In this talk we'll discuss the phenomena of wave propagation in elastic membranes and solids. Using a few simplifying assumptions, one can use Newton's equations to write down a partial differential equation for how such a medium attempts to return to equilibrium from an initial displacement or initial impulse. There are several striking consequences one can derive from these partial differential equations. The first is that even in an isotropic solid there are two different kinds of waves, each of which propagates with a distinct speed! The second is that there is a marked difference between how waves propagate through a solid as opposed to how they propagate across a membrane. In the former case the disturbance completely leaves any bounded region and heads outward unless it is reflected back by some sort of boundary. But in the latter case there is always some lingering "residue" of a wave in any bounded region, no matter how far the bulk portion of the wave has progressed. There are still open research questions concerning this curious phenomena, and we'll try to discuss these by the end of the talk.