Wave statistics in non-linear random sea (Q1775995)

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.