II. THE SPECTRAL COMPUTATION 



Two basic methods have been used in the past for computing the wave 

 spectrum. The earlier one was based on the covariance function which was 

 then niunerically Fourier transformed and smoothed. That is, if rin 

 (n = 0,1,2..., N-1) is the water level elevation above mean water level, 

 then the covariance function is: 



1 



N-k 



N-k-1 



n+k 



(1) 



n=0 



(The quantity, N, is 4,096 for the analysis of Hurricane Carla.) Usually 

 C]^ would be negligible for k larger than some value, say kjj,. Thus, 



■ '"^ ' ^' ' " ' '' " "k- 



the N numbers would be adequately summarized in the kjjj+1 values of 

 Ordinarily, kn^ is selected to be around one-tenth of N for most 

 analysis of this type. The spectral density would be obtained from the 



numerical transform of the C 



k- 



P(fr) = ^t 



E 



q=l 



q-rrr 



— + C 



riT 



(2) 



where 



^r " 2k„At 



(3) 



for r=0,l,3,..., k^ (Blackman and Tukey, 1958). The quantity. At, is the 

 timelag between successive measurements of HnC^t = 0.2 second for the 

 Hurricane Carla data) . 



The computation of equation (1) entails a loss of information (N values 

 replaced by kjj^ values). This causes pCf^^.) in equation (2) to be a smoothed 



version of the true spectral density and distorts or eliminates features 

 of the spectrum. The method of computation prevents the user from seeing 

 aspects of the spectrum which may be important. 



The second method for computing the spectral density, which has come 

 into wide favor during the last few years, is based on the application of 

 the fast Fourier transform computing algorithm to the water level eleva- 

 tions (Bergman, 1973). Complex- valued Fourier coefficients, Aj^^, are 

 obtained (for m=0,l,2,..., N-1) by: 



N-1 



Am = ^t ^ rin 



2TTmn 



- iAt 



n=0 



N-1 



E 



n=0 



2iTmn 



iV„ 



(4) 



