On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Water waves in a deep square basin

1995 ◽  
Vol 302 ◽  
pp. 65-90 ◽  
Author(s):  
Peter J. Bryant ◽  
Michael Stiassnie
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Three Dimensional ◽  
Standing Waves ◽  
Two Dimensional ◽  
Initial State ◽  
Dimensional Properties ◽  
Periodic Standing Waves ◽  
The Waves ◽  
Small Disturbances

The form and evolution of three-dimensional standing waves in deep water are calculated analytically from Zakharov's equation and computationally from the full nonlinear bounddary value problem. The water is contained in a basin with a square cross-cection, when three-dimensional properties to pairs of sides are the same. It is found that non-periodic standing waves commonly follow forms of cyclic recurrence over times. The two-dimensional Stokes type of periodic standing waves (dominated by the fundamental harmonic) are shown to be unstable to three dimensional disturbances, but over long times the waves return cyclically close to their initial state. In contrast, the three-dimensional Stokes type of periodic standing waves are found to be stabel to small disturbances. New two-dimensional periodic standing waves with amplitude maxima at other than the fundamental harmonic have been investigated recently (Bryant & Stiassnie 1994). The equivalent three-dimensional standing waves are described here. The new two-dimensional periodic standing waves, like the two-dimensional Stokes standing waves, are found to be unstable to three-dimensional disturbances, and to exhibit cyclic recurrence over long times. Only some of the new three-dimensional periodic standing waves are found to be stable to small disturbances.

scholarly journals Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 205
Author(s):  
Dan Lucas ◽  
Marc Perlin ◽  
Dian-Yong Liu ◽  
Shane Walsh ◽  
Rossen Ivanov ◽  
...  
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Gravity Waves ◽  
Deep Water ◽  
Plane Waves ◽  
Surface Elevation ◽  
Wave Interactions ◽  
Frequency Space ◽  
Target Mode ◽  
The Hilbert Transform

In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

scholarly journals Scattering of gravity waves by a periodically structured ridge of finite extent

2019 ◽  
Vol 871 ◽  
pp. 350-376 ◽  
Author(s):  
Agnès Maurel ◽  
Kim Pham ◽  
Jean-Jacques Marigo
Keyword(s):  
Gravity Waves ◽  
Time Domain ◽  
Water Waves ◽  
Classical Result ◽  
Water Channel ◽  
Three Dimensional ◽  
Dimensional Problem ◽  
Homogenized Model ◽  
The Time Domain ◽  
Finite Extent

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured ridge and at its boundaries. This provides an effective two-dimensional wave equation which is a classical result but also non-intuitive transmission conditions between the region of the ridge and the surrounding regions of constant immersion depth. Numerical results provide evidence that the scattering properties of a structured ridge can be strongly influenced by the evanescent fields, a fact which is accurately captured by the homogenized model.

Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves

2002 ◽  
Vol 450 ◽  
pp. 201-205 ◽  
Author(s):  
ELIEZER KIT ◽  
LEV SHEMER
Keyword(s):  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Velocity Potential ◽  
Surface Elevation ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Dimensional Version

A spatial two-dimensional version of the Zakharov equation describing the evolution of deep-water gravity waves is used to derive two fourth-order evolution equations, for the amplitudes of the surface elevation and of the velocity potential. The scaled form of the equations is presented.

Wave Kinematics Computed With the Nonlinear Schro¨dinger Method for Deep Water

1999 ◽  
Vol 121 (2) ◽  
pp. 126-130 ◽  
Author(s):  
K. Trulsen
Keyword(s):  
Time Series ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Wave ◽  
Surface Displacement ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Limited Bandwidth ◽  
Wave Kinematics ◽  
The Waves

The nonlinear Schro¨dinger method for water wave kinematics under two-dimensional irregular deepwater gravity waves is developed. Its application is illustrated for computation of the velocity and acceleration fields from the time-series of the surface displacement measured at a fixed horizontal position. The method is based on the assumption that the waves have small steepness and limited bandwidth.

scholarly journals Momentum Flux Spectrum of Convectively Forced Internal Gravity Waves and Its Application to Gravity Wave Drag Parameterization. Part I: Theory

2005 ◽  
Vol 62 (1) ◽  
pp. 107-124 ◽  
Author(s):  
In-Sun Song ◽  
Hye-Yeong Chun
Keyword(s):  
Gravity Waves ◽  
Momentum Flux ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Spectral Structure ◽  
Two Dimensional ◽  
Resonance Factor ◽  
Flux Spectrum ◽  
Wave Momentum

Abstract The phase-speed spectrum of momentum flux by convectively forced internal gravity waves is analytically formulated in two- and three-dimensional frameworks. For this, a three-layer atmosphere that has a constant vertical wind shear in the lowest layer, a uniform wind above, and piecewise constant buoyancy frequency in a forcing region and above is considered. The wave momentum flux at cloud top is determined by the spectral combination of a wave-filtering and resonance factor and diabatic forcing. The wave-filtering and resonance factor that is determined by the basic-state wind and stability and the vertical configuration of forcing restricts the effectiveness of the forcing, and thus only a part of the forcing spectrum can be used for generating gravity waves that propagate above cumulus clouds. The spectral distribution of the wave momentum flux is largely determined by the wave-filtering and resonance factor, but the magnitude of the momentum flux varies significantly according to spatial and time scales and moving speed of the forcing. The wave momentum flux formulation in the two-dimensional framework is extended to the three-dimensional framework. The three-dimensional momentum flux formulation is similar to the two-dimensional one except that the wave propagation in various horizontal directions and the three-dimensionality of forcing are allowed. The wave momentum flux spectrum formulated in this study is validated using mesoscale numerical model results and can reproduce the overall spectral structure and magnitude of the wave momentum flux spectra induced by numerically simulated mesoscale convective systems reasonably well.

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