A standing wave forms when a wave interferes with its reflection: fixed nodes (zero displacement) and antinodes (maximum) that don’t travel. Resonance occurs at the natural frequencies (harmonics).
The three cases, side by sideBoth ends alike (string fixed at both ends, or a pipe open at both ends) fits whole half-wavelengths, so every harmonic exists and fₙ = n·v/2L. A pipe closed at one end must have a node at the wall and an antinode at the open end, so it fits ODD quarter-wavelengths only: its fundamental is half the frequency of an open pipe of the same length, and there are no even harmonics at all.
Strings & pipes
String / pipe open both ends f_n=\frac{nv}{2L} n = 1,2,3… Fundamental at n=1. Both ends antinode (pipe) or node (string). Pipe closed one end f_n=\frac{nv}{4L} Only odd harmonics (n = 1,3,5…). Node at closed end, antinode at open end.
Counting harmonics
Fundamental (1st harmonic): the simplest fit. String = one loop (½λ), open pipe = ½λ, closed pipe = ¼λ.
Distance between adjacent nodes (or antinodes) = ½ λ.
Closed pipes are missing the even harmonics: a classic exam distinction.
Beats: two slightly different frequencies produce a slow pulsing in loudness. Beat frequency f<sub>beat</sub> = | f₁ − f₂ | Number of loudness maxima per second.