Small Room Acoustics 133or more modes in that octave band. By applying
Bonello’s method to the 23 ft × 17 ft × 9 ft room, we
obtained the graph of Fig. 6-8. The conditions of both
criteria are met. The monotonic increase of successive
octave bands confirms that the distribution of modes
is favorable.
It is possible that the critical bands of the ear should
be used instead of octave bands. Actually, octave
bands follow critical bandwidths above 500 Hz better
than do octave bands. Bonello considered critical
bands in the early stages of his work but found that
one-third octave bands better show subtle effects of
small changes in room dimensions.^5 Another question is
whether axial, tangential, and oblique modes should be
given equal status as Bonello does when their energies
are, in fact, quite different. In spite of these questions,
the Bonello criteria are used by many designers and a
number of computer programs are using the Bonello
criteria in determining the best room mode distributions.
D’Antonio et al, have suggested a technique which
calculates the modal response of a room, simulating
placing a measurement microphone in one corner of a
room then energizing the room with a flat power
response loudspeaker in the opposite corner.^6 The
authors claim that this approach yields significantly
better results than any other criteria.
Another tool which historically has been used to help
choose room dimensions is the famous Bolt footprint
shown in Fig. 6-9. Please note the chart to the right of
the footprint which limits the validity of the footprint.
The ratios of Fig. 6-9 are all referenced to ceiling height.
6.5 Modes in Nonrectangular Rooms
Nonrectangular rooms are often built to avoid flutter
echo and other unwanted artifacts. This approach is
usually more expensive, therefore it is desirable to see
what happens to modal patterns when room surfaces are
skewed. At the higher audio frequencies, the modal
density is so great that sound pressure variations
throughout a rectangular room are small and there is
little to be gained except, of course, the elimination of
flutter echoes. At lower audio frequencies, this is not the
case. The modal characteristics of rectangular rooms
can be readily calculated from Eq. 6-1. To determine
modal patterns of nonrectangular rooms, however,
requires one of the more complex methods, such as the
use of finite elements. This is beyond the scope of this
book. We, therefore, refer to the work of van Nieuwland
and Weber of the Philips Research Laboratories of Eind-
hoven, the Netherlands, on reverberation chambers.^8
In Fig. 6-10 the results of finite element calculations
are shown for 2D rectangular and nonrectangular rooms
of the same area (377 ft^2 or 35 m^2 ). The lines are
contours of equal sound pressure. The heavy lines
represent the nodal lines of zero pressure of the standing
wave. In Fig. 6-10 the 1,0,0 mode of the rectangular
room, resonating at 34.3 Hz, is compared to a 31.6 Hz
resonance of the nonrectangular room. The contours of
equal pressure are decidedly nonsymmetrical in the
latter. In Fig. 6-10 the 3,1,0 mode of the rectangular
room (81.1 Hz) is compared to an 85.5 Hz resonance in
the nonrectangular room. Increasing frequency in Fig.
6-10, the 4,0,0 mode at 98 Hz in the rectangular room is(^1) » 3
(^1) » 3
(^1) » 3 1 / 6
(^1) » 3
Figure 6-9. Room proportion criterion.
0 4000 8000 12K 16K 20K
160
140
120
100
80
60
40
20
0
Frequency—Hz
Dimension ratios: 1 : X : Y
Volume—ft^3
1.0 1.2 1.4 1.6 1.8 2.0
X
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
Y
Curve encloses dimension ratios giving
smoothest frequency response at low
frequencies in small rectangular rooms.
Range of validity