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Reflectance of Conical Cap Seen from Cylinder

From the figure, we can derive the conical cap reflectance to be

\begin{eqnarray*}
R_J(s) &=& \frac{R(s) + 2 R(s) R_{t_{\xi}}(s) + R_{t_{\xi}}(s)}{1 - R(s)R_{t_{\xi}}(s)} \\
&=& \frac{1 + (1+2s{t_{\xi}})R_{t_{\xi}}(s)}{2s{t_{\xi}}-1-R_{t_{\xi}}(s)} \\
&=& \frac{1 - (1+2s{t_{\xi}})e^{-2s{t_{\xi}}}}{2s{t_{\xi}}-1+e^{-2s{t_{\xi}}}} \\
&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{N(s)}{D(s)}
\end{eqnarray*}

For very large $ {t_{\xi}}$ , the conical cap reflectance approaches $ R_J =
-e^{-2s{t_{\xi}}}$ which coincides with the impedance of a length $ \xi=c{t_{\xi}}$ open-end cylinder, as expected.


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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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