Reverberation
Process at Low Frequencies
by
Holger
Larsen
ABSTRACT
The
reverberation process in a rectangular room was
investigated using sophisticated
instrumentation.
By averaging many reverberation decays it was possible to determine the decay
curves free of interference
allowing the details to
be
seen. The decay curves
measured show some
curvature especially at low
frequencies and as a result there are discrepancies
between reverberation
times determined from the early decay rate and from those
determined from the slope
between
-5 and -35 dB,
as suggested by ISO Standards 354
and 3382.
The
reverberation process far a frequency band is the result of the decays of the normal
modes in the band, each mode decaying with its own time constant which depends
esentially
on
the mean free path between reflections. A
formula for the mean free path
is
derived and hence a formula for the decay
curve is also derived. Good agreement was obtained
between the measured and calculated decay
curves.
Introduction
Reverberation time is an important factor in assessing the acoustic
quality of a room. It is also used in the determination of isolation,
absorption, and sound power. Reverberation time for a given frequency band
is defined as the time required for the average sound pressure level, originally
in the steady state, to decrease 60 dB after the source is stopped. Although
there are several methods by which it can be determined, the classical
method, where the sound pressure level decay is registered on a level recorder,
is still the most common and is also standardised in ISO 354 and 3382
Ref.[1, 2]. It is assumed in these standards that
the decay rate is exponential and therefore manifests itself as a straight line
when the sound pressure level is represented on a logarithmic scale.
However, on account of the interferences between the individual eigenmodes the curve traced out is not smooth, especially at
low frequencies where the number of eigenmodes are
relatively few.
Fig. 1 shows typical examples of decay curves at
low frequencies. According to the standards the reverberation time is determined
from the slope of a line drawn on the decay curve between -5 dB and -35 dB
levels below the steady state level. To improve the accuracy an average value is
determined from a number of curves obtained with different microphone and source
positions.
At
low frequencies, however, the sound pressure does not often decay exponentially
and the decay curve deviates from a straight line on a logarithmic plot.
Several authors [3, 4] have investigated this phenomenon theoretically,
where it is shown that the slope at the start of the decay curve (early decay
rate) is numerically highest and reduces as the level
decreases. It has been difficult to substantiate the theoretical treatment
with practical measurements upto the present day on
account of the uneveness of the curves mentioned
above.
Today,
however, sophisticated instrumentation and new
methods been
developed which permit reverberation
time measurements
to
be carried out with a
much higher degree of
accuracy. Such a measurement method which makes use of a
digital and a desk-top calculator is described in Technical
Review No.2-1977
[5]. The purpose of this article is to illustr measurement results and analyze the reverberation
process at low frequencies.
Measurement
Method
Fig.2
shows an instrumentation set-up for automatic measurement of reverberation time. The system is
controlled remotely by the desk-top calculator and operates after the following
procedure: the Noise Generator Type 1405 is started and sends a pink
noise signal to the Sound Power
Source Type 4205 which is used here as a power amplifier for the loudspeaker
Type HP 1001. After a steady
state sound field is builtup in the room, a
signal to the Digital Frequency Analyzer Type 2131 starts recording of the
spectra over short time intervals and soon after the sound source is
stopped, see Fig.3. During each time interval the frequency spectra are recorded over a
selected averaging time and the averaged spectrum then readout to the
desk-top calculator and stored. (Each time interval is made up of an averaging
time plus read-out time).
The
shortest time interval that can be chosen is 44 ms.
Larger time
intervals can be chosen (depending on the
averaging time see Table 1
in steps of 44 + n.22 where n is an integer. After the
spectra have been recorded and read-out for a maximum of 65 time intervals and time
lowed for calculation, the sound source is started again and the c repeated. The
time taken for each cycle is between 40 and 60 s depE
ing on the chosen parameters. The cycle is repeated as
many time; desired after which the average level in each of the time intervals
each frequency band is calculated.
The
reverberation time may now be evaluated using the calculator each
frequency band or the average levels in each time interval can
be fed to a level
recorder and the reverberation time evaluated from slope of the curves for each
frequency band.
For
the measurements a reverberation room in the Acoustics Lab tory at the Danish Technical University (D.T.H.) was used.
Twenty
diffusors of area 1,2 x 0,9
m2 were installed in the room the
dimensions of which are given in Table
2.
Measurement
Results
a/
Measurements with stationary source and microphone positions The
microphone and source positions are shown in Pig
4, For source position A
measurements were carried out for all the microphone positions while for
position B only microphone positions 1, and 2 were used. Figs.6a and b
show reverberation curves plotted on a level recorder chart for 1/3 octave frequency bands of
centre frequencies 80 Hz and 160 Hz respectively, for different source and
microphone positions. Values of 600 spectra have been averaged for each time
interval which was chosen to be 110 ms.
It
can be seen from the figures that the random fluctuations of the curves have
disappeared on account of averaging over a large number of spectra. But the
curves still show some irregularity, the magnitude of which depends on the
source and microphone positions. The irregularities would seem to be due to
the interference between the eigenmodes with
slight differences in the resonant frequencies.
The
problem of interference could be avoided if spatial averaging, in addition
to the time averaging was also carried out. In other words, the level
in each time interval would be averaged over a number of spectra and over a number of source and microphone
positions. To try this out
further measurements were taken with a rotating microphone whilst revolving
the source in the room.
b)
Measurements with rotating microphone and sound
source
Fig
.6 shows a measurement set-up where the sound source is mounted on the end of a
wooden beam fixed to a Turntable Type
3922. The sound source was rotated in a circle of radius approx. 2 m with a
rotation time of
80
s. The Microphone Boom Type 3923
was used for rotating the microphone in a circle of radius 1,5m and rotation
time 64 s.
The
reverberation curves obtained using the instrumentation described above are
shown in Figs.7a and b. It can be seen that the curves are regular (due to
the absence of interferences) but with some curvature which is especially
noticeable for low frequencies. In each time interval of 110 ms the levels have
been averaged over approx. 1600 cycles, and as each cycle takes about 1 minute
the total measurement time involved was approx.
27 hours.
From
the decay curves in Fig.7 the reverberation times T have been evaluated
according to ISO standard 354 by using the slope of the curves between the -5 d8
and -35 dB levels. The reverberation times evaluated from the slope of the early
decay rate are designated by Te.
Both
T and Te are plotted against frequency in Rg.8 and it can be that Te is much
lower than T for low frequencies.
In
order to examine the early decay rate of the reverberation process greater detail, further measurements were taken with a
time inter) 44 ms instead of 110 ms. The levels in each
lime interval were
again averaged over 1600 cycles. Fig.9 shows the reverberation
curves in Fig.10 the early decay rates obtained for 44 ms and 110 ms time
intervals are
compared.
Sound
Pressure.
Particle Velocity end Energy Density Distribution in a
Room
When
a sound source is operated in a
room, a number of different
room resonances (sometimes referred to as normal modes or eigenmodes) are excited. The type of eigenmode excited depends on how the initial wave is
reflected and returns to the point of excitation in the same phase and
direction as the initial wave. In a
rectangular room the simplest eigenmode is the
axial mode in which the component
waves travel along one axis (one dimensionall parallel
to two wall pairs as shown in
Fig. 11 a. The sound pressure
in the room varies as shown by the curve in the Figure. It can be seen
that when a moving microphone is used, the RMS value
of the sound pressure in the room will be measured
which is
√2 times lower than the values at the points
of maxima. The sound pressure variation in the room can also be illustrated by
drawing lines through points of equal pressures. As shown in Fig. 11b these
points lie in planes for axial waves.
Fig.11c
shows planes formed by points of equal particle velocities
and
it
can be seen that the planes with maximum pressures occur at the points of zero particle velocity and vice
versa.
The
energy density is uniform all over the room as shown in Fig.11d for
axial waves. At any point in the room the energy density is obtain by
determining the sum of the squares of the normalised values given in
Figs.11b and c. The
mathematical derivation of the values shown these figures is given in
Appendix
A.
Another
type of eigenmode is one in which the component wave
parallel to one pair of walls but are oblique to the other two pair dimensional)
and is termed a tangential mode, Fig.
12a. Figs 12b,
c and d show surfaces with constant pressure,
particle velocity and e density for a tangential mode. It can be seen
from Figs. 1 1 and 12 for a tangential mode, in
contrast to an axial mode, there are points the room
where the sound pressure, particle velocity, and energy
density are all zero.
The
third type of an eigenmode is one in which the
component
waves are parallel to none of the three wall pairs and is termed an
oblique mode. Fig.13 shows surfaces with
constant pressure for an oblique mode. As for the tangential
modes, the oblique modes have points the room where the
sound pressure, particle velocity and energy
density are
all zero.
Another
type of eigenmode is one in which the component wave
parallel to one pair of walls but
are oblique to the other two pair: dimensional) and is termed a tangential mode, Fig.12a. Figsl2b, d show surfaces with constant pressure, particle
velocity and e density for a
tangential mode. It can be
seen from Figs 11 and
1 for a tangential mode, in
contrast to an axial mode, there are poi the room where the sound pressure,
particle velocity, and energy sity are all
zero.
The
third type of an eigenmode is one in which the
component v are parallel to none of the three wall pairs and is termed an al
mode. Fig.13 shows surfaces with constant pressure for an
of mode. As for the tangential modes, the oblique modes have pot the room
where the sound pressure, particle velocity and energy sity are all zero.
The number of eigentones of
the
measurement
room lying in 1/3 octave
bands are represented crosses, circles and dots for axial, tangential and
oblique modes resF tively in
Fig. 14. It is time consuming
to determine the number of eigen- modes using the above
formula at higher frequencies as the number modes increases considerably. However, approximate expressions determining the
number of eigenmodes are derived in Ref.[3] quoted in Ref.[8] from which Fig. 14 is reproduced. It can be seen
that
the axial and tangential modes dominate at low frequencies while oblique modes
dominate at high frequencies.
In
statistical treatment of acoustics, it is assumed that the sound field in the room
is diffuse i.e.
a)
the energy density in the room is uniform everywhere,
b) the energy flow in all directions is the same and
c)
the phase between all waves is
random.
The above requirements are fulfilled if there are a large number of
eigenmodes in the room, which is the case at high frequencies.
At frequencies, however, the above requirements are not fulfilled therefore it
is necessary to use wave theory instead of statistical only. In the following the wave theory will be used to
explain the culture in the reverberation decay curves
at low frequencies, by treating the eigenmodes in each frequency band
individually.
Theory
When
a sound source excites a room, forced oscillations are general characterised by
the spectrum of the sound source and its position the
room. When the sound source is stopped, the oscillation pattern changed in that only the normal modes of oscillations
will now exist the room. The damping of the
oscillations takes place on account of lections from the walls and to a much
lesser extent on account of damping. The Q factor of the eigenmodes is thus determined by the sorption coefficient of
the walls and the mean free path between reflections. As
the mean free path is generally longest for the axial modes they will have a longer
reverberation time than the tangential and
oblique
modes.
Mean
Free Path
The
mean free path for an axial mode can immediately be seen to the same as the
dimension of the room in the direction of the wave.
The
above equation illustrates the sound pressure decay of a frequency band with I
number of eigenmodes and their respective
reverberation times.
Calculation
of Reverberation Decay Curves
With
the use of the formulae derived above, the reverberation decay curves for each
eigenmode and the collective reverberation curves for
63 Hz, 80 Hz and 100 Hz 1 /3 octave bands are calculated. As the absorption
coefficient a of the walls is not known, the values used are those which give
the best correlation between the calculated and the measured decay curves, see
"Absorption Coefficient" section under Discussion of
Results.
In
Table 3 the frequencies of the eigenmodes lying in the
63 Hz 1 /3 octave band are calculated using eq(1). The mean free path and
the relative steady state sound pressure for each eigenmode are evaluated from eq.(2) and (7b) respectively, while the theoretical
reverberation times are calculated from eq.(9). The
slope of the reverberation curves for each eigenmode
can be plotted using eq.(10b)
while the start of the reverberation
curves are given by the relative steady state levels. The reverberation curves
for each eigenmode is thus calculated from the equation
20log pi/pi0=-
60t /Ti- 20logpi0/pb0
and
plotted in Fig.
15.
The
collective reverberation curve for the 1/3 octave 63 Hz band be plotted either
by adding the reverberation curves on energy basis the
eight eigenmodes, or by using eq.(13) and is shown by the dashed
line in Fig.
15. The values in Fig. 15 indicated by circles are the al
measured
values for the 63 Hz 1/3 octave band and are taken
Fig.7a.
Tables
4 and 5 show similar results for the 80 Hz and 100 Hz 1 /3 octave bands. As
there are a large number of eigenmodes in these
frequency bands, the tangential and oblique modes have been grouped separately
and the average value of their mean free paths and reverberation times evaluated. The
relative steady state sound pressure level the whole
groups have also been evaluated. The reverberation cur, for the 80 Hz and 100 Hz
1 /3 octave bands are plotted in Figs. 16
and 17
and compared with the measured values.
Discussion
of Results
It
can be seen from Figs. 16 and 17 that there is good agreement
between the calculated and measured results.
There is however a principal difference, in that the
collective calculated curves have slightly less curvature than the curves obtained from
measured results. For the theoretical curves the
absorption coefficient is assumed to be independ, of
the angle of incidence. Fig.18 which is reproduced from Ref
shows typical curves for the variation of the absorption coefficient a function
of the angle of incidence.
The curves are plotted for different
values
of the acoustic impedance of the wall. 0° refers to normal incidence. The
absorption coefficient can be seen to be greater for oblique incidence than for
0° and 90° incidence. This would mean that the axial modes would be less
damped and thus have longer reverberation times than calculated while the
tangential and oblique modes will have lower reverberation times. As a result
there would be better agreement between the calculated and the measured
curves.
The
agreement is, however, not as good in Fig. 15. This could be because the
cut-off frequency of the loudspeaker was at 100 Hz and' therefore the frequency
spectrum was not flat at 63 Hz as assumed. As the level at 63 Hz was also low,
the background noise could have had some influence on the measured
curve.
From
the curves it is evident that the axial modes have a dominating
influence
on the reverberation decay process. For the 80 Hz frequency band the
reverberation curve below the -15 dB level is almost entirely determined by the
eigenmode (4,0,0) which
corresponds to an axial mode in the direction of the length of the room.
Although there are eight tangential modes they have little influence on the
reverberation curve while the oblique modes have practically
none.
Absorption
Coefficient
The
absorption coefficients a given in Table 6 have been evaluated using Sabines formula a = 0,161 V/ST. The reverberation times used
for
the
evaluation have been taken from Fig.8, both for the early de rate and for the
slope between the -5 and -35 dB levels. Also she in the table are the values assumed for Figs. 1 5, 16, and 17. It can seen
that there are significant differences between the values an( due to the fact that Sabine's equation has been derived
using a m, free path of 4V/S.
However, the average value of the mean free p will
be greater on account of the axial modes dominating at low
frequencies. For the room used here 4V/S is equal to 4,08m.
From Tables 3, 4 and 5 the mean values of R can be found
to be 4,98
meters, 4,65 m and 4,58 m for the 63 Hz, 80 Hz and 100 Hz 1/3 octave
frequency bands respectively. Using these values of the mean
free path and reverberation time from early decay rate in Sabines formula the
absorption coefficient is found to be
0,023, 0,022 and 0,021 in three bands respectively. These values are close to
the values assumed for Figs. 1
5, 16 and 17.
Energy
Content in the Eigenmodes
The
energy in an eigenmode is given
by
2 2
E=
p
/
ñc
x V
Substituting eq.(4) in (15) we obtain
2
2
E= C1 x T/ (åxåyåz)
x V/ ñC
assuming
that the sound pressure is averaged over the whole room whilst revolving the
source.
2
As (exeyez)
is equal to 4, 16 and
64
for axial, tangential and oblique modes respectively, the energy content the
axial and tangential modes relative to the oblique modes can be calculated.
Ea/E0=Ta/T0
x 64/4=Ta/T0
x 16 [17]
Et/E0
=Tt/T0
x 16/4=Tt/T0
x 4 [18]
where
Ta Tt
and To
are reverberation times for axial tangential and oblique modes respectively.
As
Ta/T0 = 2 the energy content is up to 32 times greater in an axial mode than in
an oblique mode, and more than 4 times greater in a tangential mode than in an
oblique mode as Tt >
To.
Conclusion
With
the use of a Digital Frequency Analyzer Type 2131 and a desk-top calculator it
is possible to carry out very accurate reverberation time measurements by
averaging with a movable microphone and sound source. The curvature of the
reverberation curves at low frequencies is caused on account of the axial modes
dominating the last stages of the reverberation process on account of their long
reverberation times and high energy content. This has been made clear with
relatively simple and approximate calculations of the reverberation process of
the individual eigenmodes in a frequency
band.
On
account of the curvature there is a considerable difference between the
reverberation time evaluated over 30 dB (as defined in ISO R 354 and 3382) and
that evaluated from the early decay rate up to 500 Hz
(see Fig.8) for the measurement room used here.
References
[1] ISO R 354 Measurement of absorption coefficients in a reverberation
room
[2]
ISO3382
Measurement
of reverberation time in auditoria
[3]
BRUEL, P.V Sound Insulation and Room Acoustics
Chapman & Hall.
1951
[4] KUTTRUFF, H. Eigenschaften
and Auswertung Nachhallkurven, Acustica, Vol.8,
1956
[5]
UPTON,
R Automated Measurements of Reveration Time using the
Digital Frequency Analyzer Type 2131. Technical Re
No.2-1977
[6]
BERANEK, L.L Noise and
Vibration Control. McGraw- Hill
1971
[7]
MORSE, P. M. and
Theoretical Acoustics McGraw-
Hill
1968
INGARD, K.U.
[8]
BRUEL, P.V The Enigma of Sound Power Measurements at Low Frequencies.
Technical view No.3-1
978
[9] RASMUSSEN, K. Noter til
forelaesninger i lydfelter. Modul 5105. 1973 Danmarks Tekniske Hojskole
[10]
KINSLER L.E. &
FREY
A.R. Fundamentals of Acoustics.
J.
Willey& Sons.
1962
