Interior Scattering Junctions

A so-called Kelly-Lochbaum scattering junction [301,458] 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 [458].

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'' [458]) 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