Digital Waveguide Models

In the same way they can be used to simulate traveling waves in free space (our first simple physical model), delay lines can be used to simulate one-dimensional (1D) wave propagation along strings or in acoustic tubes. Since waves can propagate simultaneously in two opposite directions along a 1D waveguide, two delay lines are necessary (wave simulation, waves on a string).

• Wave travel in an ideal (no losses or dispersion) 1D medium along the $x$ axis (such as a stretched string) is described mathematically by the one-dimension wave equation as:
  $$\frac{\partial^{2} y}{\partial t^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}},$$
  where $y$ is the direction perpendicular to the $x$ axis and $c$ is the speed of wave motion in the medium.

• A discrete time and space solution to the 1D lossless wave equation is given by
    $$y\left(t_{n},x_{m}\right) = y^{+}(n - m) + y^{-}(n + m),$$ (3)
  where the superscripts $+$ and $-$ denote wave components travelling to the right and left, respectively, with time indices $n$ and spatial indices $m$.

• The spatial sampling interval is given by $X = c T$ meters, or the distance traveled by a wave in one temporal sampling interval. In this way, each traveling-wave component moves left or right one spatial sample for each time sample.

• Physical wave variables are given by the superposition of traveling waves. In a one-dimensional system, we can use two sets of unit delay elements to model left- and right-going traveling waves and sum delay-line values at corresponding “spatial” locations to obtain physical outputs, as depicted below.

  Figure 16: Discrete-time simulation of ideal, lossless wave propagation with observation points at $x = 0$ and $x = 3 X = 3 cT$.

  

• Any ideal, lossless, one-dimensional waveguide can be simulated in this way. The model is exact at the sampling instants to within the numerical precision of the processing system.

• In many modeling contexts, the calculation of physical output values can be limited to just one or two discrete spatial locations. In this case, the unit delays are combined and represented by digital delay lines, as shown below.

  Figure 17: Digital waveguide simulation of ideal, lossless wave propagation using delay lines.