We study two-dimensional wave propagation in materials whose properties vary periodically in one direction only. High-order homogenization is carried out to derive a dispersive effective medium approximation. One-dimensional materials with constant impedance exhibit no effective dispersion. We show that a new kind of effective dispersion may arise in two dimensions, even in materials with constant impedance. This dispersion is a macroscopic effect of microscopic diffraction caused by spatial variation in the sound speed. We analyze this dispersive effect by using high-order homogenization to derive an anisotropic, dispersive effective medium. We generalize to two dimensions a homogenization approach that has been used previously for one-dimensional problems. Pseudospectral solutions of the effective medium equations agree to high accuracy with finite volume direct numerical simulations of the variable-coefficient equations.


  1. wave propogation
  2. periodic media
  3. effective dispersion
  4. diffraction
  5. homogenization

MSC codes

  1. 35B27
  2. 35L45
  3. 35P25

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1852 - 1869
ISSN (online): 1095-712X


Submitted: 23 September 2013
Accepted: 21 August 2014
Published online: 3 December 2014


  1. wave propogation
  2. periodic media
  3. effective dispersion
  4. diffraction
  5. homogenization

MSC codes

  1. 35B27
  2. 35L45
  3. 35P25



Manuel Quezada de Luna

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