Diffusion approximation to wave propagation in imperfectly periodic structures

A diffusion approximation to the random evolution modeling of wave propagation in randomly imperfect periodic laminated composites is presented. It is shown that unless the material properties of the various layers satisfy some inequality (Eqn. (2)) all waves have exponentially decaying expected amplitudes.

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Résumé On utilise une diffusion comme approximation de l'évolution aléatoire modélisant la propagation des ondes dans un matériau composite laminaire avec imperfections aléatoires. On montre qu'à moins que les propriétés des matériaux formant les différentes couches satisfassent l'inéquation (2), toutes les ondes ont une amplitude qui décroit de façon exponentielle.

     

A method of solution of some problems of elastic wave propagation in the presence of a periodic inhomogeneity

The matrix method is used to examine the propagation of harmonic P and SV waves in an inhomogeneous medium whose properties are periodic functions of the depth z.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 30–33, 1985.     

Approximate models for wave propagation across thin periodic interfaces

This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy.     

Spectral theory for the wave equation in two adjacent wedges

Consider the two adjacent rectangular wedges K₁, K₂ with common edge in the upper halfspace of ℝ³ and the operator A (=−Laplacian multiplied by different constant coefficients a₁, a₂ in K₁, K₂, respectively) acting on a subspace of ∏²_j=1L₂(K_j). This subspace should consist of those sufficiently regular functions u=(u₁,u₂) satisfying the homogeneous Dirichlet boundary condition on the bottom of the upper halfspace. Moreover, the coincidence of u₁ and u₂ along the interface of the two wedges is prescribed as well as a transmission condition relating their first one-sided derivatives. We interpret the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. In this paper it is shown (by Friedrichs' extension) that A is selfadjoint in a suitable Hilbert space. Applying the Fourier (-sine) transformations we reduce our problem with singularities along the z-axis to a non-singular Klein–Gordon equation in one space dimension with potential step. The resolvent, the limiting absorption principle and expansion in generalized eigenfunctions of A are derived (by Plancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theorem for unbounded selfadjoint operators on Hilbert spaces yields the solution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physical geometry of the problem and its spectral properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.     

Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.     

Spectral method for solving nonlinear wave equations

In this paper,we consider the following nonlinear wave equations:(■~2φ)/(■t~2)-(■~2φ)/(■x~2)+μ~2φ+v~2x~2φ+f(|φ|~2)φ=0,(■~2x)/(■t~2-(■~2X)/(■X~2)+α~2x+α~2x+v~2x|φ|~2+g(X)=0with the periodic-initial conditions:φ(x-π,t)=φ(x+π,t),x(x-π,t)=x(x+v,t),φ(x,0)=■_0(x),φ_t(x,0)=■_1(x),X(x,0)=■_0(x),x_t(x,0)=■_1(x),-∞相似文献   

Analysis of the influence of Spectral Elements for elastic wave propagation on the CFL-Condition

For the simulation of the interaction of elastic waves in CFRP plates with inhomogeneities and defects the spectral finite element method (SEM) is under investigation. The SEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature (GLq) points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. The goal of this work is to investigate the effect of SEM on the CFL-Condition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic theory for coupled thermoelastic wave propagation in rods

A systematic approach to obtain successive approximations to the coupled thermoelastic dynamic problem of a circular cylindrical rod is developed. The equations governing the first two terms of the asymptotic series are derived. The lowest order approximation gives the classical one-dimensional thermoelastic rod equations. The second-order term describes the lateral inertia correction for the coupled thermoelastic rod.

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Zusammenfassung Ein systematisches Verfahren zur Herleitung sukzessiver Näherungstheorien für das dynamische, gekoppelte, thermoelastische Problem eines kreiszylindrischen Stabs wird entwickelt. Die expliziten Differentialgleichungen der ersten zwei Näherungen werden aufgestellt. Die tiefste Ordnung ergibt die klassischen, eindimensionalen, thermoelastischen Stabgleichungen. Die zweite Ordnung stellt die Korrektur der lateralen Trägheit dar.

     

Spectral theory of Schrödinger operators with periodic complex-valued potentials

Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 85–86, April–June, 1988.     

Spectral method for Zakharov equation with periodic boundary conditions

This paper describes the spectral method for numerically solving Zakharov equation with periodic boundary conditions. This method is spectral method for spatial variable and difference method for time variable. We make error estimation of approximate solution and prove the convergence of spectral method. We had given the convergence rate. Also, we prove the stability of approximate method for initial values.Project supported by the Science Foundation of the Chinese Academy of Sciences.     

On the extrema of band functions in periodic waveguides

In the work we construct an example of a periodic differential operator whose spectrum has gaps with edges attained by the band functions at interior points of the Brillouin zone. This example is the Laplacian on a pair of infinite parallel strips with common boundary from which a periodic system of small holes is cut out. At that, on the outer boundaries the Dirichlet condition is imposed, while on the common boundary the Neumann condition is considered.