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

and this substitution in Eq. (C.143) gives

Define

where is arbitrarily chosen as the position along the cone closest to the tip. (We avoid a sample at the tip itself, where a pressure singularity may exist.) Then a section of the ideal cone can be simulated as shown in Figure C.43 (where pressure outputs are shown for and ). A particle-velocity output is formed by dividing the traveling pressure waves by the wave impedance of air. Since the wave impedance is now a function of frequency and propagation direction (as derived below), a digital filter will replace what was a real number for cylindrical tubes.

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.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University