  Exciton-polaritons in an optical microcavity can form a macroscopically
coherent state despite being an inherently driven-dissipative system. In
comparison with equilibrium bosonic fluids, polaritonic condensates possess
multiple peculiarities that make them behave differently from well-known
textbook examples. One such peculiarity is the presence of dark excitons which
are created by the pump together with optically-active particles. They can
considerably affect the spectrum of elementary excitations of the condensate
and hence change its superfluid properties. Here, we theoretically analyze the
influence of the bright and dark ``reservoir'' populations on the sound
velocity $c_s$ of incoherently-driven polaritons. Both pulsed and
continuous-wave pumping schemes characterized by essentially different
condensate-to-reservoir ratio are considered. We show that the dark exciton
contribution leads to considerable lowering of $c_s$ and to its deviation from
the square-root-like behavior on the system's chemical potential (measurable
condensate blueshift). Importantly, our model allows to unambiguously define
the density of dark excitons in the system by experimentally tracking $c_s$
against the condensate blueshift and fitting the dependence at a given
temperature.
