PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Prediction of the sound absorption of micro-perforated panels
whose holes are extended in length by tubes
John Laurence DAVY1; Mohammad FARD2; Qian ZHANG3; Yadong LYU4; Jun YANG5
1

Infrastructure Technologies, CSIRO, Australia & School of Science, College of Science, Engineering and Health,
RMIT University, Australia
2

School of Engineering, College of Science, Engineering and Health, RMIT University, Australia

3, 4, 5

The Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences,
China

ABSTRACT
Maa’s theory, which models the specific acoustic impedance of a narrow air-filled hole in front of an air
cavity using a lumped parameter model, works well for short holes at low frequencies. However, as the lengthof the hole or the frequency increases, the acoustic velocity of the air in the hole varies along the length of thehole as well as radially. It is necessary to use the transfer matrix method (TMM) rather than Maa’s lumped
parameter model to calculate the impedance of the hole and the cavity behind it. The peak sound absorptionof a micro-perforated panel (MPP) in front of an air cavity can be moved to lower frequencies by extending
the length the holes in the MPP with rigid or flexible tube bundles in the air cavity behind the MPP without
increasing the total depth of the sound absorbing system. Previously Maa’s lumped parameter model had
been used to predict the sound absorption coefficients of these tube bundle sound absorbers. These
predictions are only correct up to a certain frequency which depends on the length of the tube extended holes.
This paper predicts the sound absorption to higher frequencies using the TMM.
Keywords: Sound, Absorption, Micro-perforated panel

1. INTRODUCTION
In order to obtain better sound absorption at low frequencies, in 2001 Lyu et al. (1) introduced theapproach of extending the length of holes in a perforated panel in front of an air cavity by attachingtubes in the air cavity with the same or similar diameter to the back of the holes. In (1) and in several
later papers, Lyu et al. presented experimental results of this approach and gave some theoreticalformulae. However, they never actually published the results of any theoretical calculations. In 2011,
Zhang (2) compared the theory with experimental results. Zhang followed the approach of Maa (3) andignored wave motion along the length of the tubes. Zhang also ignored wave motion in the air cavity inthe direction of cavity depth, although Maa had included it. This was presumably because the ends ofthe tubes opening into the air cavity were not usually at one edge of the air cavity. The ignoring ofwave motion in the tubes and the air cavity meant, as Zhang commented, that her calculations would
only be accurate when the wavelength of sound was greater than 10 times the length of the tubes andgreater than 10 times the depth of the cavity. In 2016, Larner and Davy (4) pointed out that it was
sometimes necessary to consider wave motion in the holes of the panel and/or t he tubes at highfrequencies even if the length of the hole was only 10 mm. When Davy was visiting the Institute of
Acoustics of the Chinese Academy of Sciences (CAS) in 2018 under the CAS -CSIRO exchange
scheme, he suggested that they repeat Zhang’s calculations while taking account of wave motion inboth the tubes and the air cavity. The comparisons of these new calculations with Zhang ’s
1john.davy@rmit.edu.aumohammad.fard@rmit.edu.au
3zhangqian3989@dingtalk.com
4
yadong@mail.ioa.ac.cn
5 jyang@mail.ioa.ac.cn
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experimental results are reported in this paper. Lyu drew Davy’s attention to the fact that Simon (5)had also made calculations taking account of wave motion in similar sound absorbing systems.
Simon’s theoretical modelling assumed that his tube ends opened into the air cavity at one end of air
cavity, although this was not actually the case with his experimental measurements. In this paper, theposition of the ends of the tubes where they open into the air cavity is considered. Simon also used
formulae for the propagation of sound between two closely spaced plates rather than formulae for the
propagation of sound in narrow tubes as used in this paper.

2. THEORYIn this paper, the time dependence of a pure tone wave is assumed to be proportional to exp( jZt ) .
This choice effects the sign of the imaginary part of the impedances. The normalized specific acoustic
impedance z 2 encountered by a plane sound wave incident normally on a planar layer of an acoustic
medium on plane 2 of the two planes which bound the planar layer of the acoustic medium lying
between the planes numbered 2 and 1 is (6)z2 zc > zc sin(kL)  jz1 cos(kL) @ > z1 sin(kL)  jzc cos(kL) @ .
(1)
The acoustic medium has a normalized characteristic specific acoustic impedance of zc and wave
number of k. The distance between the two planes is L. The normalized specific acoustic impedance
encountered by a plane wave incident normally on plane 1 from the acoustic medium is z 1. The specificacoustic impedances are normalized by dividing them by the characteristic specific acoustic
impedance of the free acoustic medium ρ 0c 0, where ρ 0 is the ambient density of the free acoustic
medium and c0 is the speed of sound in the free acoustic medium.
If z 1 tends to infinity, dividing the numerator and denominator of Eq. (1) by z 1, shows that Eq, (1)tends to(2)z2  jzc cot(kL) .
Eq. (2) will be used for a planar layer of an acoustic medium which is rigidly terminated at plane 1. It
shows that a rigidly terminated planar layer of acoustic medium behaves like an acoustic spring at low
frequencies. If z1 = 0, Eq. (1) becomes(3)
z2 jzc tan(kL) .
Eq. (3) shows that a planar layer of an acoustic medium behaves like an acoustic mass at low
frequencies if it is freely terminated at plane 1 by a large space or half space containing a similar
acoustic medium. The open end of a hole or tube is a good approximation to free termination, although
it is necessary to add a normalized specific acoustic impedance end correction to take account of thefact that the normalized specific acoustic impedance which the end of the hole or tube experiences is
not completely zero.
For air in narrow circular holes or tubes, the normalized characteristic specific acoustic impedancez c and the wave number k differ from those of the free acoustic medium because of viscous and thermal
effects at the walls of the hole or tube (6). Define (6)sr

ZU0 K ,

(4)

where r is the radius of the circular hole or tube and ω is the angular frequency. η is the dynamic shear
viscosity of the acoustic medium (equal to 1.813 u 10 -5 Pa.s for air at 20 °C). The effective density ρ ofthe acoustic medium in a narrow circular hole or tube due to viscous effects is (6)UU0 ª«1  2 J1 s  j¬^s  j J s  j `º»¼ ,0(5)where J 0 and J 1 are Bessel functions of the first kind of order 0 and 1 respectively. The effective bulk
modulus K of an acoustic medium in a narrow circular hole or tube due to thermal effects is (6)K

J p0 ª«1  (J  1) 2 J1 Bs  j¬^Bs  j J Bs  j `º»¼ ,0(6)where γ is the adiabatic constant of the acoustic medium, p0 is the ambient pressure of the acoustic
medium and B 2 is the Prandtl number of the acoustic medium (equal to 0.71 for air at 20 °C).
The effective normalized characteristic specific acoustic impedance z c of the acoustic medium inthe hole or tube iszc

K U Uo co .

The effective speed of sound c of the acoustic medium in the hole or tube is

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(7)

c

K U.

(8)

The effective wave number k of the acoustic medium in the hole or tube isk Z c.
(9)
It is interesting to note that Maa’s (3) lumped parameter theory for the acoustic medium in a hole or
tube does not take account of the thermal effects on the effective bulk modulus predicted by equation(6). The reason for this can be seen by making the low frequency approximation of tan(kL) kL in
Eq. (3) and using Eqs. (9), (8) and (7) to obtain U o co z2 jZU L . The effective bulk modulus K has
disappeared and this equation is exactly Eq. (5) of Maa (3). In other words, because Maa assumed thatthe air in the hole or tube was incompressible, he did not need to consider the effective bulk modulus.
The perforation factor Ii is the fraction of the area occupied by an acoustic medium in ith planar
layer in any plane parallel to and in between the parallel planes 2 and 1 bounding the ith planar layer.
In this paper the perforation factor will be constant across each planar layer but may be different for the
acoustic media in different layers. The numbering i of the acoustic layers increases in the opposite
direction to which the incident planar sound wave is travelling. Continuity of acoustic pressure andacoustic volume velocity at the planar interface between two planar layers means that the normalized
specific acoustic impedances z i on each side of the planar interface satisfy
zi 1 Ii 1 zi Ii .
(10)
This means that the normalized specific acoustic impedance in an acoustic medium on ith side of theplanar interface (this will usually be z 2 in Eq. (1) for the ith planar layer of acoustic medium) is divided
by the perforation factor to convert it to the normalized specific acoustic impedance in an acoustic
medium on the (i+1)th side of the planar interface which has a perforation factor of 1. If there are two
or more acoustic media on the ith side of the planar interface, after this conversion is carried out, theirnormalized specific acoustic impedances are combined in the usual way by inverting them, summingthe inversions and inverting again. This calculated normalized specific acoustic impedance is thenmultiplied by the perforation factor of the actual acoustic medium to obtain the normalized specificacoustic impedance that plane waves in the actual acoustic medium encounter on the planar interface(z 1 in Eq. (1)) for the (i+1)th planar layer of acoustic medium.
This approach will not work if there is more than one acoustic medium on the (i+1)th side of theplanar interface because acoustic media alter the normalized specific acoustic impedance that the otheracoustic media experiences (z 1 in Eq. (1) for the (i+1)th planar layer of acoustic medium). In thecalculations described in this paper, this case has been approximated by assuming that z 1 in Eq. (1) inequation is zero. This reduces Eq. (1) to Eq. (3). Similar to above, these parallel lumped parameternormalized specific acoustic impedances are converted to impedances for a perforation ratio of 1 andcombined by inverting them, summing the inversions and inverting again. The result is then added thenormalized specific acoustic impedance on ith side of the planar interface after this impedance isconverted to the value for a perforation factor of 1 as described above.
Although this approximation has worked well, it is possible to a void the approximation. The
simplest method appears to be that of Simon (5). The transfer matrix equation relating the soundpressures and acoustic particle velocities at each side of the planar layer, from which Eq. (1) is derived,
is converted to an admittance matrix equation for each acoustic medium in the planar layer after taking
account of the perforation factor. These admittance matrices are summed and inverted to obtain thetransfer matrix for the planar layer which can be used to obtain an equation which is the equivalent of
Eq. (1) for the planar layer. A more general approach is to derive transfer matrices and coupling
matrices and combine them into a single matrix equation using the transfer matrix method as describedin chapter 11 of Allard and Atalla (6).
Consider the case of holes in a perforated panel whose length is extended by sets of tubes ofdifferent length into the air cavity behind the perforated panel. If any of the tube lengths a re shorter
than the depth of the cavity, the tube ends of different length sets of tubes opening into the air cavity
will be in different planes. This is no problem if the cavity is partitioned so that only one set of tu bes
opens into any one sub cavity and the simple calculation method described in this paper can be used.
The calculations show that, if the tubes are different enough in length, two low frequency soundabsorption coefficient peaks are obtained. This has been confirmed by experiment (7). If the air cavity
is not partitioned, experiment and theory show that only one low frequency sound absorption
coefficient peak is obtained (2, 7). Simon (5, 7) was able to make the calculations in this case because,
as stated in the introduction of this paper, he assumed that the all tubes opened into the air cavity at one

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end of the air cavity which is actually not the situation. Zhang (2) was able to make the calculations
because, as stated in the introduction of this paper, she assumed the low frequency approximation forthe specific acoustic impedance of the air cavity which does not depend on location in the air cavity. In
this paper, the specific acoustic impedances at the different planes in the air cavity, where tube ends
open into the air cavity were calculated and averaged taking into account the relative perforation ratiosof the sets of different length tubes. As discussed above, in this paper, the impedances of the air in the
different length tubes were calculated as lumped parameters assuming the tubes encountered zero
impedance at their ends and were combined as parallel elements. However low frequencies were notassumed. The full transfer matrix method could probably be used to avoid any approximations in thiscase, but the approximate methods agree reasonably well with experiment at low frequencies.
End corrections need to be applied to the ends of tubes which open into the air c avity and theexternal air space. However, because of the length of the tubes considered in this paper are large
compared to their internal radii, these end corrections have little effect. The imaginary part of the end
correction normalized specific acoustic impedance at one end of a hole or tube of internal diameter d,whose inside diameter is small compared to the wave length of sound , isIm ze hd 2 ,(11)where h equals 0.613 for an unflanged tube, 0.821 for a flanged tube, 8/3π = 0.849 for a baffled pistonand π/4 = 0.785 for a circular hole in a baffle. In this paper, the value of h for an unflanged tube has
been used for the end of a tube opening into the air cavity and the value of h for a flanged tube has beenused for the end of a tube opening into the external air space. Maa’s (3) low frequency lumped
parameter theory used the value of h for a baffled piston. When making calculations with Maa’s theory,
this paper changes Maa’s value of h to the values of h used in this paper. The real part of the end
correction is

Re ze

ZU 0K U0 c0 .

(12)This formula for the real part is only correct for the end of a hole or tube in an infinite baffle. Thus itwill not be correct for the tube ends in the air cavity. However, it has been used in that case in theabsence of a better formula because as stated above its effect is minimal because of the length of thetubes. Improved formulae for end corrections of holes and tubes which end in a baffle are discussed by
Li (8), but are not used in this paper because of their limited effect due to the length of the tubes and thefact that the tube ends do not always end in a baffle.
When the length of the tubes in the cavity are greater than or equal to the air cavity depth, this paper
assumes that the tubes are coiled, if need be, so that their ends are at one end of the cavity (but thevalue of h for an unflanged tube is still used). In this case the air cavity is treated as one cavity whencalculating its impedance. When the length of the tubes in the cavity are less than the air cavity depth,
this paper assumes that the tubes are straight and perpendicular to the planar layer. It divides the air
cavity into two sub cavities at the plane containing the tube ends which open into the air cavity andcalculates the impedance of each sub cavity at the plane containing the tube ends before combining
their impedances in parallel after correcting for the perforation ratios as described above. The
perforation ratio of the sub cavity not containing the tubes is 1. The perforation ratio of the other sub
cavity in the no sub cavity case is reduced by the external volume of the tubes in it.

3. EMPIRICAL CORRECTIONSTaking account of wave motion in the tubes and in the air cavity did predict a number of the highfrequency peaks in the experimentally measured sound absorption coefficient and in the real andimaginary parts of the normalized specific acoustic impedance. It was also hoped that taking accountof wave motion would increase the predicted sound absorption coefficient above the low frequency
peak where the predicted values are much lower than the experimental values. Examination of thepredicted and experimental imaginary part of the normalized specific acoustic impedance showed thatthe sound absorption coefficient was underestimated mainly because the magnitude of predicted
imaginary part of the normalized specific acoustic impedance was overestimated except when the
impedance was changing sign. It was decided to impose an empirical correction by limiting the
maximum value of the magnitude of the predicted imaginary part of the impedance above thefrequencyflc0 I ( DL) (2S ) ,(13)
fl is an estimate of the frequency of the low frequency sound absorption coefficient peak where the4070

imaginary part of the normalized specific acoustic impedance changes from negative to positive. I is
the perforation ratio of the tubes, D is the depth of the air cavity and L is the length of the tubes. The
value of the real part of the normalized specific acoustic impedance was also limited to make thecalculations agree better with theory. This was done as follows.

ycorr

sgn y

y

n

 ym n

1/ n

.

(14)

sgn(y) is the sign function which returns 1, 0 or -1 if its argument y is positive, zero or negative
respectively and n is an integer. y corr is the corrected calculated value, y is the original calculated value
and ym is the empirical magnitude limit of the real or imaginary part of the normalized specific acoustic
impedance. Because the value of n was chosen to be the even integer 2, the modulus signs are notactually needed for the imaginary part of the normalized specific acoustic impedance. Because the realpart of the predicted normalized specific acoustic impedance cannot be negative, the sgn function andthe modulus signs are not actually needed for the real part of the impedance. In this paper, y m has been
set to 15 for the imaginary part of the normalized specific acoustic impedance and to 45 for the realpart of the normalized specific acoustic impedance. This correction only needs to be applied to theimaginary part of the normalized specific acoustic impedance when Maa’s lumped parameter model isused for the tubes.
The second empirical correction made in this paper applies to the wave nu mber in the air cavity.
The real part of this wave number has, as is usual, been set equal to the wave number of sound in free
air. To control the magnitude of resonances in the air cavity, the imaginary part of the wave number in
the air cavity has been changed from 0 to -0.2. To make the predicted frequency at which the low
frequency sound absorption coefficient peak occurs agree better with experiment, the internal
diameters of the two tubes were increased by 0.1 mm.

4. EXPERIMENTS

Figure 1. The sound absorbing system studied in this paper. This is Fig. 2.1(a) of Zhang (2).
The sound absorbing system studied in this paper is shown in Fig. 1. The system consists of aperforated plate of thickness t1 = 2 mm with the holes arranged in a square array with the sides of thesquares having a length of b 1 = 4.8 mm. Copper ferules with an internal diameter of d 1 = 1.1 mm anda total length of l1 = 9.3 mm fill the holes in the perforated plate and their narrow flanges stand l2 = 0.2mm above the perforated plate. Rubber tubes with an internal diameter of d in = 1. 2 mm, an external
diameter of dex = 2.6 mm and a length of L 1 are inserted over the length of the copper ferules in the air
cavity. The air cavity has a depth of D = 43 or 120 mm. L 1 varied in 10 mm steps from 10 mm to 80 for
the 43 mm deep air cavity or to 160 mm for the 120 mm deep air cavity. Measurements were also madewhen b 1 = 7 mm for L 1 equals 20, 40 or 80 mm when D =43 or 120 mm. Each tube system consists of
two tubes in series. The first, in the direction of exiting the air cavity, is a tube with an internal4071

diameter of din = 1. 2 mm and a length of L2 L1  t1  l2  l1 . The second is a tube of internal
diameter of d 1 = 1.1 mm and a length of l1 = 9.3 mm. Zhang (2) also made measurements with half ofthe rubber tubes having a length of 10, 20 or 40 mm while the other half of the rubber tubes had alength 80 mm for a cavity depth of D = 43 or 120 mm.
Zhang (2) made the measurements using a two microphone impedance tube with an internal squarecross section of 40 x 40 mm and a length of 1.2 m between the face of the specimen and loudspeaker.
To avoid cross modes, the maximum frequency must be less than 4295 Hz. The maximum
measurement frequency of 3200 Hz is well below this limit. The distance of the closest microphone tothe face of the sample was 268 mm. The low and high frequency measurements were made with thetwo microphones spaced 150 mm and 50 mm apart respectively. ISO 10534-2 (9) requires that the
maximum frequency be 10 % less than the frequency at which the microphone spacing is equal to half
the wave length of sound. Thus the low and high frequency measurement maximum frequenc ies are
1031 and 3093 respectively. However in this paper, the experimental results have been graphed up to3200 Hz. This was done because it was observed that apart from when one of the microphone spacings
was close to a multiple of half the wave length of the sound, the high and low frequency measurements
agreed reasonably well. It should be noted that when the microphone spacing is close to a multiple ofhalf the wave length of the sound which occurs at 1145, 2290 and 3436 Hz, the measurement
uncertainty becomes very large.
ISO 10534-2 (9) says that “The lower frequency limit is dependent on the spacing between the
microphones and the accuracy of the analysis system but, as a general guide, the microphone spacing
should exceed 5 % of the wavelength corresponding to the lower frequency of interest.” Thus therecommended minimum measurement frequencies for the low and high frequency measurements are
115 and 345 Hz respectively. However, because some of the frequencies at which sound absorption
coefficient peaks occur are below these frequencies, measurements down to 50 Hz have been graphed.
It should be noted that the measurement uncertainty increases below the recommended minimum
measurement frequencies.

0.8

0.6

50
Theory
L1
L2
H1
H2
Maa

0.4
0.2

Normalized Real ImpedanceSound Absorption Coefficient

1.0

Theory
L1
L2
H1
H2
Maa

40
30
20
10

0

0.0
50 100 200 400 800 1600 3200
Frequency (Hz)Figure 2. The normal incident sound absorption
coefficient of a system with 160 mm long rubber
tubes in a 120 mm deep air cavity.

50

100 200 400 800 1600 3200
Frequency (Hz)Figure 3. The real part of the normalized specificacoustic impedance of a system with 160 mm longrubber tubes in a 120 mm deep air cavity.

When the spacing between the centres of the closest perforations was b 1 = 4.8 mm, there were 64
perforations in the 40 by 40 mm front perforated panel in an 8 by 8 square array. Had the holes been
spaced equally across the whole of the 40 mm in each direction, b 1 would have been 5 mm. Thus thequestion arose whether it was better to calculate the average perforation ratio across the whole of thefront perforated panel by assuming that b 1 = 5 mm or use the perforation ratio of the holes near thecentre of the front perforated panel by using b 1 = 4.8 mm. Comparison of the predicted frequency atwhich the low frequency sound absorption coefficient peak occurs with the experimental value showed4072

that b 1 = 4.8 mm was a better choice.

5. COMPARISON OF THEORY WITH EXPERIMENTFor each sound absorbing system, Zhang (2) made two low frequency measurements, L1 and L2,
and two high frequency measurements, H1 and H2. Zhang averaged the two low frequency
measurements together and averaged the two high frequency measurements together. She joined themeasurements in the two different frequency ranges together by averaging these two averages in an
overlapping frequency range in which they were both valid. However in this paper, all fourmeasurement results are displayed to give an indication of the measurement uncertainty. Note that the
departure of the low frequency measurements from the high frequency measurements for frequencies
near 1145 and 2290 Hz is due to the low frequency measurement microphone spacing being close to a
multiple of half the wavelength of sound.

1.0

15

5
Theory
L1
L2
H1
H2
Maa

-5

-15

Sound Absorption Coefficient

Normalized Imaginary Impedance

25

Theory
L1
L2
H1
H2
Maa

0.8
0.6
0.4

0.2
0.0

-25
100 200 400 800 1600 3200
Frequency (Hz)50 100 200 400 800 1600 3200
Frequency (Hz)Figure 4. The imaginary part of the normalized
specific acoustic impedance of a system with 160
mm long rubber tubes in a 120 mm deep air cavity.

Figure 5. The normal incident sound absorption
coefficient of a system with 80 mm long rubber
tubes in a 43 mm deep air cavity.

50

25
Theory
L1
L2
H1
H2
Maa

40
30
20
10

Normalized Imaginary ImpedanceNormalized Real Impedance

50

15
5
-5

Theory
L1
L2
H1
H2
Maa

-15

0

-25

50

100 200 400 800 1600 3200
Frequency (Hz)Figure 6. The real part of the normalized specificacoustic impedance of a system with 80 mm longrubber tubes in a 43 mm deep air cavity.

50

100 200 400 800 1600 3200
Frequency (Hz)Figure 7. The imaginary part of the normalized
specific acoustic impedance of a system with 80
mm long rubber tubes in a 43 mm deep air cavity.

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Figs. 2 to 7 compare the four measurement results with the theory described in this paper and withthe theory when Maa’s lumped parameter model is used for the two tubes for a cavity depth of 120 mmwith 160 mm long rubber tubes and for a cavity depth of 43 mm with 80 mm long rubber tubes. The
sound absorption coefficient and the real and imaginary parts of the normalized specific acoustic
impedance are shown. Both the theory described in this paper and Maa’s theory are able to predict the
low frequency sound absorption coefficient peak, but only the theory described in this paper is able topredict the higher frequency peaks. The theory described in this paper is able to predict the frequencyof the sound absorption coefficient peaks but does not always predict the amplitude of these peaks asaccurately as would be desired. The empirical corrections adopted in this paper have increased thepredicted sound absorption coefficients between the peaks so that these predicted values are closer tothe experimentally measured values. The empirical corrections have also brought the predicted realand imaginary parts of the normalized specific acoustic impedance closer to the experimental values.

6. CONCLUSIONS
Modelling the wave motion in the air in the bore of long tubes , rather than modelling this air as a
lumped parameter, enables the prediction of the frequencies of the higher frequency sound absorption
coefficient peaks although the prediction of the amplitude of these peaks is not always as accurate as
would be desired. However, the modelling of this wave motion was not sufficient to increase thepredicted values of the sound absorption coefficient at frequencies between the frequencies of thepeaks so that the predicted values were closer to the experimental values. It was necessary to impose
limits on the values of the real and imaginary parts of the normalized specific acoustic impedance inorder to increase these predicted values of the sound absorption coefficient. The imaginary part of the
wave number in the air cavity was also empirically set to a non-zero value in order to remove predicted
spikes due to cavity resonances.

ACKNOWLEDGEMENTS
The authors acknowledge the financial support provided by the Chinese Academy of Sciences(CAS) and CSIRO which enabled the first author to make a two week CAS-CSIRO Exchange visit to
The Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of
Sciences in Beijing in the second half of May 2018 which resulted in this paper.

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