Note that when a = 1 (or i^rmq/^rms = 2^''' from Equation 14) , Equation 13 

 becomes identical to Equation 10 and the Modified Rayleigh pdf becomes a 

 Rayleigh pdf. It should be noted that Hughes and Borgman (1987) defined H^^ 

 using a square root instead of the fourth root used here on the right side of 

 Equation 15. It makes more sense to use the fourth root since it gives a 

 height parameter with dimensions of length instead of length squared. This is 

 only a modification of notation and not of the basic model. One would simply 

 use the square root of the H in Equation 14 above. 



19. To find the cumulative distribution function of the Modified 

 Rayleigh model, one must integrate the pdf numerically. For computations used 

 in this study, integration was done using Simpson's rule in double precision 

 with integration steps of 0.001 in H/H^^^ . 



20. Examination of Equations 13 and 14 reveals that the shape of the 

 dimensionless pdf depends only on one parameter, the ratio H /H^j^^ . 

 Figure 1 shows a variety of Modified Rayleigh distributions found for select 

 values of H^q/H^^^ . The bold curve in Figure 1 is the Rayleigh pdf to which 

 the Modified Rayleigh pdf degenerates when H^^^/H^^^ = 2^'* . For complete- 

 ness, the two -wave pdf is also shown in Figure 1. 



Additional Parameters 



21. Where spectra are used for synthetic time series generation, some 

 additional parameters can be defined. These parameters are based solely on 

 spectral shape. Perhaps the most frequently used spectral scale of wave 

 height is denoted H^^ and is defined by 



Hn,o = ^ my2 (16) 



In Equation 16, m^ is the spectral zeroth moment. The more general n"^*^ 

 spectral moment m^^ is defined by 



'l> 



m„ = J^'-f" S(f) df (17) 



where f is frequency; f^ and £2 are, respectively, low and high 

 frequency bounds of the spectral definition; and S is signal variance 

 spectral density. 



11 



