Two-dimensional isotropic scattering in a semi-infinite medium

Exact results are presented for the source function, radiative flux, and intensity at the boundary of a two-dimensional, isotropically scattering, semi-infinite medium subjected to collimated or diffuse radiation. The spatial distributions of incident radiation considered are (1) cosine-varying, (2) semi-infinite step, (3) step at the origin and (4) finite strip. Two-dimensional effects are most pronounced at large albedos.     

Two-dimensional rayleigh 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 with second order Legendre phase function scattering. The incident radiation is collimated, normal to the top surface, and is dependent only on the radial coordinate. Boundary conditions which vary as a Bessel function and as a Gaussian distribution are investigated. The Gaussian distribution approximates a laser beam. Numerical results are presented in graphical and tabular forms for a Rayleigh scattering medium. The results are compared with those of isotropic scattering.     

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.     

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.     

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

Integro-differential equations are developed for the source function, flux and intensity at the boundaries of a two-dimensional finite-thick medium which scatters in a linear fashion. The incident radiation is collimated, normal to the upper surface of the medium and dependent only on the radial coordinate. Two radial distributions are investigated: (1) a Bessel function and (2) a Gaussian laser beam. The solution for the Gaussian beam is constructed from the Bessel solution. Numerical results are presented in graphical and tabular forms for both boundary conditions. Comparisons are made between forward and isotropic scattering and between the finite and semi-infinite cases.     

Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering

Exact integral equations are derived describing the source function and radiative flux in a two-dimensional, radially infinite cylindrical medium which scatters anisotropically. The problem is two-dimensional and cylindrical because of axisymmetric loading. Radially varying collimated radiation is incident normal to the upper surface while the lower boundary has no radiation incident upon it. The scattering phase function is represented by a spike in the forward direction plus a series of Legendre polynomials. The two-dimensional integral equations are reduced to a one-dimensional form by separating variables for the case when the radial variation of the incident radiation is a Bessel function. The one-dimensional form consists of a system of linear, singular Fredholm integral equations of second kind. Other more complex boundary conditions are shown to be solvable by a superposition of this basic Bessel function case. Diffusely incident radiation is also considered.     

Time-dependent radiation transfer in a semi-infinite stochastic medium with Rayleigh scattering

The time-dependent radiation transfer in a semi-infinite stochastic medium of binary Markovian mixture with Rayleigh scattering is presented. A formalism, developed to treat radiation transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average reflectivity, radiant energy and net flux are computed for specular-reflecting boundary. For the sake of comparison, we use two different weight functions in our calculations.     

Two-dimensional radiative transfer in a finite scattering planar medium

The source function, radiative flux, and intensity at the boundaries are calculated for a two-dimensional, scattering, finite medium subjected to collimated radiation. The scattering phase function is composed of a spike in the forward direction super-imposed on an isotropic background. Exact radiative transfer theory is used to formulate the problem and Ambarzumian's method is used to obtain results. Using the principle of superposition, the results for any step variation in incident radiation are expressed in terms of universal functions for the semi-infinite step case. Two-dimensional effects are most pronounced at large optical thicknesses and albedos.     

Study of Love-type wave propagation in an isotropic tri layers elastic medium overlying a semi-infinite elastic medium structure

This paper deals with the propagation of Love-type wave in a composite isotropic structure embraced of tri layers elastic medium overlying a semi-infinite elastic medium. The heterogeneity is caused due to the variation of linear, exponential, and quadratic with respect to the depth. Modified Bessel function with Debye Asymptotic Expansion approach is used to achieve closed form of dispersion equation analytically and found to be in well agreement to the classical Love wave equation. Numerical computation has been carried out to accomplish the graphical demonstration to unravel some important peculiarities of wave number associated in presence or absence of layers medium and effect of heterogeneities on phase velocity of Love-type wave.     

Two-dimensional radiative equilibrium: A semi-infinite medium subjected to a finite strip of radiation

Exact numerical results are presented for the emissive power and radiative flux at the boundary of a two-dimensional, absorbing-emitting, semi-infinite medium bounded by (1) a strip of collimated radiation and (2) a constant temperature black strip. The method of super-position is used to obtain the finite strip solutions in terms of cosine varying solutions. The infinite integrals arising in the solutions are converted to an alternating series of finite integrals. The Euler transformation is then applied to speed up convergence. Error bounds are determined whereby the two-dimensional finite strip analysis can be approximated by the simpler one-dimensional solution.     

Single and double scattering approximations for a two-dimensional cylindrical medium

Large spatial frequency expansions for the source function, radiative flux, and intensity are obtained for an isotropically scattering finite two-dimensional medium exposed to collimated radiation. With these expansions, the single and double scattering results are obtained which are valid at small optical distances away from the incident radiation. Results are presented for a circular disk, exponential distribution and a Gaussian distribution of incident radiation.     

Expansions for two-dimensional radiative transfer in an isotropically scattering semi-infinite medium

Small spatial frequency expansions for the source function and radiative flux are obtained for a purely scattering, semi-infinite, two-dimensional medium. Both collimated and diffuse boundary conditions are analyzed. With these expansions, other expansions are obtained which are valid at large optical distances away from the incident radiation. Expansions are presented for a finite strip, circular disk and a Gaussian distribution.     

横向各向同性柱状复合结构对超声波的散射

首先描述了横向各向同性复合圆柱结构的入射声场、散射声场及内部的驻波声场,然后利用转移矩阵方法导出了求解散射声场的方程组,计算了铝/各向异性界面层/纤维复合结构对斜入射声波的背向散射谱和散射截面积。将柱状复合结构中各向异性界面薄层相应的转移矩阵作渐近展开,建立了模拟这种界面薄层的弹簧模型及界面处广义边界条件。结果表明,模型中劲度常数仅依赖于界面薄层厚度及界面层媒质的弹性常数c_ll,c₁₂和c₄₄,而振子质量与c_ll,c₄₄,c₁₃和c₃₃有关。     

Two-dimensional radiative equilibrium: A semi-infinite medium subjected to cosine varying radiation

Exact numerical solutions are presented for the radiative flux and emissive power at the boundary of a semi-infinite, two-dimensional, planar, absorbing-emitting, gray medium subjected to cosine-varying collimated and cosine-varying diffuse boundary radiation, respectively. The emissive power at the boundary due to the cosine varying collimated boundary condition is shown to be a generalized H-function which is analogous to the H-function of Chandrasekhar. The nonlinear integral equation of the Chandrasekhar type is developed for the generalized H-function and solved for a wide range of the parameters. The emissive power and radiative flux at the boundary for the cosine-varying diffuse model, as well as the radiative flux for the cosine-varying collimated model, are expressed in terms of the generalized H-function and solved numerically.     

Electromagnetic scattering from a continuously inhomogeneous random medium with cylindrical symmetry

Abstract

Born's approximation is used to determine the mean value of the turbulent radar cross section (RCS) of an inhomogeneous cylindrical random medium at oblique incidence. The mean medium is taken into account by a renormalization procedure. Then, only the diffraction due to the fluctuating part of the permittivity has to be considered. The fluctuations are approximated by means of the turbulence spectrum given by Kolgomorov's theory. Furthermore, Maxwell's equations are solved in terms of fields rather than potentials. This leads us to a significant reduction of the linear system size which simplifies numerical calculations. It turns out that the fields which propagate in the mean medium are noticeably modified by that medium. Thus, the renormalization has a considerable effect on the assessment of the turbulent RCS of the wake. The influence of the direction of incidence on the RCS levels is also analysed. Finally, numerical results are given in order to compare calculations with experiment.     

Electromagnetic scattering from a continuously inhomogeneous random medium with cylindrical symmetry

Born's approximation is used to determine the mean value of the turbulent radar cross section (RCS) of an inhomogeneous cylindrical random medium at oblique incidence. The mean medium is taken into account by a renormalization procedure. Then, only the diffraction due to the fluctuating part of the permittivity has to be considered. The fluctuations are approximated by means of the turbulence spectrum given by Kolgomorov's theory. Furthermore, Maxwell's equations are solved in terms of fields rather than potentials. This leads us to a significant reduction of the linear system size which simplifies numerical calculations. It turns out that the fields which propagate in the mean medium are noticeably modified by that medium. Thus, the renormalization has a considerable effect on the assessment of the turbulent RCS of the wake. The influence of the direction of incidence on the RCS levels is also analysed. Finally, numerical results are given in order to compare calculations with experiment.     

Two-dimensional scattering of electromagnetic waves from a permeable inclusion in an anisotropic medium

Integrodifferential equations for the cross section of the scatterer and a collocation method are used to obtain a numerical solution of the problem of scattering of an H-polarized wave by an anisotropic layer of a three-layer dielectric structure. Zh. Tekh. Fiz. 68, 84–88 (January 1998)     

Transient wave propagation in a semi-infinite,transversely isotropic rod

In this study the response in a transversely isotropic, semi-infinite elastic rod is found to an input on its end that is time dependent. The solution follows closely the method developed by Skalak for isotropic rods which uses integral transforms to solve the governing equations. The solution differs from that of Skalak, apart from material properties, in that in this study mixed-mixed conditions are assumed for the end of the rod. The responses are found at a “distant” station and are expressed in terms of the axial and tangential strains.     

Free localized vibrations of a semi-infinite cylindrical shell

Free vibrations of a semi-infinite cylindrical shell, localized near the edge of the shell are investigated. The dynamic equations in the Kirchhoff-Love theory of shells are subjected to asymptotic analysis. Three types of localized vibrations, associated with bending, extensional, and super-low-frequency semi-membrane motions, are determined. A link between localized vibrations and Rayleigh-type bending and extensional waves, propagating along the edge, is established. Different boundary conditions on the edge are considered. It is shown that for bending and super-low-frequency vibrations the natural frequencies are real while for extensional vibrations they have asymptotically small imaginary parts. The latter corresponds to the radiation to infinity caused by coupling between extensional and bending modes.