Modal theory for the two-frequency mutual coherence function in random media: beam waves

Pulse propagation in a random medium is mainly determined by the two-frequency mutual coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

A new exact solution for colliding gravitational plane waves

A new exact solution of the vacuum Einstein equations describing the spacetime following the collision of two plane impulsive gravitational waves, each supporting a plane gravitational wave, is obtained. The solution has been extended prior to the instant of collision and the main features of the resulting space-time have been analyzed, using the Newman-Penrose formalism. It is shown that the result of the collision is the development of a singularity of the spacetime due to the simultaneous focussing of the two plane waves.     

The reference wave solution for a two-frequency wave propagating in a random medium

In this work a new reference wave method for solving parabolic-type equations is proposed. The performance of the method is demonstrated by applying it to the equation governing the propagation of the two-frequency mutual coherence function in a random medium. An analytic solution is presented for arbitrary correlation properties of the medium. It is shown that when approximating the transverse structure function of the medium by a quadratic form, the solution reduces to the exact result derived previously. Extensions to more general types of media are considered.     

Radiative transfer through a spherically symmetric cloud of extincting particles — exact solution of the inverse absorption problem

We derive an exact solution to the inverse absorption problem to calculate the density distribution in spherical symmetry of absorbing particles from the intensity pattern obtained for homogeneous illumination. We illustrate the capabilities of the method by the simple example of a constant density core and find the required numerical effort to be negligible. The applicability is discussed for physical problems where unknown absorption coefficients, particle size or density distributions can be determined from multi-frequency measurements of the transmission coefficient. The applications range from targets being evaporated by laser pulses to Bok globules in astrophysics.     

The exact solution of an octagonal rectangle-triangle random tiling

We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eightfold-symmetric phase. The solution is obtained using the Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling which were found by M. Widom.     

The exact probability distribution of a two-dimensional random walk

A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at timet that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: I

In this paper, the modal expansion theory is presented as a new analytical approach together with the resulting new physical parameters. In particular, the features of an arbitrary power-law structure function are investigated. The exact expression for the Gaussian spectrum is rederived. An approximate analytical expression for the two-frequency coherence function evaluated at equal positions for the Kolmogorov spectrum is presented and comparison with the numerical solution in the literature exhibits a remarkable agreement. As a result of the modal decomposition, general properties for a transversally homogeneous and isotropic medium are demonstrated, such as the exponential decay of the amplitude of the solution and the linear phase behaviour at large propagation distances.     

An exact solution of Laplace's equation for cuspidal geometry

An exact solution of Laplace's equation is obtained for a system of conducting electrodes with cuspidal symmetry. The significance of this result in predicting and verifying the equilibrium configuration of a rotationally symmetric conducting fluid subject to electrostatic stress is discussed.This work was supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-8108829     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II

Abstract

In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency mutual coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.     

Radiative transfer in a layer of magnetized plasma with random irregularities

The problem of radio wave reflection from an optically thick plane uniform layer of magnetized plasma is considered in the present work. The plasma electron density irregularities are described by a spatial spectrum of arbitrary form. The small-angle scattering approximation in invariant ray coordinates is proposed as a technique for the analytical investigation of the radiation transfer equation. The approximate solution describing the spatial and angular distribution of radiation reflected from a plasma layer is obtained. The solution obtained is investigated numerically for the case of ionospheric radio wave propagation. Two effects occur as a consequence of multiple scattering: a change in the reflected signal intensity and an anomalous refraction.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: II

In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency mutual coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.     

An exact limit of a local mixed valence model

Recently, Khomskii and Kocharjan have considered a local mixed valence model and, in a generalized linear Hartree-Fock approximation, found solutions which, when generalized to the many site problem, displayed continuous and discontinuous transitions from ground states of integral to intermediate valence. The transitions were due to an energy dependence of the virtual bound state width and persisted in the limit of zero hybridization. We examine the model of Khomskii and Kocharjan and demonstrate that in the limit of zero hybridization an exact solution may be found for the valence behaviour. The generalization of this result to the case of a small but finite concentration of localized level sites, exhibits intermediate valence only as a consequence of the pinning of the Fermi level to the narrow localized levels and the transitions between the ground states of integral and intermediate valence are continuous.     

An exact stationary solution of Einstein's equations

We give an exact stationary solution ofEinstein's empty space field equations. It represents two massless sources possessing angular momentum, and held in position by stresses. The solution conforms withMach's principle in the following sense: the spinning sources cause a rotation of the local inertial frame relative to test particles at infinity.     

An exact,closed form,solution for the flexural vibration of a thin annular plate having a parabolic thickness variation

An exact, closed form, solution is obtained for the transverse vibrations, with nodal diameters and circles, of a thin annular plate having a parabolic thickness variation. Representative numerical values for the frequency parameter and typical mode shapes are presented for three different combinations of simple boundary conditions. The corresponding exact solution for an aeolotropic annular plate of the same geometry is also presented. Aside from possible design applications, these exact, closed form, data can be used as test cases for assessing the accuracy of various approximate methods of solution. The analysis involves only the powers of the radius and is simpler than that for the constant thickness solution which involves Bessel functions.     

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.