the propagation of singularities
Pluck a guitar string sharply so it has a kink, then let go. The kink does not melt away — it splits and travels, riding along the string at the wave speed, staying just as sharp as it started. This is the exact opposite of the heat equation's instant healing. For wave-type (hyperbolic) equations, a singularity in the data is not smoothed; it is carried forward, and it moves along very specific paths. Understanding those paths is the propagation of singularities.
Precisely: for the wave equation u_tt = c^2 u_xx, d'Alembert's formula writes the solution as a right-moving piece f(x - c t) plus a left-moving piece g(x + c t). If the initial profile has a corner at a point, that corner simply slides along the characteristic lines x - c t = const and x + c t = const, never blurring. The general principle, made precise by microlocal analysis, is that singularities of solutions to hyperbolic equations travel along the characteristics (more exactly, along bicharacteristic rays in phase space) — the same curves that classify the equation and carry its information. Where the data is already smooth, the solution stays smooth; where it has a wavefront, that wavefront propagates. This is why you can hear a sharp clap as a sharp clap across a room: the wave equation refuses to round it off.
The contrast with diffusion is the whole point and a genuine qualitative dichotomy. Parabolic and elliptic equations smooth instantly and have infinite speed; hyperbolic equations preserve singularities and have finite speed c. So the location of a singularity tells you the type of physics: a discontinuity that persists and moves is a wave; one that vanishes immediately is diffusion. Honesty note: nonlinear hyperbolic equations can do even more — they can CREATE singularities (shocks) that were not in the smooth data, after which weak solutions and an entropy condition are needed.
Initial displacement of a string is a tent shape with a sharp peak; release from rest. d'Alembert splits it into two half-height tents, one sliding left and one sliding right, each keeping its sharp peak forever. The corner propagates along the characteristics x +/- c t = const, never smoothing.
Hyperbolic singularities ride the characteristics — wave behaviour, the antithesis of diffusion.
For LINEAR hyperbolic equations, singularities are only transported, never created — smooth data stays smooth. Nonlinear equations break this: smooth data can self-steepen into a shock in finite time, after which you must move to weak solutions plus an entropy condition to recover uniqueness.