Wave equation

A second-order linear partial differential equation relating a field's acceleration in time to its curvature in space. In one spatial dimension it reads ∂²y/∂t² = v²∂²y/∂x², where v is the phase velocity set by the medium.

d'Alembert showed in 1746 that every solution is of the form y(x,t) = f(x − vt) + g(x + vt) — an arbitrary shape moving right plus an arbitrary shape moving left, both at speed v, both shape-preserving. The same equation appears with different constants for strings, sound, light, elastic solids, transmission lines, plasma oscillations, and gravitational waves, making it one of the most universal equations in physics.

Brook Taylor analysed a single mode of the vibrating string in 1713. Jean le Rond d'Alembert wrote down the PDE and proved its general solution in 1746. Euler and Daniel Bernoulli fought for three decades over what counted as a valid solution; Fourier's 1807 decomposition of arbitrary shapes into sinusoids ultimately settled the debate in Bernoulli's favour.