Standing Wave Resonant Frequency Calculator
Calculate the resonant frequencies of standing waves on strings, open pipes, and closed pipes.
Shows the first 5 harmonics.
Standing waves form when two waves of the same frequency travel in opposite directions. Resonant frequencies occur when the length fits a whole number of half-wavelengths (or quarter for closed pipes):
String or open pipe (both ends same): f_n = nv/(2L) where n = 1, 2, 3, 4, 5…
Closed pipe (one end open, one closed): f_n = nv/(4L) where n = 1, 3, 5, 7… (odd harmonics only)
Where:
- f_n = frequency of nth harmonic
- v = wave speed (m/s)
- L = length of string or pipe (m)
- n = harmonic number
Wave speeds (typical):
- String: v = √(T/μ) where T = tension, μ = mass per unit length
- Air at 20°C: v ≈ 343 m/s
- Water: v ≈ 1480 m/s
Musical implications:
The harmonic series explains musical harmony. The fundamental frequency determines the pitch. Higher harmonics (overtones) determine the timbre (tone quality):
- Flute: mostly fundamental and even harmonics
- Clarinet (closed pipe): predominantly odd harmonics — sounds hollow
- Violin: rich in all harmonics — complex, warm tone
Resonance and standing waves are also responsible for:
- Chladni patterns in vibrating plates
- The resonance in concert halls and musical instrument bodies
- Laser cavity modes (standing light waves between mirrors)
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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