The solitary wave is a localized hydrodynamic phenomenon that can occur because of a balance between nonlinear cohesive and linear dispersive forces in a fluid. It has been shown theoretically, and observed experimentally, that some solitary waves have properties analogous to those of elementary particles, and the waves have therefore been named solitons. During a measurement program in the Andaman Sea near northern Sumatra, large-amplitude, long internal waves were observed with associated surface waves called tide rips. Using theoretical results from the physics of nonlinear waves, it is shown that the internal waves are solitons and their interactions with surface waves are described.

Freak waves are very large, rare events in a random ocean wave train. Here we study the numerical generation of freak waves in a random sea state characterized by the JONSWAP power spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schroedinger (NLS) equation. We identify two parameters in the power spectrum that control the nonlinear dynamics: the Phillips parameter α and the enhancement coefficient γ. We discuss how freak waves in a random sea state are more likely to occur for large values of α and γ. Our results are supported by extensive numerical simulations of the NLS equation with random initial conditions. Comparison with linear simulations are also reported.Freak waves are extraordinarily large water waves whose heights exceed by a factor of 2.2 the significant wave height of a measured wave train [1]. The mechanism of freak wave generation has become an issue of principal interest due to their potentially devastating effects on offshore structures and ships. In addition to the formation of such waves in the presence of strong currents [2] or as a result of a simple chance superposition of Fourier modes with coherent phases, it has recently been established that the nonlinear Schroedinger (NLS) equation can describe many of the features of the dynamics of freak waves which are found to arise as a result of the nonlinear self-focusing phenomena [3][4][5]. The self-focusing effect arises from the Benjamin-Feir instability [7]: a monochromatic wave of amplitude, a 0 , and wave number, k 0 , modulationally perturbed on a wavelength L = 2π/∆k, is unstable whenever ∆k/(k 0 ε) < 2 √ 2, where ε is the steepness of the carrier wave defined as ε=k 0 a 0 . The instability causes a local exponential growth in the amplitude of the wave train. This result is established from a linear stability analysis of the NLS equation [8] and has been confirmed, for small values of the steepness, by numerical simulations of the fully nonlinear water wave equations [5,6] (for high values of steepness wave breaking, which is clearly not included in the NLS model, can occur). Moreover, it is known that small-amplitude instabilities are but a particular case of the much more complicated and general analytical solutions of the NLS equation obtained by exploiting its integrability properties via Inverse Scattering theory in the θ-function representation [11,12].Even though the above results are well understood and robust from a physical and mathematical point of view, it is still unclear how freak waves are generated via the Benjamin-Feir instability in more realistic oceanic conditions, i.e. in those characterized not by a simple monochromatic wave perturbed by two small side-bands, but instead by a complex spectrum whose perturbation of the carrier wave cannot be viewed as being small. Furthermore, the focus herein is not to attempt to model ocean waves but instead to study leading order effects using the nonlinear Schroedinger equation, as suggested by [3][4][5]. Research at hi...

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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.

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A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30• at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests.Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

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We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading. DOI: 10.1103/PhysRevLett.102.114502 PACS numbers: 47.35.Bb, 47.55.NÀ An important task in the study of surface gravity waves is the determination of the probability density function of the surface wave elevation. The knowledge of the probability of the occurrence of large amplitude waves is essential for different engineering purposes such as the prediction of wave forces and structural responses or the design of offshore structures. A deep comprehension of the physical mechanisms of the generation of such waves is also a first step towards the development of an operational methodology for the probabilistic forecast of freak waves. It is well known that surface gravity waves obey nonlinear equations and, to date, a universal tool suitable for deriving the probability distribution function of a nonlinear system has not yet been developed. Fortunately, water waves are on average weakly nonlinear [1,2] and solutions can be generally written as power series, where the small parameter, in the case of deep water waves, is the wave steepness ". Strong departure from Gaussian statistics of the surface elevation can be observed if third order nonlinearities are considered. At such order it has been shown numerically [3] and theoretically [4] that, for long-crested waves, a generalization of the Benjamin-Feir instability [5] (or modulational instability [2]) for random spectra can take place [6]. This instability, that corresponds to a quasiresonant four-wave interaction in Fourier space, results in the formation of large amplitude waves (or rogue waves) [7] which affect the statistical properties of the surface elevation (see, for example, [8]). This is particularly true if the ratio between the wave steepness and the spectral bandwidth, known as the Benjamin-Feir Index (BFI), is large [4]. We mention that rogue waves have also been recently observed in optical systems [9] and in acoustic turbulence in He II [10] where giant waves are observed during an inverse cascade process.We emphasize that in many different fields of physics (plasmas [11,12], nonlinear optics [13,14], chargedparticle beam dynamics [15,16]) the modulational instability plays an important role; under suitable physical conditions a nonlinear Schrödinger equation can be derived and the modulational instability can be analyzed directly with this equation [2]. A major question which has to be addressed (and is the subject of the pre...