Extremal periodic wave profiles

As a contribution to deterministic investigations into extreme fluid surface waves, in this paper wave profiles of prescribed period that have maximal crest height will be investigated. As constraints the values of the momentum and energy integrals are used in a simplified description with the KdV model. The result is that at the boundary of the feasible region in the momentum-energy plane, the only possible profiles are the well known cnoidal wave profiles. Inside the feasible region the extremal profiles of maximal crest height are ¿cornered¿ cnoidal profiles: cnoidal profiles of larger period, cut-off and periodically continued with the prescribed period so that at the maximal crest height a corner results

Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom

The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event

Coherence and predictability of extreme events in irregular waves

This paper concerns the description and the predictability of a freak event when at a certain position information in the form of a time signal is given. The prediction will use the phase information for an estimate of the position and time of the occurrence of a large wave, and to predict the measure of phase coherence at the estimated focussing position. The coherence and the spectrum will determine an estimate for the amplitude. After adjusting for second order nonlinear effects, together this then provides an estimate of the form of a possible freak wave in the time signal, which will be described by a pseudo-maximal signal. In the exceptional case of a fully coherent signal, it can be described well by a so-called maximal signal. \ud \ud We give four cases of freak waves for which we compare results of predictions with available measured (and simulated) results by nonlinear AB-equation (van Groesen and Andonowati, 2007; van Groesen et al., 2010). The first case deals with dispersive focussing, for which all phases are (designed to be) very coherent at position and time of focussing; this wave is nearly a maximal wave. The second case is the Draupner wave, for which the signal turns out to be recorded very close to its maximal wave height. It is less coherent but can be described in a good approximation as a pseudo-maximal wave. The last two cases are irregular waves which were measured at MARIN (Maritime Research Institute Netherlands); in a time trace of more than 1000 waves freak-like waves appeared "accidentally". Although the highest wave is less coherent than the other two cases, this maximal crest can still be approximated by a pseudo-maximal wav

Extremal periodic wave profiles

As a contribution to deterministic investigations into extreme fluid surface waves, in this paper wave profiles of prescribed period that have maximal crest height will be investigated. As constraints the values of the momentum and energy integrals are used in a simplified description with the KdV model. The result is that at the boundary of the feasible region in the momentum-energy plane, the only possible profiles are the well known cnoidal wave profiles. Inside the feasible region the extremal profiles of maximal crest height are "cornered" cnoidal profiles: cnoidal profiles of larger period, cut-off and periodically continued with the prescribed period so that at the maximal crest height a corner results

Effects of delayed Kerr nonlinearity and ionozation on the filamentary ultrashort laser pulses in air

We present a systematic study of filamentary ultrashort laser pulses in air, through numerical solutions of the nonlinear Schrödinger equation for various contributions of the delayed Kerr nonlinearity. The results show that a relatively larger contribution of the delayed Kerr nonlinearity will lead to a longer stable filament. This is explained from the transfer of the nonlinear contributions from the frontier part to the back of the pulse and the counterbalanced action of the negative plasma induced nonlinearity by the delayed Kerr nonlinearity in the trailing part of the pulses. Furthermore, effect of ionization on the stability of the filament is investigated. Two formulas are used to generate the data of the ionization, i.e., the Perelemov, Popov, and Terent'ev (PPT) and the Ammosov, Delone, and Krainov formula. It is found that simulation with higher ionization rate (PPT) could generate a more stable and longer filament

A dynamic variation principle for elastic-fluid contacts applied to elastohydrodynamic lubrication theory

This paper discusses the variational structure of the line contact problem between an elastic medium and a fluid. The equations for the deformation in the elastic material, and for the flow of the viscous fluid are assumed to be determined from an elastic energy E and a power functional P respectively. Then it is shown that a variational formulation of the combined system can be given: apart from the equations in the interior of both media also the equations expressing balance of forces on the separating boundary are obtained from the power functional\ud \ud Image\ud \ud . To that end time dependent deformations are to be considered for which the velocity in the elastic medium vanishes and for which the acceleration of particles on both sides of the common boundary is equal.\ud \ud This general result is employed in the rest of the paper to a typical problem from elastohydrodynamic lubrication theory. The flow of the lubricant allows a basic variational formulation by assuming it to be dominated by viscous dissipation. The complicated resulting expressions are simplified considerably by imposing the common restriction to small deformations and by exploiting the characteristic length scales of the problem. These approximations are performed directly into the governing power and energy functional. The formulation of the approximated system becomes a genuine variational principle and produces correctly the differential expressions. Moreover, it generates in a natural way efficient numerical methods to calculate the deformation of and the pressure at the free boundary if the time variable is discretized

Sensitivity of the inverse wave crest problem

In a previous paper [Physica D 141 (3-4) (2000) 316], the inverse problem for wave crests was introduced and a solution strategy for two-wave interactions was given. Here these solutions are actually constructed, in particular for the cases with small interaction angle, moderate phase shifts, and/or symmetric interactions. Two detailed examples are presented and analyzed. The sensitivity of the method is investigated, and conclusions about the practical applicability are given

Near-coast tsunami waveguiding: phenomenon and simulations

In this paper we show that shallow, elongated parts in a sloping bottom toward the coast will act as a waveguide and lead to large enhanced wave amplification for tsunami waves. Since this is even the case for narrow shallow regions, near-coast tsunami waveguiding may contribute to an explanation that tsunami heights and coastal effects as observed in reality show such high variability along the coastline. For accurate simulations, the complicated flow near the waveguide has to be resolved accurately, and grids that are too coarse will greatly underestimate the effects. We will present some results of extensive simulations using shallow water and a linear dispersive Variational Boussinesq model.\u