  Scientists have observed and studied diffusive waves in contexts as disparate
as population genetics and cell signaling. Often, these waves are propagated by
discrete entities or agents, such as individual cells in the case of cell
signaling. For a broad class of diffusive waves, we characterize the transition
between the collective propagation of diffusive waves -- in which the wave
speed is well-described by continuum theory -- and the propagation of diffusive
waves by individual agents. We show that this transition depends heavily on the
dimensionality of the system in which the wave propagates and that disordered
systems yield dynamics largely consistent with lattice systems. In some system
dimensionalities, the intuition that closely packed sources more accurately
mimic a continuum can be grossly violated.
