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FDS for the Ideal String

Consider first the ideal (lossless, dispersionless, unforced) string wave equation:

$\displaystyle \frac{\partial^{2}y}{\partial t^{2}} = c^{2}\frac{\partial^{2}y}{\partial x^{2}}
$

Replacing the second-derivatives by their finite-difference approximations gives

\begin{eqnarray*}
y_{m}^{n-1}-2y_{m}^{n}+y_{m}^{n+1} = \frac{c^{2}T^{2}}{X^{2}}
(y_{m-1}^{n}-2y_{m}^{n}+y_{m+1}^{n})
\end{eqnarray*}

If we choose $ X = cT$ (which is most natural physically), the equation reduces further to

\begin{eqnarray*}
y_{m}^{n+1} = y_{m-1}^{n}+y_{m+1}^{n}-y_{m}^{n-1}
\end{eqnarray*}

Let's examine this recursion on the time-space grid, assuming for the moment no boundary conditions:


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Download PianoString.pdf
Download PianoString_2up.pdf
Download PianoString_4up.pdf

``Modal Synthesis of a Piano String'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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