1 . Calculation of Wave Direction and Energy Spectrum at Wave Gages . 



As noted previously, each of the input time-series pressure records con- 

 sists of 4,096 data points with a time increment of 0.25 second. To reduce 

 computational costs, modified time series are formed for analysis by averaging 

 four adjacent data points. These new time series contain 1,024 data points 

 spaced at 1.0-second intervals. This increases the aliasing period from 0.5 

 to 2.0 seconds; however, this is justified as the pressure response factor 

 for a water depth of 6 meters and a wave period of 2 seconds is approximately 

 0.005. 



The time series are analyzed using a standard fast Fourier transform (FFT) 

 program to determine the coefficients. For example, for pressure time series 

 from gage 1 



N-l / • 9 • \ 



Pl (j) = I [ ai (n) - i Dl (n)] exp(^i^ij (1) 



in which i = /-l and N is the total number of data points, T/At = 1,024, 

 where T is the time series record length of 1,024 seconds, At the time 

 increment of 1 second between samples, and j a discrete time t. where tj = 

 discrete time value = jAt. The FFT coefficients are defined in terms of the 

 pressure time series as 



N-l 

 i(n) - ib!(n) = I I Pl (j) exp (-i £Bl) 



(2) 



where the argument "n" of the Fourier coefficients a(n) and b(n) speci- 

 fies the quantity to be a discrete function of wave frequency, f , where 

 f n , a discrete frequency value, is nAf (where Af = l/T) and the a (0) term 

 represents the mean value of the time-series pressure record for wave gage 1. 

 Similar relationships exist for wave gage 2. In calculating the FFT coeffi- 

 cients, there are several options that may be employed in an attempt to reduce 

 spectral leakage which arises due to representing an aperiodic time series by 

 a periodic series. A large number of possible data windows (weighting func- 

 tions for data) have been developed to reduce the adverse effects of spectral 

 leakage (Harris, 1974). These can be expressed in the form of a weighting 

 function w( j ) , such that the modified time series p'(j) is of the form 



P'(j) = w(j) p(j) 



in which p(j) is the digitized measured pressure value at time tj = jAt, and 

 w(j) a weighting function. A characteristic of these weighting functions is 

 that they are equal to unity at the midpoint of the time series and decrease 

 to a lesser value near the two ends. In the present program, a "cosine bell" 

 weighting function is used; however, through comparisons of Pn with and 

 without this function, it was established that the effect of trie weighting 

 function was minimal (<5 percent). The cosine bell weighting function is 

 expressed by 



w(j) -|(l.O - cos ij&) (3) 



