  The scattering phase, defined as $ \log \det S ( \lambda ) / 2\pi i $ where $
S ( \lambda ) $ is the (unitary) scattering matrix, is the analogue of the
counting function for eigenvalues when dealing with exterior domains and is
closely related to Krein's spectral shift function. We revisit classical
results on asymptotics of the scattering phase and point out that it is never
monotone in the case of strong trapping of waves. Perhaps more importantly, we
provide the first numerical calculations of scattering phases for non-radial
scatterers. They show that the asymptotic Weyl law is accurate even at low
frequencies and reveal effects of trapping such as lack of monotonicity. This
is achieved by using the recent high level multiphysics finite element software
FreeFEM.
