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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 $ 1/f$ is a simple example that increases accuracy at low frequencies). Some filter-design methods are summarized in §8.6.

Exponentially Decaying Traveling Waves

Let $ g(r,\omega)$ denote the decay factor associated with propagation of a plane wave over distance $ r$ at frequency $ \omega $ rad/sec. For an ideal plane wave, there is no ``spreading loss'' (attenuation by $ 1/r$). 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:

$\displaystyle g(r_1+r_2,\omega) =
g(r_1,\omega)g(r_2,\omega)
$

This property implies that $ g$ is an exponential function of distance $ r$.3.3

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

$\displaystyle z^{-1}\leftarrow gz^{-1}
$

in the transfer function of the simulating delay line, where $ g$ denotes the attenuation associated with propagation during one sampling period ($ T$ seconds). Thus, to simulate absorption corresponding to an $ M$-sample delay, the difference equation Eq.$ \,$(2.1) on page [*] becomes

$\displaystyle y(n) = g^Mx(n-M),
$

as depicted in Fig.2.9.


Frequency-Dependent Air-Absorption Filtering

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

$\displaystyle z^{-1}\leftarrow G(z)z^{-1}
$

where $ G(z)$ denotes the filtering per sample in the propagation medium. Since air absorption cannot amplify a wave at any frequency, we have $ \left \vert G(e^{j\omega T})\right \vert\leq 1$. A lossy delay line for plane-wave simulation is thus described by

$\displaystyle Y(z) = G^M(z) z^{-M}X(z)
$

in the frequency domain, and

$\displaystyle y(n) = \underbrace{g\ast g\ast \dots \ast g \, \ast }_{\hbox{$M$\ times}} x(n-M)
$

in the time domain, where `$ \ast $' denotes convolution, and $ g(n)$ is the impulse response of the per-sample loss filter $ G(z)$. The effect of $ G(z)$ on the poles of the system is discussed in §3.7.4.

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

$\displaystyle Y(z) \propto \frac{G^M(z) z^{-M}}{r}X(z)
$

where $ r$ is the distance from $ X$ to $ Y$. We see that the spherical spreading loss factor is ``hyperbolic'' in the propagation distance $ r$, while air absorption is exponential in $ r$.


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 $ 1/r$, where $ r$ 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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Digital Waveguides
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Acoustic Wave Propagation Simulation