is suggested in Figure 8 by the widely spaced and nearly vertical standard 

 deviation curves at high H/Hj^3 and by extrapolation of the results shown in 

 Figures 5, 6, and 7. However, this has not affected adversely the statistics 

 chosen for the present tests. 



62. A second observation of the results shown in Figure 3 is that 

 beyond a certain bulk spectral bandwidth, the parameters do not appear to 

 approach Rayleigh parameters no matter how many component waves are used in 

 the spectral definition. This is most clear for the widest case, 



Af/f =1.60 , but is also slightly evident in the next- to-widest case 

 Af/f^ = 0.80 . It is notable that even the mean wave height H'-"-'^' and the 

 average of the highest one-third waves H'^'^^ also show some effect of 

 spectral broadening. One of the most extreme examples of this is shown in 

 Figure 9. Though there are over 5,000 component wave trains, the mean curve 

 deviates everywhere from the Rayleigh curve and the deviation is significant 

 at least at the level of the standard deviation of the synthetic data. 



63. Also shown in Figure 9 are the Modified Rayleigh pdf and exceedence 

 curves. These curves clearly represent the synthetic data better than the 

 Rayleigh curves. This is evident in the broader sense in Figure 4, where all 

 four measures of comparison are much closer than the Rayleigh comparison of 

 Figure 3 as long as enough component wave trains are present. This result 

 suggests that the Modified Rayleigh curve is a better model for broad spectra 

 if the added parameter H can be determined directly from the spectra. It 

 is not yet evident that this determination can be made. 



64. These tests reveal certain properties about unimodal , band-limited, 

 white spectra and the probable wave height distributions contained in the 

 corresponding time series. If the spectra contain in excess of 100 conponent 

 wave trains and do not have a width such that Af/f^ is greater than about 

 0.4, the resulting wave height distributions will conform very nearly to the 

 Rayleigh model. 



Implications for Spectral Filtering 



65. A common way to remove unwanted noise or extraneous signal from a 

 time series is by spectral filtering. In this method, a time series is 

 Fourier transformed to obtain a spectrum after which energy in specified 

 frequency bands is either reduced or eliminated according to the filter 



38 



