  The spectrum of collective excitations in Weyl materials is studied by using
consistent hydrodynamics. The corresponding framework includes the vortical and
chiral anomaly effects, as well as the dependence on the separations between
the Weyl nodes in energy $b_0$ and momentum $\mathbf{b}$. The latter are
introduced via the Chern-Simons contributions to the electric current and
charge densities in Maxwell's equations. It is found that, even in the absence
of a background magnetic field, certain collective excitations (e.g. the
helicon-like modes and the anomalous Hall waves) are strongly affected by the
chiral shift $\mathbf{b}$. In a background magnetic field, the existence of the
distinctive longitudinal and transverse anomalous Hall waves with a linear
dispersion relation is predicted. They originate from the oscillations of the
electric charge density and electromagnetic fields, in which different
components of the fields are connected via the anomalous Hall effect in Weyl
semimetals.
