for the pdf and Its intgral using Equation 4 for the exceedence curve. The 

 fundamental parameters of these time series averages are the defining spectral 

 widths and numbers of components . 



40. Mathematically, the spectra are defined by first specifying the 

 number of time steps N in a time series and a time step dt between time 

 steps. By Equation 40, the basic frequency increment of the discrete spectrum 

 is df = 1/Ndt so that the n*^** frequency of the spectrum is f„ = (n-l)df . 



It then remains to assign variance densities to each of the N/2 + 1 frequen- 

 cy bands at or below the Nyquist frequency. For band-limited white spectra, 

 four more variables are needed: a total variance, a reference center frequen- 

 cy, a bandwidth, and a number of spectral lines to which to assign energy. In 

 all of the tests discussed here, the total variance is constrained to be 

 equivalent to 0.25 m^ . This yields an H^^ = 2.0 m , which is a convenient 

 number of order one for computational purposes, but is otherwise unimportant 

 as all results are normalized with the computed RMS upcrossing wave height 



Hrms ■ 



41. The variables in the problem are then the center frequency f^ , 

 the overall bandwidth Af , and the number N of spectral lines within Af 

 which are assigned a finite variance. The bandwidth is normalized by the 

 center frequency to take the form Af/f^ as a parameter. The bandwidth is 

 then found from the product f(,'Af/f^ . The low-frequency bound of this band 

 is the discrete raw frequency band nearest to f^ - Af/2 . The high-frequency 

 bound is the discrete frequency band nearest to f^, + Af/2 . 



42. Assignment of the number of bands within the bounding frequencies 

 is determined by specifying the interval between raw frequency increments to 

 which finite variance is assigned. If every line is of finite variance, the 

 number of lines is N = Af/df . If every other line has finite variance (the 

 intervening lines having zero variance) , the number of lines is N = Af/2df . 

 In this case, the effective overall bandwidth of the spectrum is still the 

 same Af , but it is represented with only half as many lines, as if the 

 resolution bandwidth is twice as wide. If the variance in every other band of 

 the second case is twice the variance in each band of the first case, the 

 total energy represented by the spectrum remains the same. This same 

 procedure can be followed by assigning finite variance to every third line, 

 every fourth line, and so on until as few as two lines remain to represent the 

 spectrum. Two lines is clearly a lower limit for the number of lines with 



23 



