Resonant Frequency Formula
Calculate the natural resonant frequency of a vibrating string or air column.
Used in musical instrument design.
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