Driven Terminated Strings

Figure 7 depicts a digital waveguide model for a rigidly terminated ideal string excited at an interior point $x=mX$. It is easy to show using elementary block-diagram manipulations [161] that the system of Fig. 8 is equivalent. The advantage of the second form is that it can be implemented as a cascade of standard comb filter sections.⁹

Figure 7: External string excitation at a point.

Figure 8: System equivalent to that in Fig. 7.

When damping is introduced, the string model becomes as shown in Fig. 9 [156], shown for the case of displacement waves (changing to velocity or force waves would only change the signal name from $y$ to either $v$ or $f$, respectively in this simple example). Also shown are two ``pick-up'' points at $x=0$ and $x=2X$ which read out the physical string displacements $y(t_n,x_0)$ and $y(t_n,x_2)$.¹⁰ The symbol $z^{-1}$ means a one-sample delay. Because delay-elements and gains $g$ commute (i.e., can be reordered arbitrarily in cascade), the diagram in Fig. 10 is equivalent, when round-off error can be ignored. (In the presence of round-off error after multiplications, the commuted form is more accurate because it implements fewer multiplies.) This example illustrates how internal string losses may be consolidated sparsely within a waveguide in order to simplify computations (by as much as three orders of magnitude in practical simulations, since there are typically hundreds of (delay-element,gain) pairs which can be replaced by one long delay line and single gain). The same results hold for dispersion in the string: the gains $g$ simply become digital filters $G(z)$ having any desired gain (damping) versus frequency, and any desired delay (dispersion) at each frequency [162].¹¹

Figure 9: Digital waveguide string simulation with damping.

Figure 10: Damped string simulation with commuted losses.