scholarly journals Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 103
Author(s):  
Philippe Guyenne ◽  
Adilbek Kairzhan ◽  
Catherine Sulem ◽  
Boyang Xu
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Short Wave ◽  
Birkhoff Normal Form ◽  
Spatial Form ◽  
Periodic Groups ◽  
Dimensional Gravity ◽  
Nonlinear Modulation ◽  
Dysthe Equation

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

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.

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.

Integral properties of periodic gravity waves of finite amplitude

1975 ◽  
Vol 342 (1629) ◽  
pp. 157-174 ◽  
Keyword(s):  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Cnoidal Wave ◽  
The Mean ◽  
Exact Relations ◽  
Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

scholarly journals Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

1988 ◽  
Vol 29 (4) ◽  
pp. 430-439 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Wave Packet ◽  
Wave Field ◽  
Gravity Waves ◽  
Deep Water ◽  
Short Wave ◽  
Wave Action ◽  
Leading Order ◽  
Order Interaction ◽  
Short Waves ◽  
Long Wave

AbstractThe leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

Some new highest-wave solutions for deep-water waves of permanent form

1980 ◽  
Vol 100 (4) ◽  
pp. 801-810 ◽  
Author(s):  
D. B. Olfe ◽  
James W. Rottman
Keyword(s):  
Series Expansion ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Solutions ◽  
Expansion Procedure ◽  
Wave Profiles ◽  
Permanent Form ◽  
Kinetic And Potential Energies ◽  
Deep Water Waves

The classical series expansion procedure of Michell is used to calculate some new highest-wave solutions. These solutions are shown to correspond to the types of gravity waves studied recently by Chen & Saffman (1980). Results are presented for wave profiles, phase speeds, and kinetic and potential energies.

scholarly journals Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

2020 ◽  
Vol 102 (4) ◽  
pp. 2385-2398
Author(s):  
D. Eeltink ◽  
A. Armaroli ◽  
C. Luneau ◽  
H. Branger ◽  
M. Brunetti ◽  
...  
Keyword(s):  
Phase Space ◽  
Phase Shift ◽  
Symmetry Breaking ◽  
Fundamental Frequency ◽  
Deep Water ◽  
Water Waves ◽  
Plane Waves ◽  
Real Space ◽  
Separatrix Crossing ◽  
Dysthe Equation

AbstractWe theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation. In the latter, the spectrum is truncated to only the fundamental frequency and the upper and lower sidebands. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general for NLSE systems. By means of experimentally obtained phase-space trajectories, we demonstrate that forcing and damping cause a separatrix crossing during the evolution. When the system is damped, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi–Pasta–Ulam–Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix, lifting the phase-shift. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

Surfing surface gravity waves

2017 ◽  
Vol 823 ◽  
pp. 316-328 ◽  
Author(s):  
Nick E. Pizzo
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Phase Speed ◽  
Surface Gravity ◽  
Simple Criterion ◽  
Surface Gravity Wave ◽  
Surface Gravity Waves

A simple criterion for water particles to surf an underlying surface gravity wave is presented. It is found that particles travelling near the phase speed of the wave, in a geometrically confined region on the forward face of the crest, increase in speed. The criterion is derived using the equation of John (Commun. Pure Appl. Maths, vol. 6, 1953, pp. 497–503) for the motion of a zero-stress free surface under the action of gravity. As an example, a breaking water wave is theoretically and numerically examined. Implications for upper-ocean processes, for both shallow- and deep-water waves, are discussed.

scholarly journals Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 115
Author(s):  
Dmitry Kachulin ◽  
Sergey Dremov ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Nonlinear Models ◽  
Ideal Incompressible Fluid ◽  
Equation Model ◽  
System Of Nonlinear Equations ◽  
Potential Flows ◽  
Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

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