Standing Waves

How the model works

Waves on a stretched string travel at v = √(T/μ), where T is tension (N) and μ is mass per unit length (kg/m). With both ends fixed, only whole numbers of half-wavelengths fit:

λₙ = 2L / n, fₙ = n·v / (2L), n = 1, 2, 3 …

Drive the string at frequency f and the steady response is a sum of modes. Each mode n responds like a damped oscillator, with amplitude

Aₙ ∝ 1 / √((fₙ² − f²)² + (2ζ fₙ f)²)

So near any fₙ one mode dominates and a clean standing wave appears: n half-loops, n + 1 nodes including the ends.

Model: small-amplitude ideal string, light damping (ζ), driven near one end. Shown as the steady state; the start-up transient is not simulated.

Guided experiments

1. Fundamental. L = 1 m, T = 100 N, μ = 0.01 kg/m. Prediction: v = 100 m/s, so f₁ = 50 Hz. Driving at 50 Hz gives one big loop. Drive at f₁
2. Third harmonic. Drive at 150 Hz. Prediction: three loops and four nodes. Drive at 150 Hz
3. Tighten the string. Quadruple the tension to 400 N and drive at 100 Hz. Prediction: v doubles to 200 m/s, so f₁ becomes 100 Hz and one loop returns. This is how a guitarist tunes a string. T = 400 N at 100 Hz

Challenge

A guitar string is 0.65 m long with μ = 0.0012 kg/m. What tension tunes its fundamental to 110 Hz (the A string)?

References

• OpenStax, University Physics Volume 1, §16.6 “Standing Waves and Resonance”.
• HyperPhysics, “Standing waves on a string”.