SEAWAY 



45 



the long ones grow very slowly at the beginning. They 

 are at first not A'isiljle in the same sense that tidal wa\'es 

 are not visible under wind sea. 



The reader is referred to Gelci, CasaM and Vassal's 

 1956 and 1957 papers for explanation of the energy 

 spectrum properties and the ajjpropriate methods of 

 estimating the wa\'es caused by winds of \-arious strengths 

 and directions. The spectrum of the fully arisen sea is 

 finally defined as 



dU 

 dT 



"" = 209 T' 



if T < 0.31 

 dT 



5 U, and 

 == 209 T' e.xp 



(100) 



■329 



-0.315 



if 0.315 U < T, 



where dUr/dT is the spectral-energy density of the fully 

 arisen sea in ergs/cm-/sec, [/ the wind at anemometer 

 height in knots, and T the wa\-e-coniponent period in 

 seconds. 



Consideralile discussion of the ilirectional varial)ility of 

 component waves is given in the referenced papers. 

 This ciucstion is placed in Section S of this monograph. 



The spectnmi for a limited wind duration is defined, 

 in the words of Gelci, et al: "In order to define the 

 ideas, let us consider a generating area with a very large 

 fetch; the sea being originally calm, the wind U (in 

 knots) begins to blow instantaneously. At the end of a 

 relatively short time (of the order of an hoiu') the spectral 

 energy density is expressed (in ergs per cm- per second) 

 as: 



dT 



209 T' 



1 



-329 



U 



Y - exp 



ii T < 0.315 U 



dU,/dT = if 0.315 [' < T 



0.315 



/(d)} (101)'" 



"We will designate by po this function of T and U . . . 

 On the scale of meteorological .synoptic charts, it is even 

 possible to assume that this regime establishes itself in- 

 stantaneously. 



"Then, each wave component grows linearly following 

 the law: 



dUr/dT = PO + - 209 T* 

 lo 



exp 



-329 I j-, - 0.315 



m (102) 



t expressed in hours is below 18. For IS < t the regime 

 is stationary (fully arisen sea)." 



The successive spectra of the developing wave in a 40- 

 knot wind are shown in Fig. 45. 



" Here /(ff) designates a fuiietion of the tlire(;tii)nal wave dis- 

 tribution. This is not considered in this section of the mono- 

 graph. 



The author has n(.)t foinid a discussion of the fetch in 

 the ]«ipers by Clelci, Casale, and Vassal. It should be 

 emphasized that fornuilation of the spectra (as by Gelci, 

 et al, and by Darbyshire) is only a part of the problem. 

 Another important part of references cited is the detailed 

 discussion of methods of evaluating effective fetch and 

 wind strength. 



6.4 Spectra of Incompletely Developed and Decay- 

 ing Seas. Most often in nature winds of significant 

 strength do not blow long enough to develop a "fully 

 arisen sea"; i.e. waves in which the energy intake from 

 the wind is eciual to the energy dissipation and for which 

 the wave structiu'e remains constant. In wa\'e predic- 

 tion it is necessary to treat such incompletely developed 

 seas. In this connection, as indeed throughout the 

 spectrum discussions, the form of spectrum and the wave 

 height indicated liy its area must be considered sepa- 

 rately. The prespectrum methods of wave prediction of 

 Sverdrup and J\lunk (with Bretschneider's extension) in 

 the U.S.A. and Suthon and Bracelin in England, have 

 been successful in predicting the height and period of 

 significant waves, but give no indication of the distribu- 

 tion of the component wa\'es in a complex sea. It will be 

 shown later, in Section 2 of Chapter 3, that minor varia- 

 tions of the spectral curve ha\'e little significance for the 

 analysis of a ship's motion. The position of maximum 

 spectral density on the frec|uency or period scale is, 

 however, important since it determines the conditions 

 under which a ship falls into synchronism with waves. 



Xeumami's method (included in Pierson, Neumann, 

 and .lames' forecasting manual) of predicting the wave 

 conditions for a limited fetch or limited wind duration 

 was discussed in Section 6.22. In this method the shape 

 of the spectral curve at the higher frecjuency end depends 

 only on the wind strength. A limited fetch or duration 

 pro\'ides cut-off points beyond which longer periods 

 (lower frecjuencies) are assumed not to exist. This 

 method, therefore, determines principally the mean (or 

 the significant) wave height but defines the spectrum 

 shape only crudely, and in particular gi\'es no informa- 

 tion on the position of the maximum of the spectral 

 curve. 



Gelci, Casale, and Vassal's method (designated D.S.A. 

 II), described in the foregoing .section and illustrated by 

 Fig. 45 defines the entire spectrum curve for any wind 

 duration. Gelci, et al, agree with Xeumann that the 

 spectrum grows from the high-frequency (low period) 

 end. 



The effect of fetch was e\'aluated by Darbyshire in 

 connection with the coastal region spectrum (1952) but 

 not with the open-sea spectrum (1955). The author 

 was not ai)le to find in Darbyshire's references any dis- 

 cussion of the wind-duration effect. Gelci, et al, how- 

 ever, discuss Darbyshire's spectra thus: "The beha\-ior 

 of the successive spectra corresponding to duration of 

 action of 4 hours and 20 hours will be found in Fig. 

 (46).'- It will be noted that the spectrum grows first 



'- Of the figures cited, one for the wind of 40 knots is reproduced 

 here as Fig. 46. 



