hyperbolic equation
Pluck a string and a kink races along it; clap your hands and a pressure front travels outward at the speed of sound; drop a pebble and a ring expands across the pond. In all of these a disturbance keeps its shape and moves at a definite, finite speed, and it leaves the medium behind it just as it found it. The PDEs that describe propagation and waves are the hyperbolic equations, the third of the three second-order types.
A hyperbolic equation has discriminant B^2 - A C strictly positive, with the wave equation u_tt = c^2 u_xx as its prototype. The decisive feature is two distinct families of real characteristic curves — for the wave equation, the lines x - c t = const and x + c t = const — along which information actually travels at speed c. Because signals move at finite speed, a hyperbolic equation has a domain of dependence (a point's value depends only on a finite region of the initial data, a backward light cone) and a range of influence (initial data at a point affect only a finite forward cone). Sharp features and discontinuities are preserved and carried along the characteristics rather than smoothed away, the very opposite of diffusion. A hyperbolic problem typically needs two initial conditions in time (position and velocity) because it is second order in time.
Hyperbolic equations govern everything that propagates without dissipation: acoustic and electromagnetic waves, vibrations of strings, membranes and elastic solids, shallow-water and gas dynamics (where nonlinear hyperbolic equations form shocks), and the propagation of light in Maxwell's theory. The finite signal speed and the characteristic structure are not incidental — they are why causality, echoes, and wavefronts exist, and why numerical schemes for these equations must respect a speed limit (the CFL condition) to stay stable.
A plucked guitar string obeys u_tt = c^2 u_xx. A triangular pluck splits into two half-height triangles, one running left and one running right at speed c, each keeping its sharp corner — the corner is carried by a characteristic and never smooths out.
Hyperbolic equations carry sharp features along characteristics at finite speed — the exact opposite of the parabolic smoothing seen in heat flow.
Finite speed of propagation is the signature of the hyperbolic type, and it is a real physical prediction: unlike the heat equation, a hyperbolic wave that starts confined to a region stays exactly zero outside the expanding light cone until the front arrives.