Pluck a string under tension and it trembles. Let u(x, t) be the small sideways displacement of the string at position x. Newton's second law applied to a tiny element gives the most important hyperbolic PDE in physics.

The string has tension T and mass per length \rho. On a small piece, the vertical force is the difference in the tension's slope at the two ends — and for small displacements that net force is T\,u_{xx}\,\Delta x. The mass is \rho\,\Delta x and its acceleration is u_{tt}, so

The wave equation and its speed

Writing c^2 = T/\rho gives the wave equation:

The constant c = \sqrt{T/\rho} is the wave speed: tighter strings (more T) and lighter strings (less \rho) carry waves faster — exactly why tuning a guitar raises its pitch. Unlike the heat equation's single time derivative, this has u_{tt}, so it is second order in time and needs two initial conditions: the starting shape u(x, 0) and the starting velocity u_t(x, 0).

And there is no decay term. Where heat diffused away, the wave equation conserves energy: a disturbance oscillates forever rather than melting to zero.

Oscillation, not decay

The simplest motion is a single standing mode u = \cos(ct)\sin(x) (here c = 1). Advance time and the whole shape swings up and down between +\sin x and -\sin x, returning to full amplitude again and again — the energy never leaves. Compare this with the heat equation, whose modes only ever shrank.