174 



THEORY OF SEAKEEPING 



Theoretical Wave- Heave 



G 8 10 



ojg, I /Sec 



20 1.5 1.25 1.5 

 A/L 



100 



c 

 < 



T5 



Legend 

 • Average- 

 Individual Runs 



o Average-Spectra 

 X Regular Waves 



_i 51-100 



Fig. 17 Average motions phase angles compared with regular 



wave data (Model 1445, 2.9 fps broad spectrum) (from Dalzell 



and Yamanouchi, 1958) 



tioii (30) results in the complex transfer function 



, _ ^^..(ajj _ C,..(a),) -f iqy.M ,,-,,,. 



The kernel function A'^(r) of Fuchs and MacCamy (16) 

 requires the transfer function in the form 



T,M 



EM 



(34) 



while the St. 

 by 



Denis-Pierson transfer function is Kiven 



[TA^,)r- = 





(35) 



If the wave spectrum A'(co.) and the response spectrum 

 $(0;,) are measured independently, a transfer func- 

 tion given by erjuation (29) results. From equations 

 (29) and (35) a (juantitj' called the coherency (7^,) is 

 defined 



7..- 



[c„.-(co.)]^-f VhMW 



(36) 



where E{w,) and ^(co,) are the energy spectra computed 

 from ij{t) and z[t), respectively. The coherency is a 

 measure of the linear dependence of two trigonometric 

 components of the same frequency in two spectra and de- 

 termines whether the sj'stem being studied is indeed 

 linear. At the suggestion of Pier.son, Dalzell examined 

 the effect on coherency of increasing the number of 

 spectral estimates. The result was an increase in co- 

 herency between signals which had by their nature sug- 



Fig. 18 Coherencies (Model 144 5, 2.9 fps broad spectrum) 

 (from Dalzell and Yamanouchi, 1958) 



gested a strong linear relationship but which had showed 

 low coherencies originally. These results are, as yet, 

 impublished. 



Pierson (1957) has shown that theoretically 7=1 for 

 the long-crested case but that in the short-crested case 

 complications arise because the same frectuency of en- 

 counter can result in quite different phases in the re- 

 sponse, depending on the relative angle with which the 

 different wave trains encounter the ship. In general, 

 when c is large and q is small, the same frequency com- 

 ponents are in phase. When c is small and q is large, 

 they are 90 deg out of phase. When c = q. the com- 

 ponents are 45 deg out of phase. 



Dalzell and Yamanouchi (1958) studied the phase 

 relationships between waves at the midship .section of 

 a model and hea\e or pitch. The result for the aver- 

 age of six tests is shown in Fig. 17 where zero phase 

 means maximum pitch up at the bow or maximum up- 

 heave when the wave crest is at the midship section. 

 Phase lag means that maximum motion occurred after 

 the wave crest reached the midship .-section. Comparison 

 between pha.se relationships obtained in regular and ir- 

 regular wave experiments shows good agreement. 



Dalzell and Yamanouchi (1958) also plotted the coher- 

 ency (7) between the parameters just di.scussed and these 

 are shown in Fig. 18. If 7 > 0.85 is used us the limiting 

 criterion for a linear system, it is seen that heave-pitch 

 fails at the very low and \'ery high frequencies while wave- 

 heave and wave-pitch fail in a large part of the fre- 



