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Piano String Wave Equation

A wave equation suitable for modeling linearized piano strings is given by [77,45,320,519]

$\displaystyle f(t,x) = \epsilon{\ddot y}- K y''+ EIy''''+ R_0{\dot y}+ R_2 {\ddot y'} \protect$ (10.30)

where the partial derivative notation $ y'$ and $ {\dot y}$ are defined on page [*], and

\begin{eqnarray*}
f(t,x) &=& \mbox{driving force density (N/m) at position $x$\ and time $t$}\\
\epsilon &=& \mbox{mass density (kg/m)}\\
K &=& \mbox{tension force along the string axis (N)}\\
E &=& \mbox{Young's modulus (N/m$\null^2$)}\\
I &=& \mbox{radius of gyration of the string cross-section (m).}
\end{eqnarray*}

Young's modulus and the radius of gyration are defined in Appendix B.

The first two terms on the right-hand side of Eq.$ \,$ (9.30) come from the ideal string wave equation (see Eq.$ \,$ (C.1)), and they model transverse acceleration and transverse restoring force due to tension, respectively. The term $ EIy''''$ approximates the transverse restoring force exerted by a stiff string when it is bent. In an ideal string with zero diameter, this force is zero; in an ideal rod (or bar), this term is dominant [320,263,170]. The final two terms provide damping. The damping associated with $ R_0$ is frequency-independent, while the damping due $ R_2$ increases with frequency.


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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
CCRMA