Scattering Filters at the Cylinder-Cone Junction

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

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

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 $\xi\to\infty$ ,

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

Reflectance of the conical cap, seen from cylinder:

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

Transmittance to the right:

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

• $R(s)$ and $T(s)$ are first-order transfer functions, each having a single real pole at $s=c/(2\xi)$ $\Rightarrow$ unstable
• $R(s)$ and $T(s)$ identical from left and right given no wavefront area discontinuity.