July  2006, 6(4): 783-802. doi: 10.3934/dcdsb.2006.6.783

Time reversal of parabolic waves and two-frequency Wigner distribution


Department of Mathematics, University of California at Davis, Davis, CA 95616, United States


Department of Mathematics, University of California at Irvine, Irvine, CA 92697, United States

Received  February 2005 Revised  November 2005 Published  April 2006

We consider propagation and time reversal of wave pulses in a random environment. The focus of our analysis is the development of an expression for the two frequency mutual coherence function for the harmonic wave field. This quantity plays a crucial role in the analysis of many wave propagation phenomena and we illustrate by explicitly considering time reversal in the context of time pulses with a high carrier frequency. In a time-reversal experiment the wave received by an active transducer or antenna (receiver-emitter) array, is recorded in a finite time window and then re-emitted into the medium time reversed, that is, the tails of the recorded signals are sent first. The re-emitted wave pulse will focus approximately on the original source location. We use explicit expressions for the mutual coherence functions and their asymptotic approximations in the regime of long or short propagation distance and a high carrier frequency to analyze the refocusing of the wave pulse in the time reversal experiment. A novel aspect of our analysis is that we are able to characterize precisely the decoherence length in temporal frequency. This allows us to analyze for instance the time reversal experiment when the mirror has a finite aperture in time.
Citation: 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

Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147


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


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


Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049


Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367


Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems and Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63


Rodrigo I. Brevis, Jaime H. Ortega, David Pardo. A source time reversal method for seismicity induced by mining. Inverse Problems and Imaging, 2017, 11 (1) : 25-45. doi: 10.3934/ipi.2017002


Jingyu Wang, Yejuan Wang, Tomás Caraballo. Multi-valued random dynamics of stochastic wave equations with infinite delays. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021310


Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617


Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic and Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529


Kazufumi Ito, Karim Ramdani, Marius Tucsnak. A time reversal based algorithm for solving initial data inverse problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641


James Nolen. A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167


D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843


Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1


Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111


Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations and Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024


Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033


Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1


Yaiza Canzani, Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of random waves. Electronic Research Announcements, 2015, 22: 76-86. doi: 10.3934/era.2015.22.76


Corinna Burkard, Aurelia Minut, Karim Ramdani. Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities. Inverse Problems and Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445

2020 Impact Factor: 1.327


  • PDF downloads (55)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]