Correlation Between Harmonics

A type of wavelet analysis can be used to study time-correlations between harmonics. This will be illustrated with a synthetic example consisting of a mixture of a nonlinear 12-second unidirectional wave from Dean's numerical solutions, combined with a simulation of random waves with spectral peak frequency corresponding to a 12-second period. The harmonics in the random simulation are independent of each other, while those in the nonlinear numerical solution are tightly correlated over time.

The ``mother'' wavelet used in the study was selected to have the formula in time domain (with B=0.1)

 

$$w (t) = e^{-\pi B^2t^2} \cos (2\pi t)$$ (8)

The frequency domain version of this is

 

$$W (f) = \frac{e^{-\pi \left( \frac{f-1}B \right)^2} + e^{-\pi \left( \frac{f+1}B \right)^2}}{2B}$$ (9)

A form of the wavelet transform may be stated as

 

$$\int w \left( \frac{\tau - t}{T} \right) \eta (\tau) d\tau = N (t,T)$$ (10)

The T is the scale parameter and the t is the shift parameter. As applied here, the parameters exactly correspond to wave period and time in the output of a narrow band filter, with effective frequency bandwidth given by B/T, and a central frequency of f₀ = 1/T. The effective width of w(t/T) in time is the reciprocal of the frequency bandwidth, or T/B. Thus, the convolution in Eq. (10) is always ``averaging'' over the same number of cycles at each T.

The above wavelet is somewhat unconventional within traditional wavelet theory, but makes very good sense in the context of a ``filter bank'' [16]. It also has the considerable advantage of allowing a direct interpretation as the output of a narrow bandpass filter bank and of involving a scale parameter exactly equal to the standard wave period. Many engineers have always felt uncomfortable with relations stated in terms of frequency, and here, finally, is a case where the period is the natural mathematical parameter of reference.

Wavelet output is commonly presented as a contour or color-scaled map with shift parameter on the horizontal scale and scale parameter on the vertical scale. With the choices made above, this corresponds to a time horizontal scale and a period vertical scale. Such a contour map is given in Fig. 6 for the wavelet analysis of the synthetic mixture. An interesting feature in Fig. 6 is the horizontal streak of slightly high values at period equal to 6 seconds, which is the second harmonic of the dominating 12-second period running across the middle of the plot.

The time series used in the wavelet analysis was divided into four equal parts, and each was analyzed to give the spectra in Fig. 5. The ``blip'' at frequency around 1/6 persists through all of the four spectra, although many of the other bumps come and go.

The $17 \times 17$ temporal correlation matrix between the 17 period bands used to develop Fig. 6 was computed and all correlations less than 0.94 were set to zero. This is shown as a mesh plot in Fig. 7. The ``spike'' remaining at indices (6,12) clearly shows up as evidence that the second harmonic of the dominant 12-second primary wave is traveling at the same speed as the main harmonic.

Wavelet methods are often useful as supporting evidence, but output results are sometimes hard to explain. Considerable caution is required and there is still much to be learned before wavelet analysis becomes a reliable tool in engineering.

  

Figure 5: Spectral Densities for Each Fourth of the Synthetic Time Series

  

Figure 6: Wavelet Plot

  

Figure 7: Correlations in Excess of 0.94 in the Correlation Matrix of Wavelet Period Bands