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.     

Diffraction of a plane electromagnetic wave by a cylindrical wedge with a rounded apex

We reduce the rigorously formulated problem of diffraction of a plane electromagnetic wave by a perfectly conducting cylindrical wedge with a rounded apex to solving the system of linear algebraic equations of the second kind for unknown coefficients of the Fourier expansions of the diffracted-field components. The expansion coefficients are determined analytically in the long-wavelength approximation. The results of calculations of the diffracted field in the far zone are presented with a given accuracy in the case of an E-polarized wave. It is shown that the rounding of the apex of a cylindrical wedge leads to an increase in the backscattering coefficient of the structure in the long-wavelength range. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 5, pp. 447–451, May 2008     

Acoustic analysis of a rectangular cavity with general impedance boundary conditions

A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh-Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh-Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.     

Solutions of the Fokker-Planck equation describing the thermalization of neutrons in a heavy gas moderator

The thermalization of neutrons is described by a transport equation with a second order differential operator with respect to the energy. First this equation is transformed to an one-variable Fokker-Planck equation. Next an eigenfunction expansion and a polynomial expansion are used to solve the time-dependent Fokker-Planck equation. The eigenvalues are obtained either by solving a Schrödinger equation or by calculating 2×2 matrix continued fractions. Explicit results for the approach to equilibrium of a pulse of neutrons as well as the stationary distribution for 1/v absorption are presented. It is shown that the theory of the photoelectromotive force in semiconductors also leads to the same problem.     

Statistical properties of the laser emission in a low-Q cavity

We investigate theoretically the stationary statistical properties of the laser radiation in a low-Q cavity with field, polarization, and population fluctuations. Eliminating adiabatically the electric field from the Maxwell-Bloch equations, coupled Langevin equations with bothadditive andmultiplicative noises are derived and are transformed into the multivariable Fokker-Planck equation of a probability density of the light intensity and the population difference. It is solved by the expansion into orthonormal sets, and a vector recurrence equation of motion of the expansion coefficients is given whose stationary solutions are analytically obtained in theMatrix continued-fraction. The stationary distribution function of the radiation intensity are calculated with several values of control parameters. We discuss the variance of the intensity distribution, the photon-counting coefficient, and the cross-correlation between intensity and population as a function of the pump parameter, and reveal the novel and characteristic features of the bad-cavity laser system. The comparison with the good-cavity (high-Q cavity) case is also made.     

The Wigner Functions for a Spin-1/2 Relativistic Particle in the Presence of Magnetic Field

This paper provides a study of Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field. Since the Dirac equation is described as a matrix equation, it is necessary to describe the Wigner function as a matrix function in phase space. What’s more, this function is then proved to satisfy the Dirac equation with ⋆-product. Finally, by solving the ⋆-product Dirac equation, the energy levels as well as the Wigner functions for a spin-1/2 relativistic particle in the presence of magnetic field are obtained.     

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.     

Electrostatic solution and rayleigh approximation for small nonspherical particles in a spheroidal basis

The electrostatic problem for the case of axially symmetric particles is analyzed in a spheroidal basis. In this case, the wavenumber is zero and Maxwell’s equations are reduced to the Laplace equation for scalar potentials. An alternative approach involves solving integral equations that are similar to those obtained within the framework of the extended boundary conditions method. The scalar potentials are represented as expansions in terms of eigenfunctions of the Laplace equation in a spheroidal frame of reference, and unknown expansion coefficients are determined from an infinite set of linear algebraic equations (the separation of variables method). These two approaches yield exact solutions of the problem in the case of axially symmetric particles, which coincide with known solutions in particular cases. Investigation of infinite systems allowed finding the boundaries where these algorithms are valid. Numerical calculations showed that, for spheroidal Chebyshev particles (i.e., perturbed spheroids), the Rayleigh approximation based on the electrostatic solution is applicable in a wide range of the problem parameters and is in fair agreement with the results obtained using the discrete dipole approximation.     

Comparative view of the various dielectric functions based on the moment sum-rule approach

The density response function obtained by solving the equation of motion for the double time retarded commutator of the classical density fluctuation operators is expanded for large ω and the coefficients are compared with those from the large ω expansion of the density response function in the effective mean field theory. It is found that the expression for the local field correction is the same as obtained by Pathak and Vashishta following a calculation of the third frequency moment using the spectral form of the density response function. In this paper a comparative study of the various dielectric functions derived on the moment conserving scheme is presented and the results are analysed.     

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials 1. Theory

On the basis of our recent investigations concerning the mathematical structure of the hierarchy which results from the Legendre polynomial expansion of the electron velocity distribution function in Boltzmann's equation a new technique for solving this equation in multi-term even-order approximation is presented. This method is, even if more complex, the logical generalization of the well known technique for solving Boltzmann's equation by backward integration in the conventional two-term approximation. A weakly ionized, spatially homogeneous and stationary plasma with elastic and exciting electron-atom collisions is considered acted upon by a dc electric field. The technique, presented in detail, determines the distribution function in even order 2l of the expansion at the end by l-fold backward and 2l-fold forward integration of the hierarchy and by continuous connection of the resulting non-singular parts of the general solutions at low and high energies at an appropriate connection point. A first application of this method is made on a model gas for the even orders from 2 to 10 and under conditions with distinct anisotropy in the velocity space due to intensive exciting collisions. The converged macroscopic quantities and the corresponding first coefficients of the distribution expansion itself are compared with very accurate Monte Carlo simulations under the same conditions where a perfect agreement between the results obtained with both techniques was found confirming the high accuracy of the new technique to be presented.     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.     

一种新颖的用于光腔模式及光束传输模拟的特征向量法

根据有限元法单元划分的思想，提出了一种新颖的模拟光腔模式及光束传输的特征向量法. 该方法的关键之处在于基于衍射积分理论构造了一种新的光束传输矩阵，通过求解特征矩阵方程可一次性得到谐振腔的一系列特征向量，每一列特征向量即代表了腔镜上光场的一个确定模式的振幅及相位分布. 并可采用该方法模拟光场传输到腔内或腔外任意地方的场分布. 该方法将传统方法中大量的迭代过程转化成为本征积分方程特征向量的求解过程，并与初值取值无关，且可一次性求得多个模式分布，从而可方便地分析谐振腔的模式鉴别能力. 特征向量法对圆形镜共焦 关键词： 谐振腔 特征向量法 模式分布     

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger representation space that incorporates an analytic solution for the non-central electric dipole potential cosθ/r², which was believed not to belong to the class of exactly solvable potentials. Therefore, we were able to obtain a closed form solution of the three-dimensional time-independent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series in a basis composed of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.     

The hyperspherical-harmonics expansion method and the integral-equation approach to solving the few-body problem in momentum space

The Faddeev and Faddeev-Yakubovsky equations for three- and four-body systems are solved by applying the hyperspherical-harmonics expansion to them in momentum space. This coupling of two popular approaches to the few-body problem together with the use of the so-called Raynal-Revai transformation, which relates hyperspherical functions, allows the few-body equations to be written as one-dimensional coupled integral equations. Numerical solutions for these are achieved through standard matrix methods; these are made straightforward, because a second transformation renders potential multipoles easily calculable. For sample potentials and a restricted size of matrix in each case, the binding energies extracted match those previously obtained in solving the Schrödinger equation through the hyperspherical-harmonics expansion in coordinate space.Work supported in part by the National Science Foundation through grant No. PHY83-06584 and grant No. PHY87-12229     

An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher–Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.     

On the initial-boundary value problems for soliton equations

We present a novel approach to solving initial-boundary value problems on the segment and the half line for soliton equations. Our method is illustrated by solving a prototypal and widely applied dispersive soliton equation—the celebrated nonlinear Schroedinger equation. It is well known that the basic difficulty associated with boundaries is that some coefficients of the evolution equation of the (x) scattering matrix S(k, t) depend on unknown boundary data. In this paper, we overcome this difficulty by expressing the unknown boundary data in terms of elements of the scattering matrix itself to obtain a nonlinear integrodifferential evolution equation for S(k, t). We also sketch an alternative approach in the semiline case on the basis of a nonlinear equation for S(k, t), which does not contain unknown boundary data; in this way, the “linearizable” boundary value problems correspond to the cases in which S(k, t) can be found by solving a linear Riemann-Hilbert problem.     

Modeling the characteristics of the waves scattering by a group of scatterers

The pattern equations method is extended to solving the diffraction problem on a group of bodies. The problem is reduced to solving an algebraic system of equations with respect to the expansion coefficients of the scattering patterns by using a series expansion of the scattering patterns in angular spherical harmonics. The explicit (asymptotic) solution of the problem is obtained in a case when the scattering bodies are far enough from each other.     

Electrostatic response of a half-disk

This article studies the response of a half-disk exposed to an external uniform static electric field. A semianalytical method is presented for computing the potential for a geometry consisting of two conjoined half-disks with different permittivities. The method is based on analytical series expansions with coefficients obtained as a numerical solution of a matrix equation. We consider the polarizability of a single dielectric half-disk and discuss a duality relation observed in 2D polarizability. We also study the surface plasmons supported by a negative-permittivity half-disk.     

Optical bistability due to a two-photon absorption resonance with fluctuations

Using the intensity-dependent complex dielectric function for a two-photon absorption resonance we derive the Langevin equation for the fluctuating light-field in the non-linear resonator. The corresponding Fokker-Planck equation is solved by expanding the distribution function in terms of products of trigonometric functions and generalized Laguerre polynomials. The expansion coefficients are calculated using the method of matrix continued fractions. Numerical results for the stationary case are given.