Digital Waveguide Models

We may begin with the PDE for the ideal 1D wave equation:

$$y''= c^2{\ddot y}$$

where

$$\begin{eqnarray*} c &=& \mbox{traveling-wave propagation speed}\\ y(t,x) &=& \mbox{displacement at time $t$\ and position $x$} \end{eqnarray*}$$

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

• Recall that the ideal wave equation derives directly from Newton's laws $f=ma$

• In the case of vibrating strings, the wave equation is derived from first principles to be
  $$\begin{eqnarray*} Ky''&=& \epsilon {\ddot y}\\ [5pt] \mbox{(Restoring Force} &=& \mbox{Mass Density times Acceleration)}, \end{eqnarray*}$$
  

  where

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

• The left-hand side of the wave equation (the restoring force as tension times ``curvature''), was first derived by Brook Taylor of ``Taylor series'' fame

• Thus, it turns out that the propagation speed $c$ can be written in terms of the string tension $K$ and mass density $\epsilon$ as
  $$c=\sqrt{\frac{K}{\epsilon }}$$

• As has been known since d'Alembert, the 1D wave equation is obeyed by arbitrary traveling waves at speed $c$ :
  $$y(t,x) \eqsp y_r(t-x/c) + y_l(t+x/c)$$
  (Just plug $y_r(t-x/c)$ or $y_l(t+x/c)$ or any linear combination of them into the wave equation to verify this)

• Next, the traveling-waves are sampled:
  $$\begin{eqnarray*} y(nT,mX) &=& y_r(nT-mX/c) + y_l(nT+mX/c)\quad \mbox{($X\isdef cT$)}\\ [5pt] &=& y_r(nT-mT) + y_l(nT+mT)\\ [5pt] &\isdef & y^{+}(n-m) + y^{-}(n+m) \end{eqnarray*}$$
  

  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 for the case of an $N$ -sample section of string or air column