Scattering Filters at the Cylinder-Cone Junction
where is the distance to the apex of the cone, is the cross-sectional area, and is the wave impedance in open air. A cylindrical tube is the special case , giving , independent of position in the tube. Under normal assumptions such as pressure continuity and flow conservation at the cylinder-cone junction (see, e.g., ), the junction reflection transfer function (reflectance) seen from the cylinder looking into the cone is derived to be
(where is the Laplace transform variable which generalizes ) while the junction transmission transfer function (transmittance) to the right is given by
The reflectance and transmittance from the right of the junction are the same when there is no wavefront area discontinuity at the junction . Both and are first-order transfer functions: They each have a single real pole at . Since this pole is in the right-half plane, it corresponds to an unstable one-pole filter.
Reflectance of the Conical Cap