## Lossy Acoustic Propagation

Attenuation of waves by spherical spreading, as described in
§2.2.5 above, is not the only source of amplitude decay
in a traveling wave. In air, there is always significant additional
loss caused by *air absorption*. Air absorption varies with
frequency, with high frequencies usually being more attenuated than
low frequencies, as discussed in §B.7.15. Wave
propagation in *vibrating strings* undergoes an analogous
absorption loss, as does the propagation of nearly every other kind of
wave in the physical world. To simulate such propagation losses, we
can use a delay line in series with a nondispersive filter, as
illustrated in §2.2.2 above. In practice, the desired attenuation
at each frequency becomes the desired magnitude frequency-response of
the filter in Fig.2.4, and filter-design software
(typically matlab) is used to compute the filter coefficients to
approximate the desired frequency response in some optimal way. The
phase response may be linear, minimum, or left unconstrained when
damping-filter dispersion is not considered harmful. There is
typically a frequency-dependent weighting on the approximation error
corresponding to audio perceptual importance (*e.g.*, the weighting
is a simple example that increases accuracy at low frequencies).
Some filter-design methods are summarized in §8.6.

### Exponentially Decaying Traveling Waves

Let
denote the decay factor associated with
propagation of a plane wave over distance at frequency
rad/sec. For an ideal plane wave, there is no ``spreading
loss'' (attenuation by ). Under uniform conditions, the
amount of attenuation (in dB) is proportional to the distance
traveled; in other words, the attenuation factors for two successive
segments of a propagation path are *multiplicative*:

*exponential*function of distance .

^{3.3}

*Frequency-independent air
absorption* is easily modeled in an acoustic simulation by making
the substitution

### Frequency-Dependent Air-Absorption Filtering

More generally, *frequency-dependent* air
absorption can be modeled using the substitution

*filtering per sample*in the propagation medium. Since air absorption cannot amplify a wave at any frequency, we have . A lossy delay line for plane-wave simulation is thus described by

For spherical waves, the loss due to spherical spreading is of the form

### Dispersive Traveling Waves

In addition to frequency-dependent attenuation, LTI filters can
provide a *frequency-dependent delay*. This can be used to
simulate *dispersive wave
propagation*, as
introduced in §2.2.3.

### 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*.

**Next Section:**

Digital Waveguides

**Previous Section:**

Acoustic Wave Propagation Simulation