An expansion of continuum wave functions in terms of resonant states

The poles and residues of the complete outgoing Green function in the complex momentum plane are used to obtain, in the case of finite range potentials, an eigenfunction expansion of the continuum wave solution. It is found that in the region r<a the wave solution may be expressed as an infinite sum of discrete terms involving the bound, antibound and resonant states of the problem. At the boundary radius r = a a different expansion is obtained. In this case, in order to get an infinite discrete sum, one has to introduce two subtraction terms. Otherwise the expansion is given by a finite sum of discrete terms and an     

A Fourier series solution for the transverse vibration response of a beam with a viscous boundary

This paper presents the generalized Fourier series solution for the transverse vibration of a beam subjected to a viscous boundary. The model of the system produces a non-self-adjoint eigenvalue-like problem which does not yield orthogonal eigenfunctions; therefore, such functions cannot be used to calculate the coefficients of expansion in the Fourier series. Furthermore, the eigenfunctions and eigenvalues are complex valued. Nevertheless, the eigenfunctions can be utilized if the space of the operator is extended and a suitable inner product is defined. The methodology presented in this paper utilizes Hilbert space methods and is applicable in general to other problems of this type. As an adjunct to the theoretical discussion, the results from numerical simulations are presented.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

电磁理论中算符的一些性质

在本文中,把在不均匀各向异性介质中的麦克斯韦方程看作算符,它定义在一个有界区域,可以被理解为微波技术中的谐振腔。但在这腔中充填着铁氧体,等离子体或其他各向异性介质,这些介质在应用中日益重要。文中证明了在某些μ、ε和边界条件下,算符成为对称。而对称性和自伴性在本征函数展开中带来很多方便;此外我们推导了本征振动的正变性和互易定理。如果不满足对称性,引入伴谐振腔的概念,所谓伴谐振腔在几何形状上和原来的腔相同,但ε、μ和边界条件不一样。它和自伴谐振腔在正交性和互易定理上有某些相似之处。     

Diffraction of sound by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace

An exact solution is obtained to the problem of sound diffraction by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace. The problem is solved using the Helmholtz integral equation in which the field of a point source in the halfspace with an elastic boundary is used as the Green function. The diffracted field is represented as a series expansion in spherical harmonics. The expansion coefficients are determined from a set of independent algebraic systems of equations. The matrix coefficients of these systems are determined as integrals of the products of the associated Legendre polynomials on the complex plane with respect to the real and complex angles of the sound incidence on the halfspace boundary. To decrease the number of such integrals, expansions using the Klebsh-Gordon coefficients are applied. As a result, algorithms for calculating the scattered field in the halfspace are obtained.     

Radiation modes of a five-layer symmetric slab waveguide

The radiation modes of a five-layered symmetric slab waveguide are treated in detail. Since the slab is assumed to be infinitely extended in the yz plane, all field quantities are considered to be independent of y. The analysis is based on modal expansion formulation and the modes can be classified as TE and TM as well as even and odd ones. Applying the boundary conditions at interfaces, the expansion coefficients of the field components are easily calculated. In addition, the normalization factors and the orthogonality properties are determined by deriving appropriate integrals over the cross section of the waveguide. Finally, the validity of the orthogonality relations between the radiation modes and the guided ones, which propagate in such a waveguide, is proved explicitly.     

Diffraction of a plane electromagnetic wave by a slot in a conducting screen with a transverse dielectric layer

The problem of diffraction of a plane electromagnetic wave by a slot in a planar perfectly conducting arbitrary thick screen with an infinite planar dielectric layer passing through the slot transversely to the screen is solved rigorously. In each of the field existence domains (two domains on either side of the screen and the interior of the slot), the solution is represented as an expansion in piecewise harmonic or exponential modes that allow for reflection and refraction at the boundaries of the dielectric layer. It is found that a set of functions describing such modes is complete enough to construct a solution satisfying all boundary conditions of the diffraction problem. The procedures of solution construction for the case at hand and for the same diffraction structure without the dielectric layer are compared.     

混响室的混沌特性及其场统计分布

采用有限元数值方法计算了理想和实际结构混响室的本征频率,将其本征频率间隔分布与量子混沌理论不同系综的理论分布进行对比分析,证明了实际结构混响室的系综为高斯正交系综,混响室为混沌系统。在此基础上,利用随机平面波假设将理想金属混响室内电场矢量本征函数表示为随机平面波的线性叠加,进而将混响室内电场表示为与横电磁波两种可能的极化方式对应的矢量本征函数展开式。根据矢量本征函数的概率分布得到了电场任一笛卡尔坐标分量服从2自由度的Rayleigh分布,总场服从6自由度的Rayleigh分布。用ETS HI-6105光纤电场探头对某混响室进行了电场分量值测量,比较了测量数据与理论模型的累积概率分布。结果表明,试验结果与理论模型符合较好,证明了电场统计分布模型的有效性。     

SUSY QM in a One-Dimensional Box and Local Observable Quantities

We investigate several Hamiltonians for a free particle in a one-dimensional box, in the context of supersymmetric quantum mechanics. Specifically, we study this problem with the Neumann boundary condition, the periodic and antiperiodic boundary condition, and some mixed and complex boundary conditions. This is achieved by using an approach recently proposed which expresses the factorization of the partner Hamiltonians in terms of the probability density and current for the ground-state eigenfunction of one of them.     

Mathematical modeling of a diode system based on a field emitter

A mathematical model of a diode system based on a field emitter is considered. The field emitter is modeled in the following way: the tip is a conducting sphere; the “body” is a solid dielectric with a spindle surface of revolution. The anode is a part of the spherical surface and the substrate of the emitter is a spherical surface or a plane. The influence of a space charge is not taken into account. To find the electrostatic potential distribution, the methods of separation of variables and paired equations are used. The problem of finding unknown coefficients in the eigenfunction expansion of the potential is reduced to the solution of the system of equations, which includes linear algebraic equations and the Fredholm equation of the second kind. Thus, the computational problem of the distribution of electrostatic potential in the entire domain of the investigated system is solved.     

Vibrations of an elastic strip with a rough boundary

A plane problem of steady-state forced vibrations of an elastic strip whose lower boundary contains a rough segment is considered. Using Green’s functions for a strip, the problem is reduced to a system of integral equations with integrals over the rough boundary, which is solved by the boundary-element method. The inverse problem of determining the shape of the rough boundary segment from the data on the displacement field of a certain part of the upper boundary is formulated. By the linearization procedure, the inverse problem is reduced to a Fredholm integral equation of the first kind with a smooth kernel, which is solved by Tikhonov’s regularization method.     

On the feasibility of variable separation method based on Hamiltonian system for plane magnetoelectroelastic solids

The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.     

A mixed integral equation of mechanics and a generalized projection method of its solution

A mixed multidimensional integral equation containing integral operators of various types is studied. The case in which the equation has one compact, self-adjoint, and strongly positive operator (with constant limits of integration) and two non-self-adjoint integral Volterra operators (with a variable upper limit of integration) is considered. To solve the equation, an effective projection method allowing one to obtain the result in a form with explicitly distinguished principal singularities is proposed.     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

Complete eigenmode analysis of a ladder-type multiple-gap resonant cavity

A theoretical model is developed for calculating the eigenmodes of the multi-gap resonant cavity. The structure of concern is a kind of ladder-type circuit, offering the advantages of easy fabrication, high characteristic impedance(R/Q),and thermal capacity in the millimeter wave to THz regime. The eigenfunction expansion method is used to establish the field expressions for the gaps and the coupling region. Then, the match conditions at the interface are employed, which leads to a group of complicate boundary equations in the form of an infinite series. To facilitate the mathematical treatments and perform a highly efficient calculation, these boundary equations are transformed into the algebraic forms through the matrix representations. Finally, the concise dispersion equation is obtained. The roots of the dispersion equation include both the axial modes in the gaps, which include the fundamental and the high-order modes, and the cavity modes in the coupling region. Extensive numerical results are presented and the behaviors of the multi-gap resonant cavity are examined.     

Network-theoretical model of the gyrotron oscillator part I: Empty cavity oscillation modes

A network-theoretical model of the gyrotron has been elaborated which is both conspicious and rigorous. The problem of self-consistently representing the field in the presence of the electron beam is attacked by expansion into the empty structure oscillation modes. In this first part, a method is presented to determine the empty gyrotron cavity oscillation modes, i.e. theQ factors, resonance frequencies, and field distributions in axisymmetric, but otherwise quite general cavities. It is based on the field representation in terms of local normal waveguide modes. Matrix equations in these modes for different type of cavity sections (building blocks) are formulated. Along with the open resonator boundary conditions, these equations form a nonstandard eigenvalue problem; the eigenvalues are the complex eigenfrequencies, the eigenvectors correspond to the field distributions. By way of example, the method is applied to a simple and a complex cavity.     

Light scattering by dielectric particles with axial symmetry

A new method is suggested for solving the problem of scattering of a plane electromagnetic wave by dielectric particles with the axial symmetry. The method is based on the separation of fields into two parts: the axially symmetric part, which is independent of the angle of rotation, and the part that is not axially symmetric, which vanishes upon averaging over this angle. The scattering problem is solved separately for each of the parts. In the first case, scalar potentials related to the azimuthal components of electromagnetic fields are used, and in the second case, superpositions of the Debye potentials and vertical components of the Hertz vectors are used. The surface integral equations for these potentials are obtained. They are represented as expansions in the spherical wave functions. The infinite systems of linear algebraic equations are obtained for unknown expansion coefficients. Our calculations demonstrated the high efficiency of the new method.     

Stability of a pretwisted tapered cantilever beam subjected to dissipative and follower forces

Stability of a pretwisted tapered cantilever beam of rectangular cross-section subjected to a follower force at its free end is investigated. The effects of internal and external damping are included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli theory and an associated adjoint boundary value problem is introduced. Approximate values of the critical load are calculated on the basis of a suitable adjoint variational principle for several values of the geometric and material parameters of the beam. The results are shown in graphs.     

Mathematics of diffusion controlled precipitates with time-dependent boundary conditions

The boundary value problem of diffusion controlled precipitation with time-dependent boundary conditions is treated by an extension of the conventional eigenfunction expansion method. In this approach, expansion coefficients and eigenvalues become functions of time and eigenfunctions become functions of space and time. Mathematical techniques to determine the relevant quantities are set forth and the general solution describing precipitate growth under time-dependent conditions is obtained. The general solution is applied to the precipitation of plates and spheres and the growth of radii as a function of time is determined. In the latter case, comparisons are made with conventional methods and the fractional error for a particle to reach half its equilibrium is estimated.