Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide

Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis that are free of the spectrum but have both endpoints in it. A cylindrical waveguide has an intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall, and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with an arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the splitting of the band edges. This effect is known in the physical literature for waveguides of different shapes, and, in this paper, we provide a rigorous mathematical proof of the effect. Several variants of the edge splitting (alone and coupled, simple and multiple knots) are examined. Explicit formulas are obtained for a plane waveguide.     

The effective model of a stratified solid-fluid medium as a special case of the Biot model

The Biot model is compared with the effective model of a stratified periodic elastic-fluid medium, and the parameters of the transversally isotropic Biot model that turn it into the effective model are found. In the course of this comparison, some intermediate models, which are generalizations of the effective model and particular cases of the Biot model, are considered. For all of the models, the wave fronts excited by a point source are determined. The distinguishing feature of wave fronts in the intermediate models is the occurrence of double loops on some of them. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 172–195. Translated by L. Molotkov     

Propagation of sound pulses in a semiinfinite stratified medium

We consider the propagation of sound pulses due to a line source in the inhomogeneous, semi-infinite mediumy ≥ 0 with the boundary conditionφ=0 or?φ/(?y)=0 aty=0, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagation,c, is given byc ^?2=p ?qe ^?ay , wherep, q, α are real and positive andp>q. The method of dual integral transformation is used. The solution in terms of pulse propagation modes yields the diffracted pulse and the method of steepest descents gives the geometrical acoustic field.     

Propagation of waves in a randomly stratified medium: An inverse problem

We pose and solve an inverse problem of finding a coefficient in the wave equation in the inhomogeneous semispace on the scattering data of a plane wave incident from the homogeneous semispace. The unknown coefficient is a sum of a deterministic summand of one variable (the “depth” z) and a small random summand α(x, z). We look for the deterministic summand, the expectation E(α(x, z)) =: m(z), and the second moment r(x ₁ t - x ₂, z ₁, z ₂):= E(α(x ₁, z ₁)α(x ₂, z ₂)). Here the symbol E(·) stands for expectation. The stratification property of a medium means that (i) the deterministic summand depends only on z, (ii) m(z) depends only on z, and (iii) the second moment for fixed z ₁ and z ₂ depends only on x ₁ ? x ₂.     

Schwarz reflection geometry I: Continuous iteration of reflection

Differential equations are derived for a continous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry. Mathematics Subject Classification (1991):53C35, 53A30, 30D05.     

The Hochschild class of the Chern character for semifinite spectral triples

We provide a proof of Connes’ formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p?1. We employ the minimum possible hypotheses on the spectral triples.     

The mortar spectral element method in domains of operators. Part I: The divergence operator and Darcy's equations

The mortar spectral element method is a domain decompositiontechnique that allows for discretizing second- or fourth-orderelliptic equations when set in standard Sobolev spaces. Theaim of this paper is to extend this method to some problemsformulated in spaces of square-integrable functions with square-integrabledivergence. A discretization of the equations due to Darcy whichmodel the flow in porous media is proposed. The numerical analysisof the discrete problem is performed and 2D numerical experimentsare presented; they turn out to be in good agreement with thetheoretical results. Résumé: La méthode d'élémentsspectraux avec joints est une technique de décompositionde domaine permettant de discrétiser des équationselliptiques d'ordre 2 ou 4 posés dans des espaces deSobolev usuels. Le but de cet article est d'étendre cetteméthode à certains problèmes variationnelsformulés dans des espaces de fonctions de carréintégrable à divergence de carré intégrable.On propose une discrétisation des équations deDarcy qui modélisent l'écoulement dans des milieuxporeux, on en effectue l'analyse numérique et on présentedes expériences numériques en dimension 2 quis'avèrent cohérentes avec les résultatsde l'analyse. Received on 6 April 2004. revised on 24 January 2005.     

Propagation of three-dimensional electroelastic shear waves in a regularly layered medium of metal-piezoelectric type

We construct and study the dispersion equations for electroelastic three-dimensional shear waves propagating in a periodic medium formed by alternating layers of conducting and piezoelectric materials. We determine the structure of the transmission and blocking zones (the spectral regions of stability and instability) for various relations among the parameters of the layers with waves propagating along the interfaces between the layers. Two figures. Bibliography: 8 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 4–8.     

High-frequency asymptotic behavior of the electromagnetic field on the surface of a stratified medium

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 174–180, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Isolation of the Trivial Part of a Nonlinear Control System by Factorization: I

The problem of constructing aggregated systems (quotient systems) of the simplest kind for nonlinear control systems is considered. With the help of this factorization, the original control system is reduced to a decomposition that allows one to reduce the dimension of control problems.     

Non-linear reflection of a plane wave in an elastic medium

Exact formulae are derived for the reflected and refracted waves which occur for the inclined incidence of a plane horizontally polarized transverse wave of arbitrary profile on a horizontal interface between two elastic half-spaces experiencing non-linear friction when they move with respect to one another. A smooth function of general form is adopted as the friction function, which depends on the difference between the horizontal velocities of the elements of the boundaries of the half-spaces considered. It is shown that if the friction function depends non-monotonically on the relative velocity of displacement of the sides of a slit, then even when the profile of the incident wave is smooth, the reflected and refracted waves may contain strong discontinuities.     

Asymptotic analysis of a transport model of organic material through a stratified porous medium: Concentration effect

We consider the transport through capillarity of an organic material inside a porous medium, using Leverett’s model. We first prove an existence result for a weak solution of this nonlinear evolution problem, using a regularization process. We then describe the asymptotic behavior of the solution, when the permeability _kε$k_{\epsilon }$ of the porous medium is associated to a scalar function which only depends on the third variable, assuming that _kε$k_{\epsilon }$ (resp. the inverse of _kε$k_{\epsilon }$) converges to some measure _λ∗${\lambda }_{\ast }$ (resp. λ^∗${\lambda }^{\ast }$). We use Γ-convergence arguments in order to describe this asymptotic behavior. We finally characterize the asymptotic behavior of the problem, considering special choices of the permeability _kε$k_{\epsilon }$, which correspond to stratified porous media, and give a numerical test for a 1D model.     

Reconstructing parameters of a medium from incomplete spectral information

The problem of the synthesis of a stratified medium with specified amplitude and phase properties is investigated. The wave propagation in the medium is described by a system of differential equations. The synthesis problem considered in the paper relates to inverse problems of spectral analysis with incomplete spectral information. Using the contour integral method we study properties of spectral characteristics and obtain algorithms for the solution of the synthesis problem for differential equations with singularities.