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Physical Audio Signal Processing
    Digital Waveguide Theory
       Sampled Traveling Waves
          Relation to the Finite Difference Recursion

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Relation to the Finite Difference Recursion

In this section we will show that the digital waveguide simulation technique is equivalent to the recursion produced by the finite difference approximation (FDA) applied to the wave equation [453, pp. 430-431]. A more detailed derivation, with examples and exploration of implications, appears in Appendix P. Recall from (H.6) that the time update recursion for the ideal string digitized via the FDA is given by

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m)$ (H.18)

To compare this with the waveguide description, we substitute the traveling-wave decomposition $ y(n,m) = y^{+}(n-m) + y^{-}(n+m)$ (which is exact in the ideal case at the sampling instants) into the right-hand side of the FDA recursion above and see how good is the approximation to the left-hand side $ y(n+1,m) = y^{+}(n+1-m) + y^{-}(n+1+m)$. Doing this gives
$\displaystyle y(n+1,m)$ $\displaystyle =$ $\displaystyle y(n,m+1) + y(n,m-1) - y(n-1,m)$ (H.19)
  $\displaystyle =$ $\displaystyle y^{+}(n-m-1) + y^{-}(n+m+1)$  
    $\displaystyle + y^{+}(n-m+1) + y^{-}(n+m-1)$  
    $\displaystyle - y^{+}(n-m-1) - y^{-}(n+m-1)$  
  $\displaystyle =$ $\displaystyle y^{-}(n+m+1) + y^{+}(n-m+1)$  
  $\displaystyle =$ $\displaystyle y^{+}[(n+1)-m] + y^{-}[(n+1)+m]$  
  $\displaystyle \isdef$ $\displaystyle y(n+1,m)$  

Thus, we obtain the result that the FDA recursion is also exact in the lossless case, because it is equivalent to the digital waveguide method which we know is exact at the sampling points. This is surprising since the FDA introduces artificial damping when applied to lumped, mass-spring systems, as discussed earlier.

The last identity above can be rewritten as

$\displaystyle y(n+1,m)$ $\displaystyle \isdef$ $\displaystyle y^{+}[(n+1)-m] + y^{-}[(n+1)+m]$ (H.20)
  $\displaystyle =$ $\displaystyle y^{+}[n-(m-1)] + y^{-}[n+(m+1)]$  

which says the displacement at time