Digital Waveguide Scheme

We now derive the digital waveguide formulation by sampling the traveling-wave solution to the wave equation. It is easily checked that the lossless 1D wave equation $Ky''=\epsilon {\ddot y}$ is solved by any string shape $y$ which travels to the left or right with speed $c\isdeftext \sqrt{K/\epsilon }$ [6]. Denote right-going traveling waves in general by $y_r(t-x/c)$ and left-going traveling waves by $y_l(t+x/c)$, where $y_r$ and $y_l$ are assumed twice-differentiable. Then, as is well known, the general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$$y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right). \protect$$ (5)

Sampling these traveling-wave solutions yields

$$y(nT,mX) = y_r(nT- mX/c) + y_l(nT+ mX/c)$$

$$= y_r[(n-m)T] + y_l[(n+m)T]$$

$$\isdef y^{+}(n-m) + y^{-}(n+m) \protect$$ (6)

where a ``$+$'' superscript denotes a ``right-going'' traveling-wave component, and ``$-$'' denotes propagation to the ``left''. This notation is similar to that used for acoustic-tube modeling of speech [17].

Figure 2: 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.)

Figure 2 shows a signal flow diagram for the computational model of Eq. (6), which is often called a digital waveguide model (for the ideal string in this case) [25,29]. Note that, by the sampling theorem, it is an exact model so long as the initial conditions and any ongoing additive excitations are bandlimited to less than half the temporal sampling rate $f_s = 1/T$ [27, Appendix G].

Note also 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, even though spatial samples have been eliminated from explicit consideration. (The arguments of $y^{+}$ and $y^{-}$ have physical units of time.)

The left- and right-going traveling wave components are summed to produce a physical output according to

$$y(nT,mX) = y^{+}(n-m) + y^{-}(n+m) \protect$$ (7)

In Fig. 2, ``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 filter structures [17], except for the delays along the upper rail, the absence of scattering junctions, and the direct physical interpretation.