Exact expressions for radiative transfer in a three-dimensional rectangular geometry

The equations for the source function, flux, and scattered intensity normal to the surface are formulated in cartesian coordinates for a 3-D rectangular absorbing, emitting, isotropically scattering medium exposed to both diffuse and collimated radiation. Simplifications of these equations for certain important geometries and uniform loading are presented. Also, superposition of these equations and radiative equilibrium are discussed. For pure scattering, the source function at the center of the square and cubic geometries is analytically determined for the diffuse boundary condition. The generalized 3-D equations are shown to reduce to the familiar 1-D results. Also, the equations for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function are expressed in terms of the isotropic expressions.     

On the Interaction of an Alternating Electrical Field with an Inhomogeneous Magnetized Plasma

It is found out the form of the vectorial and isotropic cartesian component for the distribution function for electrons in an inhomogeneous plasma interacting with an alternative electric field, without being done any previous supposition on the temporal dependence of these quantities. For this purpose it is applied a method of succesive approximations for solving an integro-differential equation for the isotropic component of the distribution function. The scalar and vector components of the plasma spherical harmonic expansion are calculated including transient and secular terms. Some particular quantities as time-independent part of vectorial component of distribution function and of nonlinear conductivity were also obtained. Also corrected expressions for the transport coefficients in a electrical field, are obtained.     

Interface response theory of electromagnetism in composite dielectric materials

It is shown how a recent interface theory can be used for solving the Maxwell equations in any composite dielectric material. General expressions for the corresponding response functions are given. These new results are illustrated by a general application to layered isotropic dielectrics and a simple derivation of the response function of two different isotropic dielectrics separated by a planar interface.     

Effect of ordering ambiguity in constructing the Schrödinger equation on perturbation theory

This work explores the application of perturbation formalism, developed for isotropic velocity-dependent potentials, to three-dimensional Schr?dinger equations obtained using different orderings of the Hamiltonian. It is found that the formalism is applicable to Schr?dinger equations corresponding to three possible ordering ambiguities. The validity of the derived expressions is verified by considering examples admitting exact solutions. The perturbative results agree quite well with the exactly obtained ones.     

Generalized thermoelastic extensional and flexural wave motions in homogenous isotropic plate by using asymptotic method

In this paper the asymptotic method has been applied to investigate propagation of generalized thermoelastic waves in an infinite homogenous isotropic plate. The governing equations for the extensional, transversal and flexural motions are derived from the system of three-dimensional dynamical equations of linear theories of generalized thermoelasticity. The asymptotic operator plate model for extensional and flexural free vibrations in a homogenous thermoelastic plate leads to sixth and fifth degree polynomial secular equations, respectively. These secular equations govern frequency and phase velocity of various possible modes of wave propagation at all wavelengths. The velocity dispersion equations for extensional and flexural wave motion are deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate. The approximation for long and short waves along with expression for group velocity has also been obtained. The Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations and its equivalence established with that of asymptotic method. The numeric values for phase velocity, group velocity and attenuation coefficients has also been obtained using MATHCAD software and are shown graphically for extensional and flexural waves in generalized theories of thermoelastic plate for solid helium material.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

Calculation of the phase velocities of three-dimensional flexural waves in isotropic cylindrical bars and shells using the Debye and Debye-type potentials

Using the Debye and Debye-type potentials, two representations of the vector potential of the displacement vector are considered for isotropic cylindrical bars and shells. With this approach, characteristic equations are obtained for the wavenumbers of three-dimensional flexural waves propagating in the aforementioned bodies; the phase velocities of such waves are calculated.

     

Microscopic modes in a Fermi superfluid. I. Linearized kinetic equations

This is the first of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this first paper we derive closed, decoupled, linearized kinetic equations for the bogolon spin density and total density in a Fermi superfluid with fluctuating superfluid velocity, and we discuss the form of the hydrodynamic equations that result from these equations.     

Phase space geometry in scalar-tensor cosmology

We study the phase space of spatially homogeneous and isotropic cosmology in general scalar-tensor theories. A reduction to a two-dimensional phase space is performed when possible—in these situations the phase space is usually a two-dimensional curved surface embedded in a three-dimensional space and composed of two sheets attached to each other, possibly with complicated topology. The results obtained are independent of the choice of the coupling function of the theory and, in certain situations, also of the potential.     

Emissivity estimates

A variational expression is developed to estimate the overall and angular emissivities for a halfspace geometry described by the equation of transfer. In contrast to the usual variational procedure of postulating a functional and then proving correctness, the appropriate functional is derived using a Lagrange multiplier technique. With asymptotic trial functions, simple expressions for the emissivities result. This variational estimate of the overall emissivity is exceeding accurate when compared to exact results for an isotropic phase function. The angular emissivity, a more detailed quantity, is estimated with less accuracy. The formalism is not restricted to an isotropic phase function, but is valid for general anisotropic scattering. As an example of an anisotropic phase function, the case of Thomson scattering is considered. The emissivity for this phase function is found to be slightly larger than that for isotropic scattering.     

Two-dimensional isotropic scattering in a finite thick cylindrical medium exposed to a laser beam

Graphical and tabular results are presented for the back-scattered intensity from a finite two-dimensional cylindrical medium exposed to a Gaussian beam of radiation. Also, results for the source function and flux at the boundaries are presented. The influence of optical thickness and albedo are most pronounced at large optical radii. The semi-infinite results can be used to approximate the finite case for small optical radii. Ranges for single, double, and multiple scattering are discussed. For locations far from the incident beam, the results can be expressed in terms of universal functions independent of beam size. A method is presented for extending the isotropic results to the anisotropic case where the phase function is made up of a spike superimposed on an otherwise isotropic phase function.     

On Green's function for cylindrically symmetric fields of polarized radiation

Analytic expressions for Green's function describing the process of transfer of polarized radiation in homogeneous isotropic infinite medium in case of cylindrical symmetry and nonconservative scattering are obtained. The solution is based on the set of systems of Abel integral equations of the first kind obtained using the principle of superposition, and the known expression of Green's function for radiation fields with plane-parallel symmetry. Eigenvalue decompositions for the corresponding matrices of generalized spherical functions are found. Using this result the systems of Abel integral equations are diagonalized, and the final solution is obtained.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

Analysis of Two-Phase Cavitating Flow with Two-Fluid Model Using Integrated Boltzmann Equations

In the present work, both computational and experimental methods are employed to study the two-phase flow occurring in a model pump sump. The two-fluid model of the two-phase flow has been applied to the simulation of the three-dimensional cavitating flow. The governing equations of the two-phase cavitating flow are derived from the kinetic theory based on the Boltzmann equation. The isotropic RNG$k-\epsilon-k_{ca}$ turbulence model of two-phase flows in the form of cavity number instead of the form of cavity phase volume fraction is developed. The RNG $k-\epsilon-k_{ca}$ turbulence model, that is the RNG$k-\epsilon$ turbulence model for the liquid phase combined with the $k_{ca}$model for the cavity phase, is employed to close the governing turbulent equations of the two-phase flow. The computation of the cavitating flow through a model pump sump has been carried out with this model in three-dimensional spaces. The calculated results have been compared with the data of the PIV experiment. Good qualitative agreement has been achieved which exhibits the reliability of the numerical simulation model.     

Two-dimensional isotropic scattering in a semi-infinite cylindrical medium

Beginning with the integral equation for the source function, the solutions for the source function, flux and intensity at the boundary of a two-dimensional, isotropically scattering cylindrical medium are found. The incident radiation is collimated and normal to the surface of the medium and depends only on the radial coordinate. For a Bessel function boundary condition, separation of variables is used to reduce the source function integral equation to a one-dimensional equation. The resulting integral equation is shown to be the same as that for the two-dimensional planar case. Solutions for other boundary conditions are then shown to be superpositions of the Bessel function solution. Numerical results are presented for a Gaussian distribution of incident radiation which closely models a laser beam. These multiple scattering results are compared to the single scattering approximation. Also, the solution for a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.     

Two-dimensional linearly anisotropic scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium which scatters linearly. The incident radiation is collimated, normal to the top surface of the medium, and is dependent only on the radial coordinate. The radial variation is assumed to be a Bessel function or a Gaussian distribution. The Gaussian boundary condition is used to simulate a laser beam. Numerical results are presented in graphical and tabular forms for both boundary conditions. Results for forward and backward scattering phase functions are compared with those for isotropic scattering. A method is presented for extending these results to the problem of a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on a linear phase function.     

A variational approach for the classical anisotropic Heisenberg model in a crystal field

The variational approach based on the Bogoliubov inequality for the free energy is used to study the three-dimensional anisotropic Heisenberg XXZ model with a crystal field. The magnetization and the phase diagrams are obtained as a function of the parameters of the Hamiltonian. Limiting cases, such as isotropic Heisenberg, XY, and planar rotator models in two and three dimensions, are analyzed and compared to previous results obtained from analytical approximations as well as to those obtained from more reliable approaches such as series expansion and Monte Carlo simulations. A parametric procedure has been used in order to simplify the solutions of the self-consistent coupled equations.     

Analytical Description of Electron Motion in a Three-Dimensional Undulator Field

The motion of relativistic electrons in an ideal three-dimensional magnetic undulator field satisfying the stationary Maxwell equation is considered. The system of nonlinear differential equations of the electron motion is solved analytically using perturbation theory rather than the method for averaging fast oscillations of the electron trajectory (the focusing approximation), as was done in a series of previous studies. The obtained analytical expressions for the trajectories describe the behavior of particles in a three-dimensional magnetic undulator field much more accurately than the formulas obtained within the framework of the focusing approximation. The analysis of these expressions shows that the behavior of electrons in a three-dimensional undulator field is much more complicated than that described by equations obtained using the averaging method. In particular, it turns out that the electron trajectories in the undulator have a cross dependence; in this case, variations in the initial trajectory parameters in the vertical plane cause changes in the horizontal trajectory components, and vice versa. The results of calculations of the trajectories carried out using analytical expressions are close to those of numerical calculations using the Runge-Kutta method.     

Exact 3D elasticity solution for free vibrations of an eccentric hollow sphere

An exact three-dimensional elastodynamic analysis for describing the natural oscillations of a freely suspended, isotropic, and homogeneous elastic sphere with an eccentrically located inner spherical cavity is developed. The translational addition theorem for spherical vector wave functions is employed to impose the zero traction boundary conditions, leading to frequency equations in the form of exact determinantal equations involving spherical Bessel functions and Wigner 3j symbols. Extensive numerical calculations have been carried out for the first five clusters of eigenfrequencies associated with both the axisymmetric and non-axisymmetric spheroidal as well as toroidal oscillation modes for selected inner-outer radii ratios in a wide range of cavity eccentricities. Also, the corresponding three-dimensional deformed mode shapes are illustrated in vivid graphical forms for selected eccentricities. The numerical results describe the imperative influence of cavity eccentricity, mode type, and radii ratio on the vibrational characteristics of the hollow sphere. The existence of “multiple degeneracies” and the trigger of “frequency splitting” are demonstrated and discussed. The accuracy of solution is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature.