takes about 100 component waves to stay within 10 percent of a Rayleigh 



yd/lOO) 



59. The reason for this behavior can be seen in the sequence of 

 Figures 5, 6, 7, and 8, which show the synthetic, Rayleigh, and Modified 

 Rayleigh probability density and exceedence functions for the cases with f^ 

 = 0.1 Hz , Af/f^ =0.1 , and N = 2, 3, 10, and 328 component waves, respec- 

 tively. Because the synthetic curves are averages of 20 different runs, 

 standard deviations could also be computed and plotted. Standard deviations 

 are shown as dashed lines in the exceedence graphs. Figure 5 represents the 

 case of two waves. For this case, the synthetic pdf in the upper part of 

 Figure 5 looks very much like the two-wave model pdf shown in Figure 1. Also, 

 the standard deviation is very small. This result is as it should be since 

 the two-wave case has an exact solution (Equation 7) and the synthetic results 

 reflect this solution very well. 



60. When a third wave train is added, the problem becomes more compli- 

 cated, having no analytic solution. The result for a single run depends 

 strongly on the initial phases of the three component waves. If all three are 

 nearly in phase at some point in the time series, the maximum wave heights 

 will be larger than if the waves are not nearly in phase at any point in the 

 time series. Since initial phases vary from run to run, the variation in wave 

 height distribution is suggested by the standard deviation of the set of runs. 

 This is shown in Figure 6 as the dashed lines on either side of the average 

 exceedence curve in the lower part of the figure. The mean of the 20 runs is 

 higher, however, than the mean of the two -wave results of Figure 5. This 

 trend continues when the number of component waves increases to 10, as shown 

 in Figure 7. Figure 7 also shows that the smaller waves begin to conform to 

 the Rayleigh model more than the two-wave or three-wave cases. This is the 

 result summarized in Figure 3. At the point where there are 10 component 

 waves in Figure 3, averages over the highest 1/10, 1/3, and all the waves are 

 very close to the Rayleigh values. However, the pjd/ioo) still differs 

 substantially from the Rayleigh curve, evidently due to the limited number of 

 wave components . 



61. In Figure 8, the synthetic data are composed of 328 wave trains and 

 appears to conform to the Rayleigh curve over the whole plotted domain. 

 Presumably, if a longer time series had been generated, there would be a 

 region of limiting heights at some point on the high-wave tail. This effect 



33 



