Standing waves calculator: string harmonics and tension, pipe resonances with end corrections, and tuning to a note

What it does

A guitar string, a flute or a bottle you blow across all make sound the same way: a wave runs back and forth and reinforces itself only at certain frequencies, the standing waves. The lowest is the fundamental (the note you hear), and the others are its harmonics (overtones), which set the tone colour.

• Strings fixed at both ends play every harmonic: f = n·v / 2L, where the wave speed v = √(T/μ) depends on the tension T and the mass per metre μ.
• Pipes open at both ends (a flute) also play every harmonic, f = n·c / 2L′. Pipes closed at one end (a clarinet, a bottle) play only the odd ones, f = n·c / 4L′, an octave lower for the same length. L′ is the acoustic length: air vibrates a little beyond each open end, about 0.61 of the radius for a plain tube end.

The page lists the harmonics with the nearest note name and cents (hundredths of a semitone), the wavelengths, and draws the first three standing waves. In the other direction, it finds the tension for a string, or the pipe length, that gives a target note.

How to use it

1. Choose a string or air in a pipe, and whether you want the notes it plays or what gives a target note.
2. For a string: the vibrating length (scale length), then either the diameter and material or the mass per metre, and the tension in newtons.
3. For a pipe: which ends are open, the length, the inside diameter, the kind of open end, and the air temperature.
4. For a target, type a note such as E4, C#3, Bb2 (middle C is C4) or a frequency in Hz.
5. Download the harmonic list as CSV, or the report as an image, PDF or Word file.

Worked example

The made-up example string is a 0.25 mm plain steel string with a 648 mm scale length, like a guitar’s high E, at a tension of 70 N. Its mass is 0.3853 g per metre, waves run along it at 426.2 m/s, and the page shows a fundamental of 328.87 Hz (E4 −4.0 cents): slightly flat. Switch to What gives a target note with E4 and it asks for 70.32 N (7.17 kg-force).

For a pipe, choose air in a pipe: the example is 600 mm long and 19 mm across, open at both ends, at 20 °C. The end corrections add 5.83 mm at each end, so the acoustic length is 611.7 mm and the fundamental is 280.56 Hz (C♯4 +21.0 cents). Close one end and it drops by about an octave to 141.63 Hz with only odd harmonics. To play A4 the open pipe would need to be 378.4 mm long.

Tips

• Doubling the tension raises a string’s pitch by only about half an octave (a factor of √2); halving the length raises it a full octave.
• Wind instruments go sharp as they warm up: the speed of sound rises about 0.6 m/s for each °C.
• A real string is slightly stiff, so its upper harmonics run a little sharp. Pianos are tuned with “stretched” octaves for this reason.

FAQ

Why is a pipe closed at one end an octave lower? A closed end must be a node (no air movement) and an open end an antinode, so the fundamental fits a quarter of a wavelength into the pipe instead of a half.

How exact is the end correction? The 0.61 and 0.82 factors are for narrow cylinders at low frequencies. Tone holes, bells, cones and thick walls change it, so instrument makers adjust by ear. For study and planning, not a substitute for professional judgement.

Last updated 2026-11-09.