Depending on the relative magnitudes of the wavelength λ, correlation length l_c and propagation length L, wave propagation in random media leads to very different physical behaviors.
In the low-frequency regime (λ ≈ L), classical ho- mogenization for rapidly fluctuating properties (l_c << λ) predicts a wave propagation regime with slowly-fluctuating homogenized coefficients, and anisotropic behavior.
In the higher frequency regime (λ << L), the wave field is not statistically stable and energy densities are more meaningful. The appropriate mathematical tool is then the Wigner measure of the displacement field, which is a wavenumber resolved energy density.
In the stochastic regime (λ ≈ l_c ), the Wigner measure verifies a radiative transfer equation. In the long-time limit, this regime is well approximated by diffusion equations. Finally, when the am- plitude of the fluctuations of the properties are large, localization can occur. In this regime, the energy is trapped close to the source and cannot propagate away.

This talk will briefly introduce these different regimes of wave propagation in random media and illustrate their appearance in two applications. In geophysics, we will discuss the influence of the anisotropy of the slowly-fluctuating background on the radiative transfer regime. In railway engineering, we will illustrate the effect of localization on the waves generated in the surroundings of a ballasted railway track.