the superposition principle

Two people toss pebbles into a pond from different spots. Where the two sets of ripples meet, the water simply does both at once, it rises higher where two crests overlap and flattens where a crest meets a trough, and then the ripples move on unchanged. The superposition principle says that waves add up. The question it answers is: what happens where two waves overlap?

The principle states that when two or more waves overlap, the total displacement at each point at each instant is the sum of the displacements each wave would produce there on its own: y_total = y_1 + y_2 + ... After overlapping, the waves pass straight through each other and continue as if nothing had happened. This works because the underlying equations are linear, so the displacements simply add.

Superposition is the engine behind interference, standing waves, and beats, every situation where waves combine. An honest caveat: it holds when the displacements are small enough that the medium responds linearly. For very large-amplitude waves (a huge shock wave, a breaking ocean wave) the medium behaves nonlinearly and simple addition breaks down.