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Waves Seminar Series

Wednesday, October 18,  2006, 3:00 pm
Cullimore 611
New Jersey Institute of Technology

Homogenization of wave propagation with the use of nonlinear phase-space densities:

The Wigner Transform method and generalizations

Agissilaos Athanassoulis

Program in Applied and Computational Mathematics

Princeton University

Abstract

In the study of wave propagation over distances much longer than the typical wavelength phase-space methods are often used, i.e. spectral densities are used as a homogenized representation of the wavefield, and kinetic equations are constructed for their evolution.

The Wigner Transform (WT) is a nonlinear, nonparametric spectral density, first introduced by E. Wigner in 1932 in the context of quantum mechanics. In the last ten years it has been extensively used in the formulation of phase-space models for a variety of problems, including geometrical optics limits, periodic problems, nonlinear and/or random waves, and more.

In this talk we present the main features of the Wigner Transform and its use in the treatment of IVPs for linear systems of (possibly pseudo-) differential equations in the geometrical optics limit. The specific examples we work with here are the linear Schrödinger equation, and the acoustic wave equations. This approach, in some sense equivalent to more traditional geometrical optics methods, gives solutions for all times, i.e. it doesn't suffer from caustics. Due to the properties of the WT, energy, energy flux and other quantities of interest can be jointly resolved over space and wavenumber in a precise way. However, due to the interference effects of the transform, this approach is often found to be best used in the limit of zero wavelength, and not for a small finite wavelength.

We discuss how this approach can be extended to finite wavelength problems with the use of smoothed WTs, and to forced problems (in particular sources) with the use of space-time transforms. The exact equations for the phase-space densities are in general pseudodifferential, so the use of the pseudodifferential operators is essential.