Fractional diffusive waves

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave
equation. This equation is obtained from the classical D’Alembert wave equation by replacing the
second-order time derivative with a fractional derivative of order β ∈ (0, 2) and is expected to
govern evolution processes intermediate between diffusion and wave propagation when β ∈ (1, 2).
Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting
a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to
be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are
expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors
turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid
and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some
stable probability distributions are outlined.