WAVES IN RANDOM MEDIA

- 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.

- 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

- There will be several take-home exams covering the material shown in class. Interested students may also work on research-level projects.