A concept of ultra-thin low frequency perfect sound absorber is proposed and
demonstrated experimentally. To minimize non-linear effects, an high ratio of
active area to total area is used to avoid large localized amplitudes. The
absorber consists of three elements: a mass supported by a very flexible
membrane, a cavity and a resistive layer. The resonance frequency of the sound
absorber can be easily adjusted just by changing the mass or thickness of the
cavity. A very large ratio between wavelength and material thickness is
measured for a manufactured perfect absorber (ratio = 201) . It is shown that
this high sub-wavelength ratio is associated with narrowband effects and that
the increase in the sub-wavelength ratio is limited by the damping in the
system

Aurégan, Yves

English

arXiv

International audienceA concept of ultra-thin low frequency perfect sound absorber is proposed and demonstrated experimentally. To minimize non-linear effects, an high ratio of active area to total area is used to avoid large localized amplitudes. The absorber consists of three elements: a mass supported by a very flexible membrane, a cavity and a resistive layer. The resonance frequency of the sound absorber can be easily adjusted just by changing the mass or thickness of the cavity. A very large ratio between wavelength and material thickness is measured for a manufactured perfect absorber (ratio = 201). It is shown that this high sub-wavelength ratio is associated with narrowband effects and that the increase in the sub-wavelength ratio is limited by the damping in the system

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Ultra-thin low frequency perfect sound absorber with
high ratio of active area
Yves Aurégan
To cite this version:
Yves Aurégan. Ultra-thin low frequency perfect sound absorber with high ratio of active area. Applied
Physics Letters, American Institute of Physics, 2018, 113 (20), pp.201904. ￿10.1063/1.5063504￿. ￿hal-
02337017￿
Ultra-thin low frequency perfect sound absorber with high ratio of active area
Yves Aure´gan∗
Laboratoire d’Acoustique de l’Universite´ du Mans,
Centre National de la Recherche Scientifique (CNRS), Le Mans Universite´,
Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
A concept of ultra-thin low frequency perfect sound absorber is proposed and demonstrated
experimentally. To minimize non-linear effects, an high ratio of active area to total area is used
to avoid large localized amplitudes. The absorber consists of three elements: a mass supported
by a very flexible membrane, a cavity and a resistive layer. The resonance frequency of the sound
absorber can be easily adjusted just by changing the mass or thickness of the cavity. A very large
ratio between wavelength and material thickness is measured for a manufactured perfect absorber
(ratio = 201) . It is shown that this high sub-wavelength ratio is associated with narrowband effects
and that the increase in the sub-wavelength ratio is limited by the damping in the system.
Sound absorption is of great interest to engineers who
want to reduce sound reflection in room acoustics or re-
duce sound emissions. There is a need for innovative
acoustic absorbent materials, effective in low frequencies
while being able to deal with spatial constraints present
in real applications. Innovative ultra-thin materials are
also useful tools for the scientific community to manip-
ulate sound waves and obtain negative refraction [1, 2],
sub-wavelength imaging [3, 4], cloaking [5], etc.
Traditional sound absorption structures use perforated
and micro-perforated panels covering air or porous ma-
terials. [6, 7]. These materials have a low reflection of
normal incident waves at frequencies such that the wave-
length (λ = c0/f where c0 and f are the sound velocity
in air and the frequency) is about four times the thick-
ness of the material H leading to a sub-wavelength ratio
rH = λ/H ' 4. There has been a very significant re-
duction in the thickness of the absorbent materials [8]
by using space-coiling structures [9–12] of by using slow-
sound inside the material [13–15].
Nevertheless, all these structures have a front plate
with small holes leading to a very low open area ratio
(σ = area of the orifices on the total area). In this case,
since the velocity in the orifices is equal to the acoustic ve-
locity in the incident wave divided by σ, some non-linear
effects can easily occur in the orifices when the amplitude
of the incident wave becomes large enough [16]. More-
over, in many engineering applications, a grazing flow is
present and its effect on the impedance of the material
is inversely proportional to σ [17]. For instance, it was
experimentally shown [18] that the efficiency of a thin
slow-sound metamaterial with σ = 0.023 was divided by
100 in the presence of a grazing flow with a Mach number
of 0.2. Therefore, in the case of high sound levels or in
the presence of flow, the additional constraint of having
a high open area ratio must be added in the design of
structures that are thin and absorbent at low frequency.
From this perspective, the use of elastic membranes
and decorated elastic membranes as sound absorbers is
∗ yves.auregan@univ-lemans.fr
an interesting option. [19–21]. In most of the studied
structures, the membrane represents a large part of the
active surface and the added mass (platelets) is of smaller
size. The maximum absorption appears for an hybrid res-
onance [20] due to the interaction between two modes of
coupled system consisting of the membrane, the platelet
and the air cavity. By a proper arrangement of the var-
ious parameters very impressive results can be obtained
both in reflection [20] and in transmission [22]. At the
resonance frequency, the three main parameters of mem-
brane absorbers (equivalent mass, stiffness and damping)
are very sensitive to the characteristics of the membrane.
By changing one of the properties of the membrane (its
tension, its mass density,...) or the size of the added
mass, the three equivalent parameters are simultaneously
changed.
Cavity
Membrane
Mass
Resistance
FIG. 1. Sketch of an ultra-thin low frequency (UTLF) res-
onator.
This letter presents an ultra-thin low frequency
(UTLF) resonator for which the three parameters (mass,
2rigidity and damping) can be modified independently of
each other. A model has been measured and perfect ab-
sorption occurs for a sub-wavelength ratio rH = λ/H =
200 which is, up to now, the highest sub-wavelength ratio
experimentally demonstrated for a perfect absorber.
This UTLF resonator is ideally composed of three el-
ements displayed on Fig. 1: A volume of air sealed in a
cavity where the compression of the air acts as a spring,
a mass that moves like a piston in the normal direc-
tion (without letting the air trapped in the cavity pass
through with the help of a membrane) and a thin resistive
layer that dissipates energy. The main difference com-
pared to decorated elastic membranes absorbers [20] is
that the membrane plays almost no role in the frequency
of perfect absorption. The role of the very flexible mem-
brane is just to guide the motion of the mass and seal
the cavity. It will be demonstrated that the membrane
also produces some damping.
As the low frequency regime is considered (any axial
dimension is much smaller than λ = c0/f ), the con-
tinuity of the acoustic flow rate is assumed across the
resistance and the mass: vi = vm = vc where vi, vm and
vc are respectively the incident wave velocity, the velocity
of the mass and the mean velocity entering in the cavity.
For simplicity, a, the difference between the radius of the
tube and the radius of the mass, is neglected compared
to the tube radius R.
The UTLF resonator can be described by its
impedance ZUTLF (normalized by the air characteristic
impedance Z0 = ρ0c0 where ρ0 is the air density) which
is the sum of three terms since the elements are mounted
in series. The first term in the impedance is linked to the
air compressibility in the cavity of thickness B and can
be written Zc = pc/Z0vc = c0/iωB = 1/iωˆC where pc
is the uniform acoustic pressure in the cavity, ω = 2pif ,
ωˆ = ω/(2pifR) where fR is the resonance frequency of
the UTLF resonator. The compressibility term
C =
2pifRB
c0
=
2pi
rB
directly characterizes the sub-wavelength ratio rB =
λR/B at resonance. This value of the cavity impedance
is only valid at very low frequencies where ωB/c0  1.
For higher frequencies or higher cavity thickness, a more
exact value is Zc = 1/i tan(ωˆC).
The second term in the impedance is linked to the
mass motion subjected to the pressure difference ∆pm
between its two faces. The associated impedance is Zm =
∆pm/Z0vi = iωm/ρ0c0Sm = iωˆL where m = ρmSme is
the mass of the moving part, Sm = pi(R−a)2 is the mass
area, ρm is the density of the mass material and e is the
mass thickness. It can be noted that the moving mass is
connected to the fixed tube by a membrane that acts as a
spring but this effect is neglected due to the low value of
the membrane stiffness. The last part of the impedance
is linked to the resistive layer and is ZR = ∆pR/Z0vi =
Re where ∆pR is the pressure drop through the resistive
layer and Re is the normalized resistance of the resistive
layer. As a result, the UTLF impedance can be written
ZUTLF = Re + iωˆL+ 1/iωˆC.
To be a perfect absorber when ωˆ = 1, the UTLF
impedance must perfectly match the characteristic
impedance of the air. This is achieved when the two
conditions R = 1 and LC = 1 are met. In this case, the
normalized UTLF impedance can be written:
ZUTLF = 1 +
1
C
(
iωˆ +
1
iωˆ
)
(1)
showing that the impedance depends only on the sub-
wavelength ratio rB . The absorption coefficient is defined
by α = 1−|(Z−1)/(Z+1)|2 and relation (1) shows that
it is equal to 1 for ωˆ = 1.
The change rate of α around ωˆ = 1 is linked to the
slope of the imaginary part of the impedance which is
given by |dZUTLF/dωˆ| = 2/C for ωˆ = 1. Using a Tay-
lor expansion near the resonance when C is small, the
frequency bandwidth ∆f for which α > 0.75 is given by
∆f/fR = 2C/
√
3. Then, α has an increasingly sharp
peak as the sub-wavelength ratio rB increases as it is
shown in Fig. 2. In principle, there is no theoretical up-
per limit for the sub-wavelength ratio but the price to
pay to have highly sub-wavelength cavities is to have a
very small bandwidth. From a practical point of view (as
will be seen later), the increase in the sub-wavelength ra-
tio rB is limited by the losses that always become large
when rB increases.
FIG. 2. Absorption coefficient α of a perfect resonant ab-
sorber as a function of the dimensionless frequency ωˆ for two
values of the sub-wavelength ratio: rB = 10 (dashed blue)
and rB = 200 (continuous red).
Figure 3(a) shows the different elements that composes
a manufactured sample of UTLF. The first element is a
resistive layer, 1 on Fig. 3(a), consisting of a metal wire
mesh glued to a short tube (inner radius R= 15 mm,
outer radius 19 mm, 1 mm long). Many wire meshes are
available with different resistances and the normalized
resistance of the one used here was measured at Re=
0.29. The second part is the mass element, 2 on Fig.
3(a). The mass is a steel disk (ρm = 7800 kg.m
−3)with a
radius of R−a = 13 mm and a thickness of e = 3 mm. A
3slightly tensioned latex membrane 20 µm thick is glued
to a short tube (3 mm long) and then the mass is glued
to this membrane. The cavity consists of tubes of various
lengths B and a plexiglas disk to close it, 3 and 4 on Fig.
3(a).
1 2 3 4
(a)
(b)
(c)
fR
FIG. 3. Experimental results:(a) Picture of the elements of
the UTLF: 1 Resistive layer, 2Mass element, 3 Spacer tube to
make the cavity, 4 plexiglass cover. (b) Measured absorption
coefficient α of the UTLF as a function of frequency. (c) Zoom
around the perfect absorption.
When a cavity is made with a tube of thickness B=12
mm, the theoretical value of the resonance frequency is
given by
fR =
c0
2pi
√
ρ0
ρmeB
= 111 Hz.
This value can be compared to the measured frequency
given in 3(b). A nearly perfect absorption (α = 0.9991) is
obtained for fR = 107.25 Hz. The 4% error made on the
frequency value is small considering all the simplifications
that have been made in the calculations. The value of the
compressibility term is C = 2pi/269. The bandwidth for
which α > 0.75 is ∆f = 2.5 Hz = CfR. The total height
of the material H = 16 mm is the sum of the cavity height
B, mass thickness e and thickness of the resistive layer
glued on its support (1 mm). Thus, the sub-wavelength
ratio of this material is rH = λ/H = 201.6. Up to now,
this ratio between wavelength and material thickness is
the highest ever measured in an absorbent material.
The resonance frequency is very easy to adjust since
it depends only on the mass of the moving part and
on the height of the cavity. The simplicity of the ad-
justment comes from the fact that membrane effects can
be neglected to compute fR because of the low value of
its equivalent stiffness. In the device under study, the
resonance frequency of the mass element measured with
the membrane alone (without the cavity) is about 35 Hz
while it increases to 107.25 Hz in the presence of the
cavity. This indicates that the equivalent stiffness of the
membrane is 9 times lower than the one of the cavity
with a thickness of B=12 mm.
The membrane is involved in the dissipative part of
the impedance which is shown in Fig. 4(a). The blue
line in this figure gives the dissipative effect when the
resistive layer is not present. In this case, the dissipative
effects come solely from the membrane. First of all, it
can be noted that this dissipation is relatively high (the
minimum value of <(Z) is 0.71 representing 71% of the
target value). It has already been noted in [20] that even
a small dissipation in the membrane material could result
in significant absorption of the overall system. Second,
it can be noted that the dissipation curve appears to
result from two different effects. For f >140 Hz, the
dissipation increases with frequency. This effect is the
only one that is experimentally observed when there is no
cavity. This dissipation can therefore be attributed to the
mechanical damping resulting from the elongation of the
membrane when the mass moves. For f <90 Hz a second
phenomenon appears for which the resistance decreases
with frequency. It has been experimentally observed that
this second resistance increases as the cavity thickness
decreases. One possible explanation is that, when the
air trapped in the cavity becomes difficult to compress
or expand (i.e. for small volume or low frequency), it
inflates and aspirates the membrane as schematized in
Fig. 4(b), which increases the dissipative phenomena.
FIG. 4. (a) Real part of the impedance (resistance) of the
UTLF absorber without the resistive layer (blue) and with
the resistive layer (black). (b) Schematic deformation of the
membrane.
4The combination of the two previous effects induces a
minimum of resistance for a frequency close to resonance.
When this minimum is smaller than 1, it is possible to
add a resistive layer that shifts the resistance to higher
values, see the red curve in Fig. 4(b) in order to adapt the
resistance exactly to 1 at the resonant frequency. This is
what has been done to obtain the perfect absorber whose
absorption curve is presented on the Fig. 3(b). In the
present case, the addition of a resistive layer only changes
the maximum attenuation by a few percent. But for less
extreme conditions (λ/B 200), the attenuation due to
the membrane can be much lower and the addition of a
resistive layer very useful to obtain a perfect absorption.
To conclude, we have seen that it is possible to build
a UTLF absorber with very simple elements: mass sup-
ported by a very flexible membrane, cavity closed by the
membrane and resistive layer. This device can have a
large open area to minimize non-linear effects or reduce
dependence on grazing flows. The frequency of the UTLF
absorber can be easily adjusted by simply changing the
mass or thickness of the cavity. With this type of ab-
sorber, it is possible to obtain a very high ratio between
the wavelength and the thickness of the material but, as
a necessary counterpart, the bandwidth is very narrow.
The energy dissipation comes partially from the mem-
brane and partially from the resistive layer. When try-
ing to further reduce the size of the cavity, the resistance
of the absorber becomes greater than the characteristic
impedance of the air due to the dissipation in the mem-
brane. The dissipation is therefore the limiting factor in
the decrease in the size of this type of absorbers. This
remark is not specific to this absorber but can be gen-
eralized to many resonant sub-wavelength absorbers: It
is the damping that limits the sub-wavelength ratio in
resonant absorbing devices that are build with three in
series elements (mass, spring and damping) like the clas-
sical Helmholtz resonator. One possibility to overcome
this limit is to add some gain in the system by supply-
ing external energy using, for example, electro-dynamical
devices [23, 24], thermo-acoustic devices[25] or using the
flow[26] which is present in many engineering applica-
tions.
ACKNOWLEDGMENTS
This work was supported by the ”Agence Nationale
de la Recherche” international project FlowMatAc No.
ANR-15-CE22-0016-01.
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Ultra-thin low frequency perfect sound absorber with high ratio of active area

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