🎸 Standing Wave & Harmonics Calculator
Choose a string or pipe and enter the wave speed and length to list the fundamental and its harmonics.
e.g. guitar/violin string. Use ~340 m/s for sound in air, or the string's wave speed.
| Harmonic | Frequency | Wavelength |
|---|---|---|
| n = 1 (fundamental) | 340 Hz | 1 m |
| n = 2 | 680 Hz | 0.5 m |
| n = 3 | 1,020 Hz | 0.333 m |
| n = 4 | 1,360 Hz | 0.25 m |
| n = 5 | 1,700 Hz | 0.2 m |
| n = 6 | 2,040 Hz | 0.167 m |
A string fixed at both ends (and an open pipe) resonates at fₙ = n·v ÷ 2L for n = 1, 2, 3…; a pipe closed at one end supports only the odd harmonics fₙ = n·v ÷ 4L (n = 1, 3, 5…), which is why a clarinet sounds an octave lower than its length suggests. The lowest frequency (n = 1) is the fundamental; the rest are overtones that give the sound its timbre. 🔒 In your browser.
How the standing wave & harmonics calculator works
A standing wave only fits certain wavelengths, giving a series of resonant frequencies. A string fixed at both ends — and a pipe open at both ends — resonate at fₙ = n·v ÷ 2L for n = 1, 2, 3…, while a pipe closed at one end supports only the odd harmonics fₙ = n·v ÷ 4L (n = 1, 3, 5…). The tool lists the fundamental (n = 1) and the next several harmonics with their frequencies and wavelengths.
Use the wave speed for the system — about 340 m/s for sound in air, or the string's own wave speed (which depends on its tension and thickness). The closed pipe missing its even harmonics is why a clarinet plays about an octave lower than a flute of the same length and has its distinctive tone. The lowest frequency is the pitch you hear; the overtones shape the timbre.
Frequently asked questions
How do I calculate the harmonics of a string?
For a string fixed at both ends, fₙ = n·v ÷ 2L, where v is the wave speed, L the length and n = 1, 2, 3… The fundamental is n = 1; each higher harmonic is an integer multiple of it.
What is the fundamental frequency?
The lowest resonant frequency of the system (n = 1) — the pitch you perceive. For a string or open pipe it's v ÷ 2L; for a closed pipe it's v ÷ 4L, an octave lower for the same length.
Why does a closed pipe only have odd harmonics?
A closed end forces a node and the open end an antinode, and only odd multiples of the quarter-wavelength fit that pattern. So a closed pipe produces fₙ = n·v ÷ 4L for n = 1, 3, 5… — the even harmonics are missing.
What's the difference between a harmonic and an overtone?
The harmonics are the whole series (1st, 2nd, 3rd…); the overtones are all of them above the fundamental. So the 2nd harmonic is the 1st overtone. On a closed pipe, where evens are absent, the numbering of overtones and harmonics differs.
How do harmonics create an instrument's tone?
The fundamental sets the pitch, and the relative strengths of the overtones give the timbre — why a violin and a flute playing the same note sound different. The set of harmonics available (all vs odd-only) is a big part of an instrument's character.