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Signal Scattering

The digital waveguide was introduced in §2.4. A basic fact from acoustics is that traveling waves only happen in a uniform medium. For a medium to be uniform, its wave impedance3.17must be constant. When a traveling wave encounters a change in the wave impedance, it will reflect, at least partially. If the reflection is not total, it will also partially transmit into the new impedance. This is called scattering of the traveling wave.

Let $ R_1$ denote the constant impedance in some waveguide, such as a stretched steel string or acoustic bore. Then signal scattering is caused by a change in wave impedance from $ R_1$ to $ R_2$ . We can depict the partial reflection and transmission as shown in Fig.2.33.

Figure 2.33: Signal scattering at a junction of different wave impedances $ R_1$ & $ R_2$ .
\includegraphics{eps/BidirectionalDelayLineScat}

The computation of reflection and transmission in both directions, as shown in Fig.2.33 is called a scattering junction.

As derived in Appendix C, for force or pressure waves, the reflection coefficient $ k_1$ is given by

$\displaystyle k_1 = \frac{R_2-R_1}{R_2+R_1} \protect$ (3.18)

That is, the coefficient of reflection for a traveling pressure wave leaving impedance $ R_1$ and entering impedance $ R_2$ is given by the impedance step over the impedance sum. The reflection coefficient $ k_1$ fully characterizes the scattering junction.

For velocity traveling waves, the reflection coefficient is just the negative of that for force/pressure waves, or $ -k_1$ (see Appendix C).

Signal scattering is lossless, i.e., wave energy is neither created nor destroyed. An implication of this is that the transmission coefficient for a traveling pressure wave leaving impedance $ R_1$ and entering impedance $ R_2$ is given by

$\displaystyle t_1 = 1 + k_1.
$

For velocity waves, the transmission coefficient is $ t_1 = 1-k_1$ , which is perhaps more intuitive.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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