A String Fixed at both Ends: Reflections and Initialization

Real strings are of finite length and fixed at both ends. Thus, we need to account for the boundary conditions imposed at the boundaries.
If we consider a string to be fixed at a position , the boundary condition at that point for transverse displacement is for all time. From the traveling-wave solution to the wave equation, we then have $y_{r}(t - L/c) = -y_{l}(t + L/c)$ , which indicates that displacement traveling waves reflect from a fixed end with an inversion (or a reflection coefficient of -1).
The simulation of displacement wave motion in a string rigidly terminated at one end, with losses commuted with the reflection at the other end by digital filter , and initialized with a plucked string shape is diagrammed in the figure below.

Figure 19: Digital waveguide simulation of commuted lossy wave propagation on a string fixed at both ends (with pluck initialization).
$\begin{figure}\begin{center} \begin{picture}(3.6,1.5) \put(0,0){\epsfig{file=f... ...H(z)$} \put (1.76,0.71){$y(\xi, n T)$} \end{picture} \end{center} \end{figure}$
Since this system is linear, we can commute the -1 multiplier through the upper delayline and combine the two delaylines, as shown in Fig. 20, which now looks very close to the KS block diagram.

Figure 20: A further commuted version of the digital waveguide block diagram shown above.
$\begin{figure}\begin{center} \begin{picture}(3.6,1.5) \put(0,0){\epsfig{file=f... ...$H(z)$} \put (1.5,0.17){$y(\xi, n T)$} \end{picture} \end{center} \end{figure}$
While psuedo-random numbers (noise) are not an accurate physical representation of an initial string shape, the feedback comb filter quickly filters out signal energy that does not fall within the resonances. And the high signal energy creates a very spectrally rich response.