On a random time series analysis valid for arbitrary spectral shape

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 ρ and a slowly varying frequency ω = −∂φ/∂t where φ 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.

The sinking of the El Faro: predicting real world rogue waves during Hurricane Joaquin

“…For third-order nonlinear random seas the excess kurtosis comprises a dynamic component due to nonlinear quasi-resonant wave-wave interactions 11 , 62 and a Stokes bound harmonic contribution 63 . In deep water it reduces to the simple form 11 , 63 , 64 where λ 3, NB is the skewness of narrowband waves 8 .…”

Fourier amplitude distribution and intermittency in mechanically generated surface gravity waves

“…The input wave fields are Gaussian random processes (surface elevation follows a normal distribution, while For a Gaussian random process, the intensity follows an exponential function, which represents a benchmark for the Rayleigh distribution [18,44,45]. Fig.…”

Real world ocean rogue waves explained without the modulational instability

“…(A23) in45 and Methods section). In deep water it reduces to the simple form 404546 where λ 3, NB  = 3 μ m 93233. As for the dynamic component, Fedele29 recently revisited Janssen’s8 weakly nonlinear formulation for .…”