Hydrodynamic interactions are transmitted by viscous diffusion and sound propagation, and the temporal evolution of hydrodynamic interactions by both mechanisms is studied using direct numerical simulation in this paper. The hydrodynamic interactions for a system of two particles in a fluid are estimated using the velocity correlation of the particles. In an incompressible fluid, hydrodynamic interactions propagate instantaneously at the infinite speed of sound followed by a temporal evolution due to viscous diffusion. Conversely, sound propagates in a compressible fluid at a finite speed, which affects the temporal evolution of the hydrodynamic interactions through an order-of-magnitude relationship between the time scales of viscous diffusion and sound propagation. The hydrodynamic interactions are characterized by introducing the ratio of these time scales as an interactive compressibility factor. (C) 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4802038]