On a kinetic model for shallow water waves

The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter ? which leads for ? → 0 to the system of shallow water waves. The corresponding ‘equilibrium’ distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.     

Excitation of waves and mean currents in a stratified fluid due to a moving heat source

Zusammenfassung Wenn eine Wärmequelle, z. B. eine Flamme, unter einer Schicht homogener Flüssigkeit bewegt wird, so kann eine mittlere Geschwindigkeit in der Flüssigkeit verursacht werden, und zwar entgegengesetzt zur Bewegungsrichtung der Flamme.Hier wird eine geschichtete Flüssigkeit betrachtet. Besondere Aufmerksamkeit wird den Verhältnissen gewidmet, bei denen Wellen in Resonanz entstehen. Die mittlere Geschwindigkeit wird dann von den Wellen beherrscht und hat im Wesentlichen die gleiche Richtung wie die Bewegung der Wärmequelle. Die Möglichkeit einer Strahlstruktur der Bewegung ist vorhanden.     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

Unique determination of stratified steady water waves from pressure

Consider a two-dimensional stratified solitary wave propagating through a body of water that is bounded below by an impermeable ocean bed. In this work, we study how such a wave can be recovered from data consisting of the wave speed, upstream and downstream density and velocity profile, and the trace of the pressure on the bed. In particular, we prove that this data uniquely determines the wave, both in the (real) analytic and Sobolev regimes.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

Equation of steady-state capillary gravitational-gyroscopic waves on shallow water and Kelvin waves

On the basis of a spectral asymptotic method developed by the authors, a rigorous derivation is given on the equation of capillary waves on shallow water with consideration of the rotation of the fluid and its stratification. The character of the wave motions described by this equation is investigated, and the existence of capillary Kelvin waves is established. Moreover, the problem of the diffraction of these waves by a half plane is studied.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 24, pp. 207–268, 1986.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Capillary-gravity waves produced by a moving pressure distribution

Summary The initial-value problem of surface waves generated by a moving oscillatory pressure distribution is considered and the effect of surface tension on such waves is studied in detail. It is found that the surface tension modifies the critical case in a remarkable way. And also it introduces two extra waves that exist in the upstream side of the pressure distribution.

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Zusammenfassung Das Anfangswertproblem der Oberflächenwellen die durch eine bewegte oszillierende Druckverteilung erzeugt werden, wird behandelt und der Einfluss der Oberflächenspannung wird ausführlich untersucht. Es wird gefunden, dass die Oberflächenspannung den kritischen Fall in bemerkenswerter Weise beeinflusst, und auch zwei besondere Wellen stromaufwärts von der Druckstörung verursacht.

     

Regularized shallow water equations for numerical simulation of flows with a moving shoreline

A numerical algorithm for simulating free-surface flows based on regularized shallow water equations is adapted to flows involving moving dry-bed areas. Well-balanced versions of the algorithm are constructed. Test computations of flows with dry-bed areas in the cases of water runup onto a plane beach and a constant-slope beach are presented. An example of tsunami simulation is given.     

Damping of Rayleigh waves excited by a moving source

The displacement field in a layered medium excited by a moving source is investigated. The resonance occurring when the speeds of the source and of the Rayleigh wave coincide is investigated and the character of the damping in this case is clarified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 137–153, 1976.     

Internal solitary waves with trapped cores in multilayer shallow water

Theoretical and Mathematical Physics - We consider a nonlinear system of equations that in the Boussinesq approximation describes near-bottom and near-surface large-amplitude internal waves...     

The diffraction of surface waves by an elastic platform floating on shallow water

The scattering of long gravitational waves by a floating elastic plate is investigated using linear shallow-water theory. For a plate of arbitrary shape, the solution of the problem is reduced to a system of boundary integral equations. Using the example of a rectangular plate, the solution obtained is compared with existing theoretical and experimental results. The behaviour of the buckling of a rectangular plate and of a strip of constant width is compared for oblique incidence of surface waves.     

Diffraction of sound waves by a finite barrier in a moving fluid

The diffraction of a line source by an absorbing finite barrier, satisfying Myers' impedance condition [M.K. Myers, On the acoustic boundary condition in the presence of flow, J. Sound Vibration 71 (1980) 429-434] in the presence of a subsonic flow is studied. The problem is solved analytically by using Integral transforms, Wiener-Hopf technique and the asymptotic methods. The expression for the diffracted field is shown to be the sum of the fields produced by the two edges of the strip and a field due to the interaction of the two edges. The diffracted field in the far zone is determined by the method of steepest decent.     

Numerical modelling of surface water waves generated by moving underwater landslide on irregular bottom

The numerical modelling of surface water waves generated by a moving underwater landslide on irregular bottom is considered. The non-linear shallow water model is used with taking into account bottom mobility. The equations are obtained for an underwater landslide movement under the action of gravity force, buoyancy force, friction force and water resistance force. The predictor-corrector scheme [5], preserving the monotonicity of the numerical solution profiles in a linear case, is used on adaptive grids. The scheme is validated on the problem with a known analytical solution. The analysis is done for the dependencies of wave regimes on bottom slope, initial landslide depth, its length and width. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)