Digital Waveguide Model:
Ideal String Struck by a Mass

We have derived an ``analog waveguide model'' for the ideal string struck by an ideal mass. We now digitize that model.

• Delays implemented using digital delay lines
• Mass reflectance digitize using the bilinear transform
• Force waves chosen

Mass Reflectance:

$$\hat{\rho}(s) = \frac{ms}{ms+2R} = \frac{1}{1+\frac{2R}{ms}} \protect$$ (1)

Bilinear Transform:

$$s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$

where $T =$ sampling interval, yields

$$\hat{\rho}_d(z) = \frac{1}{1+\frac{2R}{m}\frac{T}{2}\frac{1+z^{-1}}{1-z^{-1}}} = g\frac{1-z^{-1}}{1-pz^{-1}}$$