We study the influence of disorder on propagation of waves in one-dimensional structures. Transmi... more We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.
We consider the Helmholtz equation in the half space and suggest two methods for determining the ... more 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.
We consider transverse propagation of electromagnetic waves through a two-dimensional composite m... more 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≪ 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.
We determine the effective conductivity of a two-dimensional composite consisting of a doubly per... more 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.
The inverse scattering problem of determining the boundary impedance from knowledge of the time h... more The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We use the finite-difference approximation for the Helmholtz equation along with the exact radiation condition for the discrete equation. The approach involves two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new approach. (Some figures in this article are in colour only in the electronic version) 1.
Abstract. Approximation of one-dimensional stochastic differential equations and their additive f... more Abstract. Approximation of one-dimensional stochastic differential equations and their additive functionals by dynamical systems with piecewise-constant random coefficients is obtained. We calculate asymptotic expansion of solution in terms of the step of discretization ∆. Key words. Itô’s stochastic equations; discretization; Lyapunov exponent; density of states. AMS classification. 58J50, 35J10, 35P99, 82B44, 35P05. 1.
H-polarized electromagnetic fields of the Bloch form. Vari-ous theoretical methods for 2D photoni... more H-polarized electromagnetic fields of the Bloch form. Vari-ous theoretical methods for 2D photonic crystals were de-Using analytical methods we develop an accurate and efficient algorithm for computation of the spectrum and eigenmodes for a veloped in [25] for sinusoidally and rectangularly modu-2D photonic crystal which comprises a periodic array of parallel lated dielectric constants and in [16, 20] for a periodic array rods of air of a square cross section embedded in a background of parallel dielectric rods of circular cross section whoselossless medium of higher dielectric constant. The numerical analy-intersections with perpendicular planes form a triangularsis of dependence of the spectral bands on the parameters of the or square lattice. Similar structures were studied theoreti-2D photonic crystal is carried out. It gives a reliable base for the optimal design of 2D photonic crystals. Q 1997 Academic Press cally and experimentally in [17, 18]. All those results (see also [5,...
We consider the propagation of acoustic, electromagnetic and elastic waves in a one-dimensional p... more 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.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We consider transverse propagation of electromagnetic waves through a two-dimensional composite m... more 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 for the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the cell size is small compared with the wavelength, but large compared with the radius a of the inclusions. Explicit formulae are obtained for asymptotic expansion of the solution of the problem in terms of the dimensionless magnitude q of the wavevector and radius a . This leads to explicit formulae for the effective dielectric tensor and the dispersion relation with the rigorously justified error of order O (( q 2 + a 2 ) 5/2 ).
We study the influence of disorder on propagation of waves in one-dimensional structures. Transmi... more We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.
We consider the Helmholtz equation in the half space and suggest two methods for determining the ... more 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.
We consider transverse propagation of electromagnetic waves through a two-dimensional composite m... more 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≪ 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.
We determine the effective conductivity of a two-dimensional composite consisting of a doubly per... more 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.
The inverse scattering problem of determining the boundary impedance from knowledge of the time h... more The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We use the finite-difference approximation for the Helmholtz equation along with the exact radiation condition for the discrete equation. The approach involves two steps. First, we reduce the problem to a well-posed system of linear equations for a modified potential. We next find the boundary impedance using the modified potential through an explicit formula. Thus, the computational part of the nonlinear problem of reconstruction of the boundary impedance is reduced to the solution of a linear system. Numerical examples are given to demonstrate efficiency of the new approach. (Some figures in this article are in colour only in the electronic version) 1.
Abstract. Approximation of one-dimensional stochastic differential equations and their additive f... more Abstract. Approximation of one-dimensional stochastic differential equations and their additive functionals by dynamical systems with piecewise-constant random coefficients is obtained. We calculate asymptotic expansion of solution in terms of the step of discretization ∆. Key words. Itô’s stochastic equations; discretization; Lyapunov exponent; density of states. AMS classification. 58J50, 35J10, 35P99, 82B44, 35P05. 1.
H-polarized electromagnetic fields of the Bloch form. Vari-ous theoretical methods for 2D photoni... more H-polarized electromagnetic fields of the Bloch form. Vari-ous theoretical methods for 2D photonic crystals were de-Using analytical methods we develop an accurate and efficient algorithm for computation of the spectrum and eigenmodes for a veloped in [25] for sinusoidally and rectangularly modu-2D photonic crystal which comprises a periodic array of parallel lated dielectric constants and in [16, 20] for a periodic array rods of air of a square cross section embedded in a background of parallel dielectric rods of circular cross section whoselossless medium of higher dielectric constant. The numerical analy-intersections with perpendicular planes form a triangularsis of dependence of the spectral bands on the parameters of the or square lattice. Similar structures were studied theoreti-2D photonic crystal is carried out. It gives a reliable base for the optimal design of 2D photonic crystals. Q 1997 Academic Press cally and experimentally in [17, 18]. All those results (see also [5,...
We consider the propagation of acoustic, electromagnetic and elastic waves in a one-dimensional p... more 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.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
We consider transverse propagation of electromagnetic waves through a two-dimensional composite m... more 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 for the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the cell size is small compared with the wavelength, but large compared with the radius a of the inclusions. Explicit formulae are obtained for asymptotic expansion of the solution of the problem in terms of the dimensionless magnitude q of the wavevector and radius a . This leads to explicit formulae for the effective dielectric tensor and the dispersion relation with the rigorously justified error of order O (( q 2 + a 2 ) 5/2 ).
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Papers by Yuri Godin