## Frequencies for a Stiff String

The equation of motion for a stiff string under tension is:$T\frac{{\partial}^{2}y}{\partial {x}^{2}}-\text{}YS{\kappa}^{2}\frac{{\partial}^{4}y}{\partial {x}^{4}}=\text{}\rho S\frac{{\partial}^{2}y}{\partial {t}^{2}}$

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 $c=\sqrt{\frac{T}{{\rho}_{\ell}}}$ 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 $v\text{}=\text{}\sqrt{\omega \kappa \sqrt{Y/{\rho}_{\ell}}}$ 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, *f _{n} = n f_{o}*. However, for a stiff string, the frequencies are not exactly integer multiples, and the string exhibits

*inharmonicity*.

Fletcher and Rossing^{[1]} define an *inharmonicity constant*

$B=\frac{{\pi}^{2}YS{\kappa}^{2}}{T{L}^{2}}=\frac{{\pi}^{2}Y\left(\pi {a}^{2}\right)\left({a}^{2}/4\right)}{T{L}^{2}}$

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

^{[2]}

${f}_{n}=n{f}_{o}\sqrt{1+B{n}^{2}}$

where*f*is the fundamental frequency of the string. If the string has clamped boundary conditions at both ends, then the

_{o}*n*-th partial of the string is given by

${f}_{n}=n{f}_{o}\sqrt{1+B{n}^{2}}(1\text{}+\text{}\frac{2}{\pi}\sqrt{B}+\frac{4}{\pi}B)$

In both cases, the frequencies of the string are no longer harmonics.