Standing Wave Harmonics Calculator
Find the harmonic frequencies and wavelengths of a standing wave on a string fixed at both ends from its length and wave speed.
How it works
fₙ = n · v ÷ (2L) · λₙ = 2L ÷ n · fundamental f₁ = v ÷ (2L)
A string fixed at both ends can only support standing waves that fit a whole number of half-wavelengths between the ends, so its allowed frequencies are fₙ = n·v/(2L). The lowest, n = 1, is the fundamental; the higher harmonics are exact integer multiples of it, which is why a plucked string sounds musical. Shortening the string or increasing the wave speed (by tightening it) raises every harmonic in proportion — the physics behind fretting and tuning a guitar.
Worked example
A 1 m string carrying waves at 200 m/s has a fundamental f₁ = 200/(2 × 1) = 100 Hz, with wavelength λ₁ = 2 × 1 = 2 m. The second harmonic is 200 Hz (λ = 1 m), the third 300 Hz, and so on in integer steps.
Sources: Standing wave — Wikipedia
Frequently asked questions
Why are only certain frequencies allowed?
The fixed ends must be nodes, so only waves that fit a whole number of half-wavelengths in the length can form a stable standing wave. That quantises the frequencies into the harmonic series fₙ = n·v/(2L).
What is the difference between a harmonic and an overtone?
The fundamental is the first harmonic. Overtones are all the higher modes: the second harmonic is the first overtone, the third harmonic the second overtone, and so on. For a string fixed at both ends they are exact integer multiples of the fundamental.
How does tightening the string change the pitch?
Tightening raises the tension, which increases the wave speed v = √(T/μ). Since every harmonic is proportional to v, all of them rise together and the pitch goes up — exactly how you tune a string instrument.
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This calculator is for educational purposes. Double-check important results before acting on them.