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Standing waves on a string fixed at both ends

Only waves that fit the string with a node at each end can stay. Pick a harmonic, then slide the bridge: the shape stands still while the energy runs both ways inside it.

Harmonic3

Length and harmonic number3 × λ/2 = L

Wavelength

Nodes · antinodes

Frequency

Pick a harmonictap one, or use ← → on the drawing

Amplitude70%

Playback speed1.00×

String length L83

Show the two traveling waves hiding inside

node: never movesantinode: swings the mostwave moving rightwave moving left

Both bridges hold the string still, so those two points are always nodes. The ghost waves above are two identical traveling waves going opposite ways; their sum is the standing wave. Lengths are in units of the full string, L₀; f₀ is its fundamental.

Standing, not traveling

In a traveling wave every crest slides along the string. Here the teal and plum crests do slide, in opposite directions, but where they meet they add to a pattern whose nodes stay put and whose loops only swing up and down, each opposite to its neighbour.

Why only certain waves fit

Each end must be a node, so a whole number of half-wavelengths must fit: L = n · λ/2, so λ = 2L / n and f = n · v / 2L. Shorten the string and every harmonic rises in pitch; raise n and the nodes crowd closer. Simplified: an ideal string with no damping.