Scattering of Plane Harmonic Waves in an Infinite Elastic Solid by an Arbitrary Configuration of Two-dimensional Obstacles

The problem of the multiple scattering of plane harmonic wavesin an infinite elastic solid by arbitrary configurations oftwo-dimensional obstacles is considered. Sets of integral equationsrelating the multiple scattering functions to the single scatteringfunctions are derived. The integral equations are replaced bya set of algebraic equations which are then solved by an iterativeprocedure. Some results are given for the scattering of a planeharmonic P or SV wave by an identical pair of parallel circularcylindrical cavities.     

Closed form solution of electromagnetic wave diffraction problem in a homogeneous bi-isotropic medium

In this study, the problem of wave scattering of an electromagnetic field in a homogeneous bi-isotropic medium by a perfectly conducting strip is theoretically analyzed. The crux of the study is a rigorous construction of a closed form solution in the complex domain. A series solution of electromagnetic plane wave diffraction problem in terms of the eigenfunctions that happen to be the generalized Gamma functions is found. In the transformed domain, the scattered field is physically interpreted by computing the convergence history, and thereby, higher order accurate solution has been obtained in complex domain in closed form. Copyright © 2014 John Wiley & Sons, Ltd.     

On the domain derivative for scattering by impenetrable obstacles in chiral media

For scattering of electromagnetic waves in a chiral medium bysome perfectly conducting inclusions, we study the dependenceof the scattered field on the boundary of the inclusions andshow its Fréchet differentiability in appropriate spaces.Further, we derive a characterization of the derivative as asolution to some corresponding chiral boundary value problem.Our proof contains a new approach to rigorously derive thischaracterization.     

Inverse scattering for planar cracks via nonlinear integral equations

We present a Newton‐type method for reconstructing planar sound‐soft or perfectly conducting cracks from far‐field measurements for one time‐harmonic scattering with plane wave incidence. Our approach arises from a method suggested by Kress and Rundell (Inv. Probl. 2005; 21 (4):1207–1223) for an inverse boundary value problem for the Laplace equation. It was extended to inverse scattering problems for sound‐soft obstacles (Mathematical Methods in Scattering Theory and Biomedical Engineering. World Scientific: Singapore, 2006; 39–50) and for sound‐hard cracks (Inv. Probl. 2006; 22 (6)). In both cases it was shown that the method gives accurate reconstructions with reasonable stability against noisy data. The approach is based on a pair of nonlinear and ill‐posed integral equations for the unknown boundary. The integral equations are solved by linearization, i.e. by regularized Newton iterations. Numerical reconstructions illustrate the feasibility of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis and solution method for problems of electromagnetic wave scattering on dielectric and perfectly conducting structures

Problems of electromagnetic wave scattering on 3D dielectric structures in the presence of bounded perfectly conducting surfaces are reduced to a system of singular integral equations. We study this system mathematically and suggest a numerical solution method.     

A Bohr Phenomenon For Elliptic Equations

In 1914 Bohr proved that there is an r (0,1) such that if apower series converges in the unit disk and its sum has modulusless than 1 then, for |z| < r, the sum of absolute valuesof its terms is again less than 1. Recently, analogous resultshave been obtained for functions of several variables. The aimof this paper is to place the theorem of Bohr in the contextof solutions to second-order elliptic equations satisfying themaximum principle. 2000 Mathematics Subject Classification: 35J15, 32A05, 46A35.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Method of Integral Equations for the Three-Dimensional Problem of Wave Reflection from an Irregular Surface

The problem on the reflection of the field of a plane H-polarized three-dimensional electromagnetic wave from a perfectly conducting interface between media which contains a local perfectly conducting inhomogeneity is considered. To construct a numerical algorithm, the boundary value problem for the system of Maxwell equations in an infinite domain with irregular boundary is reduced to a system of singular integral equations, which is solved by the approximation–collocation method. The elements of the resulting complex matrix are calculated by a specially developed algorithm. The solution of the system of singular integral equations is used to obtain an integral representation for the reflected electromagnetic field and computational formulas for the directional diagram of the reflected electromagnetic field in the far region.     

On coupled differential equations which model the evolution of interacting capillary-gravity wave modes and related questions of stability

The author considers the small-amplitude waves which may ariseon the surface of a perfect fluid due to the interaction ofthe Mth and Nth harmonics of a fundamental wave. A pair of couplednonlinear partial differential equations for the complex potentialamplitudes are derived. These model the evolution of the wavetrainup to third order. Solutions to these equations are obtainedand some of the corresponding wave profiles are sketched. Thequestion of the stability of these waves to small perturbationsis then addressed. Longitudinal, transverse, and oblique perturbationsare considered. It turns out that the stability of these wavesis dependent on the ratio M/N. Regions of stability are identified     

磁场对液体-微伸缩固体介面波传播的影响

在一个传播理想的非粘性液体半空间,和一个传播理想的微伸缩弹性固体半空间之间,研究介面处纵波的反射和透射.在两个半空间中,满足介面处必需的边界条件下,得到控制方程的适当解,是一组以不同反射和透射波振幅比表示的5个非齐次方程.以水和铝-环氧树脂合成材料介面为实际例子,用Gauss消除法的Fortran程序求解方程组.考虑存在和不存在外加横向磁场两种情况,在某些入射角范围内,计算振幅比的数值解.最后用图形给出横向磁场对不同反射和透射波振幅比的影响.     

The nearly horizontally rolling of a thick disk on a rough plane

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.      

Waves in a Perfectly Conducting Fluid Filling a Half-Space

The constant, maximal, energy preserving boundary conditionsfor the equations of magnetohydrodynamics in a perfectly conductinghalf-space give rise to two essentially different selfadjointoperators in the case when the external magnetic field is orthogonalto the boundary and exactly one such operator when the externalfield is parallel to the boundary. Neither of these problemsadmits surface waves. For a normalized external field, the generalizedeigenfunction expansion is given below. It is shown that, inthe second case, the modes are not coupled by the boundary,while for only one boundary condition for the orthogonal fieldis the wave motion essentially that of free space (in the sensethat solutions are delivered by the group which determines solutionsfor the free space problem for special initial data). The Alvnwave in the parallel field case acts as a grazing wave. Asymptoticwave motion for perturbed problems (inhomogeneous media) isinvestigated as well as local decay of energy (this is not altogethertrivial, since the operators involved are never coercive evenoff their null spaces).     

Bloch Functions in Several Complex Variables, I

Bloch functions on the unit disk are those analytic functionsf for which the quantity |f'(z)|(1 – |z|²) is bounded(for z in the disk). In this paper, Bloch functions on boundedhomogeneous domains in complex m-space are studied. The mainresult shows that many of the equivalent definitions of Blochfunctions on the unit disk are also equivalent in the generalsetting. Current Address: 39 Trinity College, Dublin 2, Ireland.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Linear Differential Equations and Logarithmic Derivative Estimates

We prove two sharp inequalities for the growth of solutionsof certain linear differential equations in the unit disk. Forthe proofs of these inequalities, we use the method of successiveapproximations and sharp estimates for the logarithmic derivativesof finite order meromorphic functions in the unit disk. Thesetechniques can also be used to give an alternate proof of awell-known result in the plane. The sharp logarithmic derivativeestimates are a corollary of general estimates, and all theseestimates have independent interest. 2000 Mathematics SubjectClassification 34M10, 30D35.     

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.     

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency e_n of a diffraction order with maximal value of e_n, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 109–135. Translated by V. V. Zalipaev.     

The convergence of iterative solutions to the Electric Field Integral Equation

The Jacobi iterative method is applied to the system of linear equations arising from the discretization of the Electric Field Integral Equation (EFIE). It is shown that the resulting matrix equation is a contraction mapping, guaranteeing monotonic mean square convergence, for any initial guess, and for a preferred choice of a relaxation parameter (). Both the criterion for convergence and for the generation of the initial guess are discussed in detail. Results are shown for the 2-dimensional TM scattering by a perfectly conducting strip which illustrates the major points of this paper. The mathematical criterion herein may be applied to any electromagnetic problem employing the EFIE for perfectly conducting surfaces.     

Reflection at a Discontinuity in a Transmission Line

An exact expression is obtained for the reflection coefficientwhen a TEM wave, propagating along a co-axial line in a regionwhere both conductors are perfectly conducting, is incidenton a region where the inner conductor consists of dielectriccoated with a resistive film whilst the outer conductor remainsperfectly conducting.