Digital Waveguide Plucked-String Model

When plucking a string, it is necessary to detect ``collisions'' between the plectrum and string. Also, more complete plucked-string models will allow the string to ``buzz'' on the frets and ``slap'' against obstacles such as the fingerboard. For these reasons, it is convenient to choose displacement waves for the waveguide string model. The reflection and transmission filters for displacement waves are the same as for velocity, namely, $\hat{\rho}_y(s) = -\hat{\rho}_f(s)$ and $\hat{\tau}_y(s) = 1-\hat{\rho}_f(s)$ .

As in the mass-string collision case, we obtain the one-filter scattering-junction implementation shown in Fig.9.23. The filter $\hat{\rho}_f(s)$ may now be digitized using the bilinear transform as previously (§9.3.1).

Figure 9.23: Displacement-wave scattering model for a spring.