Effective fractional acoustic wave equations in one-dimensional random multiscale media.
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.
Garnier
Josselin
J
Laboratoire de Probabilites et Modeles Aleatoires and Laboratoire Jacques-Louis Lions, Universite Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France. garnier@math.jussieu.fr
Solna
Knut
K