Attenuation, dispersion and finite propagation speed in viscoelastic media

TL;DR

This paper investigates how dispersion and attenuation behave in viscoelastic media, establishing limits on their growth rates to ensure finite wave propagation speed and compatibility with positive relaxation spectra.

Contribution

It derives a local dispersion relation and identifies the exact boundary between attenuation growth rates that allow finite propagation speed and those that do not.

Findings

Attenuation and dispersion functions grow sublinearly at high frequencies.

Superlinear attenuation growth is incompatible with finite propagation speed.

A parametric dispersion relation is provided.

Abstract

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum have a sublinear growth rate at very high frequencies. A local dispersion relation in parametric form is found. The exact limit between attenuation growth rates compatible and incompatible with finite propagation speed is found. Incompatibility of superlinear frequency dependence of attenuation with finite speed of propagation and with the assumption of positive relaxation spectrum is demonstrated.