Room modes in nonrectangular rooms
All figures shown are the result of measurements in two-dimensional rooms or models
of such rooms, i.e. rooms where the vertical dimension was so much less than the
horizontal ones that no normal modes in the vertical direction occurred in the
frequency range used for the experiments.
In conventional reverberation chambers pronounced standing wave patterns are
present, which means that the sound field is not diffuse. However, a diffuse sound
field is an essential assumption on which reverberation theory is based [1], [2]. One
effect of this non-diffuse sound field is that measured reverberation time depends on
the location of sound source and detecting device in the room. One major effect of
this fact is that different absorption coefficients for the same materials are measured
in different laboratories [1]. In order to smooth out the standing wave pattern
chambers of irregular shape were proposed and built.
First experimental methods to determine the standing wave patterns of nonrectangular
rooms were developed by Bolt [1]. Small models were used made from brass (0.6 –
0.95 cm thick) of 5 cm high and lateral dimensions of about 20-23 cm. Three models
were built, a parallelogram, a trapezoid, and a coupled-space model. Pure tones were
generated by an oscillator and fed to an ear phone mounted flush in the wall (at
location “S” indicated in Figs. 1-3).
Fig. 1: Mode (2,0)
Fig. 2: Mode (0,1)
Fig. 3: Mode (1,1)
Bolt concludes that in rectangular rooms there are series of modes which are related
harmonically, and therefore have coincident regions of pressure maxima and minima,
which results in very pronounced pressure variations within the room, when excited.
In nonrectangular rooms no such harmonic relation exists and the average spatial
response, i.e. the response in different locations within the room, is more uniform than
in the rectangular room [1], the superposition of different modes will result in smaller
pressure variations [3].
The pressure variation from maximum to minimum of each individual mode,
however, is about the same for different room shapes and not different from the
rectangular room [1], [3], [4].
When varying the shape of a room away from the rectangle, while keeping the plan
area constant, frequency shifts of the normal modes occur:
Fig. 4 (from [5])
Solid lines: theoretical curves
Dashed lines: best average fit
Experimental points are marked by
symbols
The distance between one pair of opposite sides is the same in both the rectangular
and the trapezoidal room. For those modes which form nodal lines (i.e. lines of equal
sound pressure, which lines are perpendicular to the direction of propagation of the
sound waves) parallel to these sides, i.e. the (0,n) modes, the frequency shift is very
small [4].
When rotating one of the short walls of a rectangular room about its mid-point
unsystematic frequency changes of the room modes occur [3]. Again, for the (0,n)
modes, where the sound waves propagate between the parallel walls, the frequency
shift is very small.
Fig. 5 (from [3])
When there are parallel walls, the variation in shape of the nodal lines has some
regularity [4]. The more irregular the room shape becomes, the more complicated
become the nodal lines.
Fig. 6 (from [4])
Sato [4] gives another explanation for the more uniform sound field in rooms with
irregular shape: “The nodal lines in the rectangular room are so regular that nodal
lines of different modes overlap one another to some extent. In such a case the normal
modes taking part in decay become extremely different in nature (axial tangential,
oblique) in different parts of the room. On the other hand, when the room shape and
so the position of the nodal lines are irregular, the superposition of decaying normal
modes would show nearly the same nature throughout any part of the room.”
The following figures show modal patterns of normal modes in a rectangular and a
nonrectangular room.
Fig. 7: Eigenmode patterns for pure sine tones (from [3])
Fig. 8 (from [4])
Eigenmode
pattern for
warble tone
The warble tone (center frequency 1260 Hz, ± 125 Hz) excites several modes at the
same time. While sound pressure in the irregular room changes in a complicated
manner in the different parts of the room, the sound pressure in the rectangular room
is distributed symmetrically and is minimum in the center of the room.
The main difference between the rectangular and the nonrectangular room is the fact
that the axial and tangential modes do not exist, only the (three-dimensional) oblique
modes are active [3], [4]. This results in a more regular distribution of the modes than
one would find in a rectangular room with optimum dimension ratios. A regular
distribution of modes avoids the occurrence of large dips or peaks in the frequency
response curve. This led to the conclusion that nonrectangular rooms are superior [3].
Milner determined optimum dimensions of a nonrectangular room with parallel floor
and ceiling to be : a = 6,30 m, b = 5,24 m, c = 10,35 m , d = 7,00 m, e = 7,91 m, with
a height of 4,93 m [6].
Due to the somewhat unexpected behaviour of the modal characteristics of
nonrectangular rooms, the resonance frequencies for each particular room shape have
to be calculated (Finite Element methods) in order to know whether the frequency
distribution is as desired [3].
Fig. 9 shows the modal pattern of the 91.6 Hz resonance of a non-rectangular 227 m3
reverberation room (Philips Research Laboratory) in a plane parallel to the floor.
Fig. 9 (from [3])
Literature
[1] Bolt, “Normal modes of vibration in room acoustics: experimental investigations
in nonrectangular enclosures”, J. of the Acoustical Society of America 1939, vol. 11,
p.184
[2] Hodgson, “When is diffuse-field theory applicable?”Applied Acoustics 1996, vol.
49, no. 3, p.197
[3] Van Nieuwland et al., “Eigenmodes in non-rectangular reverberation rooms”,
Noise control engineering 1979, Nov., p.112
[4] Sato et al., “The effect of the room shape on the sound field in rooms”, J. of the
Physical Society of Japan 1959, vol. 14, no. 3, p.365
[5] Bolt, “Perturbation of sound waves in irregular rooms”, J. of the Acoustical
Society of America 1942, vol. 13, p.65
[6] Milner, “An investigation of the modal characteristics on nonrectangular
reverberation rooms”, J. of the Acoustical Society of America 1989, vol. 85, p.772