Standing Waves & Harmonics Calculator
Work out the harmonics of a string or an air column. Solve fₙ = n·v/2L for a string fixed at both ends or an open pipe, and fₙ = n·v/4L for a closed pipe, for the frequency, length, tension, linear density or speed of sound, with the note names and a drawing of the wave.
How to Use
- Pick the system: a string fixed at both ends, a pipe open at both ends, or a pipe closed at one end.
- Pick what to solve for: the frequency, the length, the string’s tension or linear density, or the speed of sound in the pipe.
- Enter the harmonic number n (1 is the fundamental; a closed pipe only has odd ones) and the other values, each with its own unit.
- For a pipe, add its radius to include the end correction of about 0.6 × radius per open end. Leave it blank for the textbook formula.
- Read the answer in the highlighted field, the nearest note and how many cents it is off, the drawing of nodes and antinodes, and the first eight harmonics in the table.
- Press a preset for a guitar, violin, organ pipe or resonance tube, and check Show Work for every step.
Worked Example
Tension in a guitar’s high E string. A plain steel string 0.010 in thick has μ = 0.398 g/m (steel at 7,850 kg/m³, a typical figure). On a 647.7 mm scale tuned to E4 = 329.63 Hz the wave speed must be v = 2Lf = 2 × 0.6477 × 329.63 = 427.0 m/s, so T = μv² = 0.0003978 × 427.0² = 72.53 N, which is 16.31 lbf.
An organ pipe for A2. An open pipe sounding 110 Hz in air at 343 m/s holds half a wavelength: L = v ÷ 2f = 343 ÷ 220 = 1.559 m. Its harmonics are 220, 330, 440 Hz and so on, every whole-number multiple.
The common mistake: using the open-pipe formula for a closed pipe. A 50 cm pipe closed at one end holds a quarter-wavelength, not a half. f = v ÷ 2L gives 343 ÷ 1 = 343 Hz, but the right answer is f = v ÷ 4L = 343 ÷ 2 = 171.5 Hz, a whole octave lower. With a 2 cm radius the end correction (0.6 × 2 cm = 1.2 cm) lowers it again to 343 ÷ (4 × 0.512) = 167.5 Hz.
Show Work
Formulas
From Mersenne’s Strings to the 8-Foot Stop
The rule that a string’s pitch rises with tension and falls with length and weight was set down by the French friar Marin Mersenne in Harmonie universelle (1636), and the three parts of it are still called Mersenne’s laws. Around 1700 Joseph Sauveur, who also coined the word acoustics, named the still points nodes and the higher tones harmonics, and showed that a string sounds several of them at once.
The full formula came with calculus. Brook Taylor derived the frequency of a vibrating string in 1713, Jean le Rond d’Alembert wrote down the wave equation in 1747, and Daniel Bernoulli argued in 1753 that any vibration of the string is a sum of its harmonics. The open-end correction was made exact for a thin-walled pipe by Harold Levine and Julian Schwinger in 1948: about 0.61 × the radius, the 0.6 used here.
Organ builders name stops by pipe length. The bottom C of an 8-foot stop sounds C2 = 65.41 Hz, and half its wavelength in air is 343 ÷ (2 × 65.41) = 2.62 m, about 8.6 ft; the name is a round nominal length rather than an exact one. Note names here use A4 = 440 Hz, the standard pitch agreed at a London conference in 1939 and set in ISO 16.
About This Tool
This calculator handles the three textbook resonators in one place: a string clamped at both ends, a pipe open at both ends and a pipe closed at one end. It solves for whichever value you leave out, including a string’s tension or mass per metre and the speed of sound from a resonance-tube measurement, and adds the approximate end correction for pipes when you give a radius. Each answer comes with its nearest note and how many cents it is off, the first eight harmonics, and a moving drawing of the wave with its nodes and antinodes. String densities in the presets are typical values, not a maker’s specification.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Speed of Sound Calculator, Double-Slit & Diffraction Grating Calculator, and Simple Harmonic Motion Calculator.
Frequently Asked Questions
What is the formula for the frequency of a string?
fₙ = (n ÷ 2L) × √(T ÷ μ), where L is the vibrating length, T the tension and μ the mass per metre. A violin A string 328 mm long at 50 N tension with μ = 0.6 g/m has a wave speed of √(50 ÷ 0.0006) = 288.7 m/s and a fundamental of 288.7 ÷ (2 × 0.328) = 440.05 Hz.
Why does a closed pipe only have odd harmonics?
The closed end must be a node (the air cannot move) and the open end an antinode, so the pipe holds 1, 3, 5 … quarter-wavelengths. A 50 cm closed pipe at 343 m/s sounds 171.5, 514.5 and 857.5 Hz; the same pipe open at both ends sounds 343, 686 and 1,029 Hz, a full octave higher at the bottom.
What is the end correction of a pipe?
The antinode at an open end sits a little outside the pipe, about 0.6 × the radius beyond it, so the air column behaves as if it were longer. A 50 cm closed pipe of 2 cm radius acts like 51.2 cm and sounds 167.5 Hz instead of 171.5 Hz, 41 cents flat. The 0.6 is an approximation for an unflanged pipe.
How much must the tension change to raise a string by a whole tone?
Frequency goes with the square root of tension, so a whole tone (200 cents, a ratio of 2^(2/12)) needs the tension multiplied by 2^(4/12) = 1.26, about 26% more. Doubling the tension raises the pitch by √2, which is 600 cents, half an octave.
Why are some harmonics out of tune with a piano?
Harmonics are exact whole-number multiples, while a piano is tuned in equal temperament. On a guitar E string at 329.63 Hz, harmonic 3 (988.89 Hz, a B) is 2.0 cents sharp of B5, harmonic 5 (1,648.15 Hz, a G♯) is 13.7 cents flat, and harmonic 7 is 31.2 cents flat, which is why the 7th harmonic sounds noticeably off.
How do I use the Standing Waves & Harmonics Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Guitar string tension
A plain steel 0.010 in string (0.398 g/m) on a 647.7 mm (25.5 in) scale tuned to E4 = 329.63 Hz pulls with 72.53 N, or 16.31 lbf.
Organ and pan pipes
An open pipe sounding A2 = 110 Hz is 343 ÷ (2 × 110) = 1.559 m long; a closed pan-pipe tube for A4 = 440 Hz is only 19.49 cm.
School resonance tube
A 512 Hz tuning fork resonating over 15.8 cm of air in a 1.5 cm radius tube gives v = 4 × 0.167 × 512 = 342.0 m/s with the end correction, against 323.6 m/s without it.
Natural harmonics on a guitar
Touching the string at the 12th fret leaves harmonic 2, 659.26 Hz (E5); at the 7th fret harmonic 3, 988.89 Hz, a B that is 2.0 cents sharp.
Physics homework
A 1.2 m string at 80 N with μ = 5 g/m: v = √(80 ÷ 0.005) = 126.5 m/s, so f₁ = 52.70 Hz and f₄ = 210.8 Hz.
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