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Refocusing for Classical Waves in Complicated Media
Optimization| Speaker: | Prof. Leonid Ryzhik, University of Chicago |
| Location: | 593 Kerr |
| Start time: | Fri, May 17 2002, 2:10PM |
Description
We will discuss the following remarkable property of solutions $u(t,x)$ of
wave equations in random media (or ergodic billiards) with localized
initial data. Let us record $u(x,T)$ at some time $T$, restrict it to a
finite (possibly small) domain, transform it linearly in some fashion and
use the resulting signal as a new initial data for the wave equation. It
turns out that the new re-propagated solution will concentrate at the
original source location at the same time $T$ for a very large class of
signal processing. In particular this explains the time-reversal
experiments. In such experiments a signal is emitted by a localized
source, propagated through a medium and recorded on a small array of
receivers-transducers. The signal is re-emitted into the medium reversed
in time, that is, the part of the signal recorded first is re-emitted last
and vice versa. The re-propagated signal approximately refocuses back on
the original source. It is also observed that refocusing is significantly
better in a random medium. We will give an explanation for refocusing and
explain why random media are good for refocusing, as are ergodic
billiards.