Fundamental Frequency of a Vibrating String

Fundamental Frequency of a Vibrating String

Every stringed instrument — guitar, violin, piano — produces pitch according to this same relationship between string length, tension, and mass per unit length. It is why tightening a guitar string raises its pitch (increasing T raises f) and why the thick, heavy bass strings sound lower than the thin treble strings at the same tension (larger μ lowers f).The same physics governs the design of any tensioned cable or wire prone to vibration — power lines, suspension bridge cables, and guy wires all have natural vibration frequencies given by this same family of equations, which engineers check to avoid wind-induced resonance.

The frequency of the nth harmonic of a vibrating string is f = (n/2L)·sqrt(T/μ). where n_harm is the harmonic number, L_str is the string length, T_str is the string tension, mu_str is the mass per unit length, and f_string is the resulting vibration frequency.

Combine the harmonic number, string length, tension, and mass per length: the square root of tension over mass density gives the wave speed on the string, and dividing by twice the length per harmonic converts that speed into a frequency.

Results

For a 0.65 m string under 80 N tension, the second harmonic lands in the low-hundreds of hertz range, comfortably within the musical pitch range this kind of string is designed for. Increasing tension raises the frequency only with the square root of T, so doubling the pitch actually requires quadrupling the tension — which is why tuning pegs need fairly fine adjustment near the top of a string's safe range.

Assumptions & limitations

• Ideal flexible string (no bending stiffness).
• Both ends fixed (standard boundary condition for these harmonics).
• Uniform mass per unit length along the string.
• Small-amplitude vibration (linear wave equation applies).

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