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Interior Scattering Junctions

A so-called Kelly-Lochbaum scattering junction [297,447] 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 [447].

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'' [447]) 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 [297,447].

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Reflectance of a General Lumped Waveguide Termination
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Reactive Terminations