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Physical Audio Signal Processing
    Piano
       The Piano String
          Wave Equation

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Wave Equation

A wave equation suitable for modeling piano strings is given by [80,45,325,525]

$\displaystyle f(t,x) = \epsilon{\ddot y}- K y''+ EIy''''+ R_0{\dot y}+ R_2 {\ddot y'} \protect$ (6.1)

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 $x$\ ... ... 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 F.

The first two terms on the right-hand side come from the ideal string wave equation (see Eq.$ \,$(H.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. The final two terms provide damping. The damping associated with $ R_0$ is frequency-independent, while the damping due to the $ R_2$ term increases with frequency.

In [46], the damping in real piano strings was modeled using a length 17 FIR filter for the lowest strings, and a length 9 FIR filter for the remaining strings.

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Next: Stiff-String Dispersion Filter Design
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.