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SIAM Journal of Applied Mathematics. 01/2003; 63:1475-1498.
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ABSTRACT: We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coecients, which we take to be Markovian in the direction of propagation.
11/2002;
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ABSTRACT: We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schrodinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution. 1
02/2002;
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ABSTRACT: In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers-transducers, time reversed, and finally re-emitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the re-emitted 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.
02/2002;
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ABSTRACT: We give a detailed mathematical analysis of the radiative transport limit for the average phase space density of solutions of the Schrödinger equation with time dependent random potential. Our derivation is based on the construction of an approximate martingale for the random Wigner distribution.
09/2001;
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ABSTRACT: We study transport and diffusion of classical waves in two-dimensional disordered systems and in particular surface waves on a flat surface with randomly fluctuating impedance. We derive from first principles a radiative transport equation for the angularly resolved energy density of the surface waves. This equation accounts for multiple scattering of surface waves as well as for their decay because of leakage into volume waves. We analyze the dependence of the scattering mean free path and of the decay rate on the power spectrum of fluctuations. We also consider the diffusion approximation of the surface radiative transport equation and calculate the angular distribution of the energy transmitted by a strip of random surface impedance. 1 Introduction The propagation of wave energy in random media can be analyzed with radiative transport theory in the regime of weak random fluctuations and propagation distances that are long compared to the wavelength. This transition of waves to tran...
08/2000;
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ABSTRACT: We derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energy transport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscattering effect in the back direction. © 1999 American Institute of Physics.
Journal of Mathematical Physics 09/1999; 40(10):4813-4827. · 1.29 Impact Factor
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ABSTRACT: . We derive a probabilistic representation for solutions of matrix-valued transport equations that account for polarization eects. Such equations arise in radiative transport for the Stokes parameters that model the propagation of light through turbulent atmospheres. They also arise in radiative transport for seismic wave propagation in the earth's crust. The probabilistic representation involves an augmented scalar transport equation in which the polarization parameters become independent variables. Our main result is that the linear moments of the augmented transport equation with respect to the polarization variables are the solution of the matrix-valued transport equation. The augmented scalar transport equation is well suited to analyzing the hydrodynamic regime of small mean free paths. It is also well suited to getting approximate solutions by Monte Carlo simulation. Contents 1 Introduction 2 2 Scalar Transport Theory 3 3 Electromagnetism and Transport Equations 6 3.1 Radiati...
06/1999;
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ABSTRACT: We derive radiative transport equations for solutions of a Schrodinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the radiative transport equations is determined entirely by the Bloch spectrum, while the scattering part by the random fluctuations.
12/1998;
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ABSTRACT: Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove the grazing angle singularity due to the Born approximation. The results are presented in a form that is convenient both for theoretical analysis and for numerical computations.
Wave Motion.
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ABSTRACT: We propose a mathematical theory for the refocusing properties observed in time-reversal experiments, where classical waves propagate through a medium, are recorded in time, then time-reversed and sent back into the medium. The salient feature of such experiments is that the refocusing quality of the time-reversed reemitted signals is greatly enhanced when the underlying medium is heterogeneous. Based on the Wigner transform formalism, we show that random media indeed greatly improve refocusing. We analyze two different types of random media, where in the limit of high frequencies, the Wigner transform satisfies a random Liouville equation or a linear transport equation.RésuméDans des expériences de renversement temporel ; où un signal acoustique qui se propage dans un milieu sous-jacent est enregistré en temps, puis subit un renversement temporel et est réémis dans le milieu ; il a été observé que le signal refocalise d'autant mieux à l'emplacement de la source initiale que le milieu est hétérogène. Nous donnons une explication mathématique de ce phénomène fondée sur le formalisme de la transformée de Wigner. Nous obtenons deux exemples de refocalisation, en fonction du type de milieu aléatoire considéré, selon que la transformée de Wigner, à la limite des hautes fréquences, satisfait soit une équation de Liouville avec coefficients aléatoires, ou une équation de transport.
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics.
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ABSTRACT: We study transport for a scalar model of Love waves. These waves arise in the propagation of seismic waves whose energy is concentrated in the vicinity of the earth surface. We derive radiative transfer equations from first principles for the angularly resolved energy density of the Love waves in a simplified acoustic model. We consider a rough top surface with weak fluctuations at the scale of the wavelength. The transport equation accounts for the multiple scattering of the Love waves and their scattering into volume waves. We also analyze a diffusive regime when energy is universally distributed over various modes of the Love waves.
Wave Motion.
