Remarks on the Mean Free Path Problem

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Abstract
The Jaeger-Sabine-Kosten mean free path lc is not the same as the mean free paths ls and lAN found by Knudsen and by Allred and Newhouse. All have in common the same mean free directed path length ld given by the ratio of the room volume to the apparent area presented in the specified direction, but they differ in the method used to compute the average of ld over all directions: lc is an apparent-area weighted average, which has the value 4V/S for rooms of any shape; ls and lAN are unweighted solid-angle and direction-cosine averages whose values differ slightly but systematically from 4V/S. The mean directed collision frequency vd is the inverse with respect to sound speed of the mean free directed path length (vdld=c). The unweighted solid-angle average of vd is cS/4V for rooms of any shape, but the apparent-area weighted average of vd differs systematically from cS/4V in about the same way that ls differs from 4V/S. Since the rate of sound decay in rooms is directly proportional to the frequency of wall reflections, the “collision frequency” concept is nominated to succeed the “free path” concept. However, the analysis of these averaging processes indicates that significant errors in the assessment of sound decay can be introduced by using either preaveraged mean free paths or preaveraged mean collision frequencies. These particular errors can be avoided by first computing the residual sound as a function of time for each representative direction of propagation before summing up these directional contributions at the receiving point.

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