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Chapters

Physical Audio Signal Processing
    Digital Waveguide Theory
       Non-Cylindrical Acoustic Tubes
          Cylinder with Conical Cap
             Reflectance of the Conical Cap

Chapter Contents:

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Reflectance of the Conical Cap

Let ${t_{\xi}}\isdef \xi/c$ 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 $e^{-s{t_{\xi}}}$ . Similarly, the transfer function from the conical tip back to the entrance is also $e^{-s{t_{\xi}}}$ . The complete reflection transfer function from the entrance to the tip and back is then

$\displaystyle R_{t_{\xi}}(s) = - e^{-2s{t_{\xi}}}$

(C.155)

Note that this is the reflectance a distance $\xi=c{t_{\xi}}$ from a conical tip inside the cone.

We now want to interface the conical cap reflectance $R_{t_{\xi}}(s)$ 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

$\displaystyle R_J(s)$	$\displaystyle =$	$\displaystyle \frac{R(s) + 2 R(s) R_{t_{\xi}}(s) + R_{t_{\xi}}(s)}{1 - R(s)R_{t... ...}}(s)} = \frac{1 + (1+2s{t_{\xi}})R_{t_{\xi}}(s)}{2s{t_{\xi}}-1-R_{t_{\xi}}(s)}$
	$\displaystyle =$	$\displaystyle \frac{1 - (1+2s{t_{\xi}})e^{-2s{t_{\xi}}}}{2s{t_{\xi}}-1+e^{-2s{t_{\xi}}}} \isdef \frac{N(s)}{D(s)}$	(C.156)

where

$\displaystyle N(s) \isdef 1 - e^{-2s{t_{\xi}}} - 2s{t_{\xi}}e^{-2s{t_{\xi}}}$

(C.157)

is the numerator of the conical cap reflectance, and

$\displaystyle D(s) \isdef 2 s{t_{\xi}}- 1 + e^{-2s{t_{\xi}}}$

(C.158)

is the denominator. Note that for very large ${t_{\xi}}$ , the conical cap reflectance approaches $R_J = -e^{-2s{t_{\xi}}}$ which coincides with the impedance of a length $\xi=c{t_{\xi}}$ open-end cylinder, as expected.

Previous: Scattering Filters at the Cylinder-Cone Junction
Next: Stability Proof

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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.

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