both high waves and low waves to occur in succession. The autocorrelation 

 between wave periods can be helpful in identifying a tendency for the grouping 

 of periods. 



A third technique used previously by some investigatoxs to identify mod- 

 ulation characteristics in a time series involves a spectral analysis of the 

 squared time series. The spectrum shows strong modulation frequencies explic- 

 itly if they exist. It can also provide an indication of the strength of 

 modulation. 



The three techniques used to investigate field wave grouping character- 

 istics are described in more detail in the following paragraphs. 



a. Local Variance Time Series . The basic starting point in this analysis 

 is a time series of sea-surface elevations as given in equation (5). The 

 first point in a new time series is created by computing the variance of the 

 first I points in the original time series. 



Z(At) = - I y2(iAt) - 



^i=l 



- I y(iAt) 



1^1=1 



I < N 



(25) 



The second point in the new time series is the variance of the second through 

 the (I + l)'th original points. The n'th point in the new time series is 



^ I+n 

 Z(nAt) = - I y^(iAt) - 

 i=n 



J I+n 



- I y(iAt) 

 i=n 



I + n < N 



(26) 



where Z(nAt) is the new time series and I a constant. Z(nAt) , n = 1, 2 ..., 

 (N - I) represents the time variation of local variance and will be referred 

 to as the Local Variance Time Series (LVTS). 



The constant I must be chosen so wave groups will be evident in the 

 LVTS. If I is too small, the LVTS fluctuates erratically with a period on 

 the order of the period of large waves in the original time series. If I is 

 too large, high wave groups are smeared out in the LVTS. After some experi- 

 mentation, a value of I was chosen to approximate the number of data points 

 in two repetitions of the peak spectral period from the original time series, 

 y(nAt). Sedivy's (1978) and Nelson's (1980) studies of statistical properties 

 of wave groups used the same LVTS approach including the same criteria for 

 choosing I. Both investigators used a systemmatic approach to determine that 

 the optimum I is twice the peak, spectral period. 



The LVTS is processed by computer procedures developed for use with time 

 series of sea-surface elevation. The mean is removed 



z(nAt) = Z(nAt) - 



N - I 



N-I 



I Z(iAt) 

 i=l 



(27) 



where z(nAt) is the LVTS with mean removed. The standard deviation is 

 computed 



38 



