Duality methods in the theory of averaging of nonlinear variational problems are developed. The questions of a general nature that are discussed include a detailed analysis of the concept of regularity, an example of a nonregular Lagrangian, and the derivation of duality formulas that take account of the regularity problem. The main content is concerned with the averaging of variational problems with stochastic Lagrangians. Three groups of questions are investigated: 1) averaging of Lagrangians of a general form; 2) averaging of the Lagrangians of plasticity (the theory of the limit load); and 3) averaging of degenerate Lagrangians (problems with random soft or rigid inclusions). Bibliography: 13 titles.

https://iopscience.iop.org/article/10.1070/IM1987v029n01ABEH000958/pdf

Averaging of functionals of the calculus of variations and elasticity theory

An extended random medium is modeled by a set of 2-D thin Gaussian phase-changing screens with phase power spectral densities appropriate to the natural medium being modeled. Details of the algorithm and limitations on its application to experimental conditions are discussed, concentrating on power-law spectra describing refractive-index fluctuations of the neutral atmosphere. Inner and outer scale effects on intensity scintillation spectra and intensity variance are also included. Images of single realizations of the intensity field at the observing plane are presented, showing that under weak scattering the small-scale Fresnel length structure of the medium dominates the intensity scattering pattern. As the strength of scattering increases, caustics and interference fringes around focal regions begin to form. Finally, in still stronger scatter, the clustering of bright regions begins to reflect the large-scale structure of the medium. For plane waves incident on the medium, physically reasonable inner scales do not produce the large values of intensity variance observed in the focusing region during laser propagation experiments over kilometer paths in the atmosphere. Values as large as experimental observations have been produced in the simulations, but they require inner scales of the order of 10 cm. Inclusion of an outer scale depresses the low-frequency end of the intensity spectrum and reduces the maximum of the intensity variance. Increasing the steepness of the power law also slightly increases the maximum value of intensity variance.

Intensity images and statistics from numerical simulation of wave propagation in 3-D random media

The paper presents a review of the Wigner distribution function (WDF) and of some of its applications to optical problems, especially in the field of partial coherence. The WDF describes a signal in space and in spatial frequency simultaneously and can be considered the local spatial-frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of an optical signal by means of its WDF closely resembles the ray concept in geometrical optics; the WDF thus presents a link between partial coherence and radiometry. Properties of the WDF and its propagation through linear optical systems are considered; again, the description of systems by WDF’s can be interpreted directly in geometric-optical terms. Some examples are included to show how the WDF can be applied to practical problems that arise in the field of partial coherence.

/pdf/application-of-the-wigner-distribution-function-to-partially-2ha88t7wve.pdf

Application of the Wigner distribution function to partially coherent light

In this survey, we fully review almost all potential ionospheric effects on the performance of space-based radar systems (SBRs), which operate in the ambient ionosphere environment; in particular, we review the use of space-based synthetic aperture radar systems (SARs) for imaging. There are two families of effects involved. One is the effects of the background ionosphere (non-turbulent ionosphere), such as dispersion, group delay, refraction, Faraday rotation, and phase shift. The other is the effects due to ionospheric irregularities, such as refractive index fluctuation, phase perturbation, angle-of-arrival fluctuation, pulse broadening, clutter, and amplitude scintillation. These effects adversely affect SAR imaging in several respects, such as by causing image shift in the range, and degradations of the range resolution, azimuthal resolution, and/or the resolution in height (elevation). We also review ionospheric irregularity characteristics and descriptions, propagation channel statistics, ...

A survey of ionospheric effects on space-based radar

Laminate materials of the type introduced by Schulgasser [1] and Bruggeman [66,67] are studied and the extent to which they can simulate the transport properties of other composites is explored. Laminates with chirality, with especially high or low field concentrations, or which attain various bounds are constructed. The Hashin-Shtrikman bounds on the shear modulus are demonstrated to be optimal, being attained by a hierarchical laminate material. While the conductivity function of two-component composites can be simulated by laminates, an example suggests this does not extend to five-component composites. Attention is drawn to the connection between conductivity functions, Stieltjes functions, and bounds.

Modelling the Properties of Composites by Laminates

We demonstrate that a recently derived approximate solution to the fourth-moment equation that was thought to be Valid only for strong scattering is in fact valid for all values of the scattering parameter. We use a modified Kolmogorov spectrum for the index-of-refraction fluctuations and present results in two dimensions for comparison with numerical solutions. We also present results in three dimensions that should represent a quantitatively accurate, theoretical description of the atmospheric-scintillation problem.

Two-scale solution for atmospheric scintillation

In the vicinity of a caustic the intensity is very different from that calculated on the basis of geometrical acoustics. Using the basic formulation of coherence theory we show here how the correct intensity may be obtained when the source has a finite (i.e., nonzero) size. We use a two‐scale expansion to include the correction term to the geometric acoustic calculation (here already modified to include the effect of a finite source). It is shown that in the limit of a point source the results are the same as those previously obtained. Finally a numerical example is presented.

Caustic corrections using coherence theory

We use the two‐scale expansion developed previously to determine the intensity variation in the neighborhood of caustics, to show how we may calculate the scattering of acoustic radiation in a random medium with a variable speed of sound profile. Equations are derived which show how the coherence function varies along the characteristic rays (identical to the rays of geometrical acoustics in the parabolic approximation). Conditions are given for simplifying the equations for high‐frequency quasi‐isotropic scattering. The problems that arise when the medium is highly anisotropic are discussed and an approximate solution is given. It is also shown how to calculate the intensity variation due to scattering in the neighborhood of a caustic.

Scattering calculations using the characteristic rays of the coherence function

We consider here the problem of determining the mean square fluctuations in a statistically homogeneous isotropic two‐phase dielectric random medium. An expression is derived for a weighted sum of the mean square fluctuations in each phase in terms of the effective dielectric constant. From this expression bounds are derived for the mean square fluctuations in each phase. An assumption is then made to allow us to obtain exact expressions for the mean square fluctuations in a particular phase.

Field fluctuations in a two‐phase random medium

We present calculations of the scintillation index that result from a plane wave of light impinging on the atmosphere. Calculations are made using the two-scale theory and assuming a Kolmogorov structure function with an altitude-dependent structure constant. A plot of the scintillation index for typical nighttime viewing is given as a function of angle from the zenith. A further approximation is then introduced which allows the calculation of the scintillation index in terms of a single nondimensional parameter. This parameter is a direct generalization of the parameter used when the structure constant is an absolute constant.

Mark J. Beran

Papers

Two-scale solution for atmospheric scintillation

Caustic corrections using coherence theory

Scattering calculations using the characteristic rays of the coherence function

Field fluctuations in a two‐phase random medium

Scintillation index calculations using an altitude-dependent structure constant.