finite speed of propagation
Clap your hands. The sound does not reach the far wall instantly — it takes a moment, travelling at the speed of sound, and someone standing beyond that distance hears nothing until the wave arrives. Disturbances in a wave equation move at a definite, finite speed, and outside the region the wave has had time to reach, nothing has changed at all. This is finite speed of propagation, the defining temperament of hyperbolic equations.
Precisely: for the wave equation u_tt = c^2 u_xx, a change made to the data at a point can only affect points within distance c t after time t. The set of points the data at one location can ever influence is its range of influence, a cone opening at slope c; conversely the solution at a space-time point depends only on the initial data inside its domain of dependence, a backward cone of slope c. So information has a strict speed limit c, set by the equation's coefficients. The cleanest proof is an energy argument: define the energy inside a shrinking backward cone, differentiate in time, and the boundary terms have a favourable sign exactly when the cone slopes at speed c — so if the data vanishes on the base, the energy stays zero up the cone, forcing the solution to be zero there.
This is the sharpest contrast in all of PDE. The heat equation has INFINITE speed — change the data anywhere and the temperature everywhere responds the very next instant (by an exponentially tiny amount, but strictly nonzero). The wave equation has finite speed — a clean front, behind which there is signal and ahead of which there is exact silence. The dichotomy is physical, not technical: relativity demands a finite speed limit, which is why fundamental wave-like field equations are hyperbolic, while diffusion's infinite speed is understood as an idealization of an underlying random walk.
Pinch a long string at the origin to make a bump, then release. After time t the string is still exactly at rest everywhere beyond distance c t from the origin — no point further out has moved yet. The disturbance occupies a clean interval [-c t, c t] and nothing outside it.
Signals respect a strict speed limit c — outside the cone, exact silence.
Finite speed is a hyperbolic trait and goes hand in hand with the failure to smooth: because there is no instantaneous mixing, singularities are not erased but transported. It is the exact qualitative opposite of the heat equation's infinite speed and smoothing.