Resonant Frequency Formula
Reference for resonant frequency formulas for strings f = (1/2L)sqrt(T/μ) and air columns.
Covers string tension, harmonics, and instrument design applications.
Need to calculate, not just reference? Use the interactive version.
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The Formula
f_n = n × v / (2L)
This formula gives the resonant frequencies (harmonics) of a string or open pipe. The fundamental frequency (n=1) is the lowest pitch. Higher harmonics are integer multiples.
Variables
| Symbol | Meaning |
|---|---|
| f_n | Frequency of the nth harmonic (Hz) |
| n | Harmonic number (1 = fundamental, 2 = second harmonic, etc.) |
| v | Wave speed in the medium (m/s) |
| L | Length of the string or pipe (meters) |
Example 1
A guitar string is 0.65 m long with a wave speed of 285 m/s. Find the fundamental frequency.
f₁ = 1 × 285 / (2 × 0.65)
f₁ = 285 / 1.30
f₁ ≈ 219 Hz (close to the note A3)
Example 2
An open organ pipe is 1.2 m long. Speed of sound = 343 m/s. Find the first three harmonics.
f₁ = 1 × 343 / (2 × 1.2) = 143 Hz
f₂ = 2 × 143 = 286 Hz
f₃ = 3 × 143 = 429 Hz
When to Use It
Use the resonant frequency formula when:
- Designing musical instruments and tuning systems
- Calculating the pitch produced by a vibrating string or pipe
- Understanding overtones and harmonic series
- Avoiding destructive resonance in structural engineering
Key Notes
- Closed vs open pipes: A pipe closed at one end supports only odd harmonics (f = v/4L, 3v/4L, 5v/4L). Open pipes support all harmonics using f = v/2L.
- String tension affects frequency: For a vibrating string, f = (1/2L)√(T/μ), where T is tension and μ is mass per unit length. Higher tension raises pitch.
- Temperature shifts air resonance: Speed of sound in air increases with temperature (~0.6 m/s per °C), so resonant frequencies of air columns rise in warmer conditions.
- Harmonics vs overtones: The second harmonic is the first overtone. In a closed pipe, the "first overtone" is actually the third harmonic — the numbering can be confusing.
- Resonance causes amplification: When a driving frequency matches a natural resonant frequency, amplitude grows dramatically. This is exploited in musical instruments and must be avoided in bridges and buildings.