Transient Waves in Linear Dispersive Media with Dissipation: An Approach Based on the Steepest Descent Path

Francesco Mainardi, Andrea Mentrelli () and Juan Luis González-Santander ()
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Francesco Mainardi: Department of Physics and Astronomy, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy
Andrea Mentrelli: Department of Mathematics, and AM2, University of Bologna, Via Saragozza 8, 40123 Bologna, Italy
Juan Luis González-Santander: Department of Mathematics, University of Oviedo, C/Leopoldo Calvo Sotelo 18, 33007 Oviedo, Spain

Mathematics, 2025, vol. 13, issue 21, 1-13

Abstract: In the study of linear dispersive media, it is of primary interest to gain knowledge of the impulse response of the material. The standard approach to compute the response involves a Laplace transform inversion, i.e., the solution of a Bromwich integral, which can be a notoriously troublesome problem. In this paper we propose a novel approach to the calculation of the impulse response, based on the well-assessed method of the steepest descent path, which results in the replacement of the Bromwich integral with a real line integral along the steepest descent path. In this exploratory investigation, the method is explained and applied to the case study of the Klein–Gordon equation with dissipation, for which analytical solutions of the Bromwich integral are available, so as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists of replacing a Laplace transform inverse with a potentially much less demanding real line integral, the method presented here could be of general interest in the study of linear dispersive waves in the presence of dissipation, as well as in other fields in which Laplace transform inversion comes into play.

Keywords: transient waves in linear viscoelasticity; Klein-Gordon equation with dissipation; steepest descent method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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