In many situations, the wave impedance of the medium varies in a continuous manner rather than in discrete steps. This occurs, for example, in conical bores and flaring horns. In this section, based on , we will consider non-cylindrical acoustic tubes.
Waves in a horn can be analyzed as one-parameter waves, meaning that it is assumed that a constant-phase wavefront progresses uniformly along the horn. Each ``surface of constant phase'' composing the traveling wave has tangent planes normal to the horn axis and to the horn boundary. For cylindrical tubes, the surfaces of constant phase are planar, while for conical tubes, they are spherical [357,317,144]. The key property of a ``horn'' is that a traveling wave can propagate from one end to the other with negligible ``backscattering'' of the wave. Rather, it is smoothly ``guided'' from one end to the other. This is the meaning of saying that a horn is a ``waveguide''. The absence of backscattering means that the entire propagation path may be simulated using a pure delay line, which is very efficient computationally. Any losses, dispersion, or amplitude change due to horn radius variation (``spreading loss'') can be implemented where the wave exits the delay line to interact with other components.
We will first consider the elementary case of a conical acoustic tube. All smooth horns reduce to the conical case over sufficiently short distances, and the use of many conical sections, connected via scattering junctions, is often used to model an arbitrary bore shape . The conical case is also important because it is essentially the most general shape in which there are exact traveling-wave solutions (spherical waves) .
Beyond conical bore shapes, however, one can use a Sturm-Liouville formulation to solve for one-parameter-wave scattering parameters . In this formulation, the curvature of the bore's cross-section (more precisely, the curvature of the one-parameter wave's constant-phase surface area) is treated as a potential function that varies along the horn axis, and the solution is an eigenfunction of this potential. Sturm-Liouville analysis is well known in quantum mechanics for solving elastic scattering problems and for finding the wave functions (at various energy levels) for an electron in a nonuniform potential well.
When the one-parameter-wave assumption breaks down, and multiple acoustic modes are excited, the boundary element method (BEM) is an effective tool . The BEM computes the acoustic field from velocity data along any enclosing surface. There also exist numerical methods for simulating wave propagation in varying cross-sections that include ``mode conversion'' [336,13,117].
This section will be henceforth concerned with non-cylindrical tubes in which backscatter and mode-conversion can be neglected, as treated in . The only exact case is the cone, but smoothly varying horn shapes can be modeled approximately in this way.
Note that the cylindrical tube is a limiting case of a cone with its apex at infinity. Correspondingly, a plane wave is a limiting case of a spherical wave having infinite radius.
On a fundamental level, all pressure waves in 3D space are composed of spherical waves . You may have learned about the Huygens-Fresnel principle in a physics class when it covered waves . The Huygens-Fresnel principle states that the propagation of any wavefront can be modeled as the superposition of spherical waves emanating from all points along the wavefront [122, page 344]. This principle is especially valuable for intuitively understanding diffraction and related phenomena such as mode conversion (which happens, for example, when a plane wave in a horn hits a sharp bend or obstruction and breaks up into other kinds of waves in the horn).
The conical acoustic tube is a one-dimensional waveguide which propagates circular sections of spherical pressure waves in place of the plane wave which traverses a cylindrical acoustic tube [22,349]. The wave equation in the spherically symmetric case is given by
Spherical coordinates are appropriate because simple closed-form solutions to the wave equation are only possible when the forced boundary conditions lie along coordinate planes. In the case of a cone, the boundary conditions lie along a conical section of a sphere. It can be seen that the wave equation in a cone is identical to the wave equation in a cylinder, except that is replaced by . Thus, the solution is a superposition of left- and right-going traveling wave components, scaled by :
where and are arbitrary twice-differentiable continuous functions that are made specific by the boundary conditions. A function of may be interpreted as a fixed waveshape traveling to the right, (i.e., in the positive direction), with speed . Similarly, a function of may be seen as a wave traveling to the left (negative direction) at meters per second. The point corresponds to the tip of the cone (center of the sphere), and may be singular there.
In cylindrical tubes, the velocity wave is in phase with the pressure wave. This is not the case with conical or more general tubes. The velocity of a traveling may be computed from the corresponding traveling pressure wave by dividing by the wave impedance.
A discrete-time simulation of the above solution may be obtained by simply sampling the traveling-wave amplitude at intervals of seconds, which implies a spatial sampling interval of meters. Sampling is carried out mathematically by the change of variables
A more compact simulation diagram which stands for either sampled or continuous simulation is shown in Figure C.44. The figure emphasizes that the ideal, lossless waveguide is simulated by a bidirectional delay line.
As in the case of uniform waveguides, the digital simulation of the traveling-wave solution to the lossless wave equation in spherical coordinates is exact at the sampling instants, to within numerical precision, provided that the traveling waveshapes are initially bandlimited to less than half the sampling frequency. Also as before, bandlimited interpolation can be used to provide time samples or position samples at points off the simulation grid. Extensions to include losses, such as air absorption and thermal conduction, or dispersion, can be carried out as described in §2.3 and §C.5 for plane-wave propagation (through a uniform wave impedance).
The simulation of Fig.C.44 suffices to simulate an isolated conical frustum, but what if we wish to interconnect two or more conical bores? Even more importantly, what driving-point impedance does a mouthpiece ``see'' when attached to the narrow end of a conical bore? The preceding only considered pressure-wave behavior. We must now also find the velocity wave, and form their ratio to obtain the driving-point impedance of a conical tube.
Momentum Conservation in Nonuniform Tubes
Newton's second law ``force equals mass times acceleration'' implies that the pressure gradient in a gas is proportional to the acceleration of a differential volume element in the gas. Let denote the area of the surface of constant phase at radial coordinate in the tube. Then the total force acting on the surface due to pressure is , as shown in Fig.C.45.
The net force to the right across the volume element
between and is then
where, when time and/or position arguments have been dropped, as in the last line above, they are all understood to be and , respectively. To apply Newton's second law equating net force to mass times acceleration, we need the mass of the volume element
where denotes air density.
or, dividing through by ,
In terms of the logarithmic derivative of , this can be written
Note that denotes small-signal acoustic pressure, while denotes the full gas density (not just an acoustic perturbation in the density). We may therefore treat as a constant.
In the case of cylindrical tubes, the logarithmic derivative of the area variation, ln, is zero, and Eq.(C.148) reduces to the usual momentum conservation equation encountered when deriving the wave equation for plane waves [318,349,47]. The present case reduces to the cylindrical case when
If we look at sinusoidal spatial waves, and , then and , and the condition for cylindrical-wave behavior becomes , i.e., the spatial frequency of the wall variation must be much less than that of the wave. Another way to say this is that the wall must be approximately flat across a wavelength. This is true for smooth horns/bores at sufficiently high wave frequencies.
Wave Impedance in a Cone
i.e., it can be expressed in terms of its own time derivative. This is a general property of any traveling wave.
Referring to Fig.C.46, the area function can be written for any cone in terms of the distance from its apex as
We can now solve for the wave impedance in each direction, where
the wave impedance may be defined (§7.1)
as the Laplace transform of the traveling pressure divided by
the Laplace transform of the corresponding traveling velocity wave:
We introduce the shorthand
Note that for a cylindrical tube, the wave impedance in both directions is , and there is no frequency dependence. A wavelength or more away from the conical tip, i.e., for , where is the spatial wavelength, the wave impedance again approaches that of a cylindrical bore. However, in conical musical instruments, the fundamental wavelength is typically twice the bore length, so the complex nature of the wave impedance is important throughout the bore and approaches being purely imaginary near the mouthpiece. This is especially relevant to conical-bore double-reeds, such as the bassoon.
Writing the wave impedance as
Up to now, we have been defining wave impedance as pressure divided by particle velocity. In acoustic tubes, volume velocity is what is conserved at a junction between two different acoustic tube types. Therefore, in acoustic tubes, we define the wave impedance as the ratio of pressure to volume velocity
This is the wave impedance we use to compute the generalized reflection and transmission coefficients at a change in cross-sectional area and/or taper angle in a conical acoustic tube. Note that it has a zero at and a pole at .
In this case, the equivalent mass is . It would perhaps be more satisfying if the equivalent mass in the conical wave impedance were instead which is the mass of air contained in a cylinder of radius projected back to the tip of the cone. However, the ``acoustic mass'' cannot be physically equivalent to mechanical mass. To see this, consider that the impedance of a mechanical mass is which is in physical units of mass per unit time, and by definition of mechanical impedance this equals force over velocity. The impedance in an acoustic tube, on the other hand, must be in units of pressure (force/area) divided by volume velocity (velocity area) and this reduces to
The real part of the wave impedance corresponds to transportation of wave energy, the imaginary part is a so-called ``reactance'' and does not correspond to power transfer. Instead, it corresponds to a ``standing wave'' which is created by equal and opposite power flow, or an ``evanescent wave'' (§C.8.2), which is a non-propagating, exponentially decaying, limiting form of a traveling wave in which the ``propagation constant'' is purely imaginary due to being at a frequency above or below a ``cut off'' frequency for the waveguide [295,122]. Driving an ideal mass at the end of a waveguide results in total reflection of all incident wave energy along with a quarter-cycle phase shift. Another interpretation is that the traveling wave becomes a standing wave at the tip of the cone. This is one way to see how the resonances of a cone can be the same as those of a cylinder the same length which is open on both ends. (One might first expect the cone to behave like a cylinder which is open on one end and closed on the other.) Because the impedance approaches a purely imaginary zero at the tip, it looks like a mass (with impedance ). The ``piston of air'' at the open end similarly looks like a mass .
The wave impedance derivation above made use of known properties of waves in cones to arrive at the wave impedances in the two directions of travel in cones. We now consider how this solution might be generalized to arbitrary bore shapes. The momentum conservation equation is already applicable to any wavefront area variation :
Defining the spatially instantaneous phase velocity as
This reduces to the simple case of the uniform waveguide when the logarithmic derivative of cross-sectional area is small compared with the logarithmic derivative of the amplitude which is proportional to the instantaneous spatial frequency. A traveling wave solution interpretation makes sense when the instantaneous wavenumber is approximately real, and the phase velocity is approximately constant over a number of wavelengths .
Generalized Wave Impedance
The horizontal axis (taken along the boundary of the cone) is chosen so that corresponds to the apex of the cone. Let denote the cross-sectional area of the bore.
Since a piecewise-cylindrical approximation to a general acoustic tube can be regarded as a ``zeroth-order hold'' approximation. A piecewise conical approximation then uses first-order (linear) segments. One might expect that quadratic, cubic, etc., would give better and better approximations. However, such a power series expansion has a problem: In zero-order sections (cylinders), plane waves propagate as traveling waves. In first-order sections (conical sections), spherical waves propagate as traveling waves. However, there are no traveling wave types for higher-order waveguide flare (e.g., quadratic or higher) .
Since the digital waveguide model for a conic section is no more expensive to implement than that for a cylindrical section, (both are simply bidirectional delay lines), it would seem that modeling accuracy can be greatly improved for non-cylindrical bores (or parts of bores such as the bell) essentially for free. However, while the conic section itself costs nothing extra to implement, the scattering junctions between adjoining cone segments are more expensive computationally than those connecting cylindrical segments. However, the extra expense can be small. Instead of a single, real, reflection coefficient occurring at the interface between two cylinders of differing diameter, we obtain a first-order reflection filter at the interface between two cone sections of differing taper angle, as seen in the next section.
Generalized Scattering Coefficients
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 .
The Digital Waveguide Oscillator