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.