Consider a cylindrical acoustic tube adjoined to a converging conical cap, as depicted in Figure C.48a. We may consider the cylinder to be either open or closed on the left side, but everywhere else it is closed. Since such a physical system is obviously passive, an interesting test of acoustic theory is to check whether theory predicts passivity in this case.
It is well known that a growing exponential appears at the junction of two conical waveguides when the waves in one conical taper angle reflect from a section with a smaller (or more negative) taper angle [7,300,8,160,9]. The most natural way to model a growing exponential in discrete time is to use an unstable one-pole filter . Since unstable filters do not normally correspond to passive systems, we might at first expect passivity to not be predicted. However, it turns out that all unstable poles are ultimately canceled, and the model is stable after all, as we will see. Unfortunately, as is well known in the field of automatic control, it is not practical to attempt to cancel an unstable pole in a real system, even when it is digital. This is because round-off errors will grow exponentially in the unstable feedback loop and eventually dominate the output.
The need for an unstable filter to model reflection and transmission at a converging conical junction has precluded the use of a straightforward recursive filter model . Using special ``truncated infinite impulse response'' (TIIR) digital filters , an unstable recursive filter model can in fact be used in practice . All that is then required is that the infinite-precision system be passive, and this is what we will show in the special case of Fig.C.48.
Scattering Filters at the Cylinder-Cone Junction
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., ), the junction reflection transfer function (reflectance) seen from the cylinder looking into the cone is derived to be
(where is the Laplace transform variable which generalizes ) while the junction transmission transfer function (transmittance) to the right is given by
The reflectance and transmittance from the right of the junction are the same when there is no wavefront area discontinuity at the junction . 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.
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.
A transfer function is stable if there are no poles in the right-half plane. That is, for each zero of , we must have re. If this can be shown, along with , then the reflectance is shown to be passive. We must also study the system zeros (roots of ) in order to determine if there are any pole-zero cancellations (common factors in and ).
Since re if and only if re, for , we may set without loss of generality. Thus, we need only study the roots of
If this system is stable, we have stability also for all . Since is not a rational function of , the reflectance may have infinitely many poles and zeros.
Let's first consider the roots of the denominator
At any solution of , we must have
To obtain separate equations for the real and imaginary parts, set , where and are real, and take the real and imaginary parts of Eq.(C.161) to get
Both of these equations must hold at any pole of the reflectance. For
stability, we further require
, we obtain the somewhat simpler conditions
For any poles of on the axis, we have , and Eq.(C.163) reduces to
It is well known that the ``sinc function'' is less than in magnitude at all except . Therefore, Eq.(C.164) can hold only at .
We have so far proved that any poles on the axis must be at .
The same argument can be extended to the entire right-half plane as follows. Going back to the more general case of Eq.(C.163), we have
Since for all real , and since for , this equation clearly has no solutions in the right-half plane. Therefore, the only possible poles in the right-half plane, including the axis, are at .
In the left-half plane, there are many potential poles. Equation (C.162) has infinitely many solutions for each since the elementary inequality implies . Also, Eq.(C.163) has an increasing number of solutions as grows more and more negative; in the limit of , the number of solutions is infinite and given by the roots of ( for any integer ). However, note that at , the solutions of Eq.(C.162) converge to the roots of ( for any integer ). The only issue is that the solutions of Eq.(C.162) and Eq.(C.163) must occur together.
Figure C.49 plots the locus of real-part zeros (solutions to Eq.(C.162)) and imaginary-part zeros (Eq.(C.163)) in a portion the left-half plane. The roots at can be seen at the middle-right. Also, the asymptotic interlacing of these loci can be seen along the left edge of the plot. It is clear that the two loci must intersect at infinitely many points in the left-half plane near the intersections indicated in the graph. As becomes large, the intersections evidently converge to the peaks of the imaginary-part root locus (a log-sinc function rotated 90 degrees). At all frequencies , the roots occur near the log-sinc peaks, getting closer to the peaks as increases. The log-sinc peaks thus provide a reasonable estimate number and distribution in the left-half -plane. An outline of an analytic proof is as follows:
- Rotate the loci in Fig.C.49 counterclockwise by 90 degrees.
- Prove that the two root loci are continuous, single-valued functions of
(as the figure suggests).
- Prove that for
, there are infinitely many extrema
of the log-sinc function (imaginary-part root-locus) which have
negative curvature and which lie below (as the figure
suggests). The and lines are shown in the
figure as dotted lines.
- Prove that the other root locus (for the real part) has
infinitely many similar extrema which occur for (again as
the figure suggests).
- Prove that the two root-loci interlace at
(already done above).
- Then topologically, the continuous functions must cross at
infinitely many points in order to achieve interlacing at
The peaks of the log-sinc function not only indicate approximately where the left-half-plane roots occur
A hasty analysis based on the reflection and transmission filters in Equations (C.154) and (C.155) might conclude that the reflectance of the conical cap converges to at dc, since and . However, this would be incorrect. Instead, it is necessary to take the limit as of the complete conical cap reflectance :
We already discovered a root at in the denominator in the context of the preceding stability proof. However, note that the numerator also goes to zero at . This indicates a pole-zero cancellation at dc. To find the reflectance at dc, we may use L'Hospital's rule to obtain
and once again the limit is an indeterminate form. We therefore apply L'Hospital's rule again to obtain
Thus, two poles and zeros cancel at dc, and the dc reflectance is , not as an analysis based only on the scattering filters would indicate. From a physical point of view, it makes more sense that the cone should ``look like'' a simple rigid termination of the cylinder at dc, since its length becomes small compared with the wavelength in the limit.
Another method of showing this result is to form a Taylor series expansion
of the numerator and denominator:
Both series begin with the term which means both the numerator and denominator have two roots at . Hence, again the conclusion is two pole-zero cancellations at dc. The series for the conical cap reflectance can be shown to be
which approaches as .
An alternative analysis of this issue is given by Benade in .
Generalized Scattering Coefficients