Bridge Reflectance

The bridge reflectance is needed as part of the loop filter in a digital waveguide model (Chapter 6).

As derived in §C.11.1, the force-wave reflectance of $R_b(s)$ seen on the string is

$$\hat{\rho}_b(s) \eqsp \frac{R_b(s)-R_0}{R_b(s)+R_0} \eqsp \frac{s^2+\frac{1}{m}(\mu-R_0)s + \omega_0^2}{s^2+\frac{1}{m}(\mu+R_0)s + \omega_0^2} \protect$$ (10.7)

where $R_0$ denotes the wave impedance of the ideal string, and $\omega_0\isdeftext \sqrt{k/m}$ denotes the resonance frequency in radians per second. The velocity reflectance is simply minus the force reflectance (§C.11.1).