arXiv subjects

Yuri A. Godin

Publications and source records attributed to Yuri A. Godin.

13 recordsLinked to original sources

arXiv2026

Embedding transmission problems for Maxwell's equations into elliptic theory

We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem. As an application, the existence of an orthogonal basis from the eigenfunctions of the general Maxwell's problem is justified.

math.AP↗

arXiv2025

Spectral properties of operators and wave propagation in high-contrast media

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter $\varepsilon > 0$ characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of $\varepsilon = 0$. Effective operators corresponding to $\varepsilon = 0$ are described.

math.AP↗

arXiv2024

Local gaps in three-dimensional periodic media

We consider the propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape. The inclusion size $a$ is much smaller than the array period. We show that global gaps do not exist if $a$ is small enough. The notion of local gaps which depends on the choice of the wave vector $\bk$ is introduced and studied. We determine analytically the location of local gaps for the Dirichlet and transmission problems.

math.AP↗

arXiv2023

Propagation and dispersion of Bloch waves in periodic media with soft inclusions

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and cutoff frequencies are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.

math-ph↗

arXiv2022

Clusters of Bloch waves in three-dimensional periodic media

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.

math.AP↗

arXiv2020

Slowdown of group velocity in periodic waveguides

We consider the propagation of acoustic, electromagnetic and elastic waves in a one-dimensional periodic two-component material. Accurate asymptotic formulas are provided for the group velocity as a function of the material parameters when the concentration of scatterers is small or the characteristic impedances of the two media differ substantially. In the latter case, it is shown that the minimum group velocity occurs when the volume fractions of the components of the material are equal. In both asymptotic cases we show that the leading terms of the group velocity do not depend on frequency. Thus slowdown is frequency-independent and is not related to the resonance phenomena.

cond-mat.mtrl-sci↗

arXiv2019

Dispersion of waves in two and three-dimensional periodic media

We consider the propagation of acoustic time-harmonic waves in a homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius $a$ of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two and three-dimensional cases as $a \to 0$ and evaluate explicitly several first terms. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Neumann, Dirichlet and transmission boundary conditions are considered. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.

math.AP↗

arXiv2018

Dispersive and effective properties of two-dimensional periodic media

We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the dimensionless wave frequency $ν\ll 1$ that allows us to view the governing equation as a perturbation of the Laplace equation. We show that the eigenfunctions and the eigenvalues are even analytic functions of the magnitude of the quasimomentum vector and provide a rigorously justified asymptotic expansion the tensor of effective properties. We also determine explicitly a frequency correction term to the tensor of effective properties.

math-ph↗

arXiv2015

Effective properties of periodic tubular structures

A method is described to calculate effective tensor properties of a periodic array of two-phase dielectric tubes embedded in a host matrix. The method uses Weierstrass' quasiperodic functions for representation of the potential that considerably facilitates the problem and allows us to find an exact expression for the effective tensor. For weakly interacting tubes we obtain Maxwell-like approximation of the effective parameter which is in very good agreement with experimental results in considered examples.

cond-mat.mtrl-sci↗

arXiv2013

Effective complex permittivity tensor of a periodic array of cylinders

We determine the effective complex permittivity of a two-dimensional composite, consisting of an arbitrary doubly periodic array of identical circular cylinders in a homogeneous matrix, and whose dielectric properties are complex-valued. Efficient formulas are provided to determine the effective complex permittivity tensor which are in excellent agreement with numerical calculations. We also show that in contrast to the real-valued case, the real and imaginary parts of the effective complex-valued tensor can exhibit non-monotonic behavior as functions of volume fraction of cylinders, and can be either greater or less than that of the constituents.

cond-mat.mtrl-sci↗

arXiv2012

The effective conductivity of a periodic lattice of circular inclusions

We determine the effective conductivity of a two-dimensional composite consisting of a doubly periodic array of identical circular cylinders within a homogeneous matrix. We obtain an exact analytic expression for the effective conductivity tensor as well as its expansion in terms of volume fraction of the cylinders. Results are illustrated by examples.

cond-mat.mtrl-sci↗

arXiv2011

The effect of disorder on the wave propagation in one-dimensional periodic optical systems

The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent $γ$ on the frequency $ν$ and magnitude of the disorder $σ$. It is shown that in the bulk of the bands $γ\sim σ^2$, while near the band edges it has the order $γ\sim σ^{2/3}$. This dependence is illustrated by numerical simulations.

math-ph↗

arXiv2009

On the determination of the boundary impedance from the far field pattern

We consider the Helmholtz equation in the half space and suggest two methods for determining the boundary impedance from knowledge of the far field pattern of the time-harmonic incident wave. We introduce a potential for which the far field patterns in specially selected directions represent its Fourier coefficients. The boundary impedance is then calculated from the potential by an explicit formula or from the WKB approximation. Numerical examples are given to demonstrate efficiency of the approaches. We also discuss the validity of the WKB approximation in determining the impedance of an obstacle.

physics.comp-ph↗