Piano String Wave Equation

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Physical Audio Signal Processing

Stiff Piano Strings

Damping-Filter Design

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A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

A transverse wave causes a disturbance in the medium perpendicular to the direction in which the wave moves. — Click for http://en.wikipedia.org/wiki/Transverse

In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

The wave equation refers to the partial differential equation governing wave phenomena in elastic media such as air and isotropic solids. — Click for https://en.wikipedia.org/wiki/Wave_equation

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Physical Audio Signal Processing

Stiff Piano Strings

Damping-Filter Design

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

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

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 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 is frequency-independent, while the damping due 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 © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Piano String Wave Equation

``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

``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