The wave equation

Underneath the many different-looking wave phenomena — a plucked string, a sound pulse, ripples on water, an electromagnetic pulse in vacuum — sits one common mathematical structure: the wave equation, relating how a disturbance curves in space to how it curves in time, with a single constant, the wave speed, connecting the two. Any function that satisfies this equation describes a disturbance that propagates without changing shape, travelling at that fixed speed. The equation does not care what physical quantity is doing the disturbing — displacement of a string, pressure in air, electric field strength in vacuum — which is exactly why the same mathematics describes such different physical situations.

What the wave equation is not is a description that fits after the fact to any arbitrary disturbance. Solving it derives the wave speed from properties of the medium alone — for a stretched string, from its tension and its mass per unit length; for sound in a gas, from the gas's pressure and density; for light, from the vacuum permittivity and permeability that appear in Maxwell's equations. The speed is a property of the medium (or, for light, of empty space itself), fixed before any particular wave travels through it — not something a taller, louder, or higher-frequency wave can change. A big pulse and a small ripple on the same string travel at exactly the same speed, because that speed comes from the string, not from the pulse.

Recognising the same equation across mechanics, acoustics and electromagnetism is what let physicists carry intuition and solution techniques wholesale from one field to another, and it is the mathematical fact underneath Maxwell's own discovery that light is an electromagnetic wave: solving his four field equations together produced exactly this wave equation, with a speed computable from two purely electrical and magnetic constants that turned out, numerically, to equal the already-measured speed of light.