44 



THEORY OF SEAKEEPING 



50 



40 



30 



20 _ 



10 ^ 



8 10 12 

 Period T Sec. 



20 



Fig. 45 Energy spectrum of waves U(T) at wind speed of 40 

 knots and wind duration of 4, 8 and 18 hr. Spectrum marked 

 Oh is attained within 1 hr (from Gelci, Casale, and Vassal, 



1957) 



tionship derived for obser\'ed apparent wave heights and 

 periods is applicable to the component waves forming the 

 spectrum, but they take the spectrum in the logarithmic 

 form shown earlier. Writing the increment of energy as 



AUt = W 



dT _ gp 



H-r' 



dT 

 T 



and substituting IIt" resulting from eciuatiou (74! 

 obtain 



MJt = C* 



r 



T* exp 



2wl 



dT 

 T 



(9.3) 

 thej^ 



(94) 



From a comparison of ecjuation (94) with (87) and con- 

 sideration of equation (72) it follows that 



(' 



C* = 1.96 X 10- 



(95) 



It is not possible to compare this constant with that of 

 Neumann because of the difference in form of equations 

 (76) and (94). It should be noted, however, that the 

 constant C in equation (94) is nondimensional,™ while in 

 equation (76) it has the dimensions of sec~'. By using 

 (82) to convert the energj' into the quantity E and using 



™ The use of clT/T and dw/oi in order to avoid dimensionality 

 of the constant has been discussed previously by lJarl)yslure 

 (19.52). 



Longuet-Higgins' relationships from Table 6, the a^'erage 

 wave height is obtained as 



^ ^ 0275 ^,, 



and the significant wa\'e height as 



0.44 



g 



(96) 



(97) 



The corresponding relationship found by Sverdrup and 

 Munk (1947) for the fully de\'eloped seaway is 



ff _ 0.26 , 



(98) 



Walden (1956) pointed out that in using Neumann's 

 relationship Roll and Fischer evaluated the constant on 

 the basis of the envelope curve of Fig. 36, which (accord- 

 ing to Neumann) corresponds to the mean of ifo highest 

 waves. It appears more correct to derive the constant on 

 the basis of the a\-erage wa^'e height, which is about half 

 of the former. With this correction, equation (97) is re- 

 written as 



Hi/z 



0.22 



ir- 



(99) 



The wave height is now indicated to be somewhat 

 lower than in Sverdrup and Munk's e(iuation (98) but 

 checks well with Walden's (195o/.54) data derived from 

 a study of many reports of weather obser\'ation ships 

 (see section 6.5). 



To summarize, the suggestions of Roll, Fischer and 

 Walden were: 



1 To express spectra in logarithmic form so that the 

 maximum of the spectral curve would occur at the same 

 wave length whether plotted \'ersus T or A^ersus oj. 



2 By the foregoing action, to make the constant C di- 

 mensionless. 



3 To use Fig. 36 consistently for both spectrum form 

 and e\^aluation of the constant. 



4 To base this derivation on the mean, rather than on 

 the envelope curve of Fig. 36. 



6.3 Gelci, CazaSe and Vassal's Spectrum. Quoting 

 from (ielci, C'azale ;uitl \'assal (1956, 1957)(iii a free 

 translation) : "The following method of predicting waves 

 is based on the independence of the various sinusoidal com- 

 ponents. This is the hypothesis adopted by J. Darby- 

 .shire (1952) in a classical study. It is in reality dif- 

 ficult to state the limits of validity of this hypothesis. 

 Recently G. Neumxinn (1953) adopted implicitly a 

 particular hypothesis of interdependence of various sinusoi- 

 dal components: the long components grow only when 

 the lesser ones already exist. We have here reconciled 

 the independence hypothesis of Darbyshire with the re- 

 sults of G. Neumann, relating to the limiting established 

 state of .sea under constant wind action (fullv ari,sen 

 sea)." 



The foregoing reconciliation consists of the statement 

 that while all wa^^e components exist at the same time. 



