#### Reflectance of the Conical Cap

Let
denote the time to propagate across the length of
the cone in one direction. As is well known [22], the reflectance
at the tip of an infinite cone is for pressure waves. *I.e.*, it
reflects like an open-ended cylinder. We ignore any absorption losses
propagating in the cone, so that the transfer function from the entrance of
the cone to the tip is
. Similarly, the transfer function from
the conical tip back to the entrance is also
. The complete
reflection transfer function from the entrance to the tip and back is then

(C.155) |

Note that this is the reflectance a distance from a conical tip

*inside*the cone.

We now want to interface the conical cap reflectance to the cylinder. Since this entails a change in taper angle, there will be reflection and transmission filtering at the cylinder-cone junction given by Eq.(C.154) and Eq.(C.155).

From inside the *cylinder*, immediately next to the cylinder-cone
junction shown in Fig.C.48, the reflectance of the conical cap is
readily derived from Fig.C.48b and Equations (C.154) and
(C.155) to be

(C.156) |

where

(C.157) |

is the numerator of the conical cap reflectance, and

(C.158) |

is the denominator. Note that for very large , the conical cap reflectance approaches which coincides with the impedance of a length open-end cylinder, as expected.

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Stability Proof

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Scattering Filters at the Cylinder-Cone Junction