PART VI: CONCLUSION 



83. In this report, a simple examination is made of wave height 

 distributions in synthetic sea states characterized by energy spectra with 

 multiple peaks, i.e., having energy centered at two or more distinct frequen- 

 cies. Sea states with broad frequency spectra are included in this study 

 since such seas are also composed of waves of diverse frequencies. These 

 cases violate the assumptions of unimodal , narrow spectra that are formally 

 required for the conventionally used Rayleigh distribution of wave heights to 

 apply. Because of common usage of this distribution in engineering design, it 

 is important to determine what errors are incurred by these violations and if 

 an alternate model can compensate for them. 



84. Reported here are tests of the Rayleigh and Modified Rayleigh wave 

 height distribution models. These models are compared with wave heights from 

 idealized, synthetic time series having spectra with variable widths, numbers 

 of modes, and modal separations. The Modified Rayleigh model is the two- 

 parameter, deepwater asymptotic form of the Beta-Rayleigh distribution 

 introduced by Hughes and Borgman (1987) . Synthetic time series are produced 

 by inverse Fourier transform techniques and consist of 65,536 points at a 

 nominal time step of 0.5 sec, so the record simulates in excess of 9 hr of 

 sampling. Synthetic wave periods are maintained near 10 sec so the samples 

 contain typically about 3,000 waves, enough to compute some reasonably concise 

 statistics. Random phases are used in signal generation, so there is the 

 possibility of some variation between runs with otherwise constant generating 

 parameters. Hence, 20 runs were done for each case, and the results averaged 

 to produce a stabler estimate of expected behavior. 



85. The first part of this work involves determining some of the limits 

 implied by the phrase "unimodal, narrow spectrum" in the synthesis of time 

 series having Rayleigh distributions of wave heights. It is found that for 

 unimodal, band- limited, white spectra, there are constraints on both the 

 overall bandwidth and the number of component waves for a close approximation 

 to a Rayleigh process to occur. Average heights H'^'^^ and the average of 

 the highest one- third waves H'^'^' are within 2 to 3 percent of Rayleigh 

 estimates as long as bandwidths normalized by center frequencies do not exceed 

 about Af/fc = 0.4 . For h'^'^°' , it appears that about 20 component waves 

 are necessary to differ from a Rayleigh h^^'^°' by less than 10 percent. For 



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