Light scattering by a homogeneous sphere near a plane boundary II

ABSTRACT

The problem of light scattering by a homogeneous sphere above a plane boundary is considered in this paper. Hankel transformation and Erdélyi's formula are used to satisfy the boundary conditions on the plane and the determination of the unknown coefficients in the scattered field is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution involving these unknown coefficients are shown and the extinction efficiency factor is presented.     

A complementary theory of light scattering by homogeneous spheres

A theory of the scattering of electromagnetic waves by homogeneous spheres, the so-called Mie theory, is presented in a unique and coherent manner in this paper. We begin with Maxwell's equations, from which the vector wave equations are derived and solved by means of the two orthogonal solutions to the scalar wave equation. The transverse incident electric field is mapped in spherical coordinates and expanded in known mathematical functions satisfying the scalar wave equation. Determination of the unknown coefficients in the scattered and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of a sphere. Far-field solutions for the electric field are then given in terms of the scattering functions. Transformation of the electric field to the reference plane containing incident and scattered waves is carried out. Extinction parameters and the phase matrix are derived from the electric field perpendicular and parallel to the reference plane. On the basis of the independent-scattering assumption, the theory is extended to cases involving a sample of homogeneous spheres.     

The integral equation method in problems of diffraction by a finite impedance section of a medium interface

A 3D problem of reflection of a plane electromagnetic wave by a local impedance section of a wavy surface is considered. The boundary value problem for the system of Maxwell’s equations in a region with an irregular boundary is reduced to solution of systems of hypersingular integral equations. A numerical algorithm is proposed for solution of these systems. Results of numerical computations are presented.     

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.     

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

Abstract In this paper, a new kind of discrete non-reflecting boundary conditions is developed.It can be usedfor a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic waveequations and the equations for wave propagation in multi-phase media and so on.In this kind of boundaryconditions,the composition of all artifical reflected waves,but not the individual reflected ones,is consideredand eliminated.Thus, it has a uniform formula for different wave equations.The velocity C_A of the composedreflected wave is determined in the way to make the reflection coefficients minimal,the value of which depends onequations.In this psper,the construction of the boundary conditions illustrated and C_A is found,numericalresults are presented to illustrate the effectiveness of the boundary conditions.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

On the Fredholm property of the electric field equation in the vector diffraction problem for a partially screened solid

We consider a vector problem of diffraction of an electromagnetic wave on a partially screened anisotropic inhomogeneous dielectric body. The boundary conditions and the matching conditions are posed on the boundary of the inhomogeneity domain, and under passage through it, the medium parameters have jump changes. A boundary value problem for the system of Maxwell equations in unbounded space is studied in a semiclassical statement and is reduced to a system of integro-differential equations on the body domain and the screen surfaces. We show that the quadratic form of the problem operator is coercive and the operator itself is Fredholm with zero index.     

Conservation Laws For the (1+2)-Dimensional Wave Equation in Biological Environment

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator's determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.     

Investigation of electromagnetic wave diffraction from an inhomogeneous partially shielded solid

Diffraction of electromagnetic wave from a partially shielded inhomogeneous dielectric is considered. The original boundary value problem for Maxwell’s equations is shown to have at most one quasi-classical solution. The problem is reduced to a system of integro-differential equations on the solid and the screens. The matrix integro-differential operator is treated in Sobolev spaces and is shown to be a continuously invertible operator. As a result, convergence of the Galerkin method is proved in the chosen functional spaces.     

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

In this work, propagation of harmonic waves in initially stressed cylindrical viscoelastic thick tubes filled with a Newtonian fluid is studied. The tube, subjected to a static inner pressure P_i and a positive axial stretch λ, will be considered as an incompressible viscoelastic and fibrous material. The fluid is assumed as an incompressible, viscous and dusty fluid. The field equations for the fluid are obtained in the cylindrical coordinates. The governing differential equations of the tube’s viscoelastic material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. A closed form solution can be obtained for equations governing the fluid body, but due to the variability of the coefficients of resulting differential equations of the solid body, such a closed form solution is not possible to obtain. For that reason, equations for the solid body and the boundary conditions are treated numerically by the finite-difference method to obtain the effects of the thickness of the tube on the wave characteristics. Dispersion relation is obtained using the long wave approximation and, the wave velocities and the transmission coefficients are computed.     

On the unique existence of the classical solution to the problem of electromagnetic wave diffraction by an inhomogeneous lossless dielectric body

A vector problem of electromagnetic wave diffraction by an inhomogeneous volumetric body is considered in the classical formulation. The uniqueness theorem for the solution to the boundary value problem for the system of Maxwell’s equations is proven in the case when the permittivity is real and varies jumpwise on the boundary of the body. A vector integro-differential equation for the electric field is considered. It is shown that the operator of the equation is continuously invertible in the space of square-summable vector functions.     

Spectral operators generated by damped hyperbolic equations

We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators, which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.     

Three-dimensional invisible cloaks with arbitrary shapes based on partial differential equation

The method of designing electromagnetic invisible cloak is usually based on the form-invariance of Maxwell’s equations in coordinate transformation. By solving the partial differential equations (PDEs) that describe how the coordinates transform, three-dimensional (3-D) electromagnetic and acoustic invisible cloaks with arbitrary shapes can be designed provided the boundary conditions of the cloaks can be determined by the corresponding transformation. Full wave simulations based on finite element method verify the designed cloaks. The proposed method can be easily used in designing other transformation media such as matter-wave cloaks.     

Electrodynamic-mechanical boundary value problems and gauge transformations in rigid dielectrics with constitutive magnetoelectric coupling

Gauge transformations are generally applied to decouple Maxwell’s equations, introducing gauge invariant vector and scalar potentials in connection with suitable gauge differential equations. This step simplifies the analytical or numerical solution of electrodynamic boundary value problems. In dielectrics, the propagation of electromagnetic waves is usually investigated restricting the problems to simple isotropic non-functional materials. On the other hand, magnetoelectric (ME) solids are of particular interest, converting electrical to magnetic energy and vice versa. The constitutive behavior of those materials has been investigated extensively, however restricting considerations to static or quasi-static loading. The goal of this paper is to combine electrodynamics in terms of the classical Maxwell equations with the constitutive behavior of ME materials. The interacting mechanisms of ME energy conversion lead to a complex behavior and are supposed to give rise to interesting phenomena influencing wave dispersion, deflection and reflection. The coupled boundary value problem is comprehensively formulated first, including mechanical stress and strain fields as well as electromagnetically induced forces. Gauge transformations are presented to decouple the electrodynamic potential equations for anisotropic ME bodies, neglecting a mechanical compliance at that point. Weak formulations are derived as a basis for numerical discretization procedures like the finite element method and simple examples demonstrate the impact of ME coupling on the phase velocity of an electromagnetic wave.     

Absorbing boundary conditions for electromagnetic wave propagation

In this paper, the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain is developed for the Maxwell system by considering the system as a whole instead of considering each component of the electromagnetic fields individually. This boundary condition allows any wave motion generated within the domain to pass through the boundary of the domain without generating any reflections back into the interior. By approximating this theoretical boundary condition a class of local absorbing boundary conditions for the Maxwell system can be constructed. Well-posedness in the sense of Kreiss of the Maxwell system with each of these local absorbing boundary conditions is established, and the reflection coefficients are computed as a plane wave strikes the artificial boundary. Numerical experiments are also provided to show the performance of these local absorbing boundary conditions

     

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.     

On the Diffraction by Biperiodic Anisotropic Structures

This article studies the scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure consists of anisotropic optical materials and separates two regions with constant dielectric coefficients. The time harmonic Maxwell equations are transformed to an equivalent strongly elliptic variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. This guarantees the existence of quasiperiodic magnetic fields in H 1 and electric fields in H (curl) solving Maxwell's equations. The uniqueness is proved for all frequencies excluding possibly a discrete set. The analytic dependence of these solutions on frequency and incident angles is studied.     

Formalism of two potentials for the numerical solution of Maxwell’s equations

A new formulation of Maxwell’s equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.     

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.