  The ringdown phase of gravitational waves emitted by a perturbed black hole
is described by a superposition of exponentially decaying sinusoidal modes,
called quasinormal modes (QNMs), whose frequencies depend only on the property
of the black hole geometry. The extraction of QNM frequencies of an isolated
black hole would allow for testing how well the black hole is described by
general relativity. However, astrophysical black holes are not isolated. It
remains unclear whether the extra matter surrounding the black holes such as
accretion disks would affect the validity of the black hole spectroscopy when
the gravitational effects of the disks are taken into account. In this paper,
we study the QNMs of a Schwarzschild black hole superposed with a gravitating
thin disk. Considering up to the first order of the mass ratio between the disk
and the black hole, we find that the existence of the disk would decrease the
oscillating frequency and the decay rate. In addition, within the parameter
space where the disk model can be regarded as physical, there seems to be a
universal relation that the QNM frequencies tend to obey. The relation, if it
holds generically, would assist in disentangling the QNM shifts caused by the
disk contributions from those induced by other putative effects beyond general
relativity. The QNMs in the eikonal limit, as well as their correspondence with
bound photon orbits in this model, are briefly discussed.
