The disintegration of wave trains on deep water Part 1. Theory

1967 ◽  
Vol 27 (3) ◽  
pp. 417-430 ◽  
Author(s):  
T. Brooke Benjamin ◽  
J. E. Feir
Keyword(s):  
Deep Water ◽  
Wave Train ◽  
Low Frequency ◽  
Periodic Wave ◽  
Wave Number ◽  
Fundamental Wave ◽  
Side Band ◽  
Wave Trains ◽  
The Stability ◽  
Deep Water Part

The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

Stability of periodic waves in shallow water

1974 ◽  
Vol 66 (1) ◽  
pp. 81-96 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Shallow Water ◽  
Wave Train ◽  
Periodic Wave ◽  
Fundamental Wave ◽  
Cnoidal Waves ◽  
Margin Of Stability ◽  
Fundamental Wavelength ◽  
Stokes Waves ◽  
Wave Trains ◽  
Permanent Shape

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

Phase velocity effects in tertiary wave interactions

1962 ◽  
Vol 12 (3) ◽  
pp. 333-336 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
O. M. Phillips
Keyword(s):  
Phase Velocity ◽  
Continuous Spectrum ◽  
Deep Water ◽  
Wave Train ◽  
The Other ◽  
Wave Number ◽  
Wave Interactions ◽  
Single Wave ◽  
Phase Velocities ◽  
Wave Trains

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

scholarly journals Emergence of Solitons from Irregular Waves in Deep Water

2021 ◽  
Vol 9 (12) ◽  
pp. 1369
Author(s):  
Weida Xia ◽  
Yuxiang Ma ◽  
Guohai Dong ◽  
Jie Zhang ◽  
Xiaozhou Ma
Keyword(s):  
Deep Water ◽  
Rogue Wave ◽  
Wave Train ◽  
Irregular Waves ◽  
Peak Wavelength ◽  
Long Distance ◽  
Irregular Wave ◽  
Resonant Interactions ◽  
Dispersion Effects ◽  
Wave Trains

Numerical simulations were performed to study the long-distance evolution of irregular waves in deep water. It was observed that some solitons, which are the theoretical solutions of the nonlinear Schrödinger equation, emerged spontaneously as irregular wave trains propagated in deep water. The solitons propagated approximately at a speed of the linear group velocity. All the solitons had a relatively large amplitude and one detected soliton’s height was two times larger than the significant wave height of the wave train, therefore satisfying the rogue wave definition. The numerical results showed that solitons can persist for a long distance, reaching about 65 times the peak wavelength. By analyzing the spatial variations of these solitons in both time and spectral domains, it is found that the third-and higher-order resonant interactions and dispersion effects played significant roles in the formation of solitons.

On the modulation of water waves in the neighbourhood of kh ≈ 1.363

1977 ◽  
Vol 357 (1689) ◽  
pp. 131-141 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Wave Interaction ◽  
Periodic Wave ◽  
Stokes Wave ◽  
Modulation Equation ◽  
Wave Trains ◽  
The Stability

In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is h , and k is the wavenumber of the fundamental, then the Stokes wave is unstable if kh ≥ σ 0 , where σ 0 ≈ 1.363. The instability is provided by the growth of waves with a wavenumber close to k . This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at kh = cr0. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when kh is near to σ 0 . The resulting equation takes the form i A T - a 1 A ζζ - a 2 A | A | 2 + a 3 A | A | 4 + i( a 4 | A | 2 A ζ - a 5 A (| A | 2 ) ζ ) - a 6 Aψ T = 0 where ψ ζ = | A | 2 , and the a i are real numbers. [Coefficients a 3 - a 6 are given on kh ≈ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrödinger equation in the appropriate limit and is correct when a 2 = 0, i.e. at kh = σ 0 . The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that soli to ns will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

scholarly journals On Natural Modulational Bandwidth of Deep-Water Surface Waves

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 67
Author(s):  
Alexander Babanin ◽  
Miguel Onorato ◽  
Luigi Cavaleri
Keyword(s):  
Deep Water ◽  
Water Surface ◽  
Laboratory Experiments ◽  
Wind Wave ◽  
Wave Spectra ◽  
Field Observations ◽  
Wave Steepness ◽  
Side Band ◽  
Wave Trains

We suggest that there exists a natural bandwidth of wave trains, including trains of wind-generated waves with a continuous spectrum, determined by their steepness. Based on laboratory experiments with monochromatic waves, we show that, if no side-band perturbations are imposed, the ratio between the wave steepness and bandwidth is restricted to certain limits. These limits are consistent with field observations of narrow-banded wind-wave spectra if a characteristic width of the spectral peak and average steepness are used. The role of the wind in such modulation is also discussed.

scholarly journals Brief communication "On one mechanism of low frequency variability of the Antarctic Circumpolar Current"

2011 ◽  
Vol 18 (3) ◽  
pp. 361-365 ◽  
Author(s):  
O. G. Derzho ◽  
B. de Young
Keyword(s):  
Rossby Wave ◽  
Antarctic Circumpolar Current ◽  
Large Scale ◽  
Wave Train ◽  
Low Frequency ◽  
Wave Number ◽  
Mean Flow ◽  
Zonal Wave Number ◽  
Circumpolar Current ◽  
The Antarctic

Abstract. In this paper we present a simple analytical model for low frequency and large scale variability of the Antarctic Circumpolar Current (ACC). The physical mechanism of the variability is related to temporal and spatial variations of the cyclonic mean flow (ACC) due to circularly propagating nonlinear barotropic Rossby wave trains. It is shown that the Rossby wave train is a fundamental mode, trapped between the major fronts in the ACC. The Rossby waves are predicted to rotate with a particular angular velocity that depends on the magnitude and width of the mean current. The spatial structure of the rotating pattern, including its zonal wave number, is defined by the specific form of the stream function-vorticity relation. The similarity between the simulated patterns and the Antarctic Circumpolar Wave (ACW) is highlighted. The model can predict the observed sequence of warm and cold patches in the ACW as well as its zonal number.

NOISE-ENHANCED WAVE TRAIN PROPAGATION IN UNEXCITABLE MEDIA

2001 ◽  
Vol 11 (11) ◽  
pp. 2837-2843 ◽  
Author(s):  
I. SENDIÑA-NADAL ◽  
V. PÉREZ-MUÑUZURI
Keyword(s):  
Noise Intensity ◽  
Colored Noise ◽  
Wave Train ◽  
Front Propagation ◽  
Optimal Level ◽  
Periodic Wave ◽  
Correlation Times ◽  
Noise Parameters ◽  
Wave Trains ◽  
Oregonator Model

The influence of spatiotemporal colored noise on wave train propagation in nonexcitable media is investigated. This study has been performed within the framework of the Oregonator model in terms of the characteristic noise parameters. Some features seen in single front propagation, like noise induced propagation facilitation for an optimal level of the noise intensity, are also found for periodic wave trains. The main new effect is, however, an enhancement of propagation for correlation times of the noise of the order of the period of the wave train.

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