### Damped Traveling Waves

The delay line shown in Fig.2.1 on page can be used
to simulate any traveling wave, where the traveling wave must
propagate in one direction with a fixed waveshape. If a traveling
wave *attenuates* as it propagates, with the same attenuation
factor at each frequency, the attenuation can be simulated by a simple
*scaling* of the delay line output (or input), as shown in
Fig.2.3. This is perhaps the simplest example of
the important principle of *lumping distributed losses* at
discrete points. That is, it is not necessary to implement a small
attenuation for each time-step of wave propagation; the same
result is obtained at the delay-line output if propagation is
``lossless'' within the delay line, and the total cumulative
attenuation is applied at the output. The input-output
simulation is exact, while the signal samples inside the delay line
are simulated with a slight gain error. If the internal signals are
needed later, they can be tapped out using correcting gains. For
example, the signal half way along the delay line can be tapped using
a coefficient of in order to make it an exact second output.
In summary, computational efficiency can often be greatly increased at
no cost to accuracy by lumping losses only at the outputs and points
of interaction with other simulations.

Modeling traveling-wave attenuation by a scale factor is only exact
physically when all frequency components decay at the same rate. For
accurate acoustic modeling, it is usually necessary to replace the
constant scale factor by a *digital filter* which
implements *frequency-dependent attenuation*, as depicted in
Fig.2.4. In principle, a linear time-invariant (LTI) filter
can provide an independent attenuation factor at each frequency.
Section 2.3 addresses this case in more detail.
Frequency-dependent damping substitution will be used in artificial
reverberation design in §3.7.4.

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Dispersive Traveling Waves

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Traveling Waves