Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrödinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrödinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrödinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Parametric instability of gravity waves on the surface of deep water

We propose to investigate the characteristics of the parametric generation of gravity waves on the surface of a body of deep water. The threshold conditions for the onset of generation are determined, and the results are compared with the experimental data. The singularities of the excitation of parametric oscillations in a resonator are noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 182–185, May–June, 1972.The authors are grateful to V. N. Pshenichnikov for assisting with the experiments.     

On steady three-dimensional deep water weakly nonlinear gravity waves

Various kinds of steady weakly nonlinear gravity waves are examined. Corrections to the linear phase speed and the direction of modulation are calculated.     

Instability of periodic capillary-gravity waves on deep water

Pure gravity waves of finite amplitude in infinitely deep water are unstable to small disturbances in the form of modulated side-bands (Benjamin-Feir instability). These disturbances will undergo unbounded magnification if the parameter Ω associated with the frequency of the side-bands lies in the range $\text{0<Ω<√2 ka}$, where k and a are the wavenumber and amplitude of the basic wavetrain. The present paper is devoted to studying the situation for capillary-gravity waves. It is shown that the corresponding range of Ω, for which instability can arise, is smaller, see eq. 4.17, than for the pure gravity wave situation. Even pure capillary waves admit instabilities.     

Standing vortex waves in deep water

Standing vibrations of the free surface of an infinitely deep fluid exist in the presence of a weak (of the order of the wave amplitude) shear flow. As a result of the interaction with the flow the waves acquire a vorticity proportional to the cube of their amplitude.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 158–161, May–June, 1996.     

Internal gravity waves: parametric instability and deep ocean mixing

The Boussinesq approximation provides a convenient framework to describe the dynamics of stably-stratified fluids. A fundamental motion in these fluids consists of internal gravity waves, whatever the strength of the stratification. These waves may be unstable through parametric instability, which results in turbulence and mixing. After a brief review of the main properties of internal gravity waves, we show how the parametric instability of a monochromatic internal gravity wave organizes itself in space and time, using energetics arguments and a simple kinematic model. We provide an example, in the deep ocean, where such instability is likely to occur, as estimates of mixing from in situ measurements suggest. We eventually discuss the fundamental role of internal gravity wave mixing in the maintenance of the abyssal thermal stratification. To cite this article: C. Staquet, C. R. Mecanique 335 (2007).     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

均匀流中双色波传播特性的模拟研究

针对双色波浪与均匀流相互作用问题,采用时域高阶边界元方法建立自由水面满足完全非线性边界条件的数学模型。求解中采用混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶龙格库塔方法更新下一时间步的波面和速度势。通过与已发表试验结果对比,验证了本模型的准确性。通过数值计算研究了水流参数对各组成波及衍生的高阶波幅值、波浪和水流间能量交换的影响规律。     

Occurrence of standing surface gravity waves modulation in shallow water

Arising of modulations of surface gravity waves in a shallow water resonator under harmonic forcing is investigated in laboratory experiments. Different types of modulations are found, when the wave amplitude exceeds a certain threshold. Bifurcation diagram on the plane “amplitude of excitation – frequency of excitation” is determined. Numerical simulations of the Euler equations within the frameworks of the High-Order Spectral Method are performed with the purpose of reproducing the modulational regimes observed in the laboratory experiments. The simulations allowed us to determine physical mechanisms responsible for the occurrence of modulated waves.     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

Oblique water entry of a wedge into waves with gravity effect

The hydrodynamic problem of a two dimensional wedge entering waves with gravity effect is analysed based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. The stretched coordinate system in the spatial domain, which is based on the ratio of the Cartesian system in the physic space to the vertical distance the wedge has travelled into the water, is adopted based on the consideration that the decay of the effect of the impact away from the body is proportional to this ratio. The solution is sought for the total potential which includes both the incident and disturbed potentials, and decays towards the incident potential away from the body. A separate treatment at initial stage is used, in which the solution for the disturbed potential is sought to avoid the very large incident potential amplified by dividing the small travelled vertical distance of the wedge. The auxiliary function method is used to calculate the pressure on the body surface. Detailed results through the free surface elevation and the pressure distribution are provided to show the effect of the gravity and the wave, and their physical implications are discussed.     

Uniform approximation to surface gravity waves around a breakwater on water of variable depth

The linear surface gravity wavefield around a breakwater and on water of varying depth is described by a uniform asymptotic representation. The scattering of not necessarily irrotational wave packets generated by sources at arbitrary distance from the breakwater can be treated with the technique here expounded.     

Stability of periodic waves of finite amplitude on the surface of a deep fluid

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface (r, t) and the hydrodynamic potential (r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.In section 2, using a method similar to van der Pohl's method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude.In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negative-pressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d²/dk²) is different from the sign of the frequency shift due to the nonlinearity.As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves.The author wishes to thank L. V. Ovsyannikov and R. Z. Sagdeev for fruitful discussions.     

Action of an adverse current on the shape of deep water surface waves

It is well established that the shape of surface waves changes when waves meet a current. The effect of an adverse current is particularly interesting from a practical point of view. A visualization method is used to study the shape change when waves meet such a current, in a large scale laboratory flume. The corresponding observations and measurements are presented in this paper.     

Note on the modified nonlinear Schrödinger equation for deep water waves

It is shown that Zakharov's integral equation yields the modified Schrödinger equation for the particular case of a narrow spectrum.     

Diffraction of unsteady gravity waves on a wedge

We consider the occurrence and diffraction on a wedge of gravity waves which appear on the surface of an ideal incompressible liquid as a result of the activation of a periodically acting source. An exact solution is obtained. The time of the transient response is determined. Asymptotic formulas are established for the height of the free liquid surface, which are applicable for any instant of time.The author wishes to thank S. S. Voit for his guidance and assistance in carrying out this study.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.     

Surface waves in deep water due to arbitrary periodic surface pressure

Summary The expressions for the velocity potential and the surface elevation are obtained for the three dimensional motion on surface waves generated by a surface pressure F(x, y)e^{i t}, t>0, on the surface of a deep sea. The wave integrals are asymptotically evaluated for large distances and times by the method of stationary phase. By a direct passage to the limit t, the steady state solution of the problem is evaluated in exact terms. The results are illustrated for a class of physically plansible pressure distributions on the surface. Some features of the wave motions are discussed.

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Sommario Si ottengono le espressioni per il potenziale della velocità e la elevazione della superficie nel moto tridimensionale di onde superficiali generate da una pressione superficiale F(x, y)e^{i t}, t>0, sulla superficie di un mare profondo. Gli integrali di onde sono calcolati asintoticamente per grandi distanze e tempi con il metodo della fase stazionaria. Con un passaggio diretto al limite t, si calcola in termini esatti la soluzione di regime del problema. I risultati vengono illustrati per una classe di distribuzioni di pressione superficiale fisicamente plausibile, Si discutono alcune caratteristiche dei moti ondosi.