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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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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