String Vibration Calculator Formula
Understand the math behind the string vibration calculator. Each variable explained with a worked example.
Formulas Used
Wave Speed on String
wave_speed = sqrt(tension / linear_density)Frequency
frequency = (harmonic / (2 * string_length)) * sqrt(tension / linear_density)Variables
| Variable | Description | Default |
|---|---|---|
harmonic | Harmonic Number (n) | 1 |
string_length | String Length(m) | 0.65 |
tension | Tension(N) | 73 |
linear_density | Linear Density(kg/m) | 0.001 |
How It Works
Vibrating String
A string fixed at both ends supports standing waves whose frequencies depend on string length, tension, and mass per unit length.
Formula
f_n = (n / 2L) * sqrt(T / mu)
where T is tension, mu is linear mass density (kg/m), L is string length, and n is the harmonic number.
Worked Example
A guitar string: L = 0.65 m, T = 73 N, mu = 0.001 kg/m, fundamental.
- 01v = sqrt(T / mu) = sqrt(73 / 0.001) = sqrt(73000) = 270.2 m/s
- 02f = (1 / (2 * 0.65)) * 270.2
- 03f = 0.769 * 270.2
- 04f = 207.8 Hz
Frequently Asked Questions
How does tightening a guitar string change the pitch?
Increasing tension increases the wave speed on the string, which raises the frequency and thus the pitch.
Why do thicker strings produce lower notes?
Thicker strings have greater linear density (mu), which reduces wave speed and lowers the frequency.
What are harmonics on a string?
Harmonics are integer multiples of the fundamental frequency. The second harmonic is twice the fundamental, the third is three times, and so on.
Ready to run the numbers?
Open String Vibration Calculator