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Unformatted text preview: Lecture Notes. Waves in Random Media Guillaume Bal 1 January 9, 2006 1 Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027; gb2030@columbia.edu Contents 1 Wave equations and First-order hyperbolic systems 4 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Elastic and Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Schr odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 First-order symmetric hyperbolic systems . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Case of constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Plane Wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Case of spatially varying coefficients . . . . . . . . . . . . . . . . . . . . 9 1.3.4 Finite speed of propagation . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.5 Characteristic equation and dispersion relation . . . . . . . . . . . . . . 13 2 Homogenization Theory for the wave equation 15 2.1 Effective medium theory in periodic media . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Multiple scale expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.2 Homogenized equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Multidimensional case and estimates of the effective propagation speed . . . . . 21 2.2.1 Effective density tensor in the case of small volume inclusions . . . . . . 21 2.2.2 Effective density tensor in the case of small contrast . . . . . . . . . . . 24 2.3 Case of random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Variational formulation and effective parameter estimates . . . . . . . . . . . . 28 2.4.1 Classical variational formulation . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Hashin-Shtrikman bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Homogenization with boundaries and interfaces . . . . . . . . . . . . . . . . . . 31 2.5.1 The time dependent case . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.2 Plane wave reflection and transmission . . . . . . . . . . . . . . . . . . . 34 3 Geometric Optics 36 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Second-order scalar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1 High Frequency Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.2 Geometric Optics Expansion . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 The Eikonal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 First-order hyperbolic systems...
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bal - Lecture Notes. Waves in Random Media Guillaume Bal 1...

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