Abstract
Abstract Reliable design and reanalysis of coastal and offshore structures requires, amongst other things, characterisation of extreme crest elevation corresponding to long return periods, and of the evolution of a wave in space and time conditional on an extreme crest. Extreme crests typically correspond to focussed wave events enhanced by wave-wave interactions of different orders. Higher-order spectral analysis can be used to identify wave-wave interactions in time-series of water surface elevation. The bispectrum and its normalised form (the bicoherence) have been reported by numerous authors as a means to characterise three-wave interactions in laboratory, field and simulation experiments. The bispectrum corresponds to a frequency-domain representation of the third order cumulant of the time-series, and can be thought of as an extension of the power spectrum (itself the frequency-domain representation of the second order cumulant). The power spectrum and bispectrum can both be expressed in terms of the Fourier transforms of the original time-series. The Fast Fourier transform (FFT) therefore provides an efficient means of estimation. However, there are a number of important practical considerations to ensuring reasonable estimation. To detect four-wave interactions, we need to consider the trispectrum and its normalised form (the tricoherence). The trispectrum corresponds to a frequency-domain (Fourier) representation of the fourth-order cumulant of the time-series. Four-wave interactions between Fourier components can involve interactions of the type where f1 + f2 + f3 = f4 and where f1 + f2 = f3 + f4, resulting in two definitions of the trispectrum, depending on which of the two interactions is of interest. We consider both definitions in this paper. Both definitions can be estimated using the FFT, but it’s estimation is considerably more challenging than estimation of the bispectrum. Again, there are important practicalities to bear in mind. In this work, we consider the key practical steps required to correctly estimate the trispectrum and tricoherence. We demonstrate the usefulness of the trispectrum and tricoherence for identifying wave-wave interactions in synthetic (based on combinations of sinusoids and on the HOS model) and measured wave time-series.
Highlights
The power spectrum, based on Fourier analysis, has been widely used as a tool to study ocean wind waves by scientists and engineers alike, since its introduction for this purpose around 1950. Barber and Ursell (1948) published the first wave spectra, and Pierson and Marks (1952) introduced power spectrum analysis to ocean wave data analysis, following techniques pioneered by Tukey (1949)
We demonstrate the usefulness of the trispectrum and tricoherence for identifying wave-wave interactions in synthetic and measured wave time-series
The power spectrum is the Fourier transform of the second-order cumulant, the bispectrum is the Fourier transform of the third-order cumulant, the trispectrum is the Fourier transform of the fourth-order cumulants, and in general, the kthorder polyspectrum is the Fourier transform of the (k+1)th-order cumulant
Summary
The power spectrum, based on Fourier analysis, has been widely used as a tool to study ocean wind waves by scientists and engineers alike, since its introduction for this purpose around 1950. Barber and Ursell (1948) published the first wave spectra, and Pierson and Marks (1952) introduced power spectrum analysis to ocean wave data analysis, following techniques pioneered by Tukey (1949). These curves are similar to those for the HOS spectra, with similar peak values, suggesting similar interpretation – i.e. possibly four wave interactions of the type where f1 + f2 = f3 + f4, but not of the type f4 = f1 + f2 + f3 that are phase-locked. Not materially different from the HOS and laboratory examples, we provide f3 axis slice images for the Ttricoherence in Figure and V-tricoherence in Figure for f3 ≈ fp on account of the historical interest in the Draupner wave record and because we believe this is the first example of trispectral analysis of this record to be published The images in these two figures have a lower resolution than the earlier examples, due to the lower sampling frequency of the Draupner measurements and our objective to reduce sampling variability in the estimates as far as practical. A similar interpretation begs – i.e. there is evidence of four wave
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