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Modes of a stretched string
Intermediate
Allowed frequencies are integer multiples of the fundamental.
f_n = nv/2LA string fixed at both ends supports wavelengths λ = 2L/n and frequencies fₙ = n v/2L = n f₁. The fundamental is the lowest mode.
L = 0.65 m, v = 400 m/s: f₁ = 308 Hz.
Only integer multiples are allowed.
In more detail
A string fixed at both ends supports standing waves with frequencies f_n = n v / 2L. The fundamental has one half wavelength on the string, and higher modes have more.
Real-world uses
- Guitar and violin makers choose string lengths and tensions to set pitches.
- Piano strings are designed to produce a series of harmonics.
- Violin strings produce harmonics whose frequencies are whole-number multiples of the fundamental, which give their characteristic sound.
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Frequently asked questions
What is the fundamental?
The lowest frequency mode, with one antinode in the middle.
Why do shorter strings sound higher?
Because the wavelength is shorter, so the frequency is higher for the same speed.
What is the fundamental frequency?
The lowest frequency at which the string vibrates, which sets the pitch we hear.
Related topics in Waves & Sound
Beats · Standing waves · Nodes and antinodes · Harmonics and overtones · Open organ pipe · Closed organ pipe