Program
Program Garnier:
Our motivation to give this lecture is twofold.
On one hand the theory of waves propagating in randomly layered media
has been intensively studied during the last thirty years and the
results are scattered among numerous papers. It is now in a stable
state, in particular in the extremely interesting regime of separation
of scales as introduced by G. Papanicolaou and his co-authors.
On the other hand the time reversal experiments conducted
since the nineties, with ultrasonic waves by M. Fink and his group in
Paris, or in the context of ocean acoustics by W. Kuperman and his
collaborators, has attracted a lot of attention due to the surprising
effects of refocusing and enhanced resolution by multiple
scattering of the waves. These experiments, have opened the door
to numerous potential applications, in particular in the
domains of imaging and communications. A quantitative
mathematical analysis is crucial in the understanding of these
phenomena and for the development of their applications.
Wave propagation in random media is a vast field where a lot of work in various regimes have been done.
The first part of this lecture focuses on wave propagation in
one-dimensional random media. We will review the asymptotic theory for
random differential equations, including homogenization theory and
diffusion approximation. We will apply this theory to study the
transmission of waves
through a random medium. We shall also give an analysis of wave
propagation in a random waveguide, where transverse spatial effects are
important. Finally we will address wave propagation in the parabolic
regime, in which the semi-classical analysis of the Schr{\"o}dinger
equation with a random potential plays an important role.
Monday Afternoon (14h-16h15):
Lecture 1. Wave propagation in one-dimensional random media: effective medium theory.
Homogenization in random media.
Tuesday Morning (10h-12h15):
Lecture 2. Wave propagation in one-dimensional random media: the coherent wave front, the incoherent wave fluctuations, time reversal.
Diffusion approximation, asymptotic theory for random differential
equations.
Tuesday Afternoon (14h30-16h45):
Lecture 3. Wave propagation and time reversal in a random waveguide.
Wednesday Morning (10h-12h15):
Lecture 4. Wave propagation and time reversal in the parabolic regime.
Semi-classical analysis of the Schrödinger equation with a random
potential.
Abstracts
(.ps and .pdf)