Reflectance of the Conical Cap
Let denote the time to propagate across the length of the cone in one direction. As is well known , 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
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
is the numerator of the conical cap reflectance, and
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.
Scattering Filters at the Cylinder-Cone Junction