String Vibration Calculator Formula

Understand the math behind the string vibration calculator. Each variable explained with a worked example.

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Formulas Used

Wave Speed on String

wave_speed = sqrt(tension / linear_density)

Frequency

frequency = (harmonic / (2 * string_length)) * sqrt(tension / linear_density)

Variables

VariableDescriptionDefault
harmonicHarmonic Number (n)1
string_lengthString Length(m)0.65
tensionTension(N)73
linear_densityLinear 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.

harmonic = 1string_length = 0.65tension = 73linear_density = 0.001
  1. 01v = sqrt(T / mu) = sqrt(73 / 0.001) = sqrt(73000) = 270.2 m/s
  2. 02f = (1 / (2 * 0.65)) * 270.2
  3. 03f = 0.769 * 270.2
  4. 04f = 207.8 Hz

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Open String Vibration Calculator