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Mean Free Path

A rough guide to the average delay-line length is the ``mean free path'' in the desired reverberant environment. The mean free path is defined as the average distance a ray of sound travels before it encounters an obstacle and reflects. An approximate value for the mean free path, due to Sabine, an early pioneer of statistical room acoustics, is

$\displaystyle {\overline d} = 4\frac{V}{S}\qquad\hbox{(mean free path)}

where $ V$ is the total volume of the room, and $ S$ is total surface area enclosing the room. This approximation requires the diffuse field assumption, i.e., that plane waves are traveling randomly in all directions [352,47] (see §3.2.1 for a simple construction). Normally, late reverberation satisfies this assumption well, away from open doors and windows, provided the room is not too ``dead''. Regarding each delay line as a mean-free-path delay, the average can be set to the mean free path by equating

$\displaystyle \frac{{\overline d}}{cT} = \frac{1}{N} \sum_{i=1}^N M_i

where $ c$ denotes sound speed and $ T$ denotes the sampling period. This number should be treated as a lower bound because in real rooms reflections are often diffuse, especially at high frequencies. In a diffuse reflection, a single incident plane wave reflects in many directions at once.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University