Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Piano String Wave Equation

A wave equation suitable for modeling linearized piano strings is given by [77,45,320,521]

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

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

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{second moment of area (m$\null^4$)}

See, e.g., [145, p. 64] for a detailed derivation.

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.

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University