We derive analytically, and validate numerically, the dispersion renormalization and attenuation of acoustic waves propagating through quenched disordered media in the long-wavelength limit. We consider weak spatial fluctuations in elastic moduli and/or mass density and compute the disorder-induced self-energies within the leading (Born) approximation. For sufficiently weak disorder, the results depend only on the variances of the fluctuations and are therefore insensitive to the detailed form of the underlying random distribution. For spatially uncorrelated elastic disorder we obtain Rayleigh-type attenuation, $\mathrm{\Gamma }\left(q\right)\propto q^{d+1}$, together with a reduction of the sound speed. In contrast, density disorder produces Rayleigh-type attenuation but does not renormalize the acoustic dispersion to leading order. Molecular dynamics simulations and normal-mode analyses of disordered one- and two-dimensional lattices quantitatively confirm the theoretical predictions.

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• Glasses
• Random & disordered media
• Rayleigh scattering