15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference)
11 - 13 May 2009, Miami, Florida
AIAA 2009-3237
The acoustic impedance characteristics of porous foams and
fibrous materials
Fumitaka Ichihashi1, Christopher Porter2 and Asif Syed3
University of Cincinnati, Cincinnati, Ohio, 45221
This paper presents the results of an investigation into the acoustic impedance
characteristics of porous materials such as porous foams and fibrous mats. Most of previous
experimental investigations focused on the characteristics of fibrous materials only. However,
Yu, Kwan and Yasukawa did include some foam materials in their investigation. Their
experiments were conducted in an impedance tube, which allowed measurement up to 6000
Hz. The empirical correlations of Delany and Bazley were derived from measurements
conducted between 250 and 4000 Hz. The data presented by Lee and Selamet 4 had an upper
limit of just over 3000 Hz. The apparatus developed by Syed and Ichihashi at the University of
Cincinnati is capable of accurate measurements up to 9000 Hz. Therefore it allows an
investigation of the acoustic characteristics of porous materials over a much wider frequency
range.
Nomenclature
an
c
ce
f
h
i
k
l
p
pe
u
ue
=
=
=
=
=
=
=
=
=
=
=
=
A
= the amplitude of the forward propagating wave in the porous medium (Figure 1);
= the intercept of the straight line fit through the steady flow resistance (cgs Rayls) test data.
= the amplitude of the backward propagating wave (Figure 1)
= the slope of the straight line fit through the steady flow resistance (cgs Rayls) test data.
= the steady flow resistance (cgs Rayls) of a porous test sample
= the real part of the normalized acoustic impedance of a porous test sample
= the real part of the normalized characteristic impedance, *, of a porous medium
= the imaginary part of the normalized acoustic impedance of a porous test sample
B
R
RC
X
empirical constant obtained by curve fits through measured data; n =1, 2, 3, ., ., 8.
the speed of sound in air
the speed of sound in the porous medium (bulk absorber)
the acoustic frequency
the depth of the cavity behind the fibrous or foam test sample in the wave tube
1
the acoustic wave number; k = /c = (2 f /c )
the thickness or the depth of a fibrous or foam sample tested in the wave tube
the acoustic pressure oscillation
the acoustic pressure oscillation within the porous material (fibrous or foam)
the acoustic velocity oscillation
the acoustic velocity oscillation within the porous material (fibrous or foam)
1
Graduate student, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 210070, Cincinnati,
OH 45221-0070, student member AIAA.
2
Undergraduate co-op student, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 210070,
Cincinnati, OH 45221-0070, student member AIAA.
3
Research Professor, Department of Aerospace Engineering, University of Cincinnati, P.O. Box 210070, Cincinnati,
OH 45221-0070, Senior Member AIAA.
1
American Institute of Aeronautics and Astronautics
Copyright © 2009 by Asif A. Syed. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
XC
Z
*
e
*
= the imaginary part of the normalized characteristic impedance, *, of a porous medium
= the normalized acoustic impedance of a porous test sample; Z = R + i X
=
=
=
=
=
=
=
=
=
=
=
the imaginary part of
the real part of
the circular frequency, = (2 f )
the complex wave propagation constant in the porous medium at frequency, f
the normalized wave propagation constant; * = ( /k )
the density of air
the density of the porous medium (fibrous or foam)
the characteristic impedance of the porous medium (fibrous or foam). = ( e c e )
the normalized characteristic impedance; * = /( c) = RC + i X C
the resistivity (cgs Rayls/cm) of porous material (fibrous or foam)
the dimensionless frequency parameter ( f / )
I. Introduction
Current designs of acoustic liners used in the nacelles of commercial aircraft engines are based on the Helmholtz
resonator principle. They are of single or double layer design. The single layer design can provide good acoustic
suppression over a relatively narrow range of frequencies, approximately one octave, centered on its tuning
frequency. The double layer design can provide good suppression for a much wider frequency range. However, it
has been known that “bulk absorber” materials, used in acoustic liners can provide superior acoustic attenuation over
a much wider frequency range than the liners based on the resonator principle.
The most commonly known bulk absorber materials are fibrous mats or porous foams. The principle reason why
bulk absorbers are not used currently has to do with their tendency to soak up water during rain. This water soakage
will immediately cause a substantial increase in the weight of the nacelle and a loss of acoustic suppression
capability. Moreover, should they get soaked with liquid fuel; they can become a very serious fire hazard.
Recently, there has been a renewed interest in bulk absorbers because of the possibility of using hydrophobic
porous membranes at the surface of a bulk absorber liner. Therefore, a small experimental research project was
undertaken at the University of Cincinnati (UC) to investigate the impedance characteristics of suitable porous
materials. A set of material samples was acquired for this purpose. It consisted of commercially available fibrous
mats and foams of different resistivity or porosity.
Empirical correlations obtained by Delany and Bazley1 in the late 1960s had established a methodology for
designing bulk absorbers consisting of fibrous materials. Kirby and Cummings2 showed that these empirical
constants depend on the materials being investigated. The objective of the research conducted at UC was to
investigate the variation in the empirical correlation constants for different bulk absorber materials. The acoustic
measurements were conducted in the recently developed wave tube apparatus, which is capable of impedance
measurements up to 9000 Hz.
This paper describes the one-dimensional acoustic wave propagation theory on which the measurement
methodology is based. It also describes the details of the test apparatus and the measurement procedures. The
empirical correlations for the complex impedance characteristics of the various test samples are compared with those
obtained by Delany and Bazley (D&B).
II. Theoretical Notes
In the analysis presented in this paper, we assume that the porous medium illustrated in Figure 1 has uniform
porosity, and therefore uniform characteristic impedance throughout.
The diagram shows the porous material of length, l, in a wave tube. The acoustic pressure oscillation in the one
dimensional (1D) acoustic field at any location, x, in the porous medium is given by
pe ( f , x) = A exp( i x) + B exp(i x)
f
- is the acoustic frequency
2
American Institute of Aeronautics and Astronautics
(1)
- is the complex wave propagation constant in the porous medium
A and B are the complex amplitudes of the forward and the backward propagating acoustic plane waves. The values
of A and B are to be determined from the boundary conditions at the two ends of the porous medium.
bulk or porous medium
p1
A
B
u1
p2
u2
l
x
Figure 1. A schematic diagram of one-dimensional wave propagation in a segment of a
tube containing a porous material.
The corresponding acoustic particle velocity at location x is given by
e
c e ue ( f , x) = {A exp( i x) B exp(i x)}
(2)
where
ue ( f , x) is the acoustic particle velocity at frequency, f.
- is the angular frequency (2
i=
f)
1
- is the acoustic wave number ( /ce)
ce - is the speed of sound in the porous medium (bulk absorber)
e - is the density of the porous medium (bulk absorber)
The complex wave propagation constant, , can be expressed as follows
=
i
The principal characteristics to be determined by measurement are the wave propagation constant,
characteristic impedance, e c e , of the porous medium (bulk absorber).
(3)
, and the
Let us denote the characteristic impedance by the symbol .
= e ce
(4)
Let us consider that by means of a measurement we know the acoustic pressure, p1, and the acoustic velocity, u1, at
the plane x = 0. We also know the corresponding values of the acoustic pressure, p2, and the acoustic velocity, u2, at
the plane x = l. Moreover, the acoustic pressure and the acoustic particle velocity must be continuous at x =0 and at
x = l. Therefore from equations (1) and (2) at x = 0, we have
A + B = p1
A B = u1
(5)
(6)
From equations (5) and (6), we can express A and B in terms of p1 and u1, as follows.
A=
p1 + u1
2
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(7)
B=
u1
p1
(8)
2
Substituting the values of A and B in equations (1) and (2), it can be shown that at x = l, the continuity of acoustic
pressure and velocity is satisfied as follows.
p2 = cos( l) p1 i sin( l) u1
(9)
u2 = i
sin( l)
p1 + cos( l) u1
(10)
Let us assume that we make a second measurement (at the same frequency) but with a different termination
condition beyond the plane x = l. Let the second set of the measured values of the acoustic pressure and the velocity
*
*
be: p1 , u1 ,
follows
p*2 and u*2 . Then we can write a second set of equations in terms of the new measurements, as
p*2 = cos( l) p1* i sin( l) u1*
sin( l) *
p1 + cos( l) u1*
u*2 = i
(11)
(12)
Note that the astrix in equations (11) and (12) does not denote a complex conjugate.
From equations (9), (10), (11) and (12), we can write two sets of equations in matrix notation, in terms of the
measured acoustic pressures and acoustic velocities at locations 1 and 2 and the unknown quantities:
cos( l), sin( l), and
sin( l)
.
These equations can be solved, as shown below.
cos( l)
i sin( l)
=
p1
u1
*
1
*
1
p
u
" cos( l) " u1
sin( l) = *
"i
" u1
p1
*
1
p
1
p2
!
1
!
p*2
u2
u*2
(13)
(14)
From (13), we have
cos( l) =
sin( l) = i
p2 u1*
p*2 u1
p1 u1*
p1* u1
p2 p1*
u1 p1*
p1 p*2
p1 u1*
(15)
(16)
and from (14), we have
p1 u*2
cos( l) =
p1 u1*
sin( l)
=i
u2 u1*
p1 u1*
p1* u2
p1* u1
u1 u*2
u1 p1*
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(17)
(18)
From (15) and (17) we write
cos( l) = 0.5
" p2 u1*
p*2 u1
" p1 u1*
p1* u1
+
p1 u*2
p1 u1*
p1* u2 "
p1* u1 "
(19)
Manipulation of equations of (16 and (18) gives
2
p2 p1*
u1 p1*
=
p1 p*2
p1 u1*
u2 u1* u1 u*2
p1 u1* u1 p1*
=
p1 p*2 p2 p1*
u2 u1* u1 u*2
(20)
Thus, from equations (19) and (20), the values of the complex propagation constant, , and the characteristic
impedance, , can be determined.
III. The Acoustic Measurement Method
Acoustic Wave Tube
Apparatus with an array of 6
acoustic pressure transducers
Movable Piston
with single acoustic
pressure transducer
Test sample of
bulk absorber material
A
l
1
3.81 cm.
A
h
2
Section
A -A
Figure 2. A schematic diagram of the Wave Tube apparatus. It shows the setup to measure the
complex propagation characteristics in a porous material of uniform porosity.
The measurement technique employed by the authors is similar to that employed by Lee and Selamet [4]. The
acoustic excitation in the wave tube is by means of a JBL compression driver. The acoustic field in the wave tube is
measured by a set of six acoustic pressure transducers in three planes. Note that up to 4500 Hz. (approx.), the
acoustic field in the wave tube must consist of plane waves only. The first transverse mode is cut-on above 4500 Hz.
The second transverse mode is cut-on above 9000 Hz. Thus up to 9000 Hz. only the plane wave and the first
transverse mode is sensed by the six transducers shown in Figure 2. The six-transducer-array allows plane wave
measurements 5 up to a frequency of 9000 Hz. The 7th transducer is flush mounted at the center of the acoustically
hard movable piston. Because of its central location, this transducer also measures only the plane wave component
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(up to 9000 Hz.) of the acoustic field beyond the measurement plane #2. Thus, from the acoustic pressures,
measured by these six transducers the acoustic pressure, the acoustic particle velocity, and the acoustic impedance at
the measurement plane #1 are computed [5]. The acoustic pressure and the acoustic particle velocity at the
measurement plane #2 are computed from the acoustic pressure measured by the 7th transducer. Thus, up to 9000
Hz., all measurements are limited to the plane wave mode only.
First, acoustic data are measured with the piston located at a cavity depth of h = h1. Then, the test is repeated by
moving the piston to a cavity depth of h = h2. Thus the two sets of acoustic data, required in equations (13) and (14),
are obtained. From these the complex acoustic propagation constant, , and the characteristic impedance, , are
computed.
IV. Flow Resistance Testing
Foam or Fibrous
Test Sample
2.54 cm. thick
Test Sample Holder
10.0 cm. diameter
Air Flow Through
Test Sample
Plenum Chamber
of the Test Stand
Figure 3. Schematic diagram of a bulk absorber test sample installed in
the sample holder of the Raylometer test stand. A very thin wire-mesh
screen was used on both sides of the test sample to contain it within the
sample holder during the test.
The resistivity,
, of the porous material is incorporated in the dimensionless frequency parameter,
f
,
defined by Delany and Bazley [1]. Therefore, tests were conducted on the flow resistance test stand, a.k.a. the
Raylometer, at UC. The method of holding the porous test sample in place during a flow resistance test is illustrated
in Figure 3. During a test, the sample must be held in place. This is done by means of a wire-mesh screen on each
side of the test sample. The diameter of the test sample is 10 cm. The procedure for mounting a circular test sample
of a fibrous or foam material is described below.
There are three metal plates that detach from the apparatus. These are used to mount the bulk (foam or fibrous)
material. These three pieces are similar. Each consists of a square plate with a circular hole (10 cm. diameter) cut
into it. The first plate is attached to the stand. Then a fine metallic mesh is placed on the first plate to cover the
circular hole (to prevent materials from falling into the plenum). The next plate (1.0 inch thick) is placed on top of
the first plate and the bulk absorber is placed inside the cavity. A piece of mesh material is placed to cover the hole
and ensure that the bulk material is not blown out of the cavity. The third plate is placed on top of second plate.
There are thin rubber gaskets in between each of the aluminum plates and the clamps seal the configuration to
ensure zero leakage.
Some of the samples were 1.27 cm. (0.5 inch) thick. Two layers of these samples were used to fill the 2.54 cm.
(1.0 inch) deep cavity of the sample holder. After completing testing on one side, the samples were flipped upside
down to test “Side 2.” Thus each test sample/configuration was tested multiple times and data were averaged to
provide a good statistical measurement of the sample. The metallic wire-mesh screens used in the tests were very
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thin and of high porosity. The measured flow resistance for these mesh materials was less than 1 cgs Rayl at 105
cm/s. Therefore we can disregard the effects of the mesh screens in any measured flow resistance data for the bulk
absorber samples.
Typical Flow Resistance data for a 2.54 cm thick fibrous sample are shown in the plots of Figure 4. A straight
line fit through the test data is obtained to represent the flow resistance characteristics of the test sample.
300
250
200
150
100
Measured
Straight line fit
50
0
0
20
40
60
80
100
120
140
160
180
Flow Velocity, U (cm/sec)
Figure 4. Flow resistance data for Polyurethane foam sample (Item #11).
.
The resistivity, , of the porous material is defined as the flow resistance per unit thickness of the material.
Therefore, Resistivity can be computed from the measured flow resistance data as follows:
=
R(U) A + B U
A
+
=
=
l
l
l
B
U
l
(21)
where
l is the thickness or depth (cm.) of the test sample used in flow resistance measurements
A is the intercept of the straight line fit through test data shown in Figure 4
B is the slope of the straight line fit through test data shown in Figure 4
From the measured flow resistance data, the linear and the nonlinear components of the resistivity (cgs Rayl/cm)
of the test samples were determined. These data together with brief descriptions of the test samples are contained in
table 1. Notice that apart from the Polyurethane material (Item #11), all the fibrous and the foam samples are very
linear. That is, their resistivity characteristics are relatively insensitive to the magnitude of the flow velocity. Item
#11 is very nonlinear, as shown in the plot of Figure 4.
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Test
Sample
ID #
Item #8
Item #9
Item #10
Item #11
Item #16
Item #21
Item #22
Description of material, manufacturer, or
supplier
Resistivity
Intercept
Resistivity
Slope
Melamine Foam; white colored soft foam. GK
Industries, Inc.
Melamine Foam; white colored soft foam.
AMI
Melamine foam much like items #8 & #9.
However, It is of a yellow-orange color
instead of white. BASF.
Polyurethane Foam; soft foam of black/dark
grey color. GK Industries, Inc.
Fibrous mat - can be torn apart easily; most
likely due to different sheets of fiber stacked
together. Color- Yellow fiber with a white
impervious sheet as a back wall. Acoustic
Solutions Inc.
Mineral fiber of brown color; rigid fibers that
can cause splinters. Must handle carefully.
Lancaster GTB Systems Ltd, UK.
Silsoft glass. Fiber of white color, not easily
compressed. BFG Industries, Inc.
11.3
0.021
9.78
0.020
8.35
0.020
5.91
0.559
57.09
0.080
43.31
0.063
116.14
0.142
Table 1. Summary of the fibrous and foam materials tested at UC. The resistivity
data, linear (intercept) and the nonlinear (slope), are in cgs Rayl units.
V. Testing in the Wave Tube Apparatus
Before discussing the acoustic data measured by the authors, the forms of the empirical correlations obtained by
Delany and Bazley (D&B) are described. The normalized complex wave propagation constant, *, can be expressed
as follows
*
=
k
=
i
k
and the normalized complex characteristic impedance,
*
=
(22)
k
c
, can be expressed as
= Rc + i X c
c
(23)
Based on their excellent measurements, Delany and Bazley derived their empirical correlations in the following
forms.
k
= a1
a2
;
Rc = 1 + a5
where
=
k
= 1+ a3
a4
; X c = a7
f
2
, is the dimensionless frequency parameter; and k =
a6
;
(24)
a8
f
c
is the acoustic wave number. The
empirical constants derived by Delany and Bazley from their measurements are presented in Table 2.
a1
0.189
a2
a3
-0.595 0.0978
a4
-0.7
a5
a6
0.0571 -0.754
a7
a8
0.087
-0.732
Table 2. The empirical constants for the correlation derived by Delany
and Bazley 1.
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Real ( *)
Several repeat tests were conducted on each sample listed in Table 1. For each test, two measurements were
made. The first measurement was made with the
h1 : h2
h1
h2
Number of
cavity depth, h = h1. The second measurement was
combination #
(cm)
(cm)
repeat tests
made with the cavity depth, h = h2. From the data
1
1.27
2.032
6
acquired from this pair of measurements, the
2
1.27
2.286
6
3
1.27
2.54
6
complex values of and were computed. We used
4
1.27
2.794
3
five combinations of h1 and h2. For each
5
1.27
3.048
3
combination of h1 and h2, the test was repeated a
Table 3. The values of the cavity depth, h, which were
number of times. The values used for h1 and h2 and
used for different repeat tests to measure the complex
the corresponding number of repeated tests for each
values of and for a porous material sample. Total of
combination are listed in Table 3. Finally, each
24 repeat measurements were conducted for each test
sample was tested with the hard wall piston located
sample.
at h = 0. These tests provided the normal impedance
data for the material samples for
3.5
which the above measurements
were made.
3
This procedure allowed a
minimum of twenty-four repeat
2.5
measurements of the complex
Test #1
Test #2
acoustic parameters, and , for
2
Test #3
each test sample at each
Test #4
1.5
measurement frequency value.
Test #5
Test #6
From these data the mean values
1
of and were obtained at each
frequency for further analysis.
0.5
A. Repeatability of Test data
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Frequency (Hz.)
Figure 5a. Measured values of the real part of the normalized propagation
constant, *, plotted against frequency – Item #16 (fibrous).
4.5
4
3.5
3
{Imag( *) -1}
Measured values of the real
and the imaginary parts of the
normalized wave propagation
constant, *, from six different
tests are plotted against the
acoustic frequency in figures 5a
and 5b. These plots show the
quality of the repeatability of
testing with different values of the
cavity depth, h, in Figure 2. The
total number of tests carried out
for this test sample, Item #16, was
28. The minimum, the maximum,
the mean values, and the standard
deviation of the data in Figures 5a
and 5b are shown in the plots of
Figures 6a and 6b. These data
show that the repeatability of the
test data is excellent between
1000 Hz. and 8500 Hz.
Test #1
Test #2
2.5
Test #3
Test #4
Test #5
Test #6
2
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Frequency (Hz.)
Figure 5b. Measured values of {Imag( *) – 1} plotted against frequency
from 6 different measurements – Item #16 (fibrous).
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4
3.5
Real ( *)
3
Mean
Min.
2.5
Max.
Standard Deviation
2
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
-0.5
Frequency (Hz.)
Figure 6a. Measured values of the real part of the normalized propagation constant,
*, plotted against frequency – Item #16. These plots are based on data from 24
different measurements.
4.5
4
{Imag( *) -1}
3.5
3
Mean
Min.
2.5
Max
2
Standard Deviation
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
Frequency (Hz.)
6000
7000
8000
9000
Figure 6b. Item #16 (fibrous). Measured values of {Imag.( *) – 1} plotted against
frequency. These plots are based on data from 24 different measurements.
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Normalized wave propagation constant ( *)
B. Correlation of the test data
4
The process of obtaining the values of the
3.5
correlation constants, a1 through a8
is
f
*
=
=
= ( * i *) ;
explained with the help of
k
3
Figures 7a and 7b which show the real and
the imaginary parts of the measured
measured real part
2.5
(averaged) values of * and *, plotted
measured imag. part
against
the
dimensionless
frequency
2
Curve fit through real part
parameter, . For each one of these plots we
*
a2
Curve fit through imag. part
1.5
seek a curve fit of the form: y = a1 x . This
was done by using the Generalized Reduced
1
Gradient (GRG2) algorithm provided in
( * -1)
Microsoft Excel 2000 Solver. This algorithm
6
0.5
was developed by Leon Lasdon, of the
University of Texas at Austin, and Allan
0
Waren, of Cleveland State University. The
0
0.05
0.1
0.15
0.2
curve fits through the data were obtained by
Non-dimensional frequency parameter,
using the constants obtained in this manner.
The constants, a1 through a8, for items
Figure 7a. Item #16, a fibrous material. The measured values of the
#8, #9, #10, #16, #21, and #22 are presented
real and the imaginary parts of the normalized wave propagation
in Table 4. Although there is some variation
constant, *.
in the values of these constants for these
different materials, this variation is considered relatively small compared to that observed for the item #11, the
Polyurethane foam. For this reason, the data for Item #11 are presented separately in Table 5.
Normalized Characteristic Impedance: Rc & Xc
3
*
=
2
c
= Rc + i X c
1
0
0
0.05
-1
0.1
0.15
0.2
measured real part, Rc
Measured imag. part, Xc
Curve fit through Rc data at UC
-2
Curve fit through Xc data at UC
-3
Non-dimensional frequency parameter,
Figure 7b. Item #16, a fibrous material. The measured values of the
real and the imaginary parts of the normalized characteristic
impedance, *.
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Delany and
Bazley
a1
a2
a3
a4
a5
a6
a7
a8
Item #8
Item #9
Item #10
Item #16
Item #21
Item #22
0.1890
0.1618
0.1513
0.1207
0.1806
0.2138
0.2217
-0.5950
-0.7336
-0.7381
-0.7556
-0.5904
-0.5884
-0.5968
0.0978
0.1020
0.0867
0.0617
0.1042
0.1146
0.1521
-0.7000
-0.6936
-0.7110
-0.7483
-0.6499
-0.6683
-0.6269
0.0570
0.0318
0.0442
0.0207
0.0339
0.0877
0.0581
-0.7500
-0.8887
-0.7606
-0.9616
-0.8173
-0.6576
-0.7349
-0.0870
-0.0892
-0.0845
-0.0717
-0.1347
-0.1396
-0.1630
-0.7320
-0.8086
-0.8166
-0.8342
-0.5949
-0.6175
-0.6098
Table 4. The correlation constants derived from the measured data for items #8, #9, #10, #16,
#21, and #22. The power curve-fit process obtained these constants.
Delany and
Bazley
a1
a2
a3
a4
a5
a6
a7
a8
C. Computed Versus Measured Impedance Data
The computed and the
measured values of the normal
impedance are compared with
the measured data in Figures 8
through 14. As expected, the
values computed with the new
correlation constants derived by
UC, compare better with the
measured
data
than
the
computed values by using the
Delany and Bazley (D&B)
constants. For the 6 samples of
Table 3, the impedance data
computed
by
the
D&B
correlations are considered to be
in good agreement with the
measured impedance data. The
computed impedance data by
using the D&B correlations for
item #11 clearly do not compare
well with the measured data, as
shown in Figure 14.
Item# 11
0.1890
0.3827
-0.5950
-0.5621
0.0978
0.8790
-0.7000
-0.2134
0.0570
0.8287
-0.7500
-0.2225
-0.0870
-0.1060
-0.7320
-0.9093
Table 5. The correlation constants
derived from the measured data for
items #11.
2
1
0
0
-1
3000
6000
9000
R ; computed by correlation from D&B
X ; computed by constants from D&B
-2
R ; measured
X ; measured
R ; new constants by UC
-3
X ; new constants by UC
Frequency (Hz.)
-4
Figure 8. Measured and computed data for the Item # 8. The Melamine foam
test sample is 4.95 cm. (1.96 inch) deep. Broadband excitation level of 139.3 dB
OASPL at the surface of the test sample.
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2
1.5
1
0.5
0
0
3000
6000
9000
-0.5
R; computed by correlations from D&B
-1
X; computed by correlations from D&B
R; measured
-1.5
X; measured
-2
R; computed using new constants by UC
X; computed using new constants by UC
-2.5
Frequency (Hz.)
-3
Figure 9. Measured and computed data for the Item # 9. The Melamine foam test
sample is 4.95 cm. (1.96 inch) deep. Broadband excitation level of 139.2 dB OASPL
at the surface of the test sample.
2
1
0
0
3000
6000
9000
R; computed by correlations from D&B
-1
X; computed by correlations from D&B
R; measured
X; measured
-2
R; computed by constants from UC
X; computed by constants from UC
-3
Frequency
Figure 10. Measured and computed data for the Item # 10. The Melamine foam test
sample is 5.84 cm. (2.3 inch) deep. Broadband excitation level of 139.1 dB OASPL at
the surface of the test sample.
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American Institute of Aeronautics and Astronautics
2
1
0
0
3000
6000
9000
-1
R; computed by correlations from D&B
X; computed by correlations from D&B
-2
R; meaasured
X; measured
R; computed by constants from UC
-3
X; computed by constants from UC
Frequency (Hz.)
-4
Figure 11. Measured and computed data for the Item # 16. The fibrous mat test
sample is 2.29 cm. (0.9 inch) deep. Broadband excitation level of 144.1 dB
OASPL at the surface of the test sample.
2
1
0
0
3000
6000
9000
-1
R; computed by correlations from D&B
-2
X; computed by correlations from D&B
R; measured
X; measured
-3
R; computed by constants from UC
X; computed by constants from UC
-4
Frequency
Figure 12. Measured and computed data for the Item # 21. The fibrous mat test
sample is 2.29 cm. (0.9 inch) deep. Broadband excitation level of 144.5 dB OASPL
at the surface of the test sample.
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3
2
1
0
0
3000
6000
9000
-1
R; computed by correlations from D&B
X; computed by correlations from D&B
-2
R; measured
X; measured
-3
R; computed by constants from UC
Frequency (Hz.)
-4
X; computed by constants from UC
Figure 13. Measured and computed data for the Item # 22. The fibrous mat test
sample is 1.78 cm. (0.7 inch) deep. Broadband excitation level of 144.2 dB
OASPL at the surface of the test sample.
4
3
2
1
0
0
3000
6000
9000
-1
-2
R ; computed with D&B correlations
-3
X; computed with D&B correlations
R ; measured data
-4
-5
-6
X ; measured data
R ; computed with new constants from UC tests
X ; Computed with new constants from tests at UC
Frequency (Hz.)
Figure 14. Measured and computed data for the Item # 11. The Polyurethane foam
test sample is 5.14 cm. (2.025 inch) deep. Broadband excitation level of 139.1 dB
OASPL at the surface of the test sample.
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VI. Discussion and Conclusions
The correlations derived by Delany and Bazley1 are of the form: y = a1 x 2 . Thus for each correlation
a
between the real or imaginary parts of the complex parameters,
*
and
*
, against the parameter
=
f
,
we need to determine two correlation constants. This is a simple curve fit but it is by no means the most accurate
representation of the measured data. It is thought that a 3rd or 4th order polynomial may provide a much more
accurate correlation.
In the analysis presented in this paper, the nonlinearity of the materials was ignored completely. This was done
by using the linear component of the resistivity in developing the correlations presented in Tables 3 and 4. In the
case of the Polyurethane foam (Item #11), nonlinearity is clearly very significant. Therefore, in any future work, it is
suggested that the nonlinearity should be accounted for when required.
It has been shown that the wave tube apparatus and measurement technique developed at University of
Cincinnati is capable of measuring the complex wave propagation ( *) and the characteristic impedance ( *)
parameters for porous materials, such as foams and fibrous mats. The accuracy and the repeatability of test data are
very good. It has also been shown that the correlations between the values of the normalized wave propagation
constant, *, the normalized characteristic impedance, *, and the dimensionless frequency parameter, , are
different for different materials. These differences can be quite significant, as shown by the data for item #11.
Therefore, it is important that before designing an acoustic liner based on fibrous materials or porous foams, tests
should be carried out to determine the correlations between * and and between *and for the porous materials to
be used.
Acknowledgements
The authors wish to acknowledge Mr. Curtis Fox for his invaluable help and advice during the initial setup of the
wave tube test apparatus. Mr. Fox is a senior research associate in the Department of Aerospace Engineering at the
University of Cincinnati.
References
1
Delany M. E., Bazley E. N., Acoustical properties of fibrous materials, Applied Acoustics (3), 1970.
2
Kirby R., Cummings A., Prediction of the bulk acoustic properties of fibrous materials at low frequencies,
Applies Acoustics (56), 1999.
3
Yu J., Kwan H. W., Yasukawa R. D., Use of HTP ceramic foam for aeroacoustic applications, AIAA-97-1705CP.
4
Lee I., A. Selamet A., N. T. Huff N. T., Acoustic impedance of perforations in contact with fibrous material, J.
Acoust. Soc. Am 119 (5) May 2006.
5
Syed, A. A. and Ichihashi F., “The Modeling and Experimental Validation of the Acoustic Impedance of MultiDegrees-of-Freedom Liners,” paper number AIAA-2008-2927, presented at the Aeroacoustics Conference in
Vancouver, Canada, May 2008.
6
L. S. Lasdon, A. D. Warren, A. Jain, and M. Ratner, Design and testing of a generalized reduced gradient code
for nonlinear programming, ACM Trans. Math. Software 4 (1978), pp. 34--50.
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