Scattering Filters at the Cylinder-Cone Junction

As derived in §C.18.4, the wave impedance (for volume velocity) at frequency rad/sec in a converging cone is given by

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., [300]), 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 [300]. 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.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.