase relationships to wave amidships on model 

 average of 6 runs (from Dalzell and Yamanouchi, 1958) 



queiicy range. Dalzell and Yanianonchi oft'er these po.s- 

 sible reasons for 7 < 1 : 



"(a) Experimental effects such as the surj^ing of the 

 model. 



"(b) The possibility of short-crestedness resulting 

 from tank side reflections. 



"(c) The shape of the tank section. 



"(d) Noise in the records and the effect of the smooth- 

 ing process and other computational error. 



"(e) Possible minor nonlinear effects in responses." 



In other wortls the coherency for any frequency will 

 be less than 1, if: 



(a) The ratio of the amplitude of that fre(|uency com- 

 ponent in y{t) to the amplitude of the same frequency 

 component in z{t) is not constant throughout the record. 



(6) The phase relationship, at that frequency, be- 

 tween i/(t) and z{t) is not constant throughout the record. 



(c) Noise is present in either y(t} or z{t) at that fre- 

 quency. 



The measure of coherence between two seakeeping 

 parameters is perhaps the single most important con- 

 tribution of cross-spectrum analysis. If linearity 

 (througii coherency) cannot be confirmed in model tanks, 

 either the superposition theory for long-crested waves 

 is in grave doubt, or the towing tank as a reliable test 

 medium is questionable. Tests similar to those of 

 Dalzell and Yamanouchi are being conducted independ- 

 ently at the Taylor Model Basin, where the tank section 



2 4 6 6 10 12 14 



Wg, I/sec 



Fig. 20 Co- and quadrature spectra, wave and heave, modified 



to reference phase relationships to wave amidships on model — 



average of 6 runs (from Dalzell and Yamanouchi, 1958) 



is wider and deejicr, in urdcr to eliminate the possible 

 difficulties set forth in items (b) and (c). 



This discussion of cross spectra would not l)e complete 

 without some examples: Fig. lU shows the co- and (juad- 

 rature spectra for wave (amidships) and pitch of a model 

 in irregular long-crested seas. The same frequency com- 

 ponents are in phase for the lower frequencies; above 

 (x> = 10, they are 45 deg out of phase. The wave-heave 

 relationship; for the same model is shown in Fig. 20 

 where the same frecjuencies are generally in phase up to 

 oj = 8.8, equation (32), where they are 45 deg out of 

 phase. The phase difference increa.ses to 90 deg at oj = 

 10. 



3.61 Covariance-digital method. The popular digital 

 method of cross-spectrum analysis is an extension of the 

 a utoco variance — Fourier transform method outlined in 

 Sections 1-8.3 and 8.4. Consider any two seakeeping 

 parameters ij{t) and z{t)\ the covariance function is 

 given liy the conxolution integral 



RUr) = lim ^, f i/(t) Ht + r)(lt 



7-^co 1 Jo 



and the cross-spectral density function Ijeconies 



*„.-W = .-^ ( /?,„-( r)c-'"Wr 



(37) 



(38) 



Where /?„.('■) is now an odd function and the cross spec- 

 trum is written in complex notation according to (30). 

 When (38) is combined with (30) the resulting co- and 

 quadrature spectra are 



C.„,('^,) 



qj' 



_ 1 f 



T Jo 



cos a)p7 (It 



R„..(r) - /?„..(- r) 



(39) 



sin (x>,T dr 



For the digital process, where simultaneous equally 



