where a and b are constants and a a known frequency. It is necessary 

 to pick the frequencies, a , to cover the range of frequencies expected in 

 the wave record. Computational labor may be reduced by choosing frequencies 

 which give rise to orthogonal functions. This set of frequencies is defined 

 as 



2inn 



<^m (8) 



™ NAt 



where NAt is the duration of the record. 

 The time series is then represented as 



^i'^ / 2innn 2irain\ 



y(nAt) = I a cos + b sin ) (9) 



m=i V N ■" N / 



/2\ N /27nnn\ N 



(10) 



b.=(^)j^y(nAt)sin(^).l,m5(f-l) 



It is noted in the above that a constant a could be defined as 



which is simply the mean value of the time series, taken as zero in this 

 analysis. Also, the value of b»T/ is undefined because sin 2Trmn/N is 

 identically equal to zero when m is equal to N/2. 



The set of coefficients a^ and b is called the Fourier transform of 

 the time series y(nAt), n = 1, 2, 3, ...» N. Trigonometric identities can be 

 used to express equation (9) in the form 



N/2 /2Tnnn \ 

 y(nAt) = I A^ cosf— — - <t,^ (12) 



m=i ^ N ■ / 



where 



m m 



^-'t 



(13) 



When the time series represents sea-surface elevations, the coefficients 

 A^ are fundamentally related to the potential energy in the sea surface as 

 shown by Harris (1974). Hence, the Fourier transform represented in equation 

 (12) is a method for partitioning energy in the sea surface among N/2 dis- 

 crete frequencies. 



28 



