Wave statistics in non-linear random sea
type: Item
instance of: scholarly article
title: Wave statistics in non-linear random sea
zbMATH Open document ID: 1060.62131
author: Ulla Machado
author: Igor Rychlik
published in: Extremes
publication date: 2005-05-20T00:00:00Z
review text: The linear sea model \(\eta_l\) consists of \(N\) cosine waves traveling along the \(x\)-axis: \[ \eta_l(x,t)=\sum_{n=-N}^N (A_N/2) e^{i(\omega_n t-\kappa_n x)}, \] where the wave number \(\kappa_n\) and the angular frequency \(\omega_n\) are related by \(\omega_n^2=g\kappa_n\text{tanh}(h\kappa_n)\) (\(g\) being the earth acceleration and \(h\) being the water depth). The second order (quadratic) correction which describes the interaction between waves is given by \[ \eta_q(x,t)=\sum_{n,m=-N}^N (A_n A_m/ 4)E(\omega_n,\omega_m) \exp(i(\omega_n t-\kappa_n x))\exp(i(\omega_m t-\kappa_m x)), \] where the quadratic transfer function \(E\) for deep water is \(E(\omega,\widetilde\omega)=(\omega^2-\widetilde\omega^2)/(2g)\). (The general form of \(E(\omega,\widetilde\omega)\) is much more complicated). The authors consider the model \(\eta=\eta_l+\eta_q\) with \(A_n=\sigma_n(U_n-iV_n)\), where \(U_n\), \(V_n\) are i.i.d. standard normal, \(\sigma^2_n\) is the energy of the wave with frequency \(\omega_n\). The Rice-Kac formula is used for the evaluation of the crossing intensity \(\mu(u)\) of fixed level \(u\). The authors derive the joint density of \((\eta,\dot\eta)\) and use saddlepoint methods to approximate \(\mu(u)\). Numerical results are presented.
Mathematics Subject Classification ID: 62P12
Mathematics Subject Classification ID: 62G15
Mathematics Subject Classification ID: 86A05
Mathematics Subject Classification ID: 62G32
zbMATH DE Number: 2169513
zbMATH Keywords: crest distribution
zbMATH Keywords: non-Gaussian sea
zbMATH Keywords: Rice's formula
zbMATH Keywords: saddlepoint method
describes a project that uses: WAFO
MaRDI profile type: Publication
DOI: 10.1023/B:EXTR.0000025663.45811.9B
description: scientific article; zbMATH DE number 2169513