Scattering Filters at the Cylinder-Cone Junction

As derived earlier, the wave impedance at frequency $\omega$ rad/sec in a converging cone is given by

$$Z_\xi(j\omega) = \frac{\rho c}{S(\xi)} \frac{j\omega}{j\omega-c/\xi}$$ (G.112)

where $\xi$ is the distance to the apex of the cone, $S(\xi)$ is the cross-sectional area, and $\rho c$ is the wave impedance in open air. A cylindrical tube is the special case $\xi=\infty$, giving $Z_\infty(j\omega) = \rho c/S$, independent of position in the tube. Under normal assumptions such as pressure continuity and flow conservation at the cylinder-cone junction (see, e.g., [300]), the junction reflection transfer function (reflectance) seen from the cylinder looking into the cone is derived to be

$$R(s) = -\frac{c/\xi}{c/\xi - 2s}$$ (G.113)

(where $s$ is the Laplace transform variable which generalizes $s=j\omega$) while the junction transmission transfer function (transmittance) to the right is given by

$$T(s) = 1 + R(s) = -\frac{2s}{c/\xi - 2s}$$ (G.114)

The reflectance and transmittance from the right of the junction are the same when there is no wavefront area discontinuity at the junction [300]. Both $R(s)$ and $T(s)$ are first-order transfer functions: They each have a single real pole at $s=c/(2\xi)$. Since this pole is in the right-half plane, it corresponds to an unstable one-pole filter.