  We study the propagation of exact gravitational waves in the ghost-free
bimetric theory. Our focus is on type N spacetimes compatible with the
cosmological constants provided by the bigravity interaction potential, and
particularly in the single class known by allowing at least a Killing symmetry:
the AdS waves. They have the advantage of being represented by a generalized
Kerr-Schild transformation from AdS spacetime. This means a notorious
simplification in bigravity by allowing to straightforwardly compute any power
of its interaction square root matrix, opening the door to explore physically
meaningful exact configurations. For these exact gravitational waves the
complex dynamical structure of bigravity decomposes into elementary exact
massless or massive excitations propagating on AdS. We use a complexified
formulation of the Euler-Darboux equations to provide for the first time the
general solutions to the massive version of the Siklos equation which rules the
resulting AdS-waves dynamics, using an integral representation originally due
to Poisson. Inspired in this progress we tackle the subtle problem on how
matter couples to bigravity and concretely if this occurs through a composite
metric, which is hard to handle in a general setting. Surprisingly, the
Kerr-Schild ansatz brings again a huge simplification in how the related
energy-momentum tensors are calculated. This allows us to explicitly
characterize AdS waves supported in one case by a massless free scalar field
and by a wavefront-homogeneous Maxwell field in another. Considering the most
general allowed Maxwell source instead is a highly nontrivial task, that we
accomplish by exploiting again the complexified Euler-Darboux description and
taking advantage of the classical Riemann method. In fact, this allow us to
find at the end the most general configurations for any matter source.
