Application of the asymptotic solution to EM field scattering problem for creation of media with prescribed permeability

Scattering of electromagnetic (EM) waves by many small impedance particles (bodies), embedded in a homogeneous medium, is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit a→0, where a is the characteristic size of a particle and the number M(a) of the particles tends to infinity at a suitable rate. The proposed theory allows one to create a medium with a desirable spatially inhomogeneous permeability. The main new physical result is the explicit analytical formula for the permeability μ(x) of the limiting medium. The computational results confirm a possibility to create the media with various distributions of μ(x).     

Electromagnetic wave scattering by small impedance particles of an arbitrary shape

Scattering of electromagnetic (EM) waves by one and many small (ka?1) impedance particles D _m of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic formula for the field, scattered by one particle, is derived. The scattered field is of the order O(a ^2?κ ), where κ∈[0,1) is a number. This field is much larger than in the Rayleigh-type scattering. An equation is derived for the effective EM field scattered by many small impedance particles distributed in a bounded domain. Novel physical effects in this domain are described and discussed.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

Zero-dispersion limit of the short-wave-long-wave interaction equations

The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ=o(ε), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.     

Distribution of particles which produces a desired radiation pattern

A method is given for calculation of a distribution of small particles, embedded in a medium, so that the resulting medium would have a desired radiation pattern for the plane wave scattering by this medium.     

The scattering of electromagnetic wave at oblique incidence in a homogeneous chiral medium

We consider the scattering of a time-harmonic electromagnetic wave by a perfectly and imperfectly conducting infinite cylinder at oblique incidence respectively. We assume that the cylinder is embedded in a homogeneous chiral medium and the cylinder is parallel to the z axis. Since the x components and y components of electric field and magnetic field can be expressed in terms of their z components, we can derive from Maxwell's equations and corresponding boundary conditions that the scattering problem is modeled as a boundary value problem for the z components of electric field and magnetic field. By using Rellich's lemma and variational approach, the uniqueness and the existence of solutions are justified.     

Boundedness of global positive solutions of a porous medium equation with a moving localized source

In this paper a porous medium equation with a moving localized source ut=ur(Δu+af(u(x₀(t),t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, in one space dimension case, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Diffraction by an Anisotropic Disk, Perfectly Conducting along Equiangular Spirals

The problem of the diffraction of electromagnetic waves by asmall circular disk which is perfectly conducting along equiangularspirals is studied. The limiting cases where the spirals degenerateinto circles and radial segments are also discussed. Generalmodal excitation appropriate to the cylindrical geometry ofthe problem is assumed. The solution is expressed in the formof an integral representation, involving a pair of unknown functions,which is designed to satisfy Maxwell's equations, the continuityconditions outside the disk, the edge conditions and the radiationcondition. The remaining conditions, specifically, that thecurrent is directed along the spirals and that the total electricfield in the direction of the spirals vanishes, lead directlyto a pair of coupled integro-differential equations for theunknown functions. Formal power series solutions of this systemare obtained for small values of ka (k is the wave number, ais the radius of the disk). Further, these formal power seriessolutions can be rigorously justified in the case where theconductivity is along circles. The results are employed to derivethe leading terms in the power series expansions of the farfields, the total scattering cross-sections, the electric fieldon the disk, and the current density near the centre and edgeof the disk. For a normally incident plane wave, it is shownthat the scattering cross-sections of the perfectly conducting,circularly conducting and radially conducting disks are in theratio of 2: 1: 0.0039.     

Scattering by an Inhomogeneous Medium

A method is given for constructing the total field resultingfrom a plane wave scattered by an inhomogeneous medium withcompact support. The problem is reformulated as a Fredholm integralequation of the second kind, whose solutions may be representedas a convergent Neumann series, for small values of the wavenumber. A truncation error estimate of this series is given.To illustrate the method, examples of both a sphere and a prolatespheroid filled with an inhomogeneous medium are considered.In both cases the scattering amplitude is investigated as afunction of the wave number. A comparison is made between themethod presented here for finding the total field and Rorres'method (1970).     

Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.     

Homogenization of some variational inequalities with restrictions on subsets <Emphasis Type="Italic">ɛ</Emphasis>-periodically located along the domain boundary

In this paper some problems concerning homogenization of variational inequalities for the Laplace operator and the biharmonic operator with restrictions on subsets ?-periodically located along the domain boundary are studied (? is a small parameter). These subsets are collections of balls of radius a _? a. An asymptotics of solutions is obtained in the so-called critical case characterized by a specific relation between the period ? and the value a _?. Strong convergence of solutions and their gradients is also established.     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

Multiple solutions and their limiting behavior of coupled nonlinear Schrödinger systems

We study the multiplicity of positive solutions and their limiting behavior as ? tends to zero for a class of coupled nonlinear Schrödinger system in R^N. We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ? sufficiently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

Recent developments in the qualitative approach to inverse electromagnetic scattering theory

We consider the inverse scattering problem of determining both the shape and some of the physical properties of the scattering object from a knowledge of the (measured) electric and magnetic fields due to the scattering of an incident time-harmonic electromagnetic wave at fixed frequency. We shall discuss the linear sampling method for solving the inverse scattering problem which does not require any a priori knowledge of the geometry and the physical properties of the scatterer. Included in our discussion is the case of partially coated objects and inhomogeneous background. We give references for numerical examples for each problem discussed in this paper.     

The lifespan for quasi-linear wave equations with multiple-speeds in space dimensions n ≥ 3

In this paper, we consider the Cauchy problem for quasi-linear wave equations with multiple propagation speeds in space dimensions n ≥ 3. The case when the nonlinearities depending on both the unknown functions and their derivatives are studied. Based on some Klainerman-Sideris type weighted estimates and space-time L ² estimates, the lifespan for quasilinear multi-speeds wave equations of small amplitude solutions are presented.     

On the determination of Dirichlet or transmission eigenvalues from far field data

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for ?Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.     

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.