Freak Waves in the Linear Regime: A Microwave Study

European Journal of Mechanics - B/Fluids, 2003

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey -Stewartson system, the Korteweg -de Vries equation, the Kadomtsev -Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

Journal of Geophysical Research, 2004

1] The results of laboratory measurements on limiting freak waves in the presence of currents are reported. Both dispersive spatial-temporal focusing and wave-current interaction are used to generate freak waves in a partial random wave field in the presence of currents. Wave group structure, for example, spectral slope and frequency bandwidth, is found to be critical to the formation and the geometric properties of freak waves. A nondimensional spectral bandwidth is shown to well represent wave group structure and proves to be a good indicator in determining limiting freak wave characteristics. The role of a co-existing current in the freak wave formation is recognized. Experimental results confirm that a random wave field does not prevent freak wave formation due to dispersive focusing. Strong opposing currents inducing partial wave blocking significantly elevate the limiting steepness and asymmetry of freak waves. At the location where a freak wave occurs, the Fourier spectrum exhibits local energy transfer to high-frequency waves. The Hilbert-Huang spectrum, a time-frequency-amplitude spectrum, depicts both the temporal and spectral evolution of freak waves. A strong correlation between the magnitude of interwave instantaneous frequency modulation and the freak wave nonlinearity (steepness) is observed. The experimental results provide an explanation to address the occurrence and characteristic of freak waves in consideration of the onset of wave breaking.

2006

This report deals with extreme wave phenomena. Exploration of the classical wave theories are made, both on the theoratical approach and on the statistical one. The first one shows wave generation phenomenon using only Euler's equation for a perfect fluid and gravity. On the other one, the statistical approach provides us with more real observations. Both models fail to explain some rare (or not so rare ?) events: freak waves. Then we defined what is a freak wave and some of the explanations that are given. Exploration on the non linear Schrödinger equation, which is known to give birth to gigantic waves is then the path taken. This equation could be easily derived from Euler's equations. Numerical solution of this equation are provided in the last chapter. Finally, the third part deals with spectral methods and how they are used to compute very easily non linear interaction for waves. Last chapter provides also results on this. In fact, the last chapter is devoted to the results obtained, either on solving NLS, either on the computation of surface waves.

Physics Reports, 2013

Rogue waves is the name given by oceanographers to isolated large amplitude waves, that occur more frequently than expected for normal, Gaussian distributed, statistical events. Rogue waves are ubiquitous in nature and appear in a variety of different contexts. Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, microwave cavities, etc. The first part of the review is dedicated to rogue waves in the oceans and to their laboratory counterpart with experiments performed in water basins. Most of the work and interpretation of the experimental results will be based on the nonlinear Schrödinger equation, an universal model, that rules the dynamics of weakly nonlinear, narrow band surface gravity waves. Then, we present examples of rogue waves occurring in different physical contexts and we discuss the related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves. The third part of the review is dedicated to optical rogue waves, with examples taken from the supercontinuum generation in photonic crystal fibers, laser fiber systems and two-dimensional spatiotemporal systems. In particular, the extreme waves observed in a two-dimensional spatially extended optical cavity allow us to introduce a description based on two essential conditions for the generation of rogue waves: nonlinear coupling and nonlocal coupling. The first requirement is needed in order to introduce an elementary size, such as that of the solitons or breathers, whereas the second requirement implies inhomogeneity, a mechanism needed to produce the events of mutual collisions and mutual amplification between the elementary solitons or wavepackets. The concepts of ''granularity'' and ''inhomogeneity'' as joint generators of optical rogue waves are introduced on the basis of a linear experiment. By extending these concepts to other systems, rogue waves can be classified as phenomena occurring in the presence of many uncorrelated ''grains'' of activity inhomogeneously distributed in large spatial domains, the ''grains'' being of linear or nonlinear origin, as in the case of wavepackets or solitons.

2004

The occurrence probability of freak waves is formulated as a function of number of waves and surface elevation kurtosis based on the weakly non-Gaussian theory. Finite kurtosis gives rise to a significant enhancement of freak wave generation. For fixed number of waves, the estimated amplification ratio of freak wave occurrence due to the deviation from the Gaussian theory is 50%-300%.

Physica D: Nonlinear Phenomena, 2000

The mechanism of the freak wave formation related to the spatial-temporal focusing is studied within the framework of the Korteweg-de Vries equation. A method to find the wave trains whose evolution leads to the freak wave formation is proposed. It is based on the solution of the Korteweg-de Vries equation with an initial condition corresponding to the expected freak wave. All solutions of this Cauchy problem by the reversal of abscissa represent the possible forms of wave trains which evolve into the freak wave. It is found that freak waves are almost linear waves, and their characteristic Ursell parameter is small. The freak wave formation is possible also from the random wave field and the numerical simulation describes the details of this phenomenon. It is shown that freak waves can be generated not only for specific conditions, but also for relative wide classes of the wave trains. This mechanism explains the rare and short-lived character of the freak wave.

Journal of Physics: Conference Series

This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a consequence of a group event that is present in a time signal at some position and contains successive high amplitudes with different frequencies. The linear theory predicts the position and time of the maximal amplitude wave quite well by minimizing the variance of the total wave phase of the given initial signal. The formation of the freak wave is shown to be mainly triggered by the local interaction of waves evolving from the group event that already contained large local energy. In the evolution, the phases become more coherent and the local energy is focussed to develop a larger amplitude. We investigate two laboratory experimental signals, a dispersive focussing wave with harmonic background and a scaled New Year wave. Both signals generate a freak wave at the predicted position and time and the freak wave can be described by a pseudo-maximal wave with specific parameters. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

The generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are simulated employing a nonlinear fourth-order Schrödinger equation. The evolution of the freak waves in deep water are analyzed. We investigate the effect of initial wave parameters on kurtosis and occurrence of freak waves. The results show that Benjamin-Feir index (BFI) is an important parameter to identify the presence of instability. The kurtosis presents a similar spatial evolution trend with the occurrence probability of freak waves. Freak waves in a random sea state are more likely to occur for narrow spectrum and small values of significant wave height.

2017

Four focusing models for generation of freak waves are presented. An extreme wave focusing model is presented on basis of the enhanced High2Order Spectral (HOS) method and the importance of the nonlinear wave2wave interaction evaluated by comparison of the calculated results with experimental and theoretical data. Based on the modification of Longuet2Higgins model , four wave models for generation of freak waves (a. extreme wave model + random wave model ; b. extreme wave model + regular wave model ; c. phase interval modulation wave focusing model ; d. number modulation wave focusing model with the same phase) are proposed. By use of different energy distribution techniques in four models , freak wave events are obtained with different Hmax/ Hs in finite space and time.

Physical Review Letters, 2006

Here we consider a simple weakly nonlinear model that describes the interaction of two-wave systems in deep water with two different directions of propagation. Under the hypothesis that both sea systems are narrow banded, we derive from the Zakharov equation two coupled nonlinear Schrödinger equations. Given a single unstable plane wave, here we show that the introduction of a second plane wave, propagating in a different direction, can result in an increase of the instability growth rates and enlargement of the instability region. We discuss these results in the context of the formation of rogue waves.

2015

We investigate the generation mechanisms and evolution dynamics of rogue (or freak) waves using direct phase-resolved nonlinear wave-field simulations. The focus is on the understanding of the effects of nonlinear broadband wave-wave interactions on rogue wave development and characteristics. Large-scale long-time simulations of nonlinear long-crested wave-fields with various wave spectrum parameters are obtained. Based on these simulations, we find that nonlinear self-focusing of wave groups and nonlinear wave group interactions play a critical role in the formation and development of rogue waves in deep seas. Occurrence of rogue waves is closely correlated to kurtosis of the wave-field, with larger kurtosis associated with rogue waves of higher height. Moreover, occurrence of rogue waves (especially of large height) is usually correlated with broadband wave spectra. Background The occurrence of rogue/freak waves in deep seas has been observed (e.g. Haver 2000). Due to exceptionall...

2009

Freak waves, or rogue waves, are one of the fascinating manifestations of the strength of nature. These devastating walls of water appear from nowhere, are short-lived and extremely rare. Despite the large amount of research activities on this subject, neither the minimum ingredients required for their generation nor the mechanisms explaining their formation have been given. Today, it is possible

Physical Review E, 2018

Rogue waves are extreme and rare fluctuations of the wave field that have been discussed in many physical systems. Their presence substantially influences the statistical properties of an incoherent wave field. Their understanding is fundamental for the design of ships and offshore platforms. Except for very particular meteorological conditions, waves in the ocean are characterised by the so-called JONSWAP (Joint North Sea Wave Project) spectrum. Here we compare two unique experimental results: the first one has been performed in a 270-meter wave tank and the other in optical fibers. In both cases, waves characterised by a JONSWAP spectrum and random Fourier phases have been launched at the input of the experimental device. The quantitative comparison, based on an appropriate scaling of the two experiments, shows a very good agreement between the statistics in hydrodynamics and optics. Spontaneous emergence of heavy tails in the probability density function of the wave amplitude is observed in both systems. The results demonstrate the universal features of rogue waves and provide a fundamental and explicit bridge between two important fields of research. Numerical simulations are also compared with experimental results.

IEEE Transactions on Antennas and Propagation, 2000

Ray chaos, manifested by the exponential divergence of trajectories in an originally thin ray bundle, can occur even in linear electromagnetic propagation environments, due to the inherent nonlinearity of ray-tracing maps. In this paper, we present a novel (two-dimensional) test example of such an environment which embodies intimately coupled refractive wave-trapping and periodicity-induced multiple scattering phenomenologies, and which is amenable to explicit full-wave analysis. Though strictly nonchaotic, it is demonstrated that under appropriate conditions which are inferred from a comprehensive parametric database generated via the above-noted rigorous reference solution, the high-frequency wave dynamics exhibits trends toward irregularity and other peculiar characteristics; these features can be interpreted as "ray-chaotic footprints," and they are usually not observed in geometries characterized by "regular" ray behavior. In this connection, known analogies from other disciplines (particularly quantum physics) are briefly reviewed and related to the proposed test configuration. Moreover, theoretical implications and open issues are discussed, and potential applications are conjectured.