  Starting from a Langevin description of active particles that move with
constant speed in infinite two-dimensional space and its corresponding
Fokker-Planck equation, we develop a systematic method that allows us to obtain
the coarse-grained probability density of finding a particle at a given
location and at a given time to arbitrary short time regimes. By going beyond
the diffusive limit, we derive a novel generalization of the telegrapher's
equation. Such generalization preserves the hyperbolic structure of the
equation and incorporates memory effects on the diffusive term. While no
difference is observed for the mean square displacement computed from the
two-dimensional telegrapher's equation and from our generalization, the
kurtosis results into a sensible parameter that discriminates between both
approximations. We carried out a comparative analysis in Fourier space that
shed light on why the telegrapher's equation is not an appropriate model to
describe the propagation of particles with constant speed in dispersive media.
