On the distributions of wave periods, wavelengths, and amplitudes in a random wave field

Xu, Dong; Li, Xiang; Zhang, Lizhen; Xu, Ning; Lü, Hongmin

doi:10.1029/2003jc002073

Cited by 17 publications

(15 citation statements)

References 26 publications

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“…[31] Xu et al [2004] calculated by using Hilbert transformation arithmetic the local wave amplitudes and periods of surface elevation records measured in a wind wave flume. It is worth noting that the good fitting of their theoretical pdf (equation (6)) with these data cannot validate the appropriateness of this pdf in describing the joint distribution of individual wave heights and periods.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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On the improvements to the wave statistics of narrowbanded waves when applied to broadbanded waves

2006

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On the basis of the narrowband approximation the joint and marginal probability distribution functions (pdf) of local wave heights and periods of ocean surface waves have been derived and have been taken approximately as those of the crest‐to‐trough wave heights and zero upcrossing periods. Various correction schemes have been proposed to these pdf's in order to apply them to broadband waves. This paper is devoted to taking a close look at these correction schemes. The relationship between the distributions of the local and the discrete wave heights and periods is derived at first. This derivation provides a precise interpretation to the improvements due to Stansell et al. (2004) and Zheng et al. (2004). The resultant relationship is different from the one deduced by Kitano and Mase (1998), on which their improved wave height distribution rests. The unique analysis of Kitano and Mase (1998) leading to this relationship is inspected and their omission is found. The crest‐to‐trough wave height distribution proposed by Tayfun (1981) is also examined and found to be reasonable only when the spectral bandwidth is infinitesimal. The wind wave data measured in Bohai China in 1997 is analyzed here. The sensitivities of the measured (as opposed to actual) distributions to smoothing timescale and to sampling rate are explored and comparisons of various theoretical pdf's with the measured distributions are made. It is found that (1) the wave height distribution due to Stansell et al. (2004) and Zheng et al. (2004) in which the contributions of negative phase derivatives are taken into account fits best with the 20 Hz sampled, 5‐point smoothed data; (2) neglecting the contributions of negative phase derivatives results in an expression of the probability of exceeding large wave heights which is closest to the 20 Hz sampled, 5 or 9‐point smoothed data; (3) the Rayleigh distribution generally agrees well with our data; the discrepancy between it and early observations mainly comes from the low sampling rate; (4) Forristall's (1984) suggestion of employing Tayfun's (1981) distribution to predict the probability of exceeding given large wave height agrees well with both the 5 and 9‐point smoothed data within the range 1.0 ≤ H ≤ 3.0; and (5) the improved joint distribution of wave heights and periods, as well as the marginal distribution of wave periods due to Stansell et al. (2004) and Zheng et al. (2004) are unsatisfactory for broadband waves.

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…[6] Representing the wave elevation at a given position by equation (1) and assuming that it is a Gaussian process, Sun [1987] and Xu et al [2004] among others obtained the joint pdf of local wave heights and periods,…”

Section: Relationship Between the Distributions Of Local And Discretementioning

confidence: 99%

On the improvements to the wave statistics of narrowbanded waves when applied to broadbanded waves

2006

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“…of envelope and period we follow the work of Longuet-Higgins (1983), see also Xu et al (2004). The starting point is the assumption that η(t) is a stationary Gaussian process.…”

Section: Joint Distribution Of Envelope Height and Periodmentioning

confidence: 99%

“…This good agreement is not only obtained for narrow-band spectra but also for broad-band spectra. The present approach is therefore an extension of the work of Longuet-Higgins (1983) and Xu et al (2004) and, as discussed in § 4, is therefore a good starting point for the development of a statistical theory of weakly nonlinear waves. Finally, in § 5 a summary of conclusions is presented.…”

Section: Introductionmentioning

confidence: 98%

On a random time series analysis valid for arbitrary spectral shape

Janssen

2014

J. Fluid Mech.

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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.

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“…Memos (1994) and Memos and Tzanis (1994) demonstrated a joint probability density function valid for wide band spectra and high spectral width parameter and validated. On the basis of narrow band Gaussian process, Xu et al (2004) derived a joint probability density function for wave period and amplitude of sea waves. A disadvantage with these models is that they depend on the spectral width parameter and predict an exact Rayleigh distribution for amplitudes.…”

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confidence: 99%

Estimation of wave period statistics using numerical coastal wave model

et al. 2008

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We used field and model wave data to investigate that zero crossing and average wave period distribution follow Gamma distribution. Since Gamma CDF is an infinite power series, further mathematical treatment is difficult. Hence its shape parameter is approximated to the nearest integer to arrive at Erlang distribution. An expression is derived from Erlang distribution to estimate various mean wave periods and significant wave period and validated by relative root mean square (RRMS) error criteria. It is shown by mathematical logic that the significant wave period distribution follows Erlang (or Gamma) distribution and is validated. The average of one-third and one-tenth highest wave periods (T s ) estimated from Erlang distribution are in accordance with the values computed from recorded buoy and numerical coastal wave model wave period data. The predicted T s values from coastal wave model wave period data underestimates the values from buoy wave period data.

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On the distributions of wave periods, wavelengths, and amplitudes in a random wave field

Cited by 17 publications

References 26 publications

On the improvements to the wave statistics of narrowbanded waves when applied to broadbanded waves

On the improvements to the wave statistics of narrowbanded waves when applied to broadbanded waves

On a random time series analysis valid for arbitrary spectral shape

Estimation of wave period statistics using numerical coastal wave model

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