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

Here, denotes the reflection coefficient ``seen'' from left to right, and 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 . When it is caused by an externally imposed termination (such as a plectrum or piano-hammer touching the string), we have , and the reflectances may become filters instead of real values in . 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 (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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.