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Physical Audio Signal Processing
    Digital Waveguide Theory
       The Finite Difference Approximation
          FDA of the Ideal String

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FDA of the Ideal String

Substituting the FDA into the wave equation gives

$\displaystyle K\frac{y(t,x+X) - 2 y(t,x) + y(t,x-X)}{X^2} = \epsilon \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x)}{T^2} $

which can be solved to yield the following recursion for the string displacement:
$\displaystyle y(t+T,x)$ $\displaystyle =$ $\displaystyle \frac{KT^2}{\epsilon X^2} \left[ y(t,x+X) - 2 y(t,x) + y(t,x-X)\right]$  
    $\displaystyle \qquad\qquad\qquad\qquad + 2 y(t,x) - y(t-T,x) \protect$  

In a practical implementation, it is common to set $ T=1,\, X=(\sqrt{K/\epsilon })T$, and evaluate on the integers $ t=nT=n$ and $ x=mX=m$ to obtain the difference equation

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m). \protect$ (C.6)

Thus, to update the sampled string displacement, past values are needed for each point along the string at time instants $ n$ and $ n-1$. Then the above recursion can be carried out for time $ n+1$ by iterating over all $ m$ along the string.

Perhaps surprisingly, it is shown in Appendix E that the above recursion is exact at the sample points in spite of the apparent crudeness of the finite difference approximation [442]. The FDA approach to numerical simulation was used by Pierre Ruiz in his work on vibrating strings [392], and it is still in use today [74,75].

When more terms are added to the wave equation, corresponding to complex losses and dispersion characteristics, more terms of the form $ y(n-l,m-k)$ appear in (C.6). These higher-order terms correspond to frequency-dependent losses and/or dispersion characteristics in the FDA. All linear differential equations with constant coefficients give rise to some linear, time-invariant discrete-time system via the FDA. A general subclass of the linear, time-invariant case giving rise to ``filtered waveguides'' is

$\displaystyle \sum_{k=0}^\infty \alpha_k \frac{\partial^k y(t,x)}{\partial t^k} = \sum_{l=0}^\infty \beta_l \frac{\partial^l y(t,x)}{\partial x^l},$ (C.7)

while the fully general linear, time-invariant 2D case is

$\displaystyle \sum_{k=0}^\infty \sum_{l=0}^\infty \alpha_{k,l} \frac{\partial^k... ...nfty \beta_{m,n} \frac{\partial^m\partial^n y(t,x)}{\partial x^m \partial x^n}.$ (C.8)

A nonlinear example is

$\displaystyle \frac{\partial y(t,x)}{\partial t} = \left(\frac{\partial y(t,x)}{\partial x}\right)^2,$ (C.9)

and a time-varying example can be given by

$\displaystyle \frac{\partial y(t,x)}{\partial t} = t^2\frac{\partial y(t,x)}{\partial x}.$ (C.10)

Previous: The Finite Difference Approximation
Next: Traveling-Wave Solution

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About the Author: 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.


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