By Zencalcy Editorial TeamLast reviewed About us
A stretched string supports standing waves whose frequencies depend on length, tension and mass per unit length. The fundamental has a wavelength of twice the string length, and each higher harmonic adds one more half wave. This is the physics behind guitar, piano and violin tuning.
Harmonic frequencies of a stretched string.
f_n = (n / 2L) √(T / μ)
n is the harmonic number, L the vibrating length, T the tension and μ the linear mass density. The fundamental is n = 1; n = 2 is the first overtone.
| Symbol | Meaning |
|---|---|
| f_n | Frequency of the nth harmonic. |
| n | Harmonic number, a positive integer. |
| L | Vibrating length of the string. |
| T | Tension in the string. |
| μ | Mass per unit length. |
With μ = 0.005 kg/m, the wave speed is √(80/0.005) = 126.5 m/s. The fundamental is 126.5 / (2 × 0.65) = 97.3 Hz, and the second harmonic is 194.6 Hz.
Continue with the Sound Wavelength Calculator, Acoustic Impedance Calculator, or the Beat Frequency Calculator — or browse more acoustics and sound tools and all physics calculators.
Higher tension increases the wave speed. Frequency is proportional to the square root of tension, so quadrupling tension doubles the pitch.
Results are computed from the standard equations of classical mechanics. Unit conversions use the SI definitions published by NIST. Air resistance is ignored unless the calculator names a drag model.
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Content updated: September 2026