A comparison of frequency downshift models of wave trains on deep water

Carter, John D.; Henderson, D. M.; Butterfield, Isabelle

doi:10.1063/1.5063016

Cited by 16 publications

(26 citation statements)

References 30 publications

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“…1, a sign of frequency down shifting captured once O( 4 ) nonlinear terms are considered, e.g. by the so-called Dysthe or modified NLS equation [46]. Similar tilts described by high order corrections have been evidenced on single pulses propagating in a wave flume [33].…”

Section: Discussionmentioning

confidence: 63%

Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

Michel¹,

Bonnefoy²,

Ducrozet³

et al. 2020

Phys. Rev. Fluids

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We study experimentally the early stages of integrable turbulence of unidirectional deep water gravity waves. By generating partially coherent waves in a 148 m long wave flume, we observe the emergence of high-amplitude structures formed by nonlinear focusing, commonly referred to as rogue waves. This work confronts the experiment with two recent results obtained in the framework of the nonlinear Schrödinger equation (NLSE), namely that (i) these structures can be locally fitted by a Peregrine soliton and (ii) their emergence leaves a visible trace on the evolution of statistical parameters such as kurtosis. Although Peregrine solitons have been observed for almost ten years in experiments using a deterministic forcing, we report their first systematic study in hydrodynamics with a random forcing. We show that (i) yields accurate results as long as the wave steepness remains moderate, whereas (ii) is very robust and remains valid beyond the assumption of integrability. Numerical simulations of the NLSE and of the fully nonlinear dynamical equations are also performed to support these results.

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Section: Discussionmentioning

confidence: 63%

Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

Michel¹,

Bonnefoy²,

Ducrozet³

et al. 2020

Phys. Rev. Fluids

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“…First, is a "direct comparison," see Section 4, in which we consider the evolution of the second harmonic band as if it were the dominant band while neglecting the first harmonic band. The evolution of the first harmonics, while ignoring all other harmonics, was previously examined in Carter & Govan [3] and Carter et al [4]. We compare experimental measurements of the second harmonic with predictions obtained from the NLS equation and its generalizations.…”

Section: Introductionmentioning

confidence: 94%

Modeling the Second Harmonic in Surface Water Waves Using Generalizations of NLS

Potgieter¹,

Carter²,

Henderson³

2020

Preprint

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When a mechanical wavemaker at one end of a water-wave tank oscillates with a frequency, ω0, time series of downstream surface waves typically include the dominant frequency (or first harmonic), ω0, along with the second, 2ω0; third, 3ω0; and higher harmonics. This behavior is common for the propagation of weakly nonlinear waves with a narrow band of frequencies centered around the dominant frequency such as in the evolution of ocean swell, pulse propagation in optical fibers, and Langmuir waves in plasmas. Presented herein are measurements of the amplitudes of the second harmonic band from four surface water wave laboratory experiments. The measurements are compared to predictions from the Stokes expansion and from nonlinear-Schrödinger (NLS) type equations.The Stokes expansion for small-amplitude surface water waves provides predictions for the amplitudes of the second and higher harmonics given the amplitude of the first harmonic. In this expansion, the harmonics are forced waves. That is, the (frequency, wavenumber) pair for the nth harmonic, (nω0, nk0), does not satisfy the dispersion relation, and therefore travels at the phase speed of (is phase-locked to) the dominant mode. If the harmonics were free waves, then they would have frequencies, nω0, and corresponding wavenumbers that satisfy the dispersion relation. The harmonics would (typically) travel at different speeds from the dominant mode, and their amplitudes would evolve differently from the forced-wave predictions. The NLS equation and its generalizations are models for the evolution of the amplitudes of weakly nonlinear, narrow-banded waves. Their derivations provide predictions, which have corrections to those of the Stokes expansion, for the amplitudes of the forced harmonic bands given the amplitudes of the waves in the dominant band.The measurements of the amplitude evolution of the second harmonic mode presented herein are compared to predictions obtained from the Stokes expansion and from the derivation of the NLS equation and four of its generalizations, all of which assume that the second harmonic is a forced wave. The measurements are also compared to predictions from numerical computations of NLS and four of its generalizations when the second harmonic is assumed to be a free wave; for these comparisons, the narrow-banded spectrum is centered at the second harmonic. Comparisons show that although the Stokes prediction and generalized NLS formulas provide reasonably accurate predictions for the amplitude evolution of the second harmonic band, the waves behave more as free waves than as forced waves. Further, the dissipative generalizations of NLS consistently outperform the conservative ones.

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“…The paths of NLS plane-wave particles are determined by using (7) to determine asymptotic expressions for φ and η that are valid up to O(ǫ 3 ) (the details of this process are described in detail in Section 2) and then numerically integrating the system of ODEs given in equation (1). Figure 2 includes a waterfall plot showing the path of a particle that starts on the surface.…”

Section: Nls Plane-wave Solutionsmentioning

confidence: 99%

Particle Trajectories in Nonlinear Schrödinger Models

2019

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The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness.While there is an abundance of studies investigating the reconstruction of the surface profile η, and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid.In the current work, it is shown that the velocity potential φ can be reconstructed in a similar way as the free-surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.Recent years have seen a flurry of activity aimed at understanding the motion of fluid particles in free-surface flows. The problem has been studied from various point of view, including field campaigns, wave-flume experiments, asymptotics, and rigorous mathematical analysis. One of the most well known results about particle trajectories in free-surface flows is Stokes's finding that *

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A comparison of frequency downshift models of wave trains on deep water

Cited by 16 publications

References 30 publications

Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

Modeling the Second Harmonic in Surface Water Waves Using Generalizations of NLS

Particle Trajectories in Nonlinear Schrödinger Models

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