03 · 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?

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

Measurement

Harmonic n, node positions, f_n and λ_n.

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.

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

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

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.

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

Assumptions

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