Sampled Traveling Waves in any Digital Waveguide

$$\begin{array}{rclcl} x &\to& x_m&=& mX\nonumber \\ t &\to& t_n&=& nT\nonumber \end{array}$$

$\Rightarrow$

$$\begin{eqnarray*} y(t_n,x_m) &\,\mathrel{\mathop=}\,& y_r(t_n- x_m/c) + y_l(t_n... ... \\ &\,\mathrel{\mathop=}\,& y^{+}(n-m) + y^{-}(n+m) \nonumber \end{eqnarray*}$$

where we defined

$$y^{+}(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}y_r(nT) \qquad\qquad y^{-}(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}y_l(nT)$$

• ``$+$'' superscript $\implies$ right-going
• ``$-$'' superscript $\implies$ left-going
• $y_r\left[(n-m)T\right]= y^{+}(n-m) =$ output of $m$-sample delay line with input $y^{+}(n)$
• $y_l\left[(n+m)T\right]\mathrel{\stackrel{\scriptscriptstyle\mathrm{\Delta}}{\scriptstyle=}}y^{-}(n+m) =$ input to an $m$-sample delay line whose output is $y^{-}(n)$