Frequencies for a Stiff StringThe equation of motion for a stiff string under tension is:
where the first term represents the net force on the string due to the tension T, the second term involves the net force due to the bending moments and shear forces and depends on the elastic modulus (Young's modulus Y), the cross-sectional area S and the radius of gyration κ, and the quantity on the right of the equal sign represents the inertia property (mass per unit length times acceleration) of the string. For a cylindrical string, the radius of gyration equals half the radius, κ = a/2.
At low frequencies (with long wavelengths), the strings behaves as if it were completely flexible, and transverse waves on the string travel with a speed associated with normal transverse waves on a flexible string. At high frequencies (with short wavelengths) the string acts more like a stiff bar, and transverse waves show a dispersive effect, traveling with the speed of flexural bending waves which depends on frequency.
If the string was perfectly flexible, then the resonances frequencies of the string would be harmonics such that the higher frequencies would be exactly integer multiples of the fundamental frequency, fn = n fo. However, for a stiff string, the frequencies are not exactly integer multiples, and the string exhibits inharmonicity.
Fletcher and Rossing define an inharmonicity constant
to demonstrate the effect that bending stiffness has on the frequencies for the strings of a musical instrument. For a string which is "fixed" a both ends with a pinned boundary condition (also called "simply supported), the frequency of the n-th partial of the string is
where fo is the fundamental frequency of the string. If the string has clamped boundary conditions at both ends, then the n-th partial of the string is given by
In both cases, the frequencies of the string are no longer harmonics.