Modeling the complex characteristic acoustic impedance and complex wavenumber of porous materials allows the prediction of the complex specific acoustic impedance of a system consisting of porous absorbers and air cavities in front of a rigid surface. By using the transfer matrix method, the complex characteristic acoustic impedance and complex wavenumber of a porous material can be predicted by using the measured complex specific acoustic impedance of two different systems of the porous material and an air cavity, performed in a two-microphone impedance tube. Depending on the method, the material can be measured with either a rigidly terminated back plate at the back of the material, or a rigidly terminated air cavity at the back. This paper looks at why predictions using the single and double thickness method break down for thinner, less dense materials

Davy, J

Larner, D

RMIT Research Repository

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Larner, D and Davy, J 2014, 'The prediction of the complex characteristic acoustic
impedance of porous materials ', in Norm Broner, Charles Don (ed.) Proceedings of the
43rd International Congress on Noise Control Engineering: Internoise 2014, Australia, 16-19
November 2014, pp. 1-8.
https://researchbank.rmit.edu.au/view/rmit:28596
Published Version
2014 Australian Acoustical Society; Author(s)
http://trove.nla.gov.au/version/210470766
 Inter-noise 2014  Page 1 of 8 
The prediction of the complex characteristic acoustic impedance of 
porous materials 
David James Larner
1
; John Laurence Davy
2
 
1,2 
RMIT University, GPO Box 2476 Melbourne, VIC 3001 Australia 
ABSTRACT 
Modeling the complex characteristic acoustic impedance and complex wavenumber of porous 
materials allows the prediction of the complex specific acoustic impedance of a system consisting of 
porous absorbers and air cavities in front of a rigid surface. By using the transfer matrix method, the 
complex characteristic acoustic impedance and complex wavenumber of a porous material can be 
predicted by using the measured complex specific acoustic impedance of two different systems of the 
porous material and an air cavity, performed in a two-microphone impedance tube. Depending on the 
method, the material can be measured with either a rigidly terminated back plate at the back of the 
material, or a rigidly terminated air cavity at the back. This paper looks at why predictions using the 
single and double thickness method break down for thinner, less dense materials. 
 
Keywords: Impedance, wavenumber, porous materials I-INCE Classification of Subjects Number(s): 
72.7.2 
1. INTRODUCTION 
Predictions of the complex characteristic acoustic impedance and complex wavenumber of porous 
materials are useful tools to help predict the complex specific acoustic impedance of multi -layered 
systems. By knowing the thickness, complex wavenumber and complex characteristic acoustic 
impedance of a material layer, the specific acoustic impedance at the face of the layer can calculated, 
and subsequent layers can be calculated using the transfer matrix method, which allows for the 
absorption coefficient of the system to be calculated. Three prediction methods have been 
investigated; the Dunn and Davern (1) method which derives the characteristic impedance of a 
material by comparing specific acoustic impedance of the single and double thickness of the same 
material with no air cavity; the Utsuno et al. (2) method which compares the specific acoustic 
impedance of a finite thick material with two different air cavity depths behind it, and a third method; 
which is the second method when one of the air cavities has zero depth. 
2. Theory 
2.1 Prediction of the complex characteristic impedance using the Dunn and Davern 
method 
Dunn and Davern developed a method for predicting the complex characteristic acoustic 
impedance of porous materials 𝑍𝑐  by deriving equations based on the prediction of the complex 
specific acoustic impedance 𝑍0 using the transfer matrix method; 
 
𝑍0 = 𝑍𝑐
𝑍1 cos(?̃?𝑑) + 𝑗𝑍𝑐 sin(?̃?𝑑)
𝑍𝑐 cos(?̃?𝑑) + 𝑗𝑍1 sin(?̃?𝑑)
 (1) 
 
where 𝑍1  is the complex specific acoustic impedance of the previous layer, ?̃?  is the complex 
                                                        
1
 david.larner@rmit.edu.au 
2
 john.davy@rmit.edu.au 
Page 2 of 8  Inter-noise 2014 
Page 2 of 8  Inter-noise 2014 
wavenumber and 𝑑 is the thickness of the porous material. 
 
By dividing 𝑍1  through this equation, and then setting 𝑍1 = ∞  because in this case, 𝑍1  is the 
impedance of the rigidly terminated back plate, Equation 1 becomes; 
 
𝑍0 = −𝑗𝑍𝑐 cot(?̃?𝑑) (2) 
 
Using this formula, Dunn and Davern stated that when the porous material is doubled, the complex 
specific acoustic impedance becomes; 
 
𝑍02𝑑 = −𝑗𝑍𝑐 cot(2?̃?𝑑) (3) 
 
Equations 2 and 3 can be solved simultaneously to obtain both 𝑍𝑐 and ?̃?. 
 
𝑍𝑐 = √𝑍0(2𝑍02𝑑 − 𝑍0) (4) 
 
?̃? =
𝑗
2𝑑
ln
𝑍0 + 𝑍𝑐
𝑍0 − 𝑍𝑐
 (5) 
 
2.2 Prediction of the complex characteristic impedance using the Utsuno et al. method 
Utsuno et al. also developed the prediction of the complex characteristic acoustic impedance and 
complex wavenumber by using the transfer matrix method shown in Equation 1. However, this 
prediction method is more complicated, as it uses complex specific acoustic impedances of porous 
materials with different air cavity depths behind the sample.  
 
𝑍𝑐 = √
𝑍0𝑍0′(𝑍1 − 𝑍1′) − 𝑍1𝑍1′(𝑍0 − 𝑍0′)
(𝑍1 − 𝑍1′) − (𝑍0 − 𝑍0′)
 (6) 
 
?̃? =
𝑗
2𝑑
ln (
𝑍0 + 𝑍𝑐
𝑍0 − 𝑍𝑐
𝑍1 − 𝑍𝑐
𝑍1 + 𝑍𝑐
) =
𝑗
2𝑑
ln (
𝑍0′ + 𝑍𝑐
𝑍0′ − 𝑍𝑐
𝑍1′ − 𝑍𝑐
𝑍1′ + 𝑍𝑐
) (7) 
 
where the dashed and undashed symbols denote two different air cavity depths. 
 
𝑍1 and 𝑍1′ denote the specific acoustic impedance of the air cavity:  
 
𝑍1 = −𝑗𝜌0𝑐 cot(𝑘𝐷) (8) 
 
𝑍1′ = −𝑗𝜌0𝑐 cot(𝑘𝐷′) (9) 
 
where 𝜌0𝑐 is the characteristic impedance of air, 𝑘 is the wavenumber of air, and 𝐷 is the depth of 
the air cavity. It can be seen in Equation 6 that the difference between the air cavity impedances is 
important. Therefore, it was important to state that the air cavity impedances should be as different as 
possible. 
2.3 Modification of the Utsuno et al. prediction method 
To gain more complex characteristic acoustic impedance predictions of various porous materials, a 
combination of the two above prediction methods was used. One of the air cavity depth measurements 
was substituted with measurements performed when the material was backed by a rigid backing, 
making the specific acoustic impedance of 𝑍1
′ = ∞. By substituting this into Equation 6 and 7, the 
complex characteristic acoustic impedance and complex wavenumber became; 
 
𝑍𝑐 = √𝑍1(𝑍0 − 𝑍0
′ ) + 𝑍0𝑍0′ (10) 
 
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Inter-noise 2014  Page 3 of 8 
?̃? =
𝑗
2𝑑
ln (
𝑍0′ + 𝑍𝑐
𝑍0′ − 𝑍𝑐
) (11) 
 
In total, seven predictions of the specific characteristic acoustic impedance can be made from six 
measurements of single and double thickness materials. 
 
Table 1: Seven predictions used to calculate the specific characteristic acoustic impedance and complex 
wavenumber 
Method Equations used 
Dunn – Davern Equations 4 & 5 
Single layer Utsuno Equations 8 & 9 
Double layer Utsuno Equations 8 & 9 
Single Utsuno modification: 
compare no air gap & quarter wavelength air gap 
Equations 10 & 11 
Single Utsuno modification: 
compare no air gap & half wavelength air gap 
Equations 10 & 11 
Double Utsuno modification: 
compare no air gap & quarter wavelength air gap 
Equations 10 & 11 
Double Utsuno modification: 
compare no air gap & half wavelength air gap 
Equations 10 & 11 
3. Results 
Four materials were used to predict the complex characteristic acoustic impedance and complex 
wavenumber. These materials are shown in Table 1: 
 
Table 2: Materials measured in two-microphone impedance tube 
Material Flow Resistance 
(MKS Rayl) 
Thickness (mm) Density (kg.m
-3
) 
Foam 148 11.8 29 
Polyester 334 25.65 795 
Polyester 742 50.9 831 
Glass Wool 1546 49.6 396 
 
Each of these materials were measured in both the low and high frequency two-microphone 
impedance tubes, with both the single and double thickness samples measured against the back plate, 
and with a 50 and 100 mm or a 12.5 and 25 mm air cavities for the low and high frequency 
measurements respectively. From these six measurements, seven predictions can be made in each of 
the frequency ranges. The reasoning behind selecting these air cavity depths was due to the fact that 
these depths are slightly below the quarter-wavelength 𝐿𝜆
4
 and half-wavelength 𝐿𝜆
2
 lengths of 53.6 & 
107.2 mm and 26.8 & 53.6 mm for the maximum low (1600 Hz) and high frequency (6400 Hz) 
measurements 𝑓𝑚𝑎𝑥 respectively, as shown below in Equations 12 and 13. At these lengths, the 
impedance of the air cavity is close to 0 and ∞ respectively, therefore giving the largest possible 
difference in air cavity impedances. 
 
Page 4 of 8  Inter-noise 2014 
Page 4 of 8  Inter-noise 2014 
𝐿𝜆
4
=
𝑐
2𝑓𝑚𝑎𝑥
 (12) 
 
𝐿𝜆
2
=
𝑐
𝑓𝑚𝑎𝑥
 (13) 
 
 
Figure 1: The normalized low frequency real characteristic acoustic impedance of the thicker polyester 
sample. S and D denote a single or double thickness material respectively. The last four predictions show the 
Equation 10 method, where the two numbers represent the air cavity depths (mm) of the dashed and undashed 
series respectively. 
 
In Figure 1, it can be seen that with the seven prediction methods, there was very good agreement 
over the low frequency range. This shows that so far, there is no failure  in the prediction of the 
complex characteristic acoustic impedance for this material; however, it does break down with the 
other materials. 
 
Figure 2: The normalized low frequency imaginary characteristic acoustic impedance of the foam sample. S 
and D denote a single or double thickness material respectively. The last four predictions show the Equation 
10 method, where the two numbers represent the air cavity depths (mm) of the dashed and undashed series 
respectively. 
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Inter-noise 2014  Page 5 of 8 
 
Above in Figure 2, the Dunn and Davern method for predicting the complex characteristic acoustic 
impedance breaks down. As this is the only prediction method that breaks down in this material, the 
material was measured again, with a similar output in the complex specific acoustic impedance, thus 
retaining similar values of the complex characteristic acoustic impedance.  This error will be 
investigated later. 
 
 
Figure 3: The normalized low frequency imaginary wavenumber of the thin polyester sample. S and D denote 
a single or double thickness material respectively. The last four predictions show the Equation 10 method, 
where the two numbers represent the air cavity depths (mm) of the dashed and undashed series respectively. 
 
The complex wavenumber was also predicted for the materials, with some success and some errors. 
Above in Figure 3, it can be seen that the majority of predictions are very similar to each other, which 
strengthens these predictions. However, there are a few outliers, particularly with two out of the three 
single layer predictions. 
 
 
Figure 4: The normalized low frequency real wavenumber of the glass wool sample. S and D denote a single 
or double thickness material respectively. The last four predictions show the Equation 10 method, where the 
two numbers represent the air cavity depths (mm) of the dashed and undashed series respectively. 
 
Page 6 of 8  Inter-noise 2014 
Page 6 of 8  Inter-noise 2014 
For some materials, most of the predictions can give a range of results. In Figure 4 from 500 Hz 
upwards, the majority of predictions of the real part of the wavenumber for glass wool appear to agree 
with each other more. However, there are fluctuations below 500 Hz, especially with the single sample 
Utsuno et al. method. As these values dip into the negative value region, this means that the phase 
constant is negative, which makes no physical sense as the rest of the predictions are in the positive 
domain. These measurements were redone, and similar predicted outputs for the wavenumber were 
present. One positive is that it appears these values converge at frequencies above 2000 Hz, so 
predictions using high frequency two microphone impedance tube measurements were calculated.  
 
 
Figure 5: The normalized high frequency real wavenumber of the glass wool sample. S and D denote a single 
or double thickness material respectively. The last four predictions show the Equation 10 method, where the 
two numbers represent the air cavity depths (mm) of the dashed and undashed series respectively. 
 
In the higher frequencies, the majority of the prediction methods converge together, as shown in 
Figure 5. Again, there are outliers, with all outliers having something in common, having the double 
thickness samples in the prediction. The thickness of the double layered glass wool of 99.2 mm could 
be too thick for these particular predictions. 
 
4. Discussion 
One of the main points of discussion is why does some of the prediction methods break down for 
some cases and not others. In the foam case, the Dunn and Davern method breaks down, even after 
re-measurement. By taking a closer look at Equations 2 and 3, if ?̃?𝑑 ≪ 1, tan(?̃?𝑑) ≈ ?̃?𝑑, then these 
Equations become; 
 
𝑍0 =
−𝑗𝑍𝑐
?̃?𝑑
 (14) 
 
𝑍02𝑑 =
−𝑗𝑍𝑐
2?̃?𝑑
 (15) 
 
If these values are used in Equations 4 and 5, 𝑍𝑐 and ?̃? are equal to 0. This problem is due to the 
2𝑍02𝑑 − 𝑍0 term in Equation 4. Because both 𝑍02𝑑 and 𝑍0 are proportional to 𝑍𝑐, 𝑍𝑐 does not affect 
the relative standard deviation of  2𝑍02𝑑 − 𝑍0. Thus 𝑍𝑐 is assumed to be equal to one in the following 
calculations. From Equation 2, the mean values of 𝑍02𝑑 and 𝑍0 are; 
 
𝑍0̅̅ ̅ = cot(?̃?𝑑) (16) 
 
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Inter-noise 2014  Page 7 of 8 
𝑍02𝑑̅̅ ̅̅ ̅̅ = cot(2?̃?𝑑) (17) 
 
The relative standard deviations of 𝑍0 and 𝑍02𝑑  are assumed to be the same where this value is 
denoted by 𝑥 
 
𝜎𝑍0
2 = 𝑥2|𝑍0̅̅ ̅|
2 (18) 
 
𝜎𝑍02𝑑
2 = 𝑥2|𝑍02𝑑̅̅ ̅̅ ̅̅ |
2 (19) 
 
The relative standard deviation of 2𝑍02𝑑 − 𝑍0 is 
 
𝜎2𝑍02𝑑−𝑍0
|2𝑍02𝑑 − 𝑍0̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ |
=
𝑥
𝑦
 (20) 
 
where the divisor 𝑦 is given by 
 
𝑦 =
|2𝑍02𝑑 − 𝑍0̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ |
√4|𝑍0̅̅ ̅|2 + |𝑍02𝑑̅̅ ̅̅ ̅̅ |2
 (21) 
 
The smaller the value of the divisor 𝑦, the more the relative standard deviation is increased. This 
value needs to be compared to |?̃?𝑑|, as this will show if the material is thick enough to be used in the 
Dunn and Davern prediction method. To calculate |?̃?𝑑|, the angle 𝜗 must first be calculated: 
 
𝜗 = arctan [
𝑅𝑒(?̃?𝑑)
𝐼𝑚(?̃?𝑑)
] (22) 
 
which allows the calculation of |?̃?𝑑|; 
 
|?̃?𝑑| =
𝑗?̃?𝑑
𝑒𝑗𝜗
 (23) 
 
 
Figure 6: Plot of 𝑦 vs. |?̃?𝑑| at various 𝜗 values 
 
In Figure 6 shows what the common value 𝑥 of the relative standard deviation needs to be divided 
by, in order to obtain the relative standard deviation of 2𝑍02𝑑 − 𝑍0. This is assuming that the relative 
standard deviations of both 2𝑍02𝑑 and 𝑍0 are the same. Below in Table 2 shows each of the materials 
used in the predictions, with their corresponding 𝜗 and |?̃?𝑑| values. It can be seen that the foam case 
has a very low |?̃?𝑑| value, which satisfies Equations 14 and 15. By looking at Figure 6, the value of 
Page 8 of 8  Inter-noise 2014 
Page 8 of 8  Inter-noise 2014 
𝑦 is very small for the foam case, which causes an increase in the relative standard deviation of 
2𝑍02𝑑 − 𝑍0. Therefore it can be stated that for materials with low |?̃?𝑑| values, the prediction of the 
complex characteristic acoustic impedance and complex wavenumber can fluctuate due to the material 
being too thin. Therefore, the value of |?̃?𝑑| should be greater than 0.7. 
 
Table 3: Materials used in the predictions with the corresponding 𝜗 and |?̃?𝑑| values 
Material 𝜗 |?̃?𝑑| 
Foam -15° 0.6 
Polyester -17° 1.1 
Polyester -21° 2.2 
Glass Wool -29° 2.7 
 
5. CONCLUSIONS 
By comparing the seven prediction methods against each other, a quantitative prediction value of 
the complex characteristic acoustic impedance and complex wavenumber of porous materials can be 
determined. However, for some materials, these methods can break down. The Dunn and Davern 
method was analyzed, and it was determined that the value of |?̃?𝑑| for these materials should be above 
0.7. It can be recommended that for the Utsuno et al. method that the air cavity depths should be just 
under the quarter-wavelength and half-wavelength depths, as this will provide the largest difference in 
air cavity impedance without any resonances. 
6. REFERENCES 
1. Dunn IP, Davern WA. Calculation of acoustic impedance of multi-layer absorbers. Applied Acoustics. 
1986;19(5):321-34. 
2. Utsuno H, Tanaka T, Fujikawa T, Seybert AF. Transfer function method for measuring characteristic 
impedance and propagation constant of porous materials. The Journal of the Acoustical Society of America. 
1989;86(2):637-43. 
 
 


The prediction of the complex characteristic acoustic impedance of porous materials

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