User Guide

Sound System Design Reference Manual
It is easier for us to comprehend this theoretical
state of affairs if energy growth and decay are plotted
on a decibel scale. This is what has been done in the
graph. In decibel relationships, the growth of sound
is very rapid and decay becomes a straight line. The
slope of the line represents the rate of decay in
decibels per second.
How closely does the behavior of sound in a
real room approach this statistical picture? Figure 5-7
shows actual chart recordings of the decay of sound
in a fairly absorptive room. Each chart was made by
using a one-third octave band of random noise as
the test signal. A sound level meter was located in
the reverberant sound field. (In practice several
readings would be taken at a number of different
locations in the room).
The upper graph illustrates a measurement
made in the band centered at 125 Hz. Note the great
fluctuations in the steady state level and similar
fluctuations as the sound intensity decreases. The
fluctuations are sufficiently great to make any “exact”
determination of the decay rate impossible. Instead,
a straight line which seems to represent the “best fit”
is drawn and its slope measured. In this case, the
slope of the line is such that sound pressure seems
to be decaying at a rate of 30 dB per 0.27 seconds.
This works out to a decay rate of 111 dB per second.
The lower chart shows a similar measurement
taken with the one-third octave band centered at 4
kHz. The fluctuations in level are not as pronounced,
and it is much easier to arrive at what seems to be
the correct slope of the sound decay. In this instance
sound pressure appears to be decreasing at a rate of
30 dB in 0.2 seconds, or a decay rate of 150 dB per
second.
Reverberation and Reverberation Time
The term decay rate is relatively unfamiliar;
usually we talk about reverberation time . Originally,
reverberation time was described simply as the
length of time required for a very loud sound to die
away to inaudibility. It was later defined in more
specific terms as the actual time required for sound
to decay 60 decibels. In both definitions it is
assumed that decay rate is uniform and that the
ambient noise level is low enough to be ignored.
In the real world, the decay rate in a particular
band of frequencies may not be uniform and it may
be very difficult to measure accurately over a total 60
dB range. Most acousticians are satisfied to measure
the first 30 dB decay after a test signal is turned off
and to use the slope of this portion of the curve to
define the average decay rate and thus the
reverberation time. In the example just given,
estimates must be made over a useful range of only
20 dB or so. However, the height of the chart paper
corresponds to a total range of 30 dB and this makes
calculation of reverberation time quite simple. At 125
Hz a sloping line drawn across the full width of the
chart paper is equivalent to a 30 dB decay in 0.27
seconds. Reverberation time (60 dB decay) must
therefore be twice this value, or 0.54 seconds.
Similarly, the same room has a reverberation time of
only 0.4 seconds in the 4 kHz band.
In his original work in architectural acoustics,
Sabine assumed the idealized exponential growth
and decay of sound we showed in Figure 5-6.
However, his equation based on this model was
found to be inaccurate in rooms having substantial
absorption. In other words, the Sabine equation
works well in live rooms, but not in moderately dead
ones. In the 1920’s and 1930’s, a great deal of work
was done in an effort to arrive at a model that would
more accurately describe the growth and decay of
sound in all types of rooms. On the basis of the
material presented thus far, let us see if we can
construct such a model.
We start by accepting the notion of a uniform
diffuse steady state sound field. Even though the
sound field in a real room may fluctuate, and
although it may not be exactly the same at every
point in the room, some sort of overall intensity
average seems to be a reasonable simplifying
assumption.
If we can average out variations in the sound
field throughout the room, perhaps we can also find
an average distance that sound can travel before
striking one of the boundary surfaces. This notion of
an average distance between bounces is more
accurately known as the mean free path (MFP) and
is a common statistical notion in other branches of
physics. For typical rooms, the MFP turns out to be
equal to 4V/S, where V is the enclosed volume and S
is the area of all the boundary surfaces.
Since sound waves will have bounced around
all parts of the room striking all of the boundary
surfaces in almost all possible angles before being
completely absorbed, it seems reasonable that there
should be some sort of average absorption
coefficient α which would describe the total boundary
surface area. We will use the simple arithmetic
averaging technique to calculate this coefficient.
At this point we have postulated a highly
simplified acoustical model which assumes that, on
the average, the steady state sound intensity in an
actual room can be represented by a single number.
We also have assumed that, on the average, sound
waves in this room travel a distance equivalent to
MFP between bounces. Finally, we have assumed
that, on the average, each time sound encounters a
boundary surface it impinges upon a material having
a random incidence absorption coefficient denoted
5-7