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Physical Audio Signal Processing
    Digital Waveguide Theory
       Sampled Traveling Waves
          Digital Waveguide Model

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Digital Waveguide Model

Figure H.3: Digital simulation of the ideal, lossless waveguide with observation points at $ x=0$ and $ x=3X=3cT$. (The symbol ``$ z^{-1}$'' denotes a one-sample delay.)
\includegraphics[scale=0.9]{eps/fideal}

In this section, we interpret the sampled d'Alembert traveling-wave solution of the ideal wave equation as a digital filtering framework. This is an example of what are generally known as digital waveguide models [439,440,442,447,453].

The term $ y_r\left[(n-m)T\right]\isdef y^{+}(n-m)$ in Eq.$ \,$(H.16) can be thought of as the output of an $ m$-sample delay line whose input is $ y^{+}(n)$. In general, subtracting a positive number $ m$ from a time argument $ n$ corresponds to delaying the waveform by $ m$ samples. Since $ y^{+}$ is the right-going component, we draw its delay line with input $ y^{+}(n)$ on the left and its output $ y^{+}(n-m)$ on the right. This can be seen as the upper ``rail'' in Fig.H.3

Similarly, the term $ y_l\left[(n+m)T\right]\isdeftext y^{-}(n+m)$ can be thought of as the input to an $ m$-sample delay line whose output is $ y^{-}(n)$. Adding $ m$ to the time argument $ n$ produces an $ m$-sample waveform advance. Since $ y^{-}$ is the left-going component, it makes sense to draw the delay line with its input $ y^{-}(n+m)$ on the right and its output $ y^{-}(n)$ on the left. This can be seen as the lower ``rail'' in Fig.H.3.

Note that the position along the string, $ x_m = mX= m cT$ meters, is laid out from left to right in the diagram, giving a physical interpretation to the horizontal direction in the diagram. Finally, the left- and right-going traveling waves must be summed to produce a physical output according to the formula

$\displaystyle y(t_n,x_m) = y^{+}(n-m) + y^{-}(n+m)$ (H.17)

We may compute the physical string displacement at any spatial sampling point $ x_m$ by simply adding the upper and lower rails together at position $ m$ along the delay-line pair. In Fig.H.3, ``transverse displacement outputs'' have been arbitrarily placed at $ x=0$ and $ x=3X$. The diagram is similar to that of well known ladder and lattice digital