  We offer a mathematical toolkit for the study of waves propagating on a
background manifold with nonvanishing torsion. Examples include electromagnetic
and gravitational waves on a spacetime with torsion. The toolkit comprises
generalized versions of the Lichnerowicz-de Rham and the Beltrami wave
operators, and the Weitzenb\"ock identity relating them on Riemann-Cartan
geometries. The construction applies to any field belonging to a matrix
representation of a Lie (super) algebra containing an $\mathfrak{so}$
subalgebra. Using these tools, we analyze the propagation of different massless
waves in the eikonal (geometric optics) limit in a model-independent way and
find that they all must propagate at the speed of light along null torsionless
geodesics, in full agreement with the multimessenger observation
GW170817/GRB170817A. We also discuss how gravitational waves could be used as a
probe to test for torsion.
