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The Finite Difference Approximation

In the musical acoustics literature, the normal method for creating a computational model from a differential equation is to apply the so-called finite difference approximation (FDA) in which differentiation is replaced by a finite difference (see Appendix D) [481,311]. For example


$\displaystyle {\dot y}(t,x)\approx \frac{y(t,x)-y(t-T,x)}{T} \protect$ (C.2)

and

$\displaystyle y'(t,x)\approx \frac{y(t,x)-y(t,x-X)}{X} \protect$ (C.3)

where $ T$ is the time sampling interval to be used in the simulation, and $ X$ is a spatial sampling interval. These approximations can be seen as arising directly from the definitions of the partial derivatives with respect to $ t$ and $ x$. The approximations become exact in the limit as $ T$ and $ X$ approach zero. To avoid a delay error, the second-order finite-differences are defined with a compensating time shift:

$\displaystyle {\ddot y}(t,x) \approx \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2} \protect$ (C.4)

$\displaystyle y''(t,x) \approx \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$ (C.5)

The odd-order derivative approximations suffer a half-sample delay error while all even order cases can be compensated as above.

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)


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Traveling-Wave Solution
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The Ideal Vibrating String