Explore standing wave harmonics on strings and in pipes. Adjust tension, density, and boundary conditions. Hear frequency audio output.
A virtual lab for standing waves: set the length of a string or pipe, choose tension and linear density (string) or the speed of sound (pipe), pick fixed-fixed, open-open, or closed-open boundary conditions, and the tool draws each mode shape and plays a tone at its frequency. Frequencies come from fitting half-wavelengths (strings and open pipes) or quarter-wavelengths (closed pipes) into the length. The surprise for most people: the open end of a pipe is an antinode of the air motion, and a pipe closed at one end only produces odd harmonics.
A concrete input and expected output from the current implementation.
Input
String, both ends fixed: length 1.0 m, tension 100 N, linear density 0.010 kg/m. Request modes n = 1 and n = 3.
Expected output
Wave speed v = sqrt(100 / 0.010) = 100 m/s. Mode 1: f = 50 Hz, wavelength 2.0 m, 2 nodes, 1 antinode. Mode 3: f = 150 Hz, wavelength 0.667 m, 4 nodes, 3 antinodes. Audio: tones at 50 Hz and 150 Hz.
For a fixed-fixed string, f = n v / (2L), so mode 1 gives 100 / 2 = 50 Hz and mode 3 gives 150 Hz; wavelength is 2L/n, and a mode n shape always shows n + 1 nodes and n antinodes.
For a string fixed at both ends, or a pipe open at both ends, the ends are nodes, so an integer number of half-wavelengths must fit into the length, giving f = n v / (2L). A pipe closed at one end holds a node at the closed end and an antinode at the open end, which only odd quarter-wavelength patterns satisfy, giving f = (2n-1) v / (4L). The tool applies these two relations from the length and wave speed you set.
For a string, tension and linear density enter only through the wave speed v = sqrt(T / linear density), so every mode moves by the same factor when you change either; doubling tension multiplies all frequencies by sqrt(2), about 1.414. For a pipe the speed is the speed of sound, so length and boundary conditions are the only levers. The audio tone is synthesized at each mode's frequency; its timbre is not the instrument's.
The closed end must hold a node of the air motion and the open end an antinode. Even-numbered harmonic patterns would need a node at the open end or an antinode at the closed end, which the boundaries forbid, so only wavelengths 4L/1, 4L/3, 4L/5 fit. This is the correct physics, not a tool bug; it is also why a clarinet differs from a flute.
No. Frequency scales with the square root of tension: f is proportional to sqrt(T). Doubling tension multiplies every mode frequency by sqrt(2), about 1.414, which is exactly six semitones, a tritone, not an octave. To gain an octave you would need to quadruple the tension.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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