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}
$](http://www.dsprelated.com/josimages_new/pasp/img3211.png)
In a practical implementation, it is common to set
![$ T=1,\,
X=(\sqrt{K/\epsilon })T$](http://www.dsprelated.com/josimages_new/pasp/img3215.png)
![$ t=nT=n$](http://www.dsprelated.com/josimages_new/pasp/img3216.png)
![$ x=mX=m$](http://www.dsprelated.com/josimages_new/pasp/img3217.png)
Thus, to update the sampled string displacement, past values are needed for each point along the string at time instants
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ n-1$](http://www.dsprelated.com/josimages_new/pasp/img1696.png)
![$ n+1$](http://www.dsprelated.com/josimages_new/pasp/img923.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
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
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
![]() |
(C.7) |
while the fully general linear, time-invariant 2D case is
![]() |
(C.8) |
A nonlinear example is
![]() |
(C.9) |
and a time-varying example can be given by
![]() |
(C.10) |
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Traveling-Wave Partial Derivatives
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3D Sound