Time harmonic acoustic scattering in anisotropic media

The scattering problem of a plane or a point source generated wave is considered for the case where both the medium of propagation and the interior of the scatterer exhibit their own anisotropies. A particular redirected gradient operator is introduced, which carries all directional characteristics of the anisotropic medium. Once the fundamental solution is obtained, integral representations for the scattered as well as for the interior and the total fields are generated. For such media even the handling of the singularities, in generating integral representations, depends on the characteristics of the particular medium. A modified, also medium dependent, radiation condition is introduced. Detailed asymptotic analysis leads to an integral representation for the scattering amplitude. The associated energy functionals are presented and the relative cross sections are also defined. Copyright © 2005 John Wiley & Sons, Ltd.     

Integral equation methods in electromagnetic scattering from anisotropic media

We investigate scattering of time‐harmonic electromagnetic waves by an anisotropic inhomogeneous medium. The problem is equivalently transformed into a system of strongly singular integral equations. The uniqueness and existence of a solution is shown and we examine the regularity of the solution by means of integral equations. We also prove the analyticity of the scattered field with respect to the refractive matrix and give a characterization of the derivatives in terms of solutions to anisotropic scattering problems. Copyright © 2000 John Wiley & Sons, Ltd.     

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Seismic data is modeled in the high‐frequency approximation, using the techniques of microlocal analysis. We consider general, anisotropic elastic media. Our methods are designed to allow for the formation of caustics. The data is modeled in two ways. First, we give a microlocal treatment of the Kirchhoff approximation, where the medium is assumed to be piecewise smooth, and reflection and transmission occur at interfaces. Second, we give a refined view on the Born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium. The joint formulation of Born and Kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters. The latter allows the treatment of scattering up to grazing angles. The outcome of the analysis is a characterization of the singular part of seismic data. We obtain a set of pseudodifferential operators that annihilate the data. In the process we construct a Fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function. In our construction this Fourier integral operator becomes invertible. We give the conditions for invertibility for general acquisition geometry. The result is also of interest for inverse scattering in acoustic media. © 2002 John Wiley & Sons, Inc.     

Love waves in layered anisotropic media

A modified transfer-matrix method is proposed to describe Love waves in multilayered anisotropic (monoclinic) media. Dispersion relations for media consisting of one and two anisotropic elastic layers in contact with an anisotropic half-space are obtained in closed form. The conditions for Love waves to exist are analysed. Waves with horizontal transverse polarization of the non-canonical type are investigated.     

Electromagnetism on anisotropic fractal media

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.     

Covolume techniques for anisotropic media

Summary The covolume method, a new approach applicable on general meshes, is extended to discretize and numerically solve the div-curl system in anisotropic media. The covolume method gives simple schemes and good approximations to the solution of the div-curl system. It works directly with the system and utilizes dual pairs of meshes that are orthogonally related. Central to the approach is the introduction of field components tangent and normal to the edges of one of the meshes, and the employment of dual discretization on the dual mesh pairs. The discretization procedures, schemes and error analysis are presented. The convergence of the method is proved.The work was partially done while this author was at Carnegie Mellon University     

Effective media for periodic anisotropic systems

One considers the wave propagation in periodic anisotropic systems. For these systems, with the aid of a matrix method, one determines the effective media.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 130–138, 1983.     

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

Time-dependent potential scattering in chiral media

Electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium are considered. We define a chiral operator and study potential scattering relative to this operator. A spectral analysis of associated operators is obtained, based on the Plancherel theory of the Fourier transform. Using the generalised eigenfunction expansion theory, we give an integral representation of the solution. A discussion of asymptotic equality of solutions is provided and the associated wave operator introduced.     

Light scattering in an open quantum system

Using a complex spectral decomposition we study light scattering in a simple open quantum system. In particular, using Fermi's golden rule we express the photodetachment rate in terms of the quasibound states of the open system and we discuss some of its main features.     

The linear sampling method for anisotropic media

We consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency. To this end, we extend the linear sampling method from the isotropic case to the case of anisotropic medium. In the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set.     

Linear elliptic equations in composite media with anisotropic fibres

Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the Hölder and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.     

Heat equation with memory in anisotropic and non-homogeneous media

A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.     

Entire solutions of sublinear elliptic equations in anisotropic media

We study the nonlinear elliptic problem −Δu=ρ(x)f(u) in RN (N?3), lim|x|→∞u(x)=?, where ??0 is a real number, ρ(x) is a nonnegative potential belonging to a certain Kato class, and f(u) has a sublinear growth. We distinguish the cases ?>0 and ?=0 and prove existence and uniqueness results if the potential ρ(x) decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.     

A mixed bvp for flow in strongly anisotropic media

Initial-boundary value problems for a class of linear parabolic equations are considered. The anisotropy of the medium is characterised by a small parameter. The solution structure is analysed by singular perturbation methods which include the construction of outer solutions and boundary and initial layer terms. The analysis is justified by convergence results     

Aspects of electromagnetic scattering in chiral media

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon‐type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon‐type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far‐field operator that is defined on a unit sphere and contains the far‐field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.     

On an integral related to biaxially anisotropic media

An integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail. The integral has three integer, two real, and one complex parameter. The integrand involves a product of Bessel functions of different argument and order. Several generalizations are discussed.     

Localization for random perturbations of anisotropic periodic media

We prove localization for random perturbations of periodic divergence form operators of the form ∇ · a_ω · ∇ near the band edges. Here a_ω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.     

An inverse source problem for Maxwell's equations in anisotropic media

In this article, we consider nonstationary Maxwell's equations in an anisotropic medium in the (x ₁,?x ₂,?x ₃)-space, where equations of the divergences of electric and magnetic flux densities are also unknown. Then we discuss an inverse problem of determining the x ₃-independent components of the electric current density from observations on the plane x ₃?=?0 over a time interval. Our main aim is, study conditional stability in the inverse problem provided the permittivity and the permeability are independent of x ₃. The main tool is a new Carleman estimate.