Standing Waves – Spectrum of a Guitar String

Let us answer the question whether the spectrum is a unique fingerprint of the musical instrument used. It is best explained using a single vibrating string, the so-called monochord.

Plucking causes the string to vibrate. We cannot, however, hear the oscillation directly. The vibrating string must be surrounded by air, so that the air molecules can carry the vibration to our ears, at the speed of sound, in the form of fluctuations in pressure and density.

We can record the time course of the sound wave using a computer. The Fourier transform reveals the mixture of frequencies that form the sound of the string.

It is evident that there is a lowest frequency f₀ – the so-called fundamental tone. Moreover, there is an equidistant spectrum. That means, all other amplitudes have a frequency that is two times, three times, four times, or, in general, n-times the frequency of the fundamental tone. All these overtones – along with the fundamental tone – form the spectrum of the oscillating string.

Is it possible to infer the characteristics of the oscillating string, such as its length, based on this spectrum? Can we use the wave to draw unequivocal conclusions about the source?