Mathematics Colloquia and Seminars

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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.