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Scattering Filters at the Cylinder-Cone Junction

Wave impedance at frequency $ \omega$ rad/sec in a converging cone:

$\displaystyle Z_\xi(j\omega) = \frac{\rho c}{S(\xi)} \cdot \frac{j\omega}{j\omega-c/\xi}
\qquad\hbox{(converging cone impedance)}
$

where

\begin{eqnarray*}
\xi &=& \hbox{distance to the apex of the cone} \\
S(\xi) &=& \hbox{cross-sectional area of cone} \\
\rho c &=& \hbox{wave impedance in open air}
\end{eqnarray*}

In the limit as $ \xi\to\infty$ ,

$\displaystyle Z_\infty(j\omega) = \frac{\rho c}{S} \qquad \hbox{(cylindrical tube impedance)}
$

Reflectance of the conical cap, seen from cylinder:

$\displaystyle R(s) = -\frac{c/\xi}{c/\xi - 2s}
$

Transmittance to the right:

$\displaystyle T(s) = 1 + R(s) = -\frac{2s}{c/\xi - 2s}
$


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``Stability Proof for a Cylindrical Bore with Conical Cap'', 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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