The wave equation with a discontinuous coefficient depending on time only: generalized solutions and propagation of singularities

TL;DR

This paper studies how singularities propagate in a one-dimensional wave equation with a time-dependent discontinuous coefficient, using Colombeau generalized functions to handle non-smooth coefficients and initial data.

Contribution

It demonstrates existence and uniqueness of Colombeau solutions for the wave equation with discontinuous coefficients and analyzes singularity propagation and their relation to classical distributional solutions.

Findings

Singularities propagate along characteristic lines in the discontinuous coefficient case.

Colombeau solutions are associated with classical distributional solutions for initial data as distributions.

The interplay between singularity strength and propagation effects is characterized.

Abstract

This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth oefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the one-dimensional wave equation with a discontinuous coefficient depending on time. After demonstrating the existence and uniqueness of generalized solutions in the sense of Colombeau to the problem, we investigate the phenomenon of propagation of singularities, arising from delta function initial data, for the case of a piecewise constant coefficient. We also provide an analysis of the interplay between singularity strength and propagation effects. Finally, we show that in case the initial data are distributions, the Colombeau solution to the model problem is associated with the piecewise distributional solution of the corresponding transmission problem.