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Published online by Cambridge University Press: 26 June 2025
We give a mathematical analysis of the ‘'time-reversal mirror”, in what concerns phenomena described by the genuine acoustic equation with Dirichlet or impedance boundary conditions. An ideal situation is first considered, followed by the boundary-data, impedance and internal time-reversal methods. We explore the relationship between local decay of energy and accuracy of the method, and explain the positive effect of ergodicity.
1. Introduction: Principle of the Method
In all time reversal experiments, a finite time 0 < T < ∞ is chosen. At time t = 0 waves are emitted from a localized source, recorded in time (for o < t < T) by an array of receivers-transducers, time-reversed and retransmitted in the media during the time (T <t < 2T); for instance the first signal to arrive is reemitted last and the last to arrive is remitted first. In this second step (t > T) one can introduce amplification. The process is possibly repeated several times, leading in some cases to an automatic focusing on the most reflective target in a multiple target media. This has several applications in nondestructive testing, medical techniques such as lithotripsy and hyperthermia, underwater acoustics, etc. See [13]. The intuitive reasons why such a process may work are:
(1) The wave equation is invariant with respect to the symmetry t ∈ (0, T) ∈ 2T-t ∈ (T,2T).
(2) At high frequencies waves propagate as rays.
(3) Inhomogeneities, randomness and ergodicity contribute to much better refocusing.
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