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Dispersive Traveling Waves

In many acoustic systems, such as piano strings9.4.1C.6), wave propagation is also significantly dispersive. A wave-propagation medium is said to be dispersive if the speed of wave propagation is not the same at all frequencies. As a result, a propagating wave shape will ``disperse'' (change shape) as its various frequency components travel at different speeds. Dispersive propagation in one direction can be simulated using a delay line in series with a nonlinear phase filter, as indicated in Fig.2.5. If there is no damping, the filter $ A(z)$ must be all-pass [452], i.e., $ \vert A(e^{j\omega T})\vert=1$ for all $ \omega T\in[-\pi,\pi]$ .

Figure 2.5: Dispersive traveling-wave simulator. In principle, the digital filter $ A(z)$ provides an arbitrary nonnegative delay corresponding to one sample of wave propagation at each frequency.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University