Communications in Mathematical Sciences

Volume 5 (2007)

Number 1

Complete Time-Reversed Refocusing in Reflection with an Acoustic Lagrangian Model

Pages: 161 – 185



Daniel G. Alfaro Vigo

Adolfo G. S. Correia

André Nachbin


Discrete reflection-transmission acoustic models are introduced and analysed regarding their underlying physical properties. Namely, phenomena related to multiple scattering as encountered in the underlying continuous model. Moreover, the discrete models are designed so that computational experiments can be performed effciently. Each discrete model is expressed through its corresponding reflection-transmission matrix establishing a connection with lattice models encountered in the Physics literature. Their connections with the continuous acoustic model are discussed in detail. In particular we show how a Goupillaud medium can arise from a stable discretization of a more general random medium. Related physical phenomena are studied computationally. In particular reflection-transmission properties of waves in a rapidly varying random medim, which are valid over long propagation distances. By using a long computational domain, in a regime where we have Anderson localization, the energy of an incoherently scattered signal is entirely reflected back and, by time-reversal, completely recompressed into the smooth initial data. Experiments are also performed in a regime where separation of scales fails to hold. Another new result is the time-reversed refocusing, in reflection, of a wave train in the form of a bit stream. The bit stream is scrambled by reflection and unscrambled through time reversal. The robustness of the time-reversed refocusing phenomenon is outstanding.


Inhomogeneous media; asymptotic theory; time reversal

2010 Mathematics Subject Classification

35L05, 60H15, 74J20

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