As derived in §C.18.4, the wave impedance (for volume velocity)
at frequency
rad/sec in a converging cone is given by
![$\displaystyle Z_\xi(j\omega) = \frac{\rho c}{S(\xi)} \frac{j\omega}{j\omega-c/\xi}$](http://www.dsprelated.com/josimages_new/pasp/img4360.png) |
(C.152) |
where
![$ \xi$](http://www.dsprelated.com/josimages_new/pasp/img2268.png)
is the distance to the apex of the cone,
![$ S(\xi)$](http://www.dsprelated.com/josimages_new/pasp/img4361.png)
is the
cross-sectional area, and
![$ \rho c$](http://www.dsprelated.com/josimages_new/pasp/img4316.png)
is the wave
impedance in open air. A
cylindrical tube is the special case
![$ \xi=\infty$](http://www.dsprelated.com/josimages_new/pasp/img4362.png)
, giving
![$ Z_\infty(j\omega) = \rho c/S$](http://www.dsprelated.com/josimages_new/pasp/img4363.png)
, independent of position in the tube. Under
normal assumptions such as
pressure continuity and flow conservation at the
cylinder-cone junction (see,
e.g.,
[
300]), the junction reflection
transfer
function (
reflectance) seen from the cylinder looking into the cone is
derived to be
![$\displaystyle R(s) = -\frac{c/\xi}{c/\xi - 2s}$](http://www.dsprelated.com/josimages_new/pasp/img4364.png) |
(C.153) |
(where
![$ s$](http://www.dsprelated.com/josimages_new/pasp/img144.png)
is the
Laplace transform variable which generalizes
![$ s=j\omega$](http://www.dsprelated.com/josimages_new/pasp/img360.png)
)
while the junction transmission transfer function
(
transmittance) to the right is given by
![$\displaystyle T(s) = 1 + R(s) = -\frac{2s}{c/\xi - 2s}$](http://www.dsprelated.com/josimages_new/pasp/img4365.png) |
(C.154) |
The reflectance and transmittance from the right of the junction are the
same when there is no wavefront area discontinuity at the junction
[
300]. Both
![$ R(s)$](http://www.dsprelated.com/josimages_new/pasp/img153.png)
and
![$ T(s)$](http://www.dsprelated.com/josimages_new/pasp/img4366.png)
are first-order
transfer functions: They each have a single real
pole at
![$ s=c/(2\xi)$](http://www.dsprelated.com/josimages_new/pasp/img4367.png)
.
Since this pole is in the right-half plane, it corresponds to an unstable
one-pole
filter.
Next Section: Reflectance of the Conical CapPrevious Section: Cylindrical Tubes