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Solution to Lossy 1D Wave Equation

$\displaystyle y(t,x) = e^{-{\left(\mu/2\epsilon \right)}{x/c}} y_r\left(t-{x/c}\right)
+ e^{{\left(\mu/2\epsilon \right)}{x/c}} y_l\left(t+{x/c}\right)
$

Assumptions:

Components decay exponentially in direction of travel

Sampling with $ t = n T$, $ x = mX$, and $ X=cT$ gives

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

where $ g \mathrel{\stackrel{\Delta}{=}}e^{-{\mu T/2\epsilon }}$


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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
CCRMA