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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.
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Frequency-Dependent Air-Absorption Filtering
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Program for Acoustic Echo Simulation