### Summary

Up to now, we have been concerned with the simulation of
*traveling waves* in *linear, time-invariant (LTI) media*.
The main example considered was wave propagation in air, but waves on
vibrating strings behave analogously. We saw that the point-to-point
propagation of a traveling plane wave in an LTI medium can be
simulated simply using only a *delay line* and an *LTI
filter*. The delay line simulates propagation delay, while the filter
further simulates (1) an independent attenuation factor at each
frequency by means of its amplitude response (*e.g.*, to simulate air
absorption), and (2) a frequency-dependent propagation speed using its
phase response (to simulate dispersion). If there is additionally
spherical spreading loss, the amplitude may be further attenuated by
, where is the distance from the source. For more details
about the acoustics of plane waves and spherical waves, see, *e.g.*,
[318,349]. Appendix B contains a bit more about
elementary acoustics,

So far we have considered only traveling waves going in one direction.
The next simplest case is 1D acoustic systems, such as vibrating
strings and acoustic tubes, in which traveling waves may propagate in
*two* directions. Such systems are simulated using a pair of
delay lines called a *digital waveguide*.

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Physical Outputs

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Dispersive Traveling Waves