In this paper we consider an acoustic problem of wave propagation through a
discontinuous medium. The problem is reduced to the dissipative wave equation
with distributional dissipation. We show that this problem has a so-called very
weak solution, we analyse its properties and illustrate the theoretical results
through some numerical simulations by approximating the solutions to the full
dissipative model for a particular synthetic piecewise continuous medium. In
particular, we discover numerically a very interesting phenomenon of the
appearance of a new wave at the singular point. For the acoustic problem this
can be interpreted as an echo effect at the discontinuity interface of the
medium.