SISOM & ACOUSTICS 2014, Bucharest 22-23 May
COMPARATIVE ANALYSIS OF TRANSFER FUNCTION AND STANDING WAVE
METHODS IN DETERMINATION OF ACOUSTIC ABSORPTION COEFFICIENT
Marius DEACONU
National Research and Development Institute for Gas Turbines COMOTI
Bucharest, Romania
This paper describes two methods for measuring the sound absorption coefficient of the materials, the
standing waves methods according 10534-1 and the transfer function method according 10534-2 are
used. The sound absorption coefficient of porous material was measured using an impedance tube and
was compared with the results obtained with Kundt tube. The sound absorption coefficient in
impedance tube was obtained using transfer function method and for Kundt tube standing wave
method was used. Good agreement between the two methods was obtained.
Keywords: sound absorption coefficient, impedance tube, transfer function.
1. INTRODUCTION
The sound absorption coefficient of sound absorbent materials is one of the essential parameter used in
building acoustic and environmental noise reduction material. Several methods are used for determination of
sound absorption coefficient at normal incidence. In this paper, the transfer function method and standing
wave method for acoustic absorption coefficient of a fibreglass sample are presented. For transfer function
method three tubes, SCS Euro Acoustic were used and for standing wave method, a B&K Kundt tube was
used. The most popular method is the standing wave ratio (SWR) method where a traversing microphone is
used to determine the location and magnitude of successive maxima and minima of the standing-wave
pattern in a tube. Then from these reflection coefficient is deduced and the acoustic absorption coefficient is
determined. The transfer function method uses generation of plane waves in the tube by a sound source
emitting a white noise, and the sound pressure level is measured in two locations close to the sample. The
complex acoustic transfer function of the two-microphone signal in standard configuration is determined for
each tube diameter and then the corrected transfer function is determined by interchanging the positions of
the microphones. The complex acoustic reflexion coefficient and absorption coefficient are determined for
each tube. In comparative analyses, the results of the two methods and the advantages of the transfer function
in determination of acoustic absorption coefficient are presented.
2. TUBE CONSTRUCTION FOR TRANSFER FUNCTION
For transfer function method three tubes is used; small tube Φ 28, medium tube Φ 45 and large tube Φ
100 mm. For each tube, the frequency domains are determined using diameter and microphone spacing. For
28mm tube diameter a distance of 20 mm between microphones is used, for 45 mm tube diameter a distance
of 30 mm and for 100 mm tube diameter a distance of 50 mm. The tube has to be long enough to ensure the
development of plane sound waves between the sound source and the sample. The microphones have to be
positioned in the plane wave field. Non-plane waves will disappear at a distance from the source
approximately three-tube diameters. It is recommended that the microphone to be placed at a distance of at
least one tube diameter from the sound source. The lower limit depends on the distance between
microphones, which must be more than 5% of the wavelength of the lowest limiting frequency of the tube.
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(1)
Plane wave can be generated in a tube only if the excitation frequency is below the smallest acoustic
mode and the upper working frequency limits of each tube was obtained using:
(2)
For the 100mm tube a frequencies domain of 125 – 1600 Hz was determined, for 45mm tube a
frequencies domain of 1000 – 4000 Hz and for 28mm tube a domain of 3800 – 7000 Hz.
2.1. TRANSFER FUNCTION METHOD
The reference point (x = 0) is situated at the face of the sample, x1, x2 and s represent positions of the
microphones and s represent the distance between the microphones. Complex pressure at position 1 can be
expressed as summation of the incident and reflected waves at location x1, the pressure at position 2 can be
expressed as superposition of the incident and reflected waves at location x2 [2].
(3)
(4)
where k is the wave number. From expressions of P1(ω) and P2(ω) derivation of Hi(ω) and Hr(ω) is
straightforward. The reflected wave pressure amplitude Pr(ω), can be written in terms of reflection
coefficient as:
(5)
where R(ω) is the reflection coefficient, the transfer function between two microphones is given by[2]:
(6)
from which results reflection factor and then sound absorption coefficient[2] :
(7)
(8)
2.2. STANDING WAVE RATION
This method relies on the fact that there are only plane incident and reflected waves propagating along
the tube axis in the test section of the tube (the section where standing wave pattern is explored). A
loudspeaker placed at one end of the tube generates the incident plane sinusoidal sound wave. The other end
of the tube is terminated with the test sample backed with a hard reflective end. Using definition of the
standing wave ratio[3]:
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Comparative analysis of transfer function and standing wave methods in determination of acoustic absorption coefficient
(9)
The reflection factor is defined as from which sound absorption coefficient is determined[3]:
(10)
(11)
2.3. MEASUREMENTS
The impedance tube for sound absorption coefficient is presented in Figure 1. A Symphony acquisition
system is used together with two GRAS 40BP microphones with G-26AC preamplifiers mounted in tube
(M1 and M2) for measuring the both incident and reflected wave (Pi and Pr). A fibre glass sample of 15mm
thickness is installed at the hard end of the tube and a white noise is generated. The signal amplitude has to
be at least 10dB higher than the background noise at all frequencies of interest, as measured at the chosen
microphone locations. During the test, any frequency having a response value 60 dB lower than the
maximum frequency response value is rejected.
Figure 1 Impedance tube standard configuration
The acoustic pressure is measured in standard configuration (Figure 1) of the microphones and transfer
is determined using:
function
(12)
where P( ) is the Fourier transform of the temporal acoustic pressure p(t),
is auto spectrum
determined from the product
and is defined from the complex sound pressure
at
is the cross spectrum determined from the product
and is
microphone position one,
defined from the complex sound pressure
and
at two microphone position, where
represents complex conjugate.
When using the two-microphone technique, one of the following procedures for correcting the
measured transfer function data for channels mismatch must be used: repeated measurements with channels
interchanged, or predetermined calibration factor. A channel consists of a microphone, preamplifier and
analyser channel. In this study, corrections for microphone and acquisition channels mismatches are done by
interchanging microphones for every measurement on a test specimen. This procedure is highly preferred
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when a limited number of specimen are to be tested. The microphones are interchanged (Figure 2) and the
transfer function
is determined using:
(13)
The corrected transfer function
is determined using:
(14)
Figure 2 Impedance tube interchanged configuration
In below figures the real part and imaginary part of the transfer functions for each tube diameters and
the corrected transfer function are presented. Minor mismatches can be observed between microphones and
acquisition channels. In the Figure 3 and 6 are presented the transfer functions between the microphones for
45 mm and 29 mm tubes which highlights the upper frequency limits. At these frequencies appears the
smallest acoustic mode (cut-off frequency) of the tube and plane wave in not generated.
Figure 3 Transfer functions for 100mm tube
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Comparative analysis of transfer function and standing wave methods in determination of acoustic absorption coefficient
Figure 4 Transfer functions for 45 mm tube
Figure 5 Transfer functions for 29 mm tube
With corrected transfer function for each tube diameter the normal incidence reflection coefficient
(Figure 6) is determined using :
e− jk0s − H12 2 jk0 x1
R(ω ) =
e
H12 − e jk0 s
(15)
Then with the calculated reflection coefficients is determined the acoustic absorption coefficient as:
(16)
In Figure 6 are presented the acoustic absorption coefficient obtained for each tube and it can be
observed the upper limits of the tubes at frequencies of 4500Hz of 45mm tube and 7400Hz for 29 mm tube
diameter. After this for 100 mm tube a frequencies domain of 125 – 1600 Hz is established, for 45mm tube a
frequencies domain of 1000 – 4000 Hz and for 28mm tube a domain of 3800 – 7000 Hz. Then the values of
the frequency domain heads are averaged and the values of the narrow band is converted in 1/3octave for the
acoustic absorption coefficient of 15mm glass fibber sample (Figure 6).
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Figure 6 Acoustic absorption coefficient for each tube diameter
The ibreglass sample then is tested in Kundt tubes using standing wave ratio method and the
comparative analyse of the results obtained with both methods are presented in Figure 7.
Figure 7 Comparative analyse of TF and SW method
3. CONCLUSION
The acoustic absorption coefficient at normal incidence of a 15mm thickness fiberglass sample was
determined using the transfer function method and the standing waves ratio method. Good agreement
between the results of the two methods was obtained.
The two microphones transfer function technique offers numerous advantages in comparison with the
standing waves method. This is a considerable saving of time and labour as compared to the standard SWR
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Comparative analysis of transfer function and standing wave methods in determination of acoustic absorption coefficient
method using microphone traversing and discrete frequency excitation technique. Another advantage of this
method is that the errors of the human factor are reduced and the results are accurate.
The use of random excitation permits the evaluation of acoustic properties at all frequencies for a
single sample. The acoustic absorption coefficient was determined using a white signal generator which
permits a frequencies analyse resolution of 6.25e+00 Hz in a frequency domain of 125 – 10.000 Hz resulting
1601 values. This high resolution is necessary, especially for the evaluation of acoustic filters, Helmholtz
resonators with acoustic properties, which are high frequency dependent.
For transfer function method any difference in the gains or phase shift of the two
microphones/amplifier system must be known. To eliminate these mismatches a calibration test is
recommended. In this study this calibration test was made interchanging the microphones and measuring the
transfer function for each configuration. This method is accurate but is highly preferred when a limited
number of specimen are to be tested otherwise a correction factor is determined using a special calibration
specimen and the correction is valid for all the successive measurements. This procedure is performed once
and after calibration the microphones remain in place.
The two microphones random-excitation method is useful for evaluating acoustic properties at low
frequencies, where the standing wave method require long tubes.
Nevertheless, the obtained results from both methods show good agreements.
REFERENCES LIST
1. Acoustics. Determination of sound absorption coefficient and impedance in impedance tubes. Part 1 : Method using standing
wave ratio, SR EN ISO 10534–1
2. Acoustics. Determination of sound absorption coefficient and impedance in impedance tubes. Part 2 : Transfer function method,
SR EN ISO 10534-2
3. SUHANEK, M., JAMBROSIC, K., DOMITROVIC, H., Student project of building an impedance tube, Acoustics 08 Paris, 2008
4. CHUNG, J.Y., BLASER, D.A., Transfer function method of measuring in-duct acoustic properties, The Journal of the Acoustical
Society of America, 1980.
5. NEGRO, F., CREMONINI, C., PROPERZI, Milena, ZANUTTINI, R., Sound absorption coefficient of perforated plywood: an
experimental case study, World conference on timber engineering, 2010.