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On a random time series analysis valid for arbitrary spectral shape

Published online by Cambridge University Press:  23 October 2014

Peter A. E. M. Janssen*
Affiliation:
ECMWF, Shinfield Park, Reading RG2 9AX, UK
*
Email address for correspondence: p.janssen@ecmwf.int

Abstract

While studying the problem of predicting freak waves it was realized that it would be advantageous to introduce a simple measure for such extreme events. Although it is customary to characterize extremes in terms of wave height or its maximum it is argued in this paper that wave height is an ill-defined quantity in contrast to, for example, the envelope of a wave train. Also, the last measure has physical relevance, because the square of the envelope is the potential energy of the wave train. The well-known representation of a narrow-band wave train is given in terms of a slowly varying envelope function ${\it\rho}$ and a slowly varying frequency ${\it\omega}=-\partial {\it\phi}/\partial t$ where ${\it\phi}$ is the phase of the wave train. The key point is now that the notion of a local frequency and envelope is generalized by also applying the same definitions for a wave train with a broad-banded spectrum. It turns out that this reduction of a complicated signal to only two parameters, namely envelope and frequency, still provides useful information on how to characterize extreme events in a time series. As an example, for a linear wave train the joint probability distribution of envelope height and period is obtained and is validated against results from a Monte Carlo simulation. The extension to the nonlinear regime is, as will be seen, fairly straightforward.

Type
Papers
Copyright
© 2014 Cambridge University Press 

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References

Gabor, D. 1946 Theory of communication. J. Inst. Electr. Engrs: Radio Commun. Engng 93, 429441.Google Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863884.Google Scholar
Janssen, P. A. E. M. 2009 On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech. 637, 144.Google Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 120.Google Scholar
Longuet-Higgins, M. S. 1957 The statistical analysis of a random, moving surface. Phil. Trans. R. Soc. Lond. A 249, 321387.Google Scholar
Longuet-Higgins, M. S. 1983 On the joint distribution of wave periods and amplitudes in a random wavefield. Proc. R. Soc. Lond. A 389, 241258.Google Scholar
Mori, N. & Janssen, P. A. E. M. 2006 On kurtosis and occurrence probability of freak waves. J. Phys. Oceanogr. 36, 14711483.Google Scholar
Naess, A. 1982 Extreme value estimates based on the envelope concept. Appl. Ocean Res. 4, 181187.Google Scholar
Phillips, O. M. 1960 The dynamics of unsteady gravity waves of finite amplitude. Part 1. J. Fluid Mech. 9, 193217.Google Scholar
Pierson, W. J. JR & Moskowitz, L. 1964 A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. J. Geophys. Res. 69, 51815190.Google Scholar
Shum, K. T. & Melville, W. K. 1984 Estimates of the joint statistics of amplitudes and periods of ocean waves using an integral transform technique. J. Geophys. Res. 89, 64676476.CrossRefGoogle Scholar
Tayfun, M. A. & Lo, J.-M. 1990 Nonlinear effects on wave envelope and phase. ASCE J. Waterway Port Coastal Ocean Engng 116, 79100.Google Scholar
Xu, D., Li, X., Zhang, L., Xu, N. & Lu, H. 2004 On the distributions of wave periods, wavelengths and amplitudes in a random wavefield. J. Geophys. Res. 109, C05016.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190194.Google Scholar