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Element Reflectance

Imposing physical continuity constraints across the junction:

\begin{eqnarray*}
F(s) &=& F_R(s)\\
0 &=& V(s) + V_R(s)
\end{eqnarray*}

with

\begin{eqnarray*}
F(s) &=& F^+(s) + F^-(s)\\
F_R(s) &=& F_R^+(s) + F_R^-(s)\\
V(s) &=& V^{+}(s)+V^{-}(s) =\frac{F^{+}(s)}{R_0} - \frac{F^{-}(s)}{R_0}\\
V_R(s) &=& V^{+}_R(s)+V^{-}_R(s) = \left[\frac{F^{+}_R(s)}{R(s)} - \frac{F^{-}_R(s)}{R(s)}\right]
\end{eqnarray*}

we obtain the reflection transfer function (``reflectance'') of the element with impedance $ R(s)$ :

$\displaystyle \fbox{$\displaystyle S_R(s) \isdef \frac{F^{-}(s)}{F^{+}(s)} = \frac{R(s)-R_0}{R(s)+R_0}$}
$

This is the impedance step over the impedance sum, the usual force-wave reflectance at an impedance discontinuity, but now in the Laplace domain.


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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2017-06-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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