the wave equation
Pluck a guitar string, drop a stone in a pond, speak into the air, or shine a light: in every case a disturbance is created somewhere and then travels outward, keeping its shape, at a definite speed. The wave equation is the single tidy equation that describes all of this. It is the master equation of vibration and propagation.
In one space dimension it reads u_tt = c^2 u_xx. Here u(x,t) is the thing that is waving — the sideways displacement of a string, the pressure in air, one component of an electric field — and the subscripts are partial derivatives: u_tt is the acceleration in time and u_xx is the curvature in space. The equation says acceleration is proportional to curvature, with the constant c^2. Read physically (think of a string): where the string is bent sharply (large curvature), the tension pulls it back hard, so it accelerates strongly. In higher dimensions u_xx is replaced by the Laplacian, giving u_tt = c^2 (Laplacian u) — the same balance of acceleration against curvature, now in space.
It is the archetypal hyperbolic equation, and almost everything special about waves flows from this one line: signals travel at the finite speed c (nothing instant), they keep their shape rather than smearing out (no smoothing — sharp corners stay sharp and propagate), and energy is conserved. This is the exact opposite of the heat equation u_t = k u_xx, which smooths everything instantly and forgets the past. Notice the wave equation is second order in time (it has u_tt), so to start it you must give BOTH the initial shape and the initial velocity — one number is not enough, just as launching a thrown ball needs both its position and its velocity.
A taut string with c = 200 m/s. A single sharp bump in the middle splits instantly into two identical half-height bumps, one running left and one running right, each at 200 m/s, each holding its shape — exactly what u_tt = c^2 u_xx predicts.
Any initial shape splits into two travelling copies — the heart of d'Alembert's solution.
It is genuinely second order in time: prescribing only the initial displacement (and no initial velocity) is a perfectly valid but incomplete problem — you must give both u and u_t at t = 0 to determine the future.