### Conical Acoustic Tubes

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

where

*pressure*at time and radial position along the cone axis (or wall). In terms of (rather than ), Eq.(C.145) expands to

*Webster's horn equation*[357]:

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.

#### Digital Simulation

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

Define

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.

**Next Section:**

Momentum Conservation in Nonuniform Tubes

**Previous Section:**

Horns as Waveguides