  In this work we have derived the evolution equation for gravitational
perturbation in four dimensional spacetime in presence of a spatial extra
dimension. The evolution equation is derived by perturbing the \emph{effective}
gravitational field equations on the four dimensional spacetime, which inherits
non-trivial higher dimensional effects. Note that this is \emph{different} from
the perturbation of the five dimensional gravitational field equations,
existing in literature, and possess quantitatively new features. The
gravitational perturbation has further been decomposed into a purely four
dimensional part and another piece that depends on extra dimensions. The four
dimensional gravitational perturbation now admits massive propagating degrees
of freedom, owing to the existence of higher dimensions. We have also studied
the influence of these massive propagating modes on the quasi-normal mode
frequencies, signaling the higher dimensional nature of the spacetime, and have
contrasted these massive modes with the massless modes in general relativity.
Surprisingly, it turns out that the massive modes experience much smaller
damping compared to the massless modes in general relativity and may even
\emph{dominate} over and above the general relativity contribution if one
observes the ringdown phase of a black hole merger event at sufficiently late
times. Furthermore, the whole analytical framework has been supplemented by the
fully numerical Cauchy evolution problem as well. In this context we have shown
that except for minute details the overall features of the gravitational
perturbations are captured in both the Cauchy evolution as well as in the
analysis of quasi-normal modes. The implications on observations of black holes
with LIGO and proposed space missions like LISA are also discussed.
