DISCRETE AND CONTINUOUS SPECTRA 



Suppose we were to convert the fluctuating water level above a point into a 

 fluctuating voltage, pass this through a filter which responds fully to frequencies be- 

 tween / — ViSf and / -f Vi8f and not at all to others; then square the filter output and 

 average. Designate the final output by E{f) 8/. If the ocean waves consisted of one or 

 several discrete frequencies, f x , f 2 , ■ • ■; then we should expect a large output 

 E{f x ) E{f 2 ) . . . when the filter is tuned to these frequencies, and a very low output 

 otherwise. What we find in fact is that E{f) varies more or less smoothly with /, as 

 if energy were distributed continuously among all possible frequencies. No matter how 

 narrow we design the width 8f of the filter, the result is the same. This leads us to 

 suspect that the ocean wave spectrum is a continuous and not a discrete spectrum. 



In practice we must always deal with records of finite length, T, and the best 

 we can do is to analyze for harmonics of this record length. This means that we can 

 sample only for frequencies separated by T-- 1 . But this discreteness in sampling has 

 nothing whatever to do with whether the true spectrum is discrete or continuous. In 

 the former case we would find that the energy content is insignificant except for just two 

 or three lines in the immediate vicinity of the discrete frequencies present; in the latter 

 case we would find individual energy values to be badly scattered, but if T is sufficiently 

 long so that we can crowd (say) 10 or more values into the frequency range to be 

 analyzed, T — 10/Sf, then the average of these ten values is a meaningful measure of 

 the energy density in this range. 



The important thing is that all pertinent measurements of ocean waves with 

 periods shorter than the tides have been in agreement with the expected results from 

 continuous spectra, and none have given evidence for discrete spectral lines. 



One observation all of us have made suffices to dispose of the idea that the 

 ocean surface can be represented by a single-frequency, single-directional wave train. 

 Consider sun glitter on the sea surface (Cox and Munk, 1956). The glitter pattern 

 is composed of millions of tiny images of the sun, an image occurring wherever the 

 sea surface has the slope (tilt and azimuth) required to reflect the sun's rays into the 

 observer's eye. The radial grid lines in Fig. 1 give the required azimuth of tilt (to the 

 right and left of the sun), the quasi-elliptical lines the required tilt, drawn for a solar 

 elevation of 30°. Suppose the water pattern were represented by a train from a direc- 

 tion 60° to the right of the sun, with a maximum slope of 20°. The resulting glitter 

 pattern is shown in the diagram on the left of Fig. 1. It lies entirely along the 60° 



Figure 7. The glitter pattern doe to a single sinusoidal wavetrain approaching from a direction 60 c 

 to the right of the sun, and a photograph of a real glitter pattern. 



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