Standing Wave on a String (2D)

Primary Harmonic

Harmonic number n₁ 2

Amplitude A₁ 1.00

Second Harmonic (Superposition)

Add a second harmonic

Harmonic number n₂ 3

Amplitude A₂ 0.60

String

Wave speed v 4.0 m/s

Live Readout

Frequency f₁

0.0 Hz

Period T₁

0.0 s

Wavelength λ₁

0.0 m

Nodes

Elapsed time

0.0 s

Peak y(t)

0.00 m

How it works

A string of length L fixed at both ends supports only wavelengths that fit an integer number of half-cycles, λₙ = 2L/n. Each such mode oscillates as a standing wave — a product of a spatial shape and a time-varying amplitude, not a wave that travels:

y(x,t) = A · sin(nπx/L) · cos(ωt)
ωₙ = nπv/L        fₙ = ωₙ/2π = nv/2L

Nodes (permanently y=0) sit at x = kL/n for k = 0…n; antinodes — the points of maximum swing — sit halfway between them. Turning on the second harmonic sums two such modes, y = y₁ + y₂, which is exactly the principle of superposition: the resulting shape is not itself a single standing wave, but the string still returns to a mirror image of itself once every beat period.

Wave speed v — set by the string's tension and mass per length (v = √(T/μ)); raising it raises every mode's frequency without changing the node pattern.
Envelope — the faint outer curve traces ±the instantaneous maximum displacement the string could reach at each x.
Nodes — fixed points that never move, shown as markers when enabled.

String y(x,t) Envelope (max |y|) Nodes