  We derive a set of coupled equations for the gravitational and
electromagnetic perturbation in the Reissner-Nordstr\"om geometry using the
Newman Penrose formalism. We show that the information of the physical
gravitational signal is contained in the Weyl scalar function $\Psi_4$, as is
well known, but for the electromagnetic signal the information is encoded in
the function $\chi$ which relates the perturbations of the radiative Maxwell
scalars $\varphi_2$ and the Weyl scalar $\Psi_3$. In deriving the perturbation
equations we do not impose any gauge condition and our analysis contains as a
limiting case the results obtained previously for instance in Chandrashekhar's
book. In our analysis, we also include the sources for the perturbations and
focus on a dust-like charged fluid distribution falling radially into the black
hole. Finally, by writing the functions on a basis of spin weighted spherical
harmonics and the Reissner-Nordstr\"om spacetime in Kerr-Schild type
coordinates a hyperbolic system of coupled partial differential equations is
presented and numerically solved. In this way, we solve completely a system
which generates a gravitational signal as well as an
electromagnetic/gravitational one, which sets the basis to find correlations
between them and thus facilitating the gravitational wave detection via the
electromagnetic signal.
