Wave Superposition & Standing Waves

When two waves meet, they don’t collide or bounce — they simply add. That one rule, superposition, explains noise-cancelling headphones, the colours in a soap bubble, and the fixed nodes of a guitar string. Slide two waves past each other below and watch a third appear.

Waves just add

The principle of superposition says that where two or more waves overlap, the total displacement at each point is the sumof the individual displacements at that instant. Crest on crest builds a bigger crest; crest on trough cancels. And crucially — once they’ve passed through each other, both waves carry on completely unchanged.

Waves are polite: they pass straight through one another. They add up only while they overlap, then continue as if nothing happened.

Overlap two waves

In interference mode, slide the phase difference from $0$ to $2\pi$ and watch the gold resultant swell and vanish. Then switch to standing-wave mode, where two waves travelling in opposite directions lock into a pattern with fixed nodes and antinodes.

Constructive vs destructive

Whether two overlapping waves reinforce or cancel comes down to their phase difference, usually expressed through path difference:

For two equal waves of amplitude $A$, the resultant amplitude is $2A\cos(\Delta\phi/2)$ — the formula the panel shows as you drag.

Standing waves

Send a wave down a string fixed at both ends and it reflects back on itself. The forward and reflected waves — identical, travelling opposite ways — superpose into a standing wave that appears to stay put:

The $\sin(kx)$ factor fixes the shape in space. Where $\sin(kx)=0$ the string never moves — a node. Halfway between, the string swings with maximum amplitude — an antinode. Adjacent nodes are half a wavelength apart.

Worked example

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Common mistakes