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Interior Scattering Junctions

A so-called Kelly-Lochbaum scattering junction [299,450] can be introduced into the string at the fourth sample by the following perturbation

   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \rightleftharpoons}}{\mathbf{A}}$}$$\displaystyle _K=$   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$\displaystyle _K+
(1-k_l){\bm \Delta}_{5,3} +
k_r {\bm \Delta}_{5,8} +
k_l {\bm \Delta}_{6,3} +
(1-k_r){\bm \Delta}_{6,8}.

Here, $ k_l$ denotes the reflection coefficient ``seen'' from left to right, and $ k_r$ is the reflectance of the junction from the right. When the scattering junction is caused by a change in string density or tension, we have $ k_r=-k_l$ . When it is caused by an externally imposed termination (such as a plectrum or piano-hammer touching the string), we have $ k_r=k_l$ , and the reflectances may become filters instead of real values in $ [-1,1]$ . Energy conservation demands that the transmission coefficients be amplitude complementary with respect to the reflection coefficients [450].

A single time-varying scattering junction provides a reasonable model for plucking, striking, or bowing a string at a point. Several adjacent scattering junctions can model a distributed interaction, such as a piano hammer, finger, or finite-width bow spanning several string samples.

Note that scattering junctions separated by one spatial sample (as typical in ``digital waveguide filters'' [450]) will couple the formerly independent subgrids. If scattering junctions are confined to one subgrid, they are separated by two samples of delay instead of one, resulting in round-trip transfer functions of the form $ H(z^2)$ (as occurs in the digital waveguide mesh). In the context of a half-rate staggered-grid scheme, they can provide general IIR filtering in the form of a ladder digital filter [299,450].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University