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)