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Sampled Traveling Waves in any Digital Waveguide

\begin{displaymath}
\begin{array}{rclcl}
x &\to& x_m&=& mX\nonumber \\ [5pt]
t &\to& t_n&=& nT\nonumber
\end{array}\end{displaymath}

$ \Rightarrow$

\begin{eqnarray*}
y(t_n,x_m) &\,\mathrel{\mathop=}\,& y_r(t_n- x_m/c) + y_l(t_n+ x_m/c)
\\ [5pt]
&\,\mathrel{\mathop=}\,& y_r(nT- mX/c) + y_l(nT+ mX/c) \nonumber \\ [5pt]
&\,\mathrel{\mathop=}\,& y_r\left[(n-m)T\right]+ y_l\left[(n+m)T\right]\nonumber \\ [5pt]
&\,\mathrel{\mathop=}\,& y^{+}(n-m) + y^{-}(n+m) \nonumber
\end{eqnarray*}

when $ X=cT$ , where we defined

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



Subsections
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``Digitizing Strings Waves in Vibrating Strings'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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