  We study the problem of plane monochromatic scalar waves impinging upon a
Schwarzschild dirty black hole -- a Schwarzschild black hole surrounded by a
thin spherical shell of matter -- using the complex angular momentum approach.
We first recall general results concerning the differential scattering cross
section in the classical limit through a null geodesic analysis by exploring
different configurations of the shell. In particular, we show that dirty black
hole spacetimes may exhibit various critical effects for geometrical optics. We
compute the Regge pole spectrum for various shell configurations and show that
it exhibits two or three distinct branches of poles, labelled inner surface
waves, broad resonances and outer surface waves. In the latter, two
sub-families have been identified, the surface waves associated with the outer
light-ring and the creeping modes associated with the surface of the shell. We
show, using WKB analysis, that the position of the shell sets the real part of
the broad resonances while its energy-momentum and the discontinuity of the
potential at the shell's surface set their imaginary part. Next, we provide the
complex angular momentum representation of the differential scattering cross
section and examine the role of the different Regge pole branches. We compute
the differential scattering cross section for various configurations at several
frequencies and show a very good agreement with the partial-wave calculations.
Finally, we highlight the role of the critical effects, i.e., orbiting, glory,
grazing and rainbow scattering, and their impact on the differential scattering
cross section.
