Pulse waves in the viscoelastic Kelvin-Voigt model: a revisited approach

TL;DR

This paper presents a new integral form solution for the mechanical response of a viscoelastic medium modeled by Kelvin-Voigt to pulse excitations, improving computational efficiency and providing asymptotic formulas.

Contribution

It introduces a novel integral solution that avoids inverse Laplace transforms, simplifying calculations for pulse responses in Kelvin-Voigt models.

Findings

Derived simpler integral expressions for pulse responses

Provided asymptotic formulas for small and large times/distances

Enhanced computational efficiency over previous methods

Abstract

We calculate the mechanical response $r(x,t$) of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin-Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications -- particularly in seismology -- here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for $r(x,t$ as $x,t \to 0,\infty$ for both step- and delta-type pulses.