Resonance of a string

Strings under tension, as in instruments such as lutes, harps, guitars, pianos, violins and so forth, have resonant frequencies directly related to the mass, length, and tension of the string. The wavelength that will create the first resonance on the string is equal to twice the length of the string. Higher resonances correspond to wavelengths that are integer divisions of the fundamental wavelength. The corresponding frequencies are related to the speed v of a wave traveling down the string by the equation

f = {nv \over 2L}

where L is the length of the string (for a string fixed at both ends) and n = 1, 2, 3... The speed of a wave through a string or wire is related to its tension T and the mass per unit length ρ:

v = \sqrt {T \over \rho}

So the frequency is related to the properties of the string by the equation

f = {n\sqrt {T \over \rho} \over 2 L} = {n\sqrt {T \over m / L} \over 2 L}

where T is the tension, ρ is the mass per unit length, and m is the total mass.

Higher tension and shorter lengths increase the resonant frequencies. When the string is excited with an impulsive function (a finger pluck or a strike by a hammer), the string vibrates at all the frequencies present in the impulse (an impulsive function theoretically contains 'all' frequencies). Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.


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Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.






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