  A class of linear hyperbolic partial differential equations, sometimes called
networks of waves, is considered. For this class of systems, necessary and
sufficient conditions are formulated on the system matrices for the operator
dynamics to be a Riesz-spectral operator. In that case, its spectrum is
computed explicitly, together with the corresponding eigenfunctions, which
constitutes the main result of our note. In particular, this enables to
characterize easily many different concepts, such as stability. We apply our
results to characterize exponential stability of a co-current heat exchanger.
