Transient waves in linear dispersive media with dissipation: an approach based on the steepest descent path

TL;DR

This paper introduces a novel method using the steepest descent path to compute impulse responses in linear dispersive media with dissipation, offering a potentially simpler alternative to traditional Bromwich integral inversion.

Contribution

It proposes a new approach based on steepest descent paths for Laplace transform inversion, demonstrated on the Klein-Gordon equation with dissipation, improving computational efficiency.

Findings

The method accurately reproduces analytical solutions.

It simplifies the inversion process for dispersive wave problems.

Potential applicability to various fields involving Laplace transforms.

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, as to compare the numerical solutions obtained by the newly proposed method to exact ones. Since the newly proposed method, at its core, consists in 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 presence of dissipation, as well as in other fields in which Laplace transform inversion come into play.