  In this paper we study the well-posedness of the Cauchy problem for a wave
equation with multiplicities and space-dependent irregular coefficients. As in
\cite{GR:14} in order to give a meaningful notion of solution, we employ the
notion of very weak solution, which construction is based on a parameter
dependent regularisation of the coefficients via mollifiers. We prove that,
even with distributional coefficients, a very weak solution exists for our
Cauchy problem and it converges to the classical one when the coefficients are
smooth. The dependence on the mollifiers of very weak solutions is investigated
at the end of the paper in some instructive examples.
