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When waves propagate through a strongly scattering medium the energy is transferred to the incoherent wave part by scattering. The wave intensity then forms a random speckle pattern seemingly witho...

A methodology is presented for bounding the higher $L^p$ norms, $2\leq p\leq\infty$, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

This paper generalizes well-established derivations of the radiative transfer equation from first principles to model the energy density of time-dependent and monochromatic high frequency waves pro...

The classic atomic mix approximation for particle transport in a stochastic spatial medium is accurate when the material chunks in the medium are small compared to a mean free path. In this paper, we show that for charged particle transport in a stochastic medium, the atomic mix approximation is accurate when the chunk sizes are small compared to a transport mean free path. For charged particle transport, the transport mean free path is generally several orders of magnitude larger than the mean free path. Therefore, the result obtained in this paper greatly extends the known range of applicability of the atomic mix approximation. Numerical results are given that validate the asymptotic theory, and an application of the theory to a practical problem in radiation oncology is discussed.

Fractal medium models are used to model, for instance, the heterogeneous earth and the turbulent atmosphere. A wave pulse propagating through such a medium will be affected by multiscale medium fluctuations. For a class of one-dimensional fractal random media defined in terms of fractional Brownian motion we show how the wave interacts with the medium fluctuations. The modification in the pulse shape depends on the roughness of the medium and can be described in a deterministic way when the pulse is observed at its random arrival time. For very rough media the coherent wave is confined to a surface layer.

In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers, time reversed, and finally reemitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the reemitted signals at the location of the initial source is improved when the medium is heterogeneous. Contrary to intuition, multiple scattering enhances the spatial resolution of the refocused signal and allows one to beat the diffraction limit obtained in homogeneous media. This paper presents a quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media. The theory is based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit. Numerical experiments complement the theory.

This article is devoted to the mathematical analysis of various formulas giving the equivalent absorption and scattering cross section for mixed materials in linear transport theory. We begin with a general result on the treatment of high-frequency oscillations in linear transport equations which is partly based upon the velocity averaging results and is the analogue, for transport equations, of the compensated compactness class of results. The case of periodic inhomogeneities is then studied in detail; in particular we show the essential difference with periodic homogenization of diffusion equations, due to small divisor problems. These results were announced in [F. Golse, C.R. Acad. Sci. Paris Ser. I Math., 305 (1987), pp. 801--804; F. Golse, Mathematical Aspects of Fluid and Plasma Dynamics, G. Toscani, V. Boffi, and S. Rionero, eds., Lecture Notes in Math 1460, Springer-Verlag, Berlin, New York, 1991, pp. 152--169]. Finally, we treat a case of stochastic inhomogeneities in linear transport theory insp...

We address the problem of diffusion approximation of a transport process in a random mixture of diffusive and nondiffusive materials. We show that the diffusion approximation is valid provided that some hypotheses on the geometrical structure of the mixture are satisfied. The diffusion matrix is given by the Kubo formula. Some bounds on the diffusion coefficients are presented.

Homogenization is used to analyze a partial differential equation in a domain with a very rough boundary or interface, following the procedure of Kohler, Papanicolaou, and Varadhan [Boundary and interface problems in regions with very rough boundaries, in Multiple Scattering and Waves in Random Media, P. Chow, W. Kohler, and G. Papanicolaou, eds., North-Holland, Amsterdam, 1981, pp. 165--197] and Brizzi and Chalot [Homogeneisation de frontiere, Ph.D. thesis, Department of Mathematics, Universite de Nice, Nice, France, 1978]. It is shown that such a boundary or interface can be replaced by an equivalent layer within which a modified differential equation holds. The coefficients in this new equation are certain "effective parameters" such as the effective conductivity, the effective dielectric constant, the effective refractive index, etc. These coefficients are determined by the solutions of certain special problems which involve the detailed shape of the boundary or interface. This analysis is applied to ...

This paper studies the reflection of pulses from a randomly layered half space. It characterizes the statistical properties of the reflected signals at the surface in a suitable asymptotic limit in...

In 1973 A. J. Chorin [Math. Comp., 27 (1973), pp. 1–15] reported quadrature formulas for the approximation of a class of Wiener path integrals by n-fold ordinary integrals with an error Similar formulas are presented here for certain stochastic Wiener function space integrals in which n-fold integral approximations result in errors $O( n^{ - 2} )$ or $O( n^{ - 3/ 2} )$. Stochastic Wiener integrals arise in the physics of wave propagation in random media and in signal estimation problems in communication theory. The present formulas provide simple, accurate computational tools for these applications, among others. This point is illustrated by several examples in the paper.