which to represent a spectrum because the pdf is no longer even approximately 

 Rayleigh, as derived by Longuet-Higgins (1952), as expressed in Equation 7, 

 and as shown in Figure 1. Figure 2 illustrates the effect of varying the 

 number of lines with which to define a band-limited white spectrum. It also 

 shows the equivalent bandwidths and spectral densities of a "continuous" 

 spectrum having the reduced number of lines and yet retaining the same overall 

 bandwidth. 



43. For the unimodal spectra, both the overall bandwidth Af and the 

 number of lines N within this band are varied within the limits imposed by 

 time series of finite length. In all tests, time series lengths are 



N = 65,536 points. With a time step of dt = . 5 sec , this represents a 

 record of 32,768 sec or 9 hr, 6 min, 8 sec, a record much longer than is 

 normally obtained in nature. In all cases, the center frequency is kept at 

 f ^ = . 1 Hz , corresponding to 10-sec waves. Hence, each simulation contains 

 in excess of 3,000 waves, enough with which to compute some reasonably concise 

 statistics . 



44. For each simulation, 20 runs were made to establish a mean charac- 

 teristic pdf. With 20 samples, other statistics can be computed as well. For 

 the present tests, a standard deviation for the exceedence curve is also 

 computed. Since the number of waves is not exactly constant in the full 

 length of the time series, the run results are scanned to find the case with 

 the fewest waves and all other runs are truncated in time at this same number 

 of waves. In this way, all 20 runs for each case have the same number of 

 waves. When wave heights are placed in order of ascending height, there are 

 then 20 samples at exactly the same discrete exceedence probability estimate. 



Empirical Probability Estimates 



45. A set of N discrete wave heights H„ can be used to estimate the 

 pdf of the governing process. If all H^ are normalized by H^^^ (defined in 

 Equation 6) , the number J^^ which fall in the range uAH/H^.^^ to 

 (u+l)AH/H^3 can be counted and divided by N to compute the fraction (or 

 estimated probability) of this occurance . When this fraction is divided by 

 AH , a wave height range arbitrarily chosen by the investigator, and 

 multiplied by H^„3 to make the result dimensionless , the result is an 

 estimate of the normalized pdf for that data set. 



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