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Reflectance of Ideal Mass, Spring, and Dashpot

For a mass $ m$ kg, the impedance and reflectance are respectively

\begin{eqnarray*}
R_m(s) &=& ms \\ [5pt]
\,\,\Rightarrow\,\,S_m(s) &=& \frac{ms-R_0}{ms+R_0} %
\end{eqnarray*}

This reflectance is a stable first-order allpass filter, as expected, since energy is not dissipated by a mass.

For a spring $ k$ N/m, we have

\begin{eqnarray*}
R_k(s) &=& \frac{k}{s} \\ [10pt]
\,\,\Rightarrow\,\,S_k(s) &=& \frac{\frac{k}{s}-R_0}{\frac{k}{s}+R_0}
\end{eqnarray*}

also allpass as expected.

For a dashpot $ \mu$ N$ \,$ s/m, we have

\begin{eqnarray*}
R_\mu(s) &=& \mu \\ [5pt]
\,\,\Rightarrow\,\,S_\mu(s) &=& \frac{\mu-R_0}{\mu+R_0}
\end{eqnarray*}


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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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