Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

Stability of Kolmogorov spectra for surface gravity waves

Theoretical and Mathematical Physics - The Kolmogorov spectrum of waves on water is a result of cascading energy via four-wave interactions. For this spectrum in the isotropic case, we introduce...     

Three-dimensional travelling waves for nonlinear viscoelastic materials with memory

In this paper, we show the existence of nontrivial travelling wave solutions, which propagating speeds are between ones determined by the equilibrium and instantaneous elastic tensor, respectively, to the nonlinear three-dimensional viscoelastic system with fading memory.     

Generation of surface gravity waves by bottom time-repetitive pulses

We investigate the deviation of free surface, generated by two repetitive excitations of the bottom surface, within the framework of model of a liquid of finite depth. The liquid is assumed to be incompressible and inviscid, which allows us to consider the problem in the potential statement. The problem is solved on the basis of the Hankel integral transformation by the radial coordinate and Laplace integral transformation by time with subsequent numerical inversion. We present and analyze some numerical results for the case of axially symmetric disturbance of the horizontal bottom surface (underwater earthquake). We show the appearance of waves with growing amplitudes for certain values of the time delay and increase in the rate of pulse rise. We also show that an increase in the pulse sharpness (its rise with time) will cause an increase in the amplitude.     

Scattering of surface gravity waves over a pair of trenches

A numerical model based on boundary element method is developed to study the scattering of surface gravity waves over a pair of trenches of varied configurations under the assumption of small amplitude water wave theory. Both the cases of symmetric and asymmetric trenches are considered in the present study. The accuracy of the numerical results is validated by comparing the reflection and transmission coefficients with energy identity, and the known results associated with single trench available in the literature. The study reveals that wave reflection decreases in an oscillatory manner with an increase in trench width. Moreover, Bragg resonance in wave reflection is observed for wave number corresponding to waves in shallow and intermediate depths in the case of a pair of trenches. Further, Bragg reflection increases with an increase in the number of trenches. In the case of multiple trenches, subharmonic peaks in Bragg reflection are depicted and the number of subharmonic peaks between two harmonic peaks is found to be two less than the number of trenches. However, for triangular trenches, the occurrence of the subharmonic peak is invariant of the number of trenches and the same vanishes for larger trench depth. Irrespective of trench configurations, wave reflection follows certain uniform oscillatory pattern with an increase in the gap between the trenches in case of deep water.     

On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves

The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by Phillips (1960) is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips’ four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple-variables are properly defined with clear physical meanings.     

Evolution equation of propagation of surface gravity waves in the presence of bottom excitation

An evolution equation that describes the propagation of surface nonlinear dispersive waves in a fluid of finite depth under excitation of a bottom surface is derived. The method of solution is based on the method of power series and asymptotic analysis. On this basis, in a particular case, we investigate the influence of the bottom compliance in the form of a Winkler elastic base and a more general Pasternak base on the transport of wave energy.     

Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface

A new nonlinear model of the propagation of wave packets in the system “liquid layer with solid bottom–liquid layer with free surface” is considered. With the use of the method of multiple-scale expansions, the first three linear approximations of the nonlinear problem are obtained. Solutions of problem of the first approximation are constructed and analyzed in detail. It is shown that there exist internal and surface components of the wave field, and their interaction is analyzed.     

On the development of packets of surface gravity waves moving over an uneven bottom

A nonlinear Schrödinger equation with varying coefficients describing the evolution of onedimensional packets of surface gravity waves moving over an uneven bottom is derived in the paper and the disintegration of both an envelope soliton and an envelope-hole soliton as they propagate onto a shelf is studied.

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Zusammenfassung Eine nichtlineare Schrödinger Differentialgleichung mit veränderlichen Koeffizienten wird aufgestellt, welche die Entwicklung eindimensionaler Pakete von Oberflächenwellen beschreibt, die sich über einen unebenen Boden bewegen. Das Problem der Spaltung an der Schwelle wird für eine umhüllende solitäre Welle und eine konkave umhüllende solitäre Welle untersucht.

     

Remark on peakon solution to the nonlinear surface wind waves equation

We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

The effect of surface topography on weakly nonlinear roll waves

The propagation of short waves in turbulent single layer flows forming on inclined surfaces has received considerable interest in the past. It is well known that such flows on flat surfaces are unstable if the Froude number of the unperturbed uniform state exceeds a critical value. In the initial linear stage disturbances grow exponentially with propagation distance but it has been shown that weakly nonlinear effects may limit the maximum wave amplitude under strictly periodic conditions leading in turn to a train of permanent roll waves. The present study investigates how the flow behaviour is affected if the slope of the bounding surface is no longer constant but changing slowly in the streamwise direction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on gravity waves in a viscous liquid with surface tension

Summary The linear theory by Reid [1] on gravity waves in a viscous liquid with surface tension is extended. Asymptotic formulae for the dispersion relation are derived. A critical Morton number equal to 0.044789 is found. Only in liquids with smaller Morton numbers, travelling ripples may exist with group velocity exceeding phase velocity.

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Zusammenfassung Reid's lineare Theorie über Gravitationswellen in einer viskosen Flüssigkeit mit Oberflächenspannung wird erweitert. Abgeleitet werden asymptotische Formeln für das Verteilungsverhältnis und eine kritische Morton-Zahl gleich 0.044789 wird gefunden. Nur in Flüssigkeiten mit geringeren Morton-Zahlen existieren wandernde kleine Wellen mit einer Gruppengeschwindigkeit, die die Phasengeschwindigkeit überschreitet.

     

Modeling and control of surface gravity waves in a model of a copper converter

Undesirable splashing appears in copper converters when air is injected into the molten matte to trigger the conversion process. We consider here a cylindrical container horizontally placed and containing water, where gravity waves on the liquid surface are generated due to water injection through a lateral submerged nozzle. The fluid dynamics in a transversal section of the converter is modeled by a 2-D inviscid potential flow involving a gravity wave equation with local damping on the liquid surface. Once the model is established, using a finite element method, the corresponding natural frequencies and normal modes are numerically computed in the absence of injection, and the solution of the system with injection is obtained using the spectrum. If a finite number of modes is considered, this approximation leads to a system of ordinary differential equations where the input is represented by the fluid injection. The dynamics is simulated as perturbations around a constant fluid injection solution, which is the desired operating state of the system, considering that the conversion process does not have to be stopped or seriously affected by the control. The solution is naturally unstable without control and the resulting increase of amplitude of the surface waves are assimilable to the splashing inside the converter. We show numerically that a variable flow around the operating injection is able to sensibly reduce these waves. This control is obtained by a LQG feedback law by measuring the elevation of the free surface at the point corresponding to the opposite extreme to where the nozzle injection is placed.     

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary observability of gravity water waves

Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof relies on the multiplier technique, the Craig–Sulem–Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.     

Gravitational waves in the relativistic theory of gravity

We demonstrate that because of the causality condition, gravitational waves have no unphysical “ghost” states in the relativistic theory of gravity with the graviton having a rest mass.     

On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.