PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Modeling sound transmission through a periodic acoustic
metamaterial grating of finite size
Xiang YU1; Fangsen CUI1; Wei Hin Mark WONG1
1

Institute of High Performance Computing, A*Star, Singapore

ABSTRACT
Recent studies have discovered that acoustic metamaterial possesses great versatility in designing
novel acoustic systems. This study investigates into the sound transmission through a periodic acoustic
metamaterial grating of finite size. The single-layer grating is constructed by periodically arrangingsub-wavelength unit cells in a slab, as a part of a large baffle between two acoustic domains . The
metamaterial unit cell consists of an open duct decorated with coiled resonators , which intends tosuppress sound transmission using its acoustic stop-band. The space-coiling structure allows themetamaterial to operate at low frequency with a compact size. Analytical approach to predict the soundtransmission loss (STL) of the combined baffle is developed. The concept of metamaterial grating canbe applied to the design of a novel ventilation window system with improved noise insulation.
Keywords: Acoustic metamaterial, Sound transmission loss, Ventilation window.

1. INTRODUCTION
Acoustic metamaterials are a type of artificial materials that are designed to provide exotic
properties absent in nature. The study of using acoustic metamaterials to manipulate sound wave
propagation has received much attention, particularly for the purpose of low-frequency absorption andisolation beyond the restriction of dimension and weight (1-3). Acoustic metasurfaces are a newer
class of acoustic metamaterials which have surface profile and sub -wavelength thickness, allowing
on-demand tailoring of sound propagation by imposing a specific boundary.
The class of metamaterial unit with an acoustic duct can be utilized for sound isolation withoutcompletely blocking air ventilation. The duct allows air passage while the resonators attached to theside-branch can provide attenuation effect on the sound transmission. For example, a two -dimensional
array of low loss cylindrical cavities was studied, showing negative bulk modulus and dispersion
relation of acoustic bands (4). The resonant tubular array demonstrated continuous tuning ofcompressibility in a wide range (5). Conceptually, the ensemble of duct and attached resonators can beconsidered as a metamaterial unit cell. If we periodically arrange such unit cells in a slab, an acoustic
grating can be constructed to realize special acoustic functions by making use of the unusual propertyof the unit cells and also the interaction between the unit cells. Here, the acoustic grating is viewed asa subclass of acoustic metasurface where the constituting unit cells are identical rather than possessing
a certain phase relationship. As the thickness of a single-layer acoustic grating equals to that of theconstituting element, the slab structure has excellent geometric advantage benefiting from thesub-wavelength size of the unit cells.
In this study, we present a numerical method to predict the sound transmission through ametamaterial grating using semi-analytical approach. The single-layer grating is constructed byperiodically arranging sub-wavelength unit cells in a slab, as a part of a large baffle between two
acoustic domains. It is shown that a metamaterial grating can provide superior sound insulation whileallowing air to ventilate through the openings distributed on the surface. By using such a principle, afull-scale metasurface ventilation window is designed and experimentally demonstrated . Substantial
improvement in the acoustic performance shows the benefit of incorporating such metasurface concept
into the design of sound insulation components.

2. Semi-analytical numerical approachThis section briefly outlines the necessary steps to model the sound transmission of a metamaterial1971

grating system. Let us consider a two-dimensional slab comprised of a number of metamaterial unit
cells arranged periodically, forming a single-layer grating as a part of a large baffle. The metamaterial
unit cells are open acoustic ducts decorated with coiled resonators, as shown in Fig. 1. The boundariesof the unit cells and the separation between grating elements are assumed as rigid to simplify thetheoretical modeling. Using the Cartesian coordinate system, let x-axis denote the slab normal andy-axis denote the slab tangential. The sound incidence angle  is defined relative to the slab normal,
with   0 being normal incidence case and the   90 being the grazing incidence case.
Assuming the slab is excited by plane wave with incidence angle  , the incident sound pressure
field on the left-hand side of the baffle can be expressed as:pin ( x, y,  )  p0e j(t  kx cos  ky sin  )(1)where pin is the incident pressure field, p0 is the amplitude of incident wave; j 

1 ; k is thewave number k   / c0  2 f / c0 ,  is the angular frequency and c0 is the speed of sound in air.
At the slab interface where x=0, the above expression reduces to p0 ej(t  ky sin  ). The time dependencee jt can be neglected if we carry out harmonic analysis.

Figure 1. A slab comprised of metamaterial unit cells arranged periodically to form a single -layer
acoustic grating. The metamaterial unit cell is an open duct decorated with coiled resonators.
On the right-hand side of the slab, the transmitted sound through the unit cell causes an air velocity
disturbance at the outlet aperture and radiates into the receiving field. Under the baffled condition,
sound radiation from the periodic outlet apertures can be modeled as cylindrical waves radiating intoa semi-infinite free space. To determine the radiation impedance between the radiated pressure field
and the unit cell velocity, the governing Helmholtz equation is:[ 2  k 2 (r )] prad (r ,  )  0(2)22where prad is the radiated sound pressure,  is the Laplacian operator,   2  2 forx y1  2rCartesian coordinate system and  for cylindrical coordinate system, r is ther r r21972

position vector of receiving point R. By omitting the time-dependent term eabove equation is:j t, the solution to theprad (r )  aH 0(2) (kr )

(3)
(2)

where a is the amplitude determined by the strength of sound radiation source, H 0

is the zero order

Hankel of the second kind.
Assuming the outlet aperture of a unit cell is small compared to the acoustic wavelength, and thereceiving point R is located in the far field, the outlet aperture can be approximated as a point sourcewith uniform velocity. The radiation strength is a  k 0c0 q / 2 and Eq. (3) can be further written as:

prad (r ) k 0 c0 q (2)H 0 (kr )2(4)where q is to describe the strength of air velocity disturbance at the outlet aperture: dq  udy , with dhere being the differential operator, u being the normal air velocity averaged over the aperture areaho .
Assuming the radiated sound pressure field behind the baffle is a linear superposition of the soundfields generated by all the periodic unit cells. The sound pressure pn at a specific unit cell n due toexcitation from all the unit cells, including cell n itself, inversely affects the pressure continuity
relation. This coupling effect has to be apprehended to solve the coupled system response. The mutual
radiation impedance between a receiving unit cell n and an exciting unit cell m is Z nm  pn / um , andthe self-radiation impedance is Z mm  pm / um . By translating the origin of the coordinate system tothe center of the outlet aperture at the exciting cell m, i.e., let y=0, the acoustic pressure on thereceiving cell n due to an exciting cell m, provided that m  n is:pn ( yn ) k 0 c0 ho /2
um H 0(2) (ky )dy2  ho /2(5)where the integration is taken over the radiating outlet aperture at cell m from ho / 2 to ho / 2 . The
mutual radiation impedance between cell m and n is thus:

Z nm pn k 0c0 ho (2)H 0 (kd nm )um2(6)

where d nm is the separation distance between the cell n and m.
For the self-radiation impedance where m  n :Z mm pm k 0c0 h0 /2 (2)H 0 (ky )dyum
2  h0 /2(7)The metamaterial unit cell is modeled as a waveguide to connect the incident and receiving soundfields analytically described as above. The sub-wavelength property of the unit cell makes it
reasonable to consider its interaction with the adjacent acoustic fields as a one -dimensional waveguide.
For the metamaterial unit cell as shown in Fig. 1, its acoustic wave propagation in the duct isone-dimensional if the frequency range is below the cut-off frequency determined by the aperture size:f c  c0 / 2ho . The one-dimensional waveguide can be described by the four-pole transfer matrix,
which relates the pressure and velocity conditions on both ends of a unit cell as: p1   A B   p2  u   C D   u  2
 1 (8)The four-pole parameters can be obtained either analytically or numerically, depending on thecomplexity of the unit cell geometry. After the unit cell parameters in Eq. 8 and the radiation

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impedances in Eqs. 6 & 7 are obtained, the system response can be determined by coupling the incident
and receiving field together using a sub-structuring approach (6). Then the sound transmission loss ofthe single-layer metamaterial can be calculated.
3.

TL of a 2D metamaterial grating

As a numerical example, the sound transmission characteristics of a metamaterial grating
composed of unit cells as shown in Fig. 2 are studied using the proposed approach. The STL of a single
baffled unit cell is first calculated. In figure 2, the STL shows a sharp peak between 700 and 900 Hz,
owing to the stop-band of the unit cell. In this frequency range, the coiled resonators along the duct
impose strong impedance mismatch along the wave propagation direction, thus the incoming wave isstrongly reflected back to the incident field. The unit cell is blocking noise transmission, while at thesame time offering a transparent path for air movement.
Then, the STL of the acoustic grating system comprised of five identical unit cells is calculated.
First, we consider normal sound incidence and the unit cells are closely adjacent to each other. With d= 0 m, the STL in Fig. 2 shows a similar sharp peak, whereas TL at other frequencies are more flattenedcompared to the single unit cell STL. This indicates that with more periodic arrangement of identicalunit cells, the metamaterial grating is able to provide the same noise isolation ability as the unit cells.
The metamaterial block can then be scaled up into a linear or planar structure to provide ideal acousticfunction.

Figure. 2. Unit cell geometry with three coiled resonators (in cm) and STL of a single unit cell andacoustic gratings comprised of five unit cells.

4.

Design of metamaterial ventilation window

Using such design principle, a metamaterial ventilation window is designed and experimentallydemonstrated at the room scale. Using the center opening of a conventional window combination, a
3x3 metamaterial array is designed. Each unit cell has a two-layer resonant chamber as sketched in Fig.
3 (a), where an opening occupying a quarter of the unit cell surface area is designed to allow airventilation. The dimension of the chamber is tuned using numerical model to optimize the attenuation
performance for traffic noise frequency near 1000 Hz (7). The fabricated prototype is shown in Fig. 3(b), the structure made of 2 mm acrylic panels provides good transparency.

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Figure 3. (a) Design and dimensions of a meta-material ventilation window; (b) The fabricated prototype.
The performance of the meta-material window is first measured in linear frequency range by using
tonal noise excitation to verify the design and prediction. Fig. 4(a) shows the predicted transmission
loss and the measured data, where good agreement is found in both frequency range and attenuationlevel. Thus the numerical model can provide an accurate tool to design such kind of ventilation
window for different applications with different noise features. In Fig. 4(b), the Sound Reduction
Index (SRI) of the metamaterial window is compared with that of a normal window, as described in Fig.
4(b). The SRI is measured with either metamaterial or normal window opened alone, and bothwindows are opened with a similar ventilation area. Measurement is conducted according to ISO
10140. The improvement by metamaterial window is shown to be very significant, and it provides anenhanced attenuation region between 800 Hz and 1600 Hz, exactly as we designed. The Single Number
Quantity (SNQ) rated by ISO 717 is 21.7 dB for the metamaterial window and 14.6 dB for the normalwindow, i.e., 7.1 dB higher. Using broadband white noise and real traffic noise recording, it wasverified that meta-material window can perform 6-8 dB better than normal window. Therefore, themetamaterial window is concluded to be capable of improving the noise attenuation by 7 dB.

Figure 4. (a) Comparison between predicted attenuation performance (design) and actual measured data; (b)Comparison between the measured SRI of the metamaterial window and the normal window with the same
opening area.
5.

CONCLUSIONSThis paper has presented a numerical approach to model the sound transmission through an acoustic
grating system comprised of sub-wavelength unit cells. The unit cell was constructed based on an
acoustic duct decorated with coiled resonators. A number of unit cells were arranged periodically in aslab to form an acoustic grating. Numerical results showed that the proposed metamaterial grating canprovide desired sound insulation in the design frequency range without entirely blocking the air flow.
The design concept has been applied to the design and tuning of a metamaterial ventilation window
prototype. Experimental measurement demonstrated the validity of the proposed numerical approach.
Improved sound reduction performance has been achieved by using metamaterial window design.

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