OCEAN WAVES 



Such a surface can be easily understood from Fig. 3. It's made up 

 of a large number of randomly superimposed, simple harmonic, pro- 

 gressive sine waves of different amplitudes, each traveling in a 

 different direction and each having a different frequency. 



Here's another way to think about this all-important concept, 

 let's step back to the physical picture of waves for a moment. As 

 the wind blows and as the waves grow, turbulent variations and 

 varying amounts of internal eddy viscosity— as well as the interaction 

 of one wave with another— set an individual limit to the growth of 

 each individual component wave train in the developing sea. They 

 do this by initiating the energy-dissipating, breaking, or "white-cap" 

 process, whenever a momentarily high crest reaches an unstable con- 

 figuration. (In the classical theory, this occurs when the height-to- 

 length ratio of waves in a train reaches the critical value of l/7th at 

 which point in a wave's development its crest is sharply peaked, as 

 shown in the margin. But on the real ocean this limiting value is 

 not knowa) Thus, the total energy present in all of the waves on a 

 developing sea progressively distributes itself over a range of fre- 

 quencies, each frequency characterizing a particular wave train. As 

 the waves continue to grow and as new trains continually develop, 

 this range extends more and more to shorter frequency— or longer 

 period — waves. In brief, a spectrum of ocean waves is formed (Fig. 

 6), in which— for any given wind velocity and for fully developed 

 waves — the energy distribution over the band of wave frequencies 

 from 0.4 X 10-' to 3 cps is distinctive. 



Fig. 6. When wind-generated ocean waves reach 

 maximum height their energy, whrch Is proportional 

 to their mean-square height, is distributed over a 

 narrow frequency band that vanes with the wind 

 velocity, as shown. Such distinctive energy spectra 

 underlie wave-forecasting systems. 



Spectral filters and wave forecasts 



This spectrum enables us to resolve the total variance of the mean- 

 square sea-surface elevation (the "energy," or total area under spec- 

 tral curves such as in Fig. 6) into contributions traveling in different 

 directions and having different frequencies. Or put another way, if 

 the waves on the sea surface are put through a filter — either in re- 

 cording or in subsequent analysis — so that only those waves traveling 

 in a small range of directions and occupying a small band of 

 frequencies are left on the model "sea surface," we can specify this 

 fraction or component of the variance. 



The problem of finding an adequate way to estimate the spectrum 

 of a statistically invariant Gaussian process is not simple. And it 

 arises in many fields besides ocean waves such as turbulence, seismic 

 analysis, electronics, and weather prediction. Happily, it was solved 

 in 1949, by John Tukey of Princeton. 



Once the problem was solved, we were able to develop ways to 

 forecast swell quantitatively; we were able to analyze rather frag- 

 mentary wave data and discover the many different spectra that 

 occur in nature; and we could predict with some sophistication the 

 effects of wave refraction in shallow water. 



The idea of spectral filtering just mentioned comes up in still 

 another important way, in connection with operational wave-and-swell 

 forecasting. About 10 years ago, for example, Gerhard Neumann of 

 New York University found it was possible to derive a theoretical 

 expression for the characteristic frequency spectrum of wind waves 

 from thousands of visual wave observations made with a stop watch. 

 This theoretical spectral equation gave no information about different 

 directions of wave travel. It depended on only two variables: (1) the 

 wind velocity; and (2) either the distance over water that the wind 

 blew with constant velocity and direction, also called the "fetch," or 

 the duration of the wind. And it provided a m.oderately accurate way 

 to forecast the spectrum of waves in deep water as a function of the 

 past history of the weather over the ocean. Fig. 7 summarizes some 

 of the salient features of this prediction system. It forecasts— among 

 other things— the average of the heights of the one-third- highest 

 waves that will be running, and an average wave period. 



45 



Fig. 7. Salient features of the wave forecasting system 

 in current use by the U. S. Navy are shown on this 

 diagram. Forecast is a function of the weather over 

 the ocean and depends on only two variables — wind 

 velocity, and either the fetch or the duration of the 

 wind. Intersection of either of the latter curves with 

 the appropriate wind velocity curve yields forecast 

 nformation shown 



0.15 

 Frequency (cps) 



0.20 



13 



