Waves that appear to stand still

Why do some points never move in a standing wave?

How many internal nodes will the second harmonic have?

  1. Select standing waves and inspect the fundamental.
  2. Increase harmonic number from 1 to 2 and 3.
  3. Count nodes and compare their spacing with λ/2.

What to measure

Harmonic n, node positions, f_n and λ_n.

The idea underneath

Equal counterpropagating waves make a stationary amplitude envelope. Fixed ends must remain nodes, allowing only integer numbers of half-wavelengths. These modes explain string-instrument pitches and help engineers avoid damaging structural resonances.

y=2A sin(nπx/L)cos(2πf_n t); f_n=nv/(2L); λ_n=2L/n

L is length in m; n is a positive integer; v in m/s; f_n in Hz.

Build the argument

  1. Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
  2. Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
  3. Two waves travelling in opposite directions form nodes when their sum is zero for all times.

Worked example 1

For L=2 m and v=4 m/s, f₁=1 Hz and λ₁=4 m.

Worked example 2

For n=3, f₃=3 Hz and λ₃=4/3 m, with two internal nodes.

Applications

The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.

Common misconception

The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.

A further investigation

At fixed v, what happens to the fundamental if length doubles?

Check your understanding

How many internal nodes occur in the second harmonic?

  • One
  • Two
  • None
Explanation

The nth mode has n−1 internal nodes. Doubling length halves f₁ because f₁=v/(2L).

Model limits

Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.

University Physics Volume 1, 16.6 Standing Waves and Resonance