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.     

Propagation of surface gravitational waves over two parallel bottom inhomogeneities

On the basis of the method of power series we construct a solution of the two-dimensional problem of propagation of surface gravitational waves in water in the presence of two bottom trenches or projections. The problem is studied in the context of the shallow water model. We carry out an analysis of the effect of the trench parameters and their location on the amplitude of the reflected wave as functions of the wave number. Four figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 126–132.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

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.

---

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.

     

Balance-characteristic scheme as applied to the shallow water equations over a rough bottom

The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems.     

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.     

Long‐time dynamics of KdV solitary waves over a variable bottom

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?_tu = ??_x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H¹(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.     

Linear waves at the free surface of a saturated porous media

The principal aim of this paper is to study the propagation of linear waves at the free surface of a saturated porous media. This problem is formulated as an eigenvalue problem with complex eigenvalues, and the solution is given in term of an orthogonal eigenfunction expansion, whose completeness has been taken for granted in the literature regarding the problem as a classical Sturm-Liouville problem, which is not the case due to the complex nature of the eigenvalues. The main purpose of the present work is to prove the completeness of the eigenfunctions for all possible physical values of the parameters involved, even for some values of the parameters, where previous numerical works have found abnormal behavior of the eigenvalues. In those cases if we mistakenly consider the problem as a Sturm-Liouville one, as has been done before, the eigenfunction expansion will not hold, but indeed we will prove that it does.On leave from Institute de Mecánica de los Fluidos, Universidad Central de Venezuela, Caracas, Venezuela.     

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.     

Homogenization in the problem of long water waves over a bottom site with fast oscillations

The system of equations of gravity surface waves is considered in the case where the basin’s bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom’s oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors’ works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.     

Shallow water waves over an uneven bottom and an inhomogeneous KP equation

Three-dimensional shallow water waves over an uneven bottom are considered. The depth is assumed to be slow in variation. As a model, an inhomogeneous Kadomtsev-Petviashvili equation is presented. Some reductions of this equation are used to describe deformation of a line soliton due to the depth change. The model equation is valid for a wide class of two-dimensional nonlinear waves in inhomogeneous systems.     

Linear stationary control systems over a Boolean semiring: Geometric properties and the isomorphism problem

In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability, identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in this class of systems.     

Influence of the bottom undulation on surface waves in thin film flow

We study gravity-driven viscous thin films flowing down an undulated plane. Applying the integral boundary-layer method we derive a set of two coupled PDEs for the film thickness and the flow rate. The steady state solution shows linear and nonlinear resonance. Based on this analytical solution we carry out a stability analysis with respect to surface waves and study wave generation and annihilation for time dependent flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero-error stationary coding over stationary channels

Summary For a type of stationary ergodic discrete-time finite-alphabet channel more general than the stationary totally ergodic ¯d-continuous channel of Gray, Ornstein and Dobrushin, it is shown that a stationary, ergodic source with entropy less than capacity can be transmitted over the channel with zero probability of error using stationary codes for encoding and decoding. This result generalizes the result of Gray et al. [3] that Bernoulli sources can be transmitted with zero error at rates below capacity over a totally ergodic ¯d-continuous channel.Research of author supported by NSF Grant MCS-78-21335 and the Joint Services Electronics Program under Contract N00014-79-C-0424.     

Linear stationary control systems over the Boolean semiring and their graphs of modules

We consider linear stationary dynamical systems over the Boolean semiring. We analyze the properties of complete observability, identifiability, attainability, and controllability of a system. We define the notion of the “graph of modules” of totally controllable totally attainable Boolean linear stationary systems by analogy with spaces of modules in the case of systems over fields. The above-mentioned graph is described in the simplest case of one-dimensional inputs and outputs. We prove the weak connectedness of this oriented graph.     

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)     

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.     

Linear theory of wave generation by a moving bottom

The computation of long wave propagation through the ocean obviously depends on the initial condition. When the waves are generated by a moving bottom, a traditional approach consists in translating the ‘frozen’ sea bed deformation to the free surface and propagating it. The present study shows the differences between the classical approach (passive generation) and the active generation where the bottom motion is included. The analytical solutions presented here exhibit some of the drawbacks of passive generation. The linearized solutions seem to be sufficient to consider the generation of water waves by a moving bottom. To cite this article: D. Dutykh et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Linearized solution of a flow over a nonuniform bottom

A linearized theory is presented for determining the shape of the free surface of a running stream which is disturbed by some irregularities lying on the bottom. The bottom is represented in integral form using Fourier's double-integral theorem. Then following Lamb [3], a linear free-surface profile is obtained for the supercritical and subcritical cases.The results are plotted for the two cases of the flow for different shapes of the bottom, and different values of the Froude number. The effect of the Froude number, the bottom height and the shape of the bottom are discussed.