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
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.151) and Eq. (C.152).
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.151) and
(C.152) to be