Abstract
Abstract Reliable design and reanalysis of coastal and offshore structures require, among other things, characterization of extreme crest elevation corresponding to long return periods. Extreme crests typically correspond to focused wave events enhanced by wave–wave interactions of different orders—third-order, four-wave interactions dominating in deep water (Janssen, P. A. E. M., 2003, “Nonlinear Four-Wave Interactions and Freak Waves,” J. Phys. Oceanogr., 33(4), pp. 863–884). Higher-order spectral (HOS) analysis can be used to identify wave–wave interactions in time-series of water surface elevation; trispectral analysis is needed to detect third-order, four-wave interactions. 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—the T- and V-trispectrum (with corresponding tricoherences), respectively. It is shown that the T-tricoherence is capable of detecting phase-locked four-wave interactions of the type f4 = f1 + f2 + f3 when these are simulated with simple sinusoids, but such interactions were not detected in HOS model simulations and field data. It is also found that high V-tricoherence levels are detected at frequencies at which four-wave interactions of the type f1 + f2 = f3 + f4 are expected, but these may simply indicate combinations of independent pairs of Fourier components that happen to satisfy the frequency relationship. Preliminary analysis shows that using a cumulant-based trispectrum (Kravtchenko-Berejnoi, V., Lefeuvre, F., Krasnosel'skikh, V. V., and Lagoutte, D., 1995, “On the Use of Tricoherent Analysis to Detect Nonlinear Wave–Wave Interactions,” Signal Process., 42(3), pp. 291–309) may improve identification of wave–wave interactions. These results highlight that caution needs to be exercised in interpreting trispectra in terms of specific four-wave interactions occurring in sea states and further research is needed to establish whether this is in fact possible in practice.
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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