A single equation,

describes an astonishing range of the physical world. It governs a guitar string trembling after you pluck it. It governs sound pushing through the air from your speaker to your ear. Written for electric and magnetic fields instead of a string's displacement, it governs light itself racing through empty space. Anything that vibrates or propagates as a wave is, underneath, obeying this one line.

What sets it apart from its cousin, the heat equation u_t = \alpha u_{xx}, is small on the page but enormous in consequence: heat has a single time derivative, u_t; the wave equation has a second one, u_{tt}. That one extra derivative is the difference between a system that smooths out and forgets, and a system that rings and remembers. Let's see exactly where the equation comes from.

The same equation, with the same second time-derivative, keeps reappearing under new names as you look further afield: u can be the height of a water surface after you drop a stone in a pond, the pressure of air compressing and rarefying as sound passes through it, or the shaking of the ground radiating outward from an earthquake's epicentre as a seismic wave. Change what u measures and what sets the value of c, and you've swapped from a guitar string to an ocean, an eardrum, or a planet — but the differential equation governing how the disturbance spreads doesn't change at all.

From Newton's second law to a PDE

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.

And there is no decay term. Where heat diffuses — a lump smooths itself away and its peak sinks over time — the wave equation conserves the disturbance: a bump doesn't melt, it travels, keeping essentially its own shape as it moves along at the fixed speed c. In that sense it is much closer in spirit to the transport equation, which also just carries a shape along without distorting it — the wave equation is what you get when the carried quantity is free to slosh back and forth rather than drift steadily in one direction.

Worked example: a plucked string splits in two

Pull a guitar string up into a triangular "tent" shape at its centre and let go from rest. What does the wave equation say happens next? Long before the disturbance reaches the fixed ends, the string behaves as though it were infinitely long, and the single pluck does something wonderfully simple: it splits into two half-height copies of the original tent, one gliding right and one gliding left, each moving at exactly the speed c = \sqrt{T/\rho}.

You can see why using only energy and symmetry, without solving anything yet. Releasing the string from rest with no initial push means whatever moves right must be balanced by an equal amount moving left — there's no reason to favour one direction. And since the two half-height pulses start exactly on top of each other, they add back up to the full-height tent at the instant of release, which is exactly the shape you started with. As time goes on, the two half-tents separate, the gap between their peaks growing at the combined rate 2c.

This "one bump splits into two travelling pulses" behaviour is not a special property of triangular tents — it is the wave equation's single most important structural fact, and it has an exact closed-form formula: d'Alembert's solution writes every solution on an infinite line as u(x,t) = F(x - ct) + G(x + ct), a right-moving shape plus a left-moving shape, and shows exactly how to build F and G out of the starting shape and starting velocity.

Worked example: how tight a string, how fast a wave?

The speed formula c = \sqrt{T/\rho} is not just decoration — it lets you actually compute a wave speed. Suppose a steel string is stretched with tension T = 100\ \text{N} and has mass per unit length \rho = 0.01\ \text{kg/m} (a realistic guitar-string ballpark). Then

Now tighten the string until the tension doubles to T = 400\ \text{N} with the same mass per length. The new speed is c = \sqrt{400/0.01} = \sqrt{40\,000} = 200\ \text{m/s} — double the speed from quadrupling the tension, because c depends on the square root of T, not on T directly. That square root is exactly why a guitar's tuning pegs need a noticeable turn to raise the pitch just a little: pitch rises with wave speed, but wave speed only creeps up as the square root of how hard you're winding the string.

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.