American Institute of Mathematical Sciences

Advanced
September  2007, 8(2): 473-492. doi: 10.3934/dcdsb.2007.8.473

Homogenization in random media and effective medium theory for high frequency waves

Guillaume Bal 1,

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027

Received  October 2006 Revised  March 2007 Published  June 2007

We consider the homogenization of the wave equation with high frequency initial conditions propagating in a medium with highly oscillatory random coefficients. By appropriate mixing assumptions on the random medium, we obtain an error estimate between the exact wave solution and the homogenized wave solution in the energy norm. This allows us to consider the limiting behavior of the energy density of high frequency waves propagating in highly heterogeneous media when the wavelength is much larger than the correlation length in the medium.
Keywords: random media, high frequency waves., Homogenization.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[1]

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

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[3]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[4]

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
[5]

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

Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29
[7]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[8]

Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko, Serhiy Yanchuk. Frequency locking of modulated waves. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 847-875. doi: 10.3934/dcds.2011.31.847
[9]

Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[10]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020215
[11]

María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004
[12]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91
[13]

Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks & Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523
[14]

Yuanjia Ma. The optimization algorithm for blind processing of high frequency signal of capacitive sensor. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1399-1412. doi: 10.3934/dcdss.2019096
[15]

Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055
[16]

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

[17]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020329
[18]

Yves Capdeboscq, Haun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228
[19]

Guillaume Bal, Lenya Ryzhik. Stability of time reversed waves in changing media. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 793-815. doi: 10.3934/dcds.2005.12.793
[20]

Marc Briane, David Manceau. Duality results in the homogenization of two-dimensional high-contrast conductivities. Networks & Heterogeneous Media, 2008, 3 (3) : 509-522. doi: 10.3934/nhm.2008.3.509

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences