Examples of the influence of the geometry on the propagation of progressive waves

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: A computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54–59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003].     

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)     

Propagation of surface waves on irregular bed topography

This paper deals with the exact solution of surface gravity waves in an ocean with irregular bed topography. In order to obtain water surface elevation and run-up of infra-gravity waves when the bed is either wavy or exponential, closed form solutions are obtained. Numerical computations indicate that when solitary wave or sinusoidal wave conditions are applied at the boundary, water surface elevation attains near Gaussian profile.     

The influence of coriolis force on MHD waves

The influence of coriolis force on the propagation of MHD waves in compressible medium is investigated by using linear stability analysis. In a combined effect we find that, while compressibility gives rise to an acoustic wave, coriolis force results in a modified Alfvén wave.     

High-frequency asymptotics of waves trapped by underwater ridges and submerged cylinders

As is well-known, underwater ridges and submerged horizontal cylinders can serve as waveguides for surface water waves. For large values of the wavenumber k   in the direction the ridge, there is only one trapped wave (this was proved in Bonnet-Ben Dhia and Joly [Mathematical analysis of guided water waves, SIAM J. Appl. Math. 53 (1993) 1507–1550]. We construct asymptotics of these trapped waves and their frequencies as k→∞$k\to \infty$ by means of reducing the initial problem to a pair of boundary integral equations and then by applying the method of Zhevandrov and Merzon [Asymptotics of eigenfunctions in shallow potential wells and related problems, Amer. Math. Soc. Trans. 208 (2) (2003) 235–284], in order to solve them.     

The influence of a crack in a medium on the interference waves

The interference Kr-waves in an elastic half-space with a crack are considered. The processes of dispersion and attenuation of such waves are described and theoretical seismograms are constructed. A number of frequency and velocity characteristics are discussed, which can help to identify the waves in mining investigations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 110–115, 1989.     

Numerical modelling of free-surface shallow flows over irregular topography with complex geometry

A well-balanced Godunov-type finite volume algorithm is developed for modelling free-surface shallow flows over irregular topography with complex geometry. The algorithm is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. Unstructured triangular grids are used to achieve the adaptability of the grid to the geometry of the problem and to facilitate localised refinement. The numerical fluxes are calculated using HLLC approximate Riemann solver, and the MUSCL-Hancock predictor–corrector scheme is adopted to achieve the second-order accuracy both in space and in time where the solutions are continuous, and to achieve high-resolution results where the solutions are discontinuous. The novelties of the algorithm include preserving well-balanced property without any additional correction terms and the wet/dry front treatments. The good performance of the algorithm is demonstrated by comparing numerical and theoretical results of several benchmark problems, including the preservation of still water over a two-dimensional hump, the idealised dam-break flow over a frictionless flat rectangular channel, the circular dam-break, and the shock wave from oblique wall. Besides, two laboratory dam-break cases are used for model validation. Furthermore, a practical application related to dam-break flood wave propagation over highly irregular topography with complex geometry is presented. The results show that the algorithm can correctly account for free-surface shallow flows with respect to its effectiveness and robustness thus has bright application prospects.     

Anomalous waves as an object of statistical topography: Problem statement

Based on ideas of statistical topography, we analyze the boundary-value problem of the appearance of anomalous large waves (rogue waves) on the sea surface. The boundary condition for the sea surface is regarded as a closed stochastic quasilinear equation in the kinematic approximation. We obtain the stochastic Liouville equation, which underlies the derivation of an equation describing the joint probability density of fields of sea surface displacement and its gradient. We formulate the statistical problem with the stochastic topographic inhomogeneities of the sea bottom taken into account. It describes diffusion in the phase space, and its solution must answer the question whether information about the existence of anomalous large waves is contained in the quasilinear equation under consideration.     

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε²| log ε|^3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e., with boundary fragments of curvature O(ε⁻¹). Bibliography: 33 titles.     

Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies

We consider problems about interaction of surface waves with fixed and freely floating bodies. We propose a new boundary value problem with integro–differential boundary conditions. This problem has physical sense only in the case of a totally submerged body, but, in the case of symmetry in the geometric or physical sense, the nonemptiness of its discrete spectrum provides the existence of eigenvalues of both problems for a fixed body and a freely floating body (in the last case, under some additional requirements which make the comparison principle incomplete). Bibliography: 25 titles. Illustrations: 5 figures.     

Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry

We show that there is a close relation between standing-wave solutions for the FitzHugh-Nagumo system where 0<a<1/2 and δ γ=β ² ∈ (0,a), and the following combinatorial problem: (*) Given K points Q ₁ , . . . , Q _K ∈R ^N with minimum distance 1, find out the maximum number of times that the minimum distance 1 can occur. More precisely, we show that for any given positive integer K, there is a δ _K >0 such that for 0<δ<δ _K , there exists a standing-wave solution (u _δ ,ν _δ ) to the FitzHugh-Nagumo system with the property that u _δ has K spikes Q ^δ ₁,. . .,Q ^δ _K and approaches an optimal configuration in (*), where .     

Experimental studies on the adhesion strength of a polymer-metal bond depending on the contact zone geometry

Theoretical investigation of the stress-strain state at special points of constructions, e.g., in adhesion joints, has demonstrated that the stress state in the contact zone strongly depends on the configuration of the zone. In order to verify these results, some experiments were carried out. Cylindrical polymer specimens having a circular groove at one of its ends were cemented to steel mushrooms and tested for abruption in tension and cantilever bending. The experiments showed that the adhesive strength depended on the profile of the joint and had a maximum at the optimum value of the angle of the joint and a minimum when the groove was absent. The ratio between the maximum and minimum strength values was 1.5 in tension and 2.5 in bending. The greater effect in the latter case can be explained by the high gradient of the stresses in bending. The experimental results confirmed the possibility of controlling the adhesive strength by changing the contact geometry in accord with the stress-strain calculations.Institute of the Mechanics of Continuous Media, Russian Academy of Sciences, Perm', Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 493–498, July–August, 1999.     

An influence of vertical structure of density and flows on free internal waves

In the framework of general linear theory we study the influence of a thin stratification of fields of density and velocity of plane-parallel shear flow on dispersion curves and vertical profiles of modes of free internal gravitational waves. It is shown that an accounting of thin structural singularities can essentially change kinematic characteristics of such waves.Translated from Dinamicheskie Sistemy, No. 8, pp. 99–107, 1989.     

A wave method of suppressing multiple waves for any complex subsurface geometry

A wave method of suppression of multiple waves in which no a priori data about the medium’s structure are required and the depth-velocity medium’s model is considered unknown, is presented. The method is constructed to completely suppress the multiple waves of a layer in a half-space. The efficiency of the method for arbitrary 3D plane-layered media is demonstrated both theoretically and numerically. Applications of the method to real media with a substantial decrease in the multiple wave amplitudes without distortion of the dynamics of useful reflections are given.