  We consider the scalar anisotropic wave equation. Recently a convergence
analysis for radial perfectly matched layers (PML) in the frequency domain was
reported and in the present article we continue this approach into the time
domain. First we explain why there is a good hope that radial complex scalings
can overcome the instabilities of PML methods caused by anisotropic materials.
Next we discuss some sensitive details, which seem like a paradox at the first
glance: if the absorbing layer and the inhomogeneities are sufficiently
separated, then the solution is indeed stable. However, for more general data
the problem becomes unstable. In numerical computations we observe
instabilities regardless of the position of the inhomogeneities, although the
instabilities arise only for fine enough discretizations. As a remedy we
propose a complex frequency shifted scaling and discretizations by Hardy space
infinite elements or truncation-free PMLs. We show numerical experiments which
confirm the stability and convergence of these methods.
