User Guide

Sound System Design Reference Manual
by a single number, α. Only one step remains to
complete our model. Since sound travels at a known
rate of speed, the mean free path is equivalent to a
certain mean free time between bounces.
Now imagine what must happen if we apply our
model to the situation that exists in a room
immediately after a uniformly emitting sound source
has been turned off. The sound waves continue to
travel for a distance equal to the mean free path. At
this point they encounter a boundary surface having
an absorption coefficient of α and a certain
percentage of the energy is lost. The remaining
energy is reflected back into the room and again
travels a distance equal to the mean free path before
encountering another boundary with absorption
coefficient α. Each time sound is bounced off a new
surface, its energy is decreased by a proportion
determined by the average absorption coefficient α.
If we know the proportion of energy lost with
each bounce and the length of time between
bounces, we can calculate the average rate of decay
and the reverberation time for a particular room.
Example: Consider a room 5m x 6m x 3m, as
diagrammed in Figure 5-8. Let us calculate the decay
rate and reverberation time for the octave band
centered at 1 kHz.
The volume of the room is 90 cubic meters, and
its total surface area is 126 square meters; therefore,
the MFP works out to be about 3 meters.
The next step is to list individually the areas
and absorption coefficient of the various materials
used on room surfaces.
The total surface area is 126 square meters;
the total absorption (Sα) adds up to 24.9 absorption
units. Therefore, the average absorption coefficient
(α) is 24.9 divided by 126, or .2.
If each reflection results in a decrease in
energy of 0.2, the reflected wave must have an
equivalent energy of 0.8. A ratio of 0.8 to 1 is
equivalent to a loss of 0.97 decibel per reflection. For
simplicity, let us call it 1 dB per reflection.
Since the MFP is 2.9 meters, the mean free
time must be about 0.008 seconds (2.9/334 = 0.008).
We now know that the rate of decay is
equivalent to 1 dB per 0.008 seconds. The time for
sound to decay 60 dB must, therefore, be:
60 x 0.008 = 0.48 seconds.
The Eyring equation in its standard form is
shown in Figure 5-9. If this equation is used to
calculate the reverberation of our hypothetical room,
the answer comes out 0.482 seconds. If the Sabine
formula is used to calculate the reverberation time of
this room, it provides an answer of 0.535 seconds or
a discrepancy of a little more than 10%.
5-9
Figure 5-10. Reverberation time chart, SI units