Digital Waveguide Modeling Elements

As mentioned above, digital waveguide models are built out of digital delay-lines and filters (and nonlinear elements), and they can be understood as propagating and filtering sampled traveling-wave solutions to the wave equation (PDE), such as for air, strings, rods, and the like [437,441]. It is noteworthy that strings, woodwinds, and brasses comprise three of the four principal sections of a classical orchestra (all but percussion). The digital waveguide modeling approach has also been extended to propagation in 2D, 3D, and beyond [522,399,526,403]. They are not finite-difference models, but paradoxically they are equivalent under certain conditions (Appendix E). A summary of historical aspects appears in §A.9.

As mentioned at Eq.(1.1), the ideal wave equation comes directly from Newton's laws of motion ($f=ma$ ). For example, in the case of vibrating strings, the wave equation is derived from first principles (in Chapter 6, and more completely in Appendix C) to be

$$\begin{eqnarray*} Ky''&=& \epsilon {\ddot y}\\ [5pt] \mbox{(Restoring Force Density} &=& \mbox{Mass Density times Acceleration)}, \end{eqnarray*}$$

where

$$\begin{eqnarray*} K& \isdef & \mbox{string tension}\\ \epsilon & \isdef & \mbox{linear mass density}. \end{eqnarray*}$$

Defining $c=\sqrt{K/\epsilon }$ , we obtain the usual form of the PDE known as the ideal 1D wave equation.

$$y''= \frac{1}{c^2}{\ddot y} \protect$$ (2.15)

where $y(t,x)$ is the string displacement at time $t$ and position $x$ . (We omit the time and position arguments $(t,x)$ when they are the same for all signal terms in the equation.) For example, $y$ can be the transverse displacement of an ideal stretched string or the longitudinal displacement (or pressure, velocity, etc.) in an air column. The independent variables are time $t$ and the distance $x$ along the string or air-column axis. The partial-derivative notation is more completely written out as

$$\begin{eqnarray*} {\ddot y}& \isdef & \frac{\partial^2}{\partial t^2} y(t,x)\\ [5pt] y''& \isdef & \frac{\partial^2}{\partial x^2} y(t,x). \end{eqnarray*}$$

As has been known since d'Alembert [100], the 1D wave equation is obeyed by arbitrary traveling waves at speed $c$ :

$$y(t,x) \eqsp y_r(t-x/c) + y_l(t+x/c)$$

To show this, just plug $y_r(t-x/c)$ or $y_l(t+x/c)$ (or any linear combination of them) into the wave equation Eq.(1.15). Thus, $c=\sqrt{K/\epsilon }$ is the traveling-wave propagation speed expressed in terms of the string tension $K$ and mass density $\epsilon$ .

In digital waveguide modeling, the traveling-waves are sampled:

$$\begin{eqnarray*} y(nT,mX) &=& y_r(nT-mX/c) + y_l(nT+mX/c)\qquad \mbox{(set $X=cT$)}\\ [5pt] &=& y_r(nT-mT) + y_l(nT+mT)\\ [5pt] &\isdef &y^{+}(n-m) + y^{-}(n+m) \end{eqnarray*}$$

where $T$ denotes the time sampling interval in seconds, $X=cT$ denotes the spatial sampling interval in meters, and $y^{+}$ and $y^{-}$ are defined for notational convenience.

An ideal string (or air column) can thus be simulated using a bidirectional delay line, as shown in Fig.1.13 for the case of an $N$ -sample section of ideal string or air column. The ``$R$ '' label denotes its wave impedance (§6.1.5) which is needed when connecting digital waveguides to each other and to other kinds of computational physical models (such as finite difference schemes). While propagation speed on an ideal string is $c=\sqrt{K/\epsilon }$ , we will derive (§C.7.3) that the wave impedance is $R=\sqrt{K\epsilon }$ .

Figure 1.13: A digital waveguide--a sampled traveling-wave simulation for waves in ideal strings or acoustic tubes having wave impedance $R$ .

Figure 1.14 (from Chapter 6, §6.3), illustrates a simple digital waveguide model for rigidly terminated vibrating strings (more specifically, one polarization-plane of transverse vibration). The traveling-wave components are taken to be displacement samples, but the diagram for velocity-wave and acceleration-wave simulation are identical (inverting reflection at each rigid termination). The output signal $y(nT,\xi)$ is formed by summing traveling-wave components at the desired ``virtual pickup'' location (position $x=\xi$ in this example). To drive the string at a particular point, one simply takes the transpose [452] of the output sum, i.e., the input excitation is summed equally into the left- and right-going delay-lines at the same $x$ position (details will be discussed near Fig.6.14).

Figure 1.14: Digital waveguide model of a rigidly terminated ideal string, with a displacement output indicated at position $x=\xi$ . Rigid terminations reflect traveling displacement, velocity, or acceleration waves with a sign inversion. Slope or force waves reflect with no sign inversion.

In Chapter 9 (example applications), we will discuss digital waveguide models for single-reed instruments such as the clarinet (Fig.1.15), and bowed-string instruments (Fig.1.16) such as the violin.

Figure: Digital waveguide model of a single-reed, cylindrical-bore woodwind, such as a clarinet (copy of Fig.9.39).

Figure: Digital waveguide model for a bowed-string instrument, such as a violin (copy of Fig.9.52).