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.
Scattering Filters at the Cylinder-Cone JunctionAs derived in §C.18.4, the wave impedance (for volume velocity) at frequency rad/sec in a converging cone is given by
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 CapLet 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.
Stability ProofA 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
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
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.
- 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 .
reflectance has no poles in the strict right-half plane. For passivity, we also need to show that its magnitude is bounded by unity for all on the axis. We have
reflectance has at least one root at . In this subsection we investigate this ``dc behavior'' of the cone more thoroughly. 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