### 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]:

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

*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