  The explicit form of perturbation equation for the $\Psi_4$ Weyl scalar,
containing the matter source terms, is derived for general type D spacetimes.
It is described in detail the particular case of the Schwarzschild spacetime
using in-going penetrating coordinates. As a practical application, we focused
on the emission of gravitational waves when a black hole is perturbed by a
surrounding dust-like fluid matter. The symmetries of the spacetime and the
simplicity of the matter source allow, by means of a spherical harmonic
decomposition, to study the problem by means of a one dimensional numerical
code.
