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As derived in §C.18.4, the wave impedance (for volume velocity)
at frequency
rad/sec in a converging cone is given by
![$\displaystyle Z_\xi(j\omega) = \frac{\rho c}{S(\xi)} \frac{j\omega}{j\omega-c/\xi}$](img4430.png) |
(C.173) |
where
is the distance to the apex of the cone,
is the
cross-sectional area, and
is the wave impedance in open air. A
cylindrical tube is the special case
, giving
, independent of position in the tube. Under
normal assumptions such as pressure continuity and flow conservation at the
cylinder-cone junction (see, e.g.,
[302]), the junction reflection transfer
function (reflectance) seen from the cylinder looking into the cone is
derived to be
![$\displaystyle R(s) = -\frac{c/\xi}{c/\xi - 2s}$](img4434.png) |
(C.174) |
(where
is the Laplace transform variable which generalizes
)
while the junction transmission transfer function
(transmittance) to the right is given by
![$\displaystyle T(s) = 1 + R(s) = -\frac{2s}{c/\xi - 2s}$](img4435.png) |
(C.175) |
The reflectance and transmittance from the right of the junction are the
same when there is no wavefront area discontinuity at the junction
[302]. Both
and
are first-order
transfer functions: They each have a single real pole at
.
Since this pole is in the right-half plane, it corresponds to an unstable
one-pole filter.
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