American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems, 2005, 12 (5) : 793-815. doi: 10.3934/dcds.2005.12.793
[1]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[2]

Rémi Carles, Clotilde Fermanian-Kammerer, Norbert J. Mauser, Hans Peter Stimming. On the time evolution of Wigner measures for Schrödinger equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 559-585. doi: 10.3934/cpaa.2009.8.559
[3]

Albert Fannjiang, Knut Solna. Time reversal of parabolic waves and two-frequency Wigner distribution. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 783-802. doi: 10.3934/dcdsb.2006.6.783
[4]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[5]

Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic and Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[6]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[7]

Wolfgang Wagner. A random cloud model for the Wigner equation. Kinetic and Related Models, 2016, 9 (1) : 217-235. doi: 10.3934/krm.2016.9.217
[8]

Thierry Colin, Pierre Fabrie. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 671-690. doi: 10.3934/dcds.1998.4.671
[9]

Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202
[10]

Wenxian Shen, Zhongwei Shen. Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1193-1213. doi: 10.3934/cpaa.2016.15.1193
[11]

Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations and Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042
[12]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499
[13]

Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669
[14]

Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058
[15]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[16]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[17]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[18]

Wolfgang Wagner. A random cloud model for the Schrödinger equation. Kinetic and Related Models, 2014, 7 (2) : 361-379. doi: 10.3934/krm.2014.7.361
[19]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic and Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[20]

Josselin Garnier. Ghost imaging in the random paraxial regime. Inverse Problems and Imaging, 2016, 10 (2) : 409-432. doi: 10.3934/ipi.2016006

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy