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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 [445, pp. 430-431]. A more detailed derivation, with examples and exploration of implications, appears in Appendix E. Recall from (C.6) that the time update recursion for the ideal string digitized via the FDA is given by

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m). \protect$ (C.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
$\displaystyle y(n+1,m)$ $\displaystyle =$ $\displaystyle y(n,m+1) + y(n,m-1) - y(n-1,m)$ (C.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

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

which says the displacement at time $ n+1$ , position $ m$ , is the superposition of the right-going and left-going traveling wave components at positions $ m-1$ and $ m+1$ , respectively, from time $ n$ . In other words, the physical wave variable can be computed for the next time step as the sum of incoming traveling wave components from the left and right. This picture also underscores the lossless nature of the computation.

This results extends readily to the digital waveguide mesh (§C.14), which is essentially a lattice-work of digital waveguides for simulating membranes and volumes. The equivalence is important in higher dimensions because the finite-difference model requires less computations per node than the digital waveguide approach.

Even in one dimension, the digital waveguide and finite-difference methods have unique advantages in particular situations, and as a result they are often combined together to form a hybrid traveling-wave/physical-variable simulation [354,355,223,124,123,225,265,224]. In this hybrid simulations, the traveling-wave variables are called ``W variables'' (where `W' stands for ``Wave''), while the physical variables are caled ``K variables'' (where `K' stands for ``Kirchoff''). Each K variable, such as displacement $ y(nT,mX)$ on a vibrating string, can be regarded as the sum of two traveling-wave components, or W variables:

$\displaystyle y(nT,mX) = y_r(nT-mX/c) + y_l(nT+mX/c) $

Conversion between K variables and W variables can be non-trivial due to the non-local dependence of one set of state variables on the other, in general. A detailed examination of this issue is given in Appendix E.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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