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Relation of Sampled D'Alembert Traveling Waves to the Finite Difference Approximation

Recall FDA result [based on $ {\dot x}(n) \approx x(n)-x(n-1)$]:

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

Traveling-wave decomposition (exact in lossless case at sampling instants):

$\displaystyle y(n,m) = y^{+}(n-m) + y^{-}(n+m)

Substituting into FDA gives

y(n+1,m) &=& y(n,m+1) + y(n,m-1) - y(n-1,m) \\
&=& y^{+}(n-m...
...onumber \\
&\mathrel{\stackrel{\Delta}{=}}& y(n+1,m) \nonumber

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``Distributed Modeling in Discrete Time'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University