15th International Congress on Sound and Vibration
6-10 July 2008, Daejeon, Korea
THE EFFECT OF SAMPLE EDGE CONDITIONS ON STANDING
WAVE TUBE MEASUREMENTS OF ABSORPTION AND
TRANSMISSION LOSS
Kwanwoo Hong1 and J. Stuart Bolton1
1
Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University
140 S. Intramural Drive, West Lafayette, IN, 47907-2031, USA
[email protected]
Abstract
The acoustical properties of isotropic, elastic porous materials are now conventionally specified
by a set of nine, frequency-independent macroscopic parameters: e.g., flow resistivity,
tortuosity, viscous and thermal characteristic lengths, etc. When these properties are known, it
is possible to predict the absorption performance of the material, for example, in arbitrary
geometries. Conversely, it has become popular to infer the macroscopic parameters of porous
materials by finding the set of parameters that results in an optimal match of measurements and
prediction. The software packages FOAM-X and COMET/Trim, for example, offer inverse
characterization features of this type. However, here it is shown that that it may not be possible
to represent large and small diameter samples of the same material as measured in standing
wave tubes by using a single set of parameters. In practice, measurements in standing wave
tubes can be significantly affected by sample edge effects, particularly gaps around the sample
resulting from minor damage of the sample during cutting. Here it will be illustrated that it is
necessary to model the sample inhomogeneity resulting from edge damage if both large and
small tube results are to be modelled by using a single, consistent set of parameters.
1. INTRODUCTION
In this paper we are concerned with the effect of sample edge conditions on normal incidence
absorption coefficients and transmission losses as measured in standing wave tubes. In
particular, it will be shown that damage to the edge of a sample due to cutting may cause a
discrepancy between small and large tube results in the frequency range in which they would be
expected to overlap. As a result, small and large tube results cannot be reproduced by using a
single set of material parameters in combination with a homogeneous finite element model.
However, by allowing the sample edge to have different properties than the main body of the
sample, i.e., by making the model inhomogeneous, it is possible to fit both large and small tube
results simultaneously with a single set of parameters. This behavior may have implications
regarding the estimation of poroelastic material properties by inverse method, e.g., by using the
software packages FOAM-X or COMET/Trim, since it is usually assumed that the material
properties are homogeneous and that the acoustical data input to the parameter estimation
procedure is unaffected by finite size sample effects: i.e., edge effects.
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2. ABSORPTION COEFFICIENT AND TRANSMISSION LOSS
MEASUREMENTS
A two-microphone arrangement was used to measure the normal incidence sound absorption
coefficient; in particular, a Brüel and Kjær two-microphone standing wave tube type 4206 was
used, as shown in Figure 1. The procedure followed ASTM E1050 [1]; a 2.9 cm diameter tube
was used for the frequency range from 500 Hz to 6400 Hz, and a 10 cm diameter tube was used
for the frequency range from 100 Hz to 1600 Hz. A loudspeaker at one end of the tube was used
to generate the required random signals. Absorption measurements were made using a 2 cm air
space behind the sample.
Figure 1. Two-microphone setup with 2 cm air backing space.
The normal incidence sound transmission loss was measured by using the four-microphone
standing wave, Brüel and Kjær 4206T, tube shown in Figure 2 [2,3]. As in the two-microphone
setup, two different diameter tubes were used in the low and high frequency ranges. The
downstream section was terminated by an approximately anechoic termination that was created
by loosely packing the standard sample holder with 3M Thinsulate sound absorbing material.
Mic 1
Mic 2
Mic 3
A
Mic 4
C
B
Speaker
D
d
x2
x3
x1
x4
x
Figure 2. Four-microphone setup.
The material used in this work was a polyurethane foam supplied by Bridgestone. All the
samples were 5 mm thick, and were carefully cut to fit snuggly, but without distortion into the
2.9 cm and 10 cm diameter tube: there were no visible leaks around the edges of the samples.
The density of the foam was 64.7 kg/m3. Ten small and large diameter samples were cut and
measured. The measurement results shown in Figure 3 are the averages of the absorption
coefficients and transmission losses (and the corresponding standard deviations) measured in
the small and large tubes. It can be seen that there is a significant discrepancy between the large
and small tube results in the region where they might be expected to overlap: that discrepancy
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was the focus of the work described below. In particular, note that the first absorption peak
appears at a lower frequency when measured in the large tube than it does when measured in the
small tube, and that the transmission loss measured in the large tube is significantly larger than
the small tube result in the region above 700 Hz.
1
10
Transmission Loss [dB]
Absorption Coefficient
0.8
12
Large Tube
+STD
-STD
Small Tube
+STD
-STD
0.6
0.4
0.2
0 2
10
8
Large Tube
+STD
-STD
Small Tube
+STD
-STD
6
4
2
10
0 2
10
3
Frequency [Hz]
(a) Absorption Coefficient
10
3
Frequency [Hz]
(b) Transmission Loss
Figure 3. Absorption coefficient and transmission loss measurement comparison.
3. FINITE ELEMENT MODELS
To help explain the origin of the discrepancy noted above, finite element models of the
absorption and transmission loss measurements were created. The simulations were performed
by using the code COMET/Safe supplied by Comet Technology Corporation. This software is
based on a finite element implementation of the Biot theory [4] for wave propagation in elastic
porous materials. Specifically, it implements both the u-U and p-U versions of the Biot theory
[5,6]. All of the finite element models used in this paper involved axisymmetric elements as
described by Kang et al. [7]. Axisymmetric finite element models were created to match both
the large and small tube absorption and transmission loss measurements. All these models were
homogeneous: i.e., all elements were given the same properties. The model boundary
conditions consisted of: a unit amplitude velocity to represent the loudspeaker, a zero normal
velocity condition to represent the hard backing in the absorption case, and an impedance
boundary condition to represent the anechoic termination in the transmission loss case. In
addition, the foam elements’ solid phase displacement at the circumferential edge was set to
zero in both the radial and axial directions, as was the radial component of fluid phase
displacement. The latter three boundary conditions implement the sample edge constraint
condition, in which it is assumed that the foam is bonded to the inner surface of the tube [8]. A
manual inverse characterization was then performed to extract sets of nine elastic porous
material properties that could fit both the experimental absorption and transmission loss data as
closely as possible. The acoustical properties of isotropic, elastic porous materials are now
conventionally specified by a set of nine, frequency-independent macroscopic parameters: flow
resistivity, tortuosity, viscous and thermal characteristic lengths, porosity, Young’s modulus,
Poisson’s ratio, bulk density, and loss factor. The manual inverse characterization process was
started by checking the sensitivity of the absorption and transmission loss data to each of the
nine material properties. The results of the sensitivity check for three material properties for the
small tube case are shown in Figures 4 and 5. The fitting process was first applied to the small
tube data, and those material properties were then used to predict the large tube results, and then
the process was repeated starting from the large tube data and ending with a prediction of the
small tube results. The resulting sets of material properties are listed in Table 1 (note that the
porosity was fixed at 0.99 and was not varied in the inverse characterization). Predictions of the
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absorption coefficients and transmission losses made using those parameters are plotted in
Figures 6. In Table 1, the first row of data is the inverse characterization result based on the
small tube experimental results, and predictions made using these parameters are labeled “set
for small tube” in Figure 6. The second row of material properties in Table 1 are based on the
large tube experimental results, and the corresponding predictions are labeled “set for large
tube” in the Figure 6. It can be seen that the estimated material properties based on the small
tube experiment allow the small tube absorption and transmission loss to be predicted very
accurately, but that the large tube results are not well predicted with that parameter set. In the
same way, the material properties based on the large tube experiment allow accurate estimation
of the large tube absorption and transmission loss (although some discrepancy remains in the
large tube absorption), but the small tube result are predicted inaccurately. Further, recall, as
discussed in connection with Figure 3 that the large tube absorption coefficient peak was
shifted down by about 300 Hz compared to the small tube absorption peak location. Also the
large tube transmission loss measured in the small tube was less than that measured in the large
tube by approximately 2 dB at 1600 Hz. Those features were not reproduced by the
homogenous finite element model. In Figure 7, the large and small tube finite element
predictions (based on the small tube parameter set) are plotted. It can be seen that the predicted
absorption coefficients for the small and large tube cases are nearly identical up to 1600 Hz, and
while there is about a 1 dB difference between the predicted small and large tube transmission
losses, that difference is not as large as the measured difference. Thus it was concluded from
these various observations that a single set of material properties could not be used to predict
both the small and large tube experimental result when using a homogenous finite element
model.
Tortuosity
Viscous characteristic length
1
0.8
0.8
0.8
0.6
0.4
0.2
0
2000
3000
4000
5000
0.6
0.4
0.2
50,000
140,000
200,000
1000
Absorption Coefficient
1
Absorption Coefficient
Absorption Coefficient
Flow resistivity
1
0
6000
2000
3000
4000
5000
0.4
0.2
1.0
2.5
4.0
1000
0.6
0
6000
1E-5
3E-5
7E-5
1000
2000
Frequency [Hz]
Frequency [Hz]
(a) Flow resistivity
3000
4000
5000
6000
Frequency [Hz]
(b) Tortuosity
(c) Viscous characteristic length
Figure 4. Sensitivity of absorption coefficient.
Flow resistivity
Tortuosity
10
Viscous characteristic length
14
10
12
6
4
2
0
50,000
140,000
200,000
1000
2000
3000
4000
Frequency [Hz]
(a) Flow resistivity
5000
Transmissoin Loss [dB]
8
Transmissoin Loss [dB]
Transmissoin Loss [dB]
8
10
8
6
4
1.0
2.5
4.0
2
6000
0
1000
2000
3000
4000
5000
6000
6
4
2
0
Frequency [Hz]
(b) Tortuosity
1E-5
3E-5
7E-5
1000
2000
3000
4000
5000
6000
Frequency [Hz]
(c) Viscous characteristic length
Figure 5. Sensitivity of transmission loss.
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Table 1. Material properties for single layer model. Top row: small tube set. Bottom row: large tube set.
1
1
0.8
0.8
Absorption Coefficient
Absorption Coefficient
Flow
Viscous
Thermal
Resistivity
characteristic characteristic
Tortuosity
(MKS
Length
Length
Rayls/m)
(m)
(m)
-5
140,000
2.5
1.0*10
8.0*10-5
-5
200,000
3.5
1.0*10
8.0*10-5
0.6
0.4
0.2
0
400
600
800
Poisson’s
ratio
Loss
factor
50,000
50,000
0.48
0.49
0.25
0.25
0.6
0.4
0.2
Experiment
Set for Small Tube
Set for Large Tube
200
Young’s
Modulus
(Pa)
0
1000 1200 1400 1600
Experiment
Set for Small Tube
Set for Large Tube
1000
2000
Frequency [Hz]
(a) Absorption coefficient in large tube
5000
6000
12
10
Transmissoin Loss [dB]
8
Transmissoin Loss [dB]
4000
(b) Absorption coefficient in small tube
10
6
4
2
0
3000
Frequency [Hz]
400
600
800
6
4
2
Experiment
Set for Small Tube
Set for Large Tube
200
8
0
1000 1200 1400 1600
Frequency [Hz]
Experiment
Set for Small Tube
Set for Large Tube
1000
2000
3000
4000
5000
6000
Frequency [Hz]
(c) Transmission loss in large tube
(d) Transmission Loss in small tube
1
10
0.8
8
Transmissoin Loss [dB]
Absorption Coefficient
Figure 6. Results of first inverse characterization.
0.6
0.4
0.2
6
4
2
Small Tube
Large Tube
0 2
10
10
Small Tube
Large Tube
0 2
10
3
Frequency [Hz]
(a) Absorption coefficient
10
3
Frequency [Hz]
(b) Transmission loss
Figure 7. Absorption coefficient and transmission loss simulation comparison.
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4. EFFECT OF EDGE CONSTRAINT
The circumferential edge constraint of a poroelastic sample has a significant impact both on its
absorption and transmission loss when measured in a standing wave tube, as first noted by
Beranek [9] in the context of absorption measurements. For example, Ingard et al. [10]
demonstrated the effect of edge constraint on the normal incidence absorption coefficient of
polyurethane foam. They found significant differences between the cases in which the foam
was held tightly by the tube walls or was cut so the sample fit loosely in the tube. They noted
that a tight contact between sample and tube wall could alter the acoustically induced motion of
the solid part of porous material, thus altering the frequency dependence of the absorption
coefficient. The same effect was more recently noted by Pilon et al. [11] and Kino and Ueno
[12]. In contrast, Cummings [13] discussed the effects of air gaps around a sample on its
absorption coefficient. It was noted that air gaps tended to have a greater significance when the
material has a relatively high flow resistivity. His theoretical model and experiment data
showed that air gaps would appear to provide acoustic leakage flow paths around the
circumferential edge of the sample. Pilon et at. [14] have also discussed these effects. The edge
effect in the present case can be seen by comparing the small tube and large tube experiment
results presented in Figure 3. In particular, the location of the first absorption coefficient peak is
shifted to a higher frequency in the small tube case when compared to the large tube result. It
can be inferred that the edge effect has a more significant impact in the small tube case since the
sample size in that case is much smaller than the large tube sample. In the transmission loss
results, the small tube sample yields a lower transmission loss than the large tube result. That
observation can be explained if there is a leakage flow path around the circumferential edge of
the sample, that leakage being more significant in the small tube case due to the relatively small
sample size. The acoustic leakage could be explained by the formation of small air gaps or
regions of reduced flow resistivity around the sample that are caused by damage around the
edge of the sample due to the cutting process. To model this effect without introducing complex
geometry or additional boundary conditions, the sensitivity results presented earlier are helpful.
Note that the observed absorption peak shift and transmission loss change can be modeled by
changing material properties, particularly around the edge of the sample: i.e., by creating an
inhomogenerous model. As illustrated Figure 4, the absorption peak can be shifted by
controlling the tortuosity and viscous characteristic length. The overall level and slope of
transmission loss can also be adjusted by changing flow resistivity, tortuosity, and viscous
characteristic length. The finite element model described in section 3 featured homogeneous
elastic porous elements: i.e., all elements possessed the same properties. A second finite
element model was built in which a layer of elastic porous elements around the circumferential
edge were introduced to allow for the effect of edge leakage or damage. The edge elements
were given different material properties than the remainder of the elements. The thickness of
the edge layer was 2 mm for both small and large tube model. The same inverse
characterization process was performed based on the small tube experiment results, and the
resulting material property estimates are listed in Table 2. In that table, the first row of values
applies to the core part of the model, while the second row applies to the circumferential edge
elements. The listed material properties were then used to estimate the large tube absorption
and transmission loss. In contrast with the results obtained using the homogeneous model, the
second model results in significantly improved agreement with the large tube experimental
result. That is, a single set of material properties reproduced both the small and large tube
absorption coefficient and transmission losses quite closely: see Figure 8. In Figure 9, the small
and large tube predictions are superimposed. It can be seen that the difference in the locations of
the absorption peaks (about 300 Hz) is very close to the difference seen in the experiment result.
The transmission loss discrepancy is still not quite as large as in the experimental result, but it is
larger than that produced by the first model. It is possible that the large tube results could be
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further improved by adjusting the stiffness of the edge elements.
Table 2. Material properties for multi layer model.
1
1
0.8
0.8
Absorption Coefficient
Absorption Coefficient
Flow
Viscous
Thermal
Resistivity
characteristic characteristic
Tortuosity
(MKS
Length
Length
Rayls/m)
(m)
(m)
200,000
3.0
-5
8.0*10-5
1.0*10
100,000
2.0
0.6
0.4
0.2
Young’s
Modulus
(Pa)
Poisson’s
ratio
Loss
factor
50,000
0.48
0.4
0.25
0.6
0.4
0.2
Experiment
SAFE
0
200
400
600
800
Experiment
SAFE
0
1000 1200 1400 1600
1000
2000
Frequency [Hz]
(a) Absorption coefficient in large tube
4000
10
10
8
8
6
4
2
6000
6
4
2
Experiment
SAFE
0
5000
(b) Absorption coefficient in small tube
Transmissoin Loss [dB]
Transmissoin Loss [dB]
3000
Frequency [Hz]
200
400
600
800
Experiment
SAFE
0
1000 1200 1400 1600
Frequency [Hz]
1000
2000
3000
4000
5000
6000
Frequency [Hz]
(c) Transmission loss in large tube
(d) Transmission Loss in small tube
1
10
0.8
8
Transmissoin Loss [dB]
Absorption Coefficient
Figure 8. Result of second inverse characterization.
0.6
0.4
0.2
0 2
10
Experiment: Small Tube
SAFE: Small Tube
Experiment: Large Tube
SAFE: Large Tube
10
6
4
2
0 2
10
3
Frequency [Hz]
Experiment: Small Tube
SAFE: Small Tube
Experiment: Large Tube
SAFE: Large Tube
10
3
Frequency [Hz]
(a) Absorption coefficient
(b) Transmission loss
Figure 9. Absorption coefficient and transmission loss simulation comparison.
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5. CONCLUSIONS
It has been shown here that the discrepancy frequently noted between absorption coefficients
and transmission losses measured in large and small tubes may be a consequence of apparently
minor damage to the edge of samples during the cutting process. It was suggested that for the
relatively dense foam considered here, the sample cutting reduces the flow resistivity at the
sample edge, thus introducing a partial leak around the sample; the effect of this damage is
relatively more important for small than large diameter samples. Further, it was concluded that
if material parameters that accurately represent a material’s properties are to be inferred from
acoustical measurements, this edge effect must be explicitly accounted for.
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[1]
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[3]
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sample”, Journal of Sound and Vibration 151, 63-75 (1991).
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