Small Room Acoustics 131The ratio of the dimensions determine the distribution
of modes. The ratio is determined by letting the smallest
dimension be 1 and dividing the other dimensions by
the smallest. Obviously, the cube with its ratio of 1:1:1
results in an acoustic disaster. Even though a 12 ft
(3.66 m) cube will sound differently from a 30 ft (9 m)
cube, both rooms will exhibit the same modal distribu-
tion and it is the distribution that overwhelmingly deter-
mines the low frequency performance of a small room.
The second room determined by the dimension of
common building materials has a ratio of 1:1.5:2.0. The
ratio is determined by letting the smallest dimension be
1 and dividing the other dimensions by the smallest. So
an 8 ft by 16 ft by 24 ft room would have the ratio of
1:2:3. The third room, which seems reasonably good at
this point, has a ratio of 1:1.89:2.56. From this we can
see that in order to have a reasonable modal distribution
one should avoid whole number ratios and avoid dimen-
sions that have common factors.6.3.1 Comparison of Modal PotencyTo this point we have only considered the axial modes.
The three types of modes, axial, tangential, and oblique
differ in energy level. Axial modes have the greatest
energy because there are the shortest distances and
fewest surfaces involved. In a rectangular room tangen-
tial modes undergo reflections from four surfaces, and
oblique modes six surfaces. The more reflections the
greater the reflection losses. Likewise the greater the
distance traveled the lower the intensity. Morse and
Bolt^5 state from theoretical considerations that, for a
given pressure amplitude, an axial wave has four times
the energy of an oblique wave. On an energy basis this
means that if we take the axial waves as 0 dB, the
tangential waves are –3 dB and the oblique waves are
–6 dB. This difference in modal potency will be even
more apparent in rooms with significant acoustical
treatment. In practice it is absolutely necessary to calcu-
late and consider the axial modes. It is a good idea to
take a look at the tangential modes because they can
sometimes be a significant factor. The oblique modes
are rarely potent enough in small rooms to make a
significant contribution to the performance of the room.6.3.2 Modal BandwidthAs in other resonance phenomena, there is a finite band-
width associated with each modal resonance. The band-
width will, in part, determine how audible the modes
are. If we take the bandwidth as that measured at the
half-power points (3 dB or 1/ ), the bandwidth is^2(6-3)where,
'f is the bandwidth in hertz,
f 2 is the upper frequency at the 3 dB point,
f 1 is the lower frequency at the 3 dB point,
kn is the damping factor determined principally by the
amount of absorption in the room and by the volume
of the room. The more absorbing material in the room,
the greater kn.If the damping factor kn is related to the reverbera-
tion time of a room, the expression for 'f becomes^3(6-4)where,
T is the decay time in seconds.From Eq. 6-4 a few generalizations may be made.
For decay times in the range of 0.3 to 0.5 s, typical of
what is found in small audio rooms, the bandwidth is in
the range 4.4 to 7.3 Hz. It is a reasonable assumption
that most audio rooms will have modal bandwidths of
the order of 5 Hz. Referring back to Table 6-6 it can be
seen that in a few instances there are modes that are
within 5 Hz of each other. These modes will fuse into
one and occasionally some beating will be audible as
the modes decay. Modal frequencies which are sepa-
rated on both sides by 20 Hz or more will not fuse at all,
and will be noticeable as well, although not as notice-
able as a double or triple mode. Consider a room with
the dimensions of 18 ft × 13 ft × 9 ft (5.48 m ×
3.96 m × 2.74 m). The axial frequencies are listed in
Table 6-7. There are some frequencies which double,
such as 62 Hz and 125 Hz. These are obvious problems.Figure 6-7. Number of modes and frequencies for a room
23 ft × 17 ft × 9 ft. (From AcousticX.)2'ff 2 –= f 1
kn
S---- -='f 6.91
ST----------=2.2
T------ -=