where Af is 0.02 cps. With the above values the resolution rrequency if 

 and the Nyquist frequency fn become 0.04 and 4.0 cps, respectively. 



It is apparent from Figure 14 that both spectra have significant 

 harmonics, but that the most prominent harmonics are associated with the 

 spectrum of the breaking wave because it has the least sinusoidal shape. 

 The spectral peaks are Identified by numbers giving the order of the 

 harmonic, where one is the first order or fundamental, two, the second 

 order or first harmonic, etc. For this analysis, with a Nyquist frequency 

 of 4.0 cps, it Is clear that any harmonic above order four will have a 

 frequency greater than the Nyquist. Thus the fifth order harmonic, with a 

 frequency of 4.4 cps, folds over and appears as a ghost signature with a 

 frequency of fg = 8.0 - 4.4 = 3.6 cps. For these breaking waves there Is 

 identifiable energy folded over for harmonics as high as the ninth order. 



2. Two-Dimenslona I Power Spectra 



The spectrum of a single time series as given by equations (6) and 

 (7) is one-dimensional and applies to the situation where a single staff Is 

 essentially a "blind" sensor. If synoptic time series are obtained at two 

 or more points in space and there is coherence between points, then the 

 data may be interpreted as having other dimensions commensurate with the 

 geometry of the sensor array. The analysis requires the computation of the 

 autocovar iance and spectra of each series (equations (5) and (6)) as well 

 as the cross-correlation (two-dimensional covarlance) between time series, 

 and their cosine and quadrature spectra. 



The two cross correlations between time series rii(t) and n2(''') can be 

 defined as 



and 



Ci2(t) = ^ni(t) . n2(t - t) y (9) 



C2i(t) = <( n2( + ) • ni(t - t) )> (10) 



where x, n, and t are defined in equation (5) and the subscripts I and 2 

 refer to time series I and 2. 



The cosine spectra and quadrature spectra for these two time series 

 become 



Pl2(f) = Jn CCi2(t) + C2i(t)] • COS(27rfT) • dx (II) 



and 



)l2(f) = CCi2(t) - C2i(t)] • sln(2TTfx). dx 



(12) 



26 



