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The Stiff String

Stiffness in a vibrating string introduces a restoring force proportional to the bending angle of the string. As discussed further in §C.6, the usual stiffness term added to the wave equation for the ideal string yields

$\displaystyle \epsilon {\ddot y}= Ky''- \kappa y''''.
$

When this wave equation is solved in terms of traveling wavesC.6), it emerges that high-frequency wave components travel faster than low-frequency components. In other words, wave propagation in stiff strings is dispersive. (See §C.6 for details.)

Stiff-string models are commonly used in piano synthesis. In §9.4, further details of string models used in piano synthesis are described (§9.4.1).

Experiments with modified recordings of acoustic classical guitars indicate that overtone inharmonicity due to string-stiffness is generally not audible in nylon-string guitars, although just-noticeable-differences are possible for the 6th (lowest) string [226]. Such experiments may be carried out by retuning the partial overtones in a recorded sound sample so that they become exact harmonics. Such retuning is straightforward using sinusoidal modeling techniques [362,459].



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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