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October  2005, 12(5): 793-815. doi: 10.3934/dcds.2005.12.793

Stability of time reversed waves in changing media

Guillaume Bal 1, and  Lenya Ryzhik 2,

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027, United States
2. 

Department of Mathematics, University of Chicago, Chicago IL, 60637, United States

Received  April 2004 Revised  September 2004 Published  February 2005

We analyze the refocusing properties of time reversed waves that propagate in two different media during the forward and backward stages of a time-reversal experiment. We consider two regimes of wave propagation modeled by the paraxial wave equation with a smooth random refraction coefficient and the Itô-Schrödinger equation, respectively. In both regimes, we rigorously characterize the refocused signal in the high frequency limit and show that it is statistically stable, that is, independent of the realizations of the two media. The analysis is based on a characterization of the high frequency limit of the Wigner transform of two fields propagating in different media.
The refocusing quality of the backpropagated signal is determined by the cross correlation of the two media. When the two media decorrelate, two distinct de-focusing effects are observed. The first one is a purely absorbing effect due to the loss of coherence at a fixed frequency. The second one is a phase modulation effect of the refocused signal at each frequency. This causes de-focusing of the backpropagated signal in the time domain.
Keywords: changing media., Wigner transforms, waves in random media, Time reversal, Itô-Schrödinger equation, paraxial equations.
Mathematics Subject Classification: 35L05, 60H25, 35Q4.
Citation: Guillaume Bal, Lenya Ryzhik. Stability of time reversed waves in changing media. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 793-815. doi: 10.3934/dcds.2005.12.793
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