  Superfluidity describes the ability of quantum matter to flow without
friction. Due to its fundamental role in many transport phenomena, it is
crucial to understand the robustness of superfluid properties to external
perturbations. Here, we theoretically study the effects of speckle disorder on
the propagation of sound waves in a two-dimensional Bose-Einstein condensate at
zero temperature. We numerically solve the Gross-Pitaevskii equation in the
presence of disorder and employ a superfluid hydrodynamic approach to elucidate
the role of the compressibility and superfluid fraction on the propagation of
sound. A key result is that disorder reduces the superfluid fraction and hence
the speed of sound; it also introduces damping and mode coupling. In the limit
of weak disorder, the predictions for the speed of sound and its damping rate
are well reproduced by a quadratic perturbation theory. The hydrodynamic
description is valid over a wide range of parameters, while discrepancies
become evident if the disorder becomes too strong, the effect being more
significant for disorder applied in only one spatial direction. Our predictions
are well within the reach of state-of-the-art cold-atom experiments and carry
over to more general disorder potentials.
