  We analytically determine the quasinormal mode (QNM) frequencies of a black
hole with quadrupole moment in the eikonal limit using the light-ring method.
The generalized black holes that are discussed in this work possess arbitrary
quadrupole and higher mass moments in addition to mass and angular momentum.
Static collapsed configurations with mass and quadrupole moment are treated in
detail and the QNM frequencies associated with two such configurations are
evaluated to linear order in the quadrupole moment. Furthermore, we touch upon
the treatment of rotating systems. In particular, the generalized black hole
that we consider for our extensive QNM calculations is a completely collapsed
configuration whose exterior gravitational field can be described by the
Hartle-Thorne spacetime [Astrophys. J. 153, 807-834 (1968)]. This collapsed
system as well as its QNMs is characterized by mass $M$, quadrupole moment $Q$
and angular momentum $J$, where the latter two parameters are treated to first
and second orders of approximation, respectively. When the quadrupole moment is
set equal to the relativistic quadrupole moment of the corresponding Kerr black
hole, $J^2/(Mc^2)$, the Hartle-Thorne QNMs reduce to those of the Kerr black
hole to second order in angular momentum $J$. Using ringdown frequencies, one
cannot observationally distinguish a generalized Hartle-Thorne black hole with
arbitrary quadrupole moment from a Kerr black hole provided the dimensionless
parameter given by $|QMc^2-J^2|c^2/(G^2M^4)$ is sufficiently small compared to
unity.
