Two-Dimensional Wave Spectrum 



The purpose of this paper is to summarize what we now think we know about 

 the directional spectrum of waves at sea and to discuss how we are trying to 

 predict this directional spectrum at 519 points in the North Atlantic Ocean. 



FULLY DEVELOPED WIND SEAS 



If the wind blows with constant speed and direction for a long enough time 

 over an initially calm ocean area, if this ocean area is big enough, if no waves 

 propagate into this area from outside it, and if the turbulent features of the wind 

 do not change, then a fully developed wind sea should be observed over part of 

 this area, and wave observations made in this fully developed wind sea should 

 all be samples that have come from the same population. The spectral esti- 

 mates, S(oj,d), made from these observations should display sampling variabil- 

 ity in terms of departures from some unknown population spectrum S(aj, 0). 



There are only a few available estimates of S(co, 6). There are, however, 

 now available in useful form about 500 estimates of S(aj) that were obtained 

 from the analysis of waves recorded at a fixed point as a function of time. 

 These estimates are given by Moskowitz, Pierson, and Mehr (1962,1963) and by 

 Pickett (1962). 



Of these 500 spectral estimates, S(cS), about 40 were found by Moskowitz 

 (1964) to correspond to fully developed seas for winds from 20 to 40 knots. All 

 of the others could not be simply defined by the wind speed measured at the 

 time the wave record was made. As one example for winds near 20 knots, the 

 waves are usually higher than those expected for a fully developed sea because 

 components left over from previously higher winds and components from swell 

 are present. 



Given some form for S(co,d) to describe the spectra of fully developed wind 

 seas, one is therefore a long way from describing the spectrum that will be es- 

 timated at a particular point at sea because the wind will not have been constant 

 in speed and direction and waves from a distance may have propagated into the 

 area. The spectrum for a fully developed wind sea is however a fundamental 

 building block in attempts to describe the spectra that will occur in more com- 

 plex situations. 



KNOWN PROPERTIES OF THE SPECTRUM 



Directional Spectra and Frequency Spectra 



The directional spectrum of waves can be thought of as being written in the 

 form 



S(aj,^) = S(a;) [i(co,e)] (1) 



where i(co,e) in turn can be written as 



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