  The ringdown of a perturbed black hole contains fundamental information about
space-time in the form of Quasi Normal Modes (QNM). Modifications to general
relativity, or extended profiles of other fields surrounding the black hole, so
called "black hole hair", can perturb the QNM frequencies. Previous works have
examined the QNM frequencies of spherically symmetric "dirty" black holes -
that is black holes surrounded by arbitrary matter fields. Such analyses were
restricted to static systems, making the assumption that the metric
perturbation was independent of time. However, in most physical cases such
black holes will actually be growing dynamically due to accretion of the
surrounding matter. Here we develop a perturbative analytic method that allows
us to compute for the first time the time dependent QNM deviations of such
growing dirty black holes. Whilst both are small, we show that the change in
QNM frequency due to the accretion can be of the same order or larger than the
change due to the static matter distribution itself, and therefore should not
be neglected in such calculations. We present the case of spherically symmetric
accretion of a complex scalar field as an illustrative example, but the method
has the potential to be extended to more complicated cases.
