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

Sound Absorption Characteristics of a Back-Perforated Honeycomb
Panel with an Air-Layer
Kyungsoo KONG1; Tetsuya SAKUMA2; Naohisa INOUE3
123

University of Tokyo, Japan

ABSTRACT
A honeycomb panel has a high potential in noise control engineering as well as in any other engineering
fields. This paper demonstrates that a Back-Perforated Honeycomb Panel(BPHP) placed with a back air-layer
exhibits unique absorption characteristics composed of the combination of panel- and resonator-typemechanisms. A BPHP is theoretically modeled as a two-mass system composed of panel and resonator’s neckmasses, and multiple air stiffness. First, resonance mechanisms are clarified from the model. It is revealed
that the panel-type resonance can arise from the panel mass and air stiffness corresponding to the total
thickness of the panel and air-layer, even if a BPHP is placed considerably closely to the back wall. Next,
normal- and random- incidence absorption coefficients are calculated with changing several parameters suchas perforation diameters, hole pitches, panel thickness, back air-layer thickness and so on. Furthermore, a
supplementary numerical calculation confirms the validity of the proposed theoretical model. Through thecase studies, several charts are arranged for effective design of BPHPs as low-frequency sound absorber.
Keywords: Sound absorbing material, Sound absorption coefficient, Honeycomb panel, Perforated plate

1. INTRODUCTION
With the expansion of acoustic design, a variety of acoustic materials have been actively developed
for high performance, lightness, durability, weather resistance, recycling and so on. One of suchmaterials is a honeycomb plate (sandwich panel having a honeycomb core) that possesses lightweightand high rigidity at the same time. This feature is advantageous for structural or exterior materials invarious fields such as transport vehicle, construction and so on. In the field of acoustics, there havebeen various utilizations proposed based on interesting mechanisms and ideas. For instance, ahoneycomb cores are used as the wave guide in the double-leaf panels [1,2], which leads to increaseof the insulation performances. Another implementation is that a Helmholtz resonator is composed byreplacing the one surface of a honeycomb sandwich panel by a micro-perforated plate(MPP) [3]. The
latter is becoming a practical use because it can be easily realized by perforating the plate surface ofthe honeycomb plate instead of using MPP. This type panel is noted PHP in the following.
PHP is usually installed with the perforated surface facing the indoor side to make use of resonatortype absorption. On the other hand, the authors have found that when PHP is installed with an air layer
directing its perforated side to the air layer side, a unique sound absorption characteristic can beobtained by the composite mechanism of the resonator type and plate vibration type. We named thisstructure as Back-Perforated Honeycomb-Panel (BPHP). This paper presents a simple theoreticalmodel based on a lumped parameter system to calculate the absorption coefficient for BPHP. Basedon this model, resonance mechanisms and absorption characteristics are clarified in detail. The
proposed model is also validated by the numerical analysis. Finally, through the case studies, several
charts are arranged for effective design of BPHPs as low-frequency sound absorber.

1

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4709384469@edu.k.u-toyko.ac.jp

2

sakuma@k.u-tokyo.ac.jp

3inoue@env-acoust.k.u-tokyo.ac.jp

2. THEORETICAL ANALYSIS
2.1 Formulation
Figure 1 shows the definition of various parameters and the lumped constant model. The equationsof motion to the whole of PHP and to the neck mass are given by the following Eqs. 1 and 2,
respectively. The continuity condition between the normal particle velocity of incidence field and thevibration velocity of the PHP is given by Eq. 3.
(1)
(2)
(3)
where P i and P r are the incident and reflected wave amplitude, respectively. ms is the area density of
PHP. m n is the effective area density of the neck, and c sn is the viscous damping coefficient of theneck, which is given as follows based on the Allard–Ingard model as follows.
,,
where S n is a hole area, and St , S h is a honeycomb area of internal and external measure, respectively.
r and ln are hole radius and neck length. ߟ and ߩ0 are viscosity coefficient and density of air. The
stiffness constants ksn, k nc, k scn and ksc are expressed as follows.

where c 0 is the speed of sound. The stiffness constants are reasonably approximated because thedimensions of each part are sufficiently smaller than the wavelength. It should be noted that the abovementioned inertia and resistance corrections are derived for the hole on the motionless baffle.
Therefore, the validity of using these corrections will be verified in the next section in comparison
with the detailed model by numerical analysis. In this paper, unit amplitude incident sound pressure(P i = 1) is given, and the simultaneous equation of Eqs. 1 to 3 is solved by the direct method to
calculate the complex reflectance.

Figure 1 – Problem settings for the theoretical calculation of absorption coefficients of a BPHP.

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2.2 Resonance Mechanism and Frequencies
The BPHP system without damping and under free vibration is given as the matrix form as follows.

Two resonance frequencies of this system are obtained as the frequency at which the determinant of
the left side matrix becomes 0. Hereinafter, these frequencies will be referred to as f1 and fh (fl < fh),
respectively. Figure 2 shows the changes of the resonant frequency fl, fh with respect to the panel
perforation ratio. The asymptotic curves obtained by approximation in terms of the perforation ratio
are also shown. It can be seen that the equivalent resonance systems are different before and after acertain perforation ratio (߶ ᇱ).
(4)In ߶ ൏ ߶ ᇱ, two resonances are brought by the single resonator (fll) and by the plate resonance (fhl)with Lc as an air layer. On the other hand, in ߶ ൏ ߶ ᇱ, two resonances are brought by the resonatorneck (mass) and two springs of the resonator cavity and the back air-layer (spring)(fhh) and by theplate resonance with total air layer Ls+L c (flh).
-2

Perforation ratio [ ]

10

-3

10

-4

10

-5

10

63

125

250
500
1k
2k
Frequency [Hz]Figure 2 – Relation between the perforation ratio and eigen-frequencies of a BPHP.

lnln

ΓRigidΩA

ΓSlip

ΓRigidLs

Lc

100 mmR1

Lsx(a) Perspective

Lc
y
z

Γv

ΓVBL

R2

Δx = 20 mm ΓIF

ΩAΩEΓRigidΓSlip

ΓRigid(b) Domain notation

Figure 3 – Problem settings for the theoretical calculation of absorption coefficients of a BPHP.

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3. COMPARISON WITH FINITE ELEMENT MODEL
3.1 Formulation of the Numerical Analysis
The finite element analysis is performed by setting an impedance tube problem as shown in Figure
3. The formulation of the analysis is briefly noted.
‫ڦ‬Acoustic Field : Let us consider the following weak-form of Helmholtz equation for the acoustic
field in the impedance tube, resonator neck, back air-layer and cavity of the honeycomb panel.
(5)
where p and k are the sound pressure and wave number in the air, respectively. μȀ߲݊ denotes theoutward normal derivative to the sound field.
‫ ڦ‬Elastic body vibration field : Let us consider the following weak-form of the elastodynamic
equation for the vibration field of the honeycomb plate.
(6)
where ‫ ܝ‬, ߩ E, ߱ , n are a three-dimensional displacement of an elastic body, density, angularfrequency, and an outward vector with respect to an elastic body vibration field. ߝ, ߪ is a strain
tensor and a stress tensor, and each component is defined as follows.

where ߣ, ߤ are Lamƴ ’s first and second parameters, respectively.
‫ڦ‬Boundary and Continuous conditions : It is essential for this problem to take the effect of the viscousboundary layer around the neck into account. Considering the influence of the translational motion of
the panel, we give the following conditions at the boundary of the acoustic field.

(7)

where t, b and n are the tangent, binormal and principal normal vector of the local orthogonal
coordinate system set on the boundary. v n is the normal direction velocity of the vibratory boundaryȞv, ‫׏‬tan is the nabla operator in the tangential plane, and c V is the square root of the kinematic viscosityof air. On the other hand, the following conditions are considered at the elastic body vibration fieldboundary.
(8)
Although the direct stress fn determined by the deformation is unknown before analysis, the normal
direction displacement on the slip boundary ȞSlip is zero (u࣭n = 0). So, the boundary integral termof Eq. 5 becomes zero.
‫ ڦ‬Identification of the normal-incidence absorption coefficient : By using the transfer function
method, the normal incidence absorption coefficient can be calculated from the sound pressure p1 , p2
at two points R 1, R 2 shown in Fig. 3 (b).

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3.2 Comparison of Lumped Constant and Finite Element Models
The normal-incidence absorption coefficients are calculated for different hole radius r and backair-layer Lc . The following parameters are fixed: p = 20 mm, ms = 4.2 kg/m2 , Ls = 30 mm, ts = 0.5 mm,
ln = 1 mm. Figure 4 shows the results of the lumped constant model and the finite element model
together with the resonance frequencies (fll, flh, fhl, fhh). Two peaks are observed in the absorption
coefficient approximately around flh and fhh. When the hole radius is 0.25 mm, the Helmholtz
resonance frequency(fll) of the PHP unit becomes close to the plate resonance frequency(flh), and thepeak frequencies of the absorption coefficient tend to shift to the low/high frequency range withrespect to flh, fhh, respectively. And it can be seen that fll and flh are closer as the back air-layer isthinner within the range of the aperture ratio ߶ ൏ ߶ ᇱ from the equation (4). The tendency of theresults of the lumped parameter model agrees well with that of the finite element model in any
conditions. However, under the condition of (a), (d) with a hole radius of 0.25 mm, the numericalanalysis value has a smaller peak value of the absorption coefficient than the theoretical value, andthe degree of shift with respect to flh, fhh is also small. Under the the condition of (a), (d), since thediameter of the hole and the viscosity boundary thickness in the vicinity of flh are approximately thesame, it is considered that both models include an error in the movement (inertia/ attenuation
correction) of the neck air. In addition, the influence of the viscous boundary layer on the bottom
surface of the acoustic tube (ȞVBL) is relatively small when the thickness of the back air-layer is 1 mm.
1(a) r = 0.25 mm, Lc = 1 mm

0.9
0.8
0.7

flh

fll

fhl

fhh

(d) r = 0.25 mm, Lc = 20 mm

flh

fll

fhl

0.6
0.5

fhh

0.4
0.3
0.2

Normal-Incidence Absorption coefficient [ ]0.101(b) r = 0.50 mm, Lc = 1 mm

0.9(e) r = 0.50 mm, Lc = 20 mm

0.8
0.7

flh

0.6

fll

fhl

fhh

flh

fhl

fll

fhh

0.5
0.4
0.3
0.2
0.1
0
1
(c) r = 0.75 mm, Lc = 1 mm

0.9(f) r = 0.75 mm, Lc = 20 mm

0.8
0.7
0.6
0.5

flh

0.4

fll

fhl

fhh

flh

fhl

fll

fhh

250

500

1k

0.3
0.2
0.1
0

63

125

250

500

1k

2k

4k 63
125
Frequency [Hz]2k

4k

Figure 4 – Normal-incidence absorption coefficients calculated for the BPHP with different
perforation radius and back air-layer thickness combinations.
Solid-black and dotted-gray lines calculated by the lumped-constant model and the finite element
model, respectively. FEM results are shown up to 2 kHz.

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4. PARAMETRIC STUDY
Since the validity of the lumped constant model has been shown in the previous section, this section
discusses the basic characteristics and design method of BPHP based on the lumped constant model.
1(a) r = 0.25 mm, Lc = 1 mm

0.9
0.8

(d) r = 0.25 mm, Lc = 20 mm

Incidence angle
0°
30°
60°
80°

0.7
0.6
0.5
0.4

Random-inc.

0.3
0.2
0.1
0
1

Absorption coefficient [ ]0.9(b) r = 0.50 mm, Lc = 1 mm(e) r = 0.50 mm, Lc = 20 mm(c) r = 0.75 mm, Lc = 1 mm(f) r = 0.75 mm, Lc = 20 mm

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

63

125

250

500

1k

2k

4k 63
125
Frequency [Hz]

250

500

1k

2k

4k

Figure 5 – Oblique- and random-incidence absorption coefficients calculated for the BPHP withdifferent perforation radius and back air-layer thickness combinations.
4.1 Oblique- and random-incident absorption coefficients
The oblique- and random-incidence sound absorption coefficients were calculated under the same
conditions as in Section 3.2. The random incident absorption coefficient was calculated by averaging
the oblique incident absorption coefficients calculated at each incident angle based on the Paris
equation. The results are shown in Figure 5.
Except for the resonance near flh in the back air-layer 1 mm (a ~ c), the spring constant kc of theback air-layer increases with the increase of the incident angle, and the peak of the absorption
coefficient shifts to the high frequency range. Because flh is substantially determined by the PHP
thickness when the back air-layer is thin. On the other hand, when the back air-layer is thick, thedegree of shift of the plate resonance also depends on the neck diameter, and the larger the neckdiameter, the more remarkable. In addition, the shift of the peak of the absorption coefficient at fh isabout the same under any conditions. In the random incident absorption coefficient, since the weightof the incident angle 45ι is the largest at the time of total averaging, the peak on the bass region side3471

can be seen around flh at about 45ι incident angle. When the shift of the resonance frequency due tothe incident angle is small (a ~ d), the peak value is also approximated to the value at 45ι incidence.
BPHP is characterized in that even if the back air-layer is very narrow, plate resonance absorption
equivalent to the PHP thickness can be obtained. At that time, it is easy to apply, because the peakfrequency of the absorption coefficient at random incidence is almost the same as that at normal
incidence. On the other hand, when the shift is large (e ~ f), there is a tendency to slightly broaden
the bandwidth.
4.2 Discussion on panel design and installation
Based on the same parameter settings in Section 3.2 as a basic condition, a plurality of parameters
was changed to calculate peak values and peak Q values of the normal-incident absorption coefficientat the resonance frequency fl on the low frequency range. The frequency at which the absorption
coefficient is a half value of the peak value is difficult to express explicitly from the equation (1) ~(3), so it was calculated by iterative calculation by iterative method. When the half width is 1/D octave
band width, the Q value is about 1.41D. After that, assuming three situations, we will consider thedesign and installation of the BPHP.
(a)

10

(b)

9

35

8
4

8

25
20

8

15
20 0

10
5

200

8

Area-density of the panel : ms [kg/m2]

40

4

Honeycomb thickness : L s [mm]

100

45

30

200 12

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12

8
100

7
6
8

8

5

8

200

4

8

3

200

2

12

12

0.2

1.0

1

200
4 300

5

10

20

25

30

200

35

4

40

45

50

(c)

0.8

0.9

1

Colormap of Absorption Coefficient

0.1

0.2

0.3

0.4

0.5

0.6

0.7

8

20
160
4
8

15
150

10
160

140

Back air-layer thickness : Lc [mm]15Back air-layer thickness : Lc [mm]

0
25

4

4

Perforation radius : r [mm]
29

12

100

4

50

4

8

0.2

150

1

1 30

5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Perforation radius : r [mm]Figure 6 – Absorption coefficient at the eigen-frequency, fl, calculated with changing (a) Ls and r,(b) ms and Lc and (c) Ls, Lc and r under the total depth Ls + Lc = 30 mm.
(Solid and dotted contour lines are drawn for f1 and Q factor, respectively.)
4.2.1 Higher performance of panels
Figure 6(a) shows the result when changing panel thickness Ls and hole radius r. Although thechange of the resonance frequency according to the hole radius is slight, the absorption coefficient
becomes large at a hole radius of about 0.3 mm ~ 0.4 mm. Also, as the Ls is larger, the resonancefrequency can be designed to be in the low frequency range, and the Q value can be further reduced.
4.2.2 Installation of existing panels
Figure 6(b) shows the results when changing the back air-layer thickness L c and the panel surface

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density ms . Even if Lc is about 1 mm, the absorption coefficient is very large, and it can be installed
in close contact with the back wall. It is possible to lower the resonance frequency by increasing Lc,
ms. However, when Lc is increased, the change in Q value is small but the peak value is small.
Conversely, when ms is increased, the change in peak value is small but the Q value is large.
4.2.3 When there is a restriction in the additional thickness
Figure 6(c) shows the results when changing the panel thickness Ls , the back air-layer thickness Lc
and the hole radius r. However, the distance from the panel surface to the back wall (the sum of L s
and Lc ) is constant at 30 mm. Under this condition, it can be seen that the contours of the absorption
coefficient, the resonance frequency, and the Q value change nearly in parallel. In addition, it isdifficult to considerably change the resonance frequency in the range where the absorption coefficientis large. Therefore, it is considered that setting a parameter having a high absorption coefficient anda low Q value allows absorption coefficient to the lower frequency range. As an example, in the caseof Ls = 10 mm, Lc = 20 mm, r = 0.2 mm, the resonance frequency is 160 Hz, the absorption coefficientis 1, and the Q value is 4 or so. At this time, absorption coefficient of 0.5 or more can be obtained
within the half width (140 Hz ~ 180 Hz).

5. CONCLUSIONSIn this paper, we constructed a simple theoretical model with lumped constant system to calculate
the absorption coefficient for the BPHP. As a result of comparing the normal incidence absorption
coefficient with the lumped constant model and the detailed model by the finite element method, it
was confirmed that both corresponded well. After that, through parametric studies, we summarized
the characteristics of the oblique-incidence and random-incidence absorption coefficients, andfoundational absorption performance design. Verification by actual measurement is required in the
future.

REFERENCES
1. M. Toyoda, M. Tanaka and D. Takahashi, Reduction of acoustic radiation by perforated board andhoneycomb layer system, Appl. Acoust., 68, 71-85 (2007)
2. M. Toyoda and D. Takahashi, Sound transmission through a microperforated-panel structure withsubdivided air cavities, J. Acoust. Soc. Am, 124, 3594-3603 (2008)
3. M. Toyoda, K. Sakagami, D. Takahashi and M. Morimoto, Effect of a honeycomb on the sound absorption
characteristics of panel-type absorbers, Appl. Acoust., 72, 943-948 (2011)
4. J.F. Allard and N. Atalla, Propagation of sound in porous media, 2nd ed., Wiley (2009)3473