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Volume 127, Issue 1, January 2010
- AEROACOUSTICS, ATMOSPHERIC SOUND 
127(2010); http://dx.doi.org/10.1121/1.3263608View Description Hide Description
This paper considers multiple scattering of waves propagating in a non-lossy one-dimensional random medium with short- or long-range correlations. Using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation, which corresponds to an effective medium with a frequency-dependent attenuation that obeys a power law with an exponent between 0 and 2. The exponent is related to the Hurst parameter of the medium, which is a characteristic parameter of the correlation properties of the fluctuations of the random medium. Moreover the frequency-dependent attenuation is associated with a special frequency-dependent phase, which ensures that causality and Kramers–Kronig relations are satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.