#### 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.

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Cylindrical Tubes

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