E6901 Handout
Handout for APMA E6901 - Fall 2005
WAVES IN RANDOM MEDIA
World Wide Web:
http://www.columbia.edu/~gb2030
Instructor: Guillaume Bal
Office: Mudd 206
Office Hours: by appointment
Office Phone: (212) 854 4731
e-mail:
gb2030@columbia.edu
Class meeting: Th 12:45-2:45 Mudd 214.
IMPORTANT CHANGE IN SCHEDULE:
The first class will be held on Th. September 15th.
Course Structure:
The objective of this course is to derive macroscopic models for the
energy density of high frequency waves propagating in heterogeneous
media. Low frequency waves in heterogeneous (i.e. highly oscillatory)
media are treated with homogenization techniques. High frequency waves
in slowly varying media (low frequency media) are treated by
geometrical optics. High frequency waves in high frequency media are a
lot harder to model because of the multiple interactions of the waves
with the underlying structure. In the so-called weak coupling regime,
we'll see how to derive radiative transfer equations for the
phase-space energy density of the waves. One of the main tools in such
a derivation will be the Wigner transform, which will be
analyzed in detail.
The course will be fairly mathematical as it will involve classical
tools in analysis and probability theory. However I plan to insist on
the physical processes involved (and the various wave interactions
with the underlying medium at different scales), which should appeal
to a broader physical and engineering audience.
Syllabus:
Here is a (very) tentative Syllabus:
- Lecture 1. Wave solutions
Existence and Uniqueness theory. Dispersion relation. Fundamental
solutions in homogeneous media.
- Lecture 2. Low frequency waves
Homogenisation in periodic and random media. Scattering off
small fluctuations.
- Lecture 3. High frequency regime and geometrical optics (WKB)
- Lecture 4. Amplitude and phase fluctuations in random media
- Lecture 5. Diffusion approximation and application
for amplitude and phase fluctuations
- Lectures 6&7. Wigner transforms
Properties of the Wigner transform. Necessary pseudo-differential
calculus (slowly varying case). High frequency limit with slowly
varying potential. Careful derivation for Schroedinger and acoustics
equations.
- Lectures 8&9. Radiative Transfer equations: weak coupling
regime
Formal expansions (including changing media). Start with Schroedinger
equation and then extend to wave equation. Introduce theory of
spatio-temporal Wigner transforms. Diffusion approximation (time
permitting).
- Lecture 10. Paraxial approximation and regularization in time
Formal derivation of paraxial approximation, which is a useful
approximation to the full wave equation. Introduction of Markov
potentials and regularizations in time. Mathematically rigorous
derivation of radiative transfer equations and generalizations to
changing media.
- Lecture 11. Statistical stability
Formal derivation of Ito-Schroedinger equation. Transport equations
and statistical stability. Comparison with case of paraxial
approximation and case of random Liouville.
- Lecture 12. Time Reversal
- Lecture 13. Numerical simulations.
Numerical simulations in random media. Modification of the kinetic
equations to account for discretization (dispersive)
effects. Numerical comparison of wave simulations with transport and
diffusion simulations.
Textbooks:
I will try to provide a set of lecture
notes. Here are useful bibliographical references, which I encourage
you to read as soon as you can. All the papers I have contributed to
are on my webpage http://www.columbia.edu/~gb2030/pubs.html.
- Lecture notes by J. Rauch on geometric optics: http://www.math.lsa.umich.edu/~rauch/oldnlgonotes.pdf
(I will use these very well written lecture notes in the first
lectures).
- L.Ryzhik, G.C.Papanicolaou, and J.B.Keller,
Transport equations for elastic and other waves
in random media, Wave Motion, 24, pp. 327-370, 1996
- G.Bal, Kinetics of scalar wave fields in random media, to appear
in Wave Motion, 2005,
http://www.columbia.edu/~gb2030/PAPERS/ScalarTtrans.pdf
- G.Bal and L.Ryzhik,
Time Reversal and Refocusing in Random Media,
SIAM J. Appl. Math., 63(5), pp. 1475-1498, 2003
- G.Bal and R.Verastegui, Time Reversal in Changing Environment,
SIAM Multiscale Model. Simul. , 2(4), pp. 639-661, 2004
- G.Bal, G. Papanicolaou and L. Ryzhik, Self-averaging in time
reversal for the parabolic wave equation, Stochastics and
Dynamics , 2(4), pp. 507-532, 2002
Grading:
- There will be several take-home exams covering the material shown
in class. Interested students may also work on research-level
projects.