Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli

TL;DR

This paper characterizes wave dispersion and attenuation in linear viscoelastic media with completely monotonic relaxation moduli using spectral measures, deriving explicit formulas and relations like Kramers-Kronig.

Contribution

It provides a comprehensive spectral measure framework for wave behavior in viscoelastic media with explicit integral expressions and dispersion-attenuation relations.

Findings

Wave dispersion and attenuation are fully characterized by a spectral measure.

Explicit integral formulas are derived for specific relaxation models.

The Kramers-Kronig relation is established within this framework.

Abstract

It is shown that viscoelastic wave dispersion and attenuation in a viscoelastic medium with a completely monotonic relaxation modulus is completely characterized by the phase speed and the dispersion-attenuation spectral measure. The dispersion and attenuation functions are expressed in terms of a single dispersion-attenuation spectral measure. An alternative expression of the mutual dependence of the dispersion and attenuation functions, known as the Kramers-Kronig dispersion relation, is also derived from the theory. The minimum phase aspect of the filters involved in the Green's function is another consequence of the theory. Explicit integral expressions for the attenuation and dispersion functions are obtained for a few analytical relaxation models.