  The existence of the classical black hole solutions of the
Einstein-Yang-Mills-Higgs equations with non-abelian Yang-Mills-Higgs hair,
which we will also refer to as "magnetic monopole black hole solutions",
implies that not all classical stationary magnetically charged black holes can
be uniquely described by their asymptotic characteristics. In fact, in a
certain domain of parameters, there exist different spherically-symmetric,
non-rotating and asymptotically-flat classical black hole solutions of the EYMH
equations which have the same ADM mass and the same magnetic charge but
significantly different geometries in the near-horizon regions. (These are
black hole solutions which are described by a Reissner-Nordstr\"om metric on
the one hand and the hairy magnetic monopole black hole solutions which are
described by a metric which is not of Reissner-Nordstr\"om form on the other
hand.) One can experimentally distinguish such black holes with same asymptotic
characteristics but different near-horizon geometries classically by probing
the near-horizon regions of the black holes. We argue that one way to probe the
near-horizon region of a black hole which allows to distinguish magnetically
charged black holes with same asymptotic characteristics but different
near-horizon geometries is by classical scattering of waves. Using the example
of a minimally-coupled massless probe scalar field scattered by magnetically
charged black holes which can be obtained as solutions of the EYMH equations
with a Higgs triplett and gauge group SU(2) in the limit of an infinite Higgs
self-coupling constant we show how, in this case, the scattering cross sections
differ for the magnetically charged black holes with different near-horizon
geometries but same asymptotic characteristics. We find in particular that the
characteristic glory peaks in the cross sections are located at different
scattering angles.
