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

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