Incorporating Damping Into The Model

To incorporate damping into the digital waveguide model, we need to increase one of the gain blocks slightly from -1 to $-g$, where $\vert g\vert < 1$. The improved model is depicted in Figure 4.

Figure 4: Basic digital waveguide of a vibrating string with damping

Let's now consider the effect of the damping more precisely. Every time a wave travels around the digital waveguide, its amplitude $a(t)$ is scaled by $g$. After $p$ trips around the waveguide, the amplitude is scaled by $g^p$. Since each trip lasts $NT$ seconds, the amplitude is scaled by $g^{\frac{\tau}{NT}}$ in $\tau$ seconds. If we let $\tau$ be the time constant as defined in the monochord laboratory assignment, then we have

$$g^{\frac{\tau}{NT}} = e^{-1}$$ (5)

and so

$$\tau = \frac{-NT}{ln(g)}$$ (6)

Finally, we can say that the amplitude of the waves flowing around the waveguide model $a(t)$ decreases approximately exponentially with time:

$$a(t) = Ae^{-t/\tau}$$ (7)

where $a(0)=A$ is the amplitude at time $t=0$.