Damping & Stiffness

Real wave propagation along strings involves losses and dispersion.
Propagation losses are generally frequency-dependent. In general, damping will increase with frequency.
Losses in strings have been investigated by Valette (1995), with the main mechanisms involving the viscous effect of the surrounding air, viscoelasticity and thermoelasticity of the string material, and internal friction representing both macroscopic rubbing (in multi-stranded and overwound strings) and inter-molecular effects (in monofilament strings).
Frequency-dependent losses could be determined on a per sample basis and implemented between each sample of delay in a digital waveguide model. However, such a scheme would be inefficient and the design of appropriate digital filters could be difficult.
Instead, we take advantage of the linear and time-invariant nature of the waveguide structure to commute and implemented the losses at just a few locations within the model.
In order to maintain stability (passivity), $\left\vert\hat{G}(e^{j \omega T})\right\vert \leq 1$ .
A string fixed at both ends can be implemented with two delay lines, a discrete-time filter $\hat{G}(z)$ representing losses and inversion at the bridge, and an inversion representing reflection from the nut, as shown below. Additional simplifications are possible.

Figure 6: Digital waveguide simulation of dispersive wave propagation on a string fixed at both ends.
$\begin{figure}\begin{center} \begin{picture}(3.6,1.5) \put(0,0){\epsfig{file=f... ...G}(z)$} \put (1.7,0.71){$y(\xi, n T)$} \end{picture} \end{center} \end{figure}$
String stiffness produces dispersion, or a variation of wave propagation speed with frequency. The use of an allpass filter in the string loop is effective in accurately simulating such dispersion (to be discussed in a subsequent week).