Two-Dimensional Wave Spectrum 



sea. If one becomes seriously interested in using this spectrum for applications 

 in naval hydrodynamics, he should check the opinions of those workers in this 

 field who may not agree with this belief. 



Known Properties of S( w, 9) 



With a handful of directional spectrum estimates available, it is not sur- 

 prising that not much is known about S(co,0). The available estimates have been 

 given by Cote et al (1960), Longuet-Higgins et al (1963) (and in other publications 

 describing the same data), Cartwright (1963), and Cartwright and Smith (1964). 

 It was assumed in Cote et al that S(a., 6) was zero outside of the range 

 -7r/2 < 6 < 7t/2 where 6 - is the direction toward which the wind was blowing. 

 There were good reasons for this assumption and the data bore them out, but it 

 could not be proved that S(w, 6) was zero outside the above range. 



Analyses by Longuet-Higgins et al (1963) were able to obtain the first five 

 values in Eq. (2), that is \/2-n, aj(a)), h^(oS), a^icS) , and h^{co). The results 

 suggested that S{co, 8) was not zero for spectral components traveling opposite 

 to the wind. However, a new device, called a clove rleaf buoy, developed at the 

 National Institute of Oceanography now yields ^^i^^') and b3(aj) . Indeed, that 

 part of S(co,9) outside of -V2 < d < n/i is small. The preponderance of the 

 available evidence now is that little or no spectral energy in a fully developed 

 wind sea will be associated with spectral components traveling opposite to the 

 wind. 



All of the available directional spectrum estimates also indicate that s>(co,9) 

 is more strongly peaked for low frequencies and that it broadens with increas- 

 ing frequency. One possible explanation for this effect is contained in the theo- 

 ries of Phillips (1957), which suggest that S{co,e) should have two peaks that 

 move further apart with increasing frequency. None of the presently available 

 directional spectrum estimates have the resolution and the degrees of freedom 

 necessary to resolve the question of whether or not this bi-modal form occurs. 

 An experiment could be designed to resolve this question by the combined use of 

 both stereo-photogrammetric techniques and the latest buoy developed at the 

 National Institute of Oceanography. 



For some applications of the power spectrum, it is desirable to be able to 

 describe the sea surface as a function of distance instead of as a function of 

 time at a point. This involves the transformation from an co, d representation 

 to an l,vi representation where ^t^ + m^ - k^, k = a-^/g, -t = oj^ cos 9/g and 

 m = 0)2 sin 6/g. A discussion in Ocean Wave Spectra suggested that k did not 

 seem to be given by w^/g, but since then Mr. Cartwright of the National Institute 

 of Oceanography has informed me that subsequent analyses all verify this linear 

 representation between wave number and frequency to within the present accu- 

 racy of the available data. This result does not eliminate the problem com- 

 pletely as nonlinear effects of a more subtle nature are present. It will be a 

 long time before these nonlinear effects are completely understood. 



For information purposes, in our attempts to forecast waves for the North 

 Atlantic, the form given by Cote et al (1960) has been used. In the notation of 



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