rapidly. In this way, statistically stable data sets can be synthesized with 

 reasonable efficiency. 



Unimodal Data 



37. In using synthetic data as described above, some assurance must be 

 gained that the procedure used replicates, in fact, the process it intends to 

 synthesize. In the present case, a narrow-banded process containing a large 

 number of spectral lines should yield a Rayleigh distribution of wave heights. 

 This distribution is the result predicted by Longuet-Higgins (1952) . Although 

 ideally a Rayleigh process contains an infinite number of spectral lines, it 

 is a practical impossibility to synthesize such a process using discrete 

 computational techniques. However, it should be possible to approximate the 

 process if a sufficiently large number of lines is used. Thus, an important 

 question in the current context is how many lines it takes for a spectrum of a 

 given width to approximate closely a Rayleigh process. 



38. A second question is related to the first. That question is how 

 narrow a spectrum must be before a Rayleigh process is realized. The deriva- 

 tion given by Longuet-Higgins (1952) simply states that the process must have 

 a narrow spectrum, but does not give a practical definition of narrowness. 



39. Part of the investigation described here is devoted to examining 

 these two questions. The idea is to find what is necessary to simulate a 

 Rayleigh process in a unimodal spectrum and then combine several of these 

 independently Rayleigh processes to investigate wave height distributions from 

 processes with multimodal spectra. The approach for this part is to generate 

 time series from band- limited white spectra having variable bandwidths and 

 different numbers of spectral lines, with each spectral component being 

 assigned a random initial phase. Due to the random phasing process, a given 

 sample time series can have a highly variable maximum wave height depending on 

 how nearly all the components are in phase at some point in the synthetic time 

 series. To get a better estimate of a characteristic maximum height, as well 

 as other high-wave statistics, an average of the wave height distributions of 

 several time series, having the same generating parameters (except for 

 phasing) and truncated at the same number of waves, is computed. This average 

 is compared with both the Rayleigh model. Equations 10 and 12 for the pdf and 

 exceedence curves, respectively, and the Modified Rayleigh model. Equation 13 



22 



