[Perfect acoustic absorption in deep sub-wavelength structures for the ventilation problems with degenerate resonators]
Vicente Romero García1 , Noé Jiménez2 , Olivier Richoux1 , Georgios Theocharis1 , Jean Philippe Groby1 and Vincent Pagneux1(1) Laboratoire d’Acoustique de l?Université du Mans, UMR CNRS 6613
(2) Instituto de Instrumentación para Imagen Molecular I3M-CSIC

Abstract
In this work we report a mechanisms providing symmetric perfect acoustic absorption by sub-wavelength structures in the ventilation problem. While in the problem of reflection the mechanism consists in critically couplinga single resonance independently of its nature, the problem of transmission becomes more complicated and adegenerate resonator with symmetric and antisymmetric resonances should be designed with both resonances
critically coupled. The system analyzed in this work is made by a panel with a periodic distribution of slits, theupper wall of which are loaded by Helmholtz Resonators. The propagation in the slit is highly dispersive dueto the presence of the resonators, producing a slow sound propagation before the resonance, and down shifting
it to low frequencies. By controlling the geometry of the resonator, the visco-thermal losses can be tuned inorder to compensate the leakage of the system and fulfill the critical coupling condition so activating the perfectabsorption of sound. In the case of transmission, a double slit is needed, one acting as symmetric and the otherone as antisymmetric resonators.
Keywords: Acoustic metamaterials, Acoustic metasurfaces, Perfect absorption

The ability to perfectly absorb an incoming wave field by a sub-wavelength material is advantageous for several
applications in wave physics. This challenge requires to solve a complex problem: reducing the geometricdimensions of the structure while increasing the density of states at low frequencies and finding the goodconditions to match the impedance to the background medium. A successful approach for increasing the densityof states at low frequencies with reduced dimensions is the use of metamaterials. Recently, several possibilities
based on these systems made of open lossy resonant building blocks have been proposed to design soundabsorbing structures which can present simultaneously sub-wavelength dimensions and strong acoustic absorption[1, 2]. Among them, Helmholtz resonators (HRs) have been shown as potential candidates to solve the problem
due to the tunable possibilities they offer [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
These open resonant lossy systems, at the resonant frequency, are characterized by both the leakage rate ofenergy (i.e., the coupling of the resonant elements with the propagating medium) and the intrinsic losses of theresonator. The interaction of an incoming wave with this lossy resonant structure, in particular, the impedancematching with the background field, can be achieved when the energy leakage perfectly compensates the inherent losses. This is known as the critical coupling condition, allowing the energy to be trap around the resonant
elements and generating a maximum of energy absorption. In the case of reflecting system (one port systems),
either symmetric or antisymmetric resonances that are critically coupled can be used to obtain perfect absorption of energy by a perfect trapping of energy around the resonators. In the case of transmission systems,
degenerate critically coupled resonators with symmetric and antisymmetric resonances or systems with broken
symmetry can be used to perfectly absorb the incoming energy. In this work we report the possibility to create
degenerate resonators.
This methodology has been shown as an efficient tool to design broadband acoustic absorbers in the low frequency range. In our case, fine tuning of the losses [11, 12] and of the geometric characteristics [13, 14] ofthe sub-wavelength resonators leads to the crossing of the complex zeros of the eigenvalues of the scattering2278

matrix with the real axis which signifies the perfect absorption condition. This methodology has been also used
to design efficient broadband absorbers in the low frequency regime.
Let us consider a two-port, one-dimensional and reciprocal scattering process. The relation between the amplitudes of the incoming (a, d), and outcoming (b, c) waves, on both sides of the scatterer Σ, is given by  caT R+a= S( f )
=,(1)
bd
R− Td
where S( f ) is the scattering matrix (S-matrix), f is the incident wave frequency, T is the complex transmission
coefficient, R− and R+ are the complex reflection coefficients for left (−) and right (+) incidence, respectively.
In this work, the time dependence convention of the harmonic regime is e−iωt , and it will be omitted in the− + 1/2
following. The eigenvalues ofare expressed as λ1,2 =T√± [R R ]  and the eigenvectors of the the S-matrix√+R+ R− , R+ . Therefore, the ratio of thesystem are v1 = (v11 , v12 ) = R , − R+ R− and v2 = (v21 , v22 ) =eigenvector components v1i and v2i is v2i /v1i = (−1)i (R− /R+ )1/2 . A zero eigenvalue of the S-matrix corresponds
to the case in which the incident waves can be completely absorbed (b = c = 0). This, called coherent perfectabsorption (CPA) [?], happens when T = ±[R− R+ ]1/2 and the incident waves a, d correspond to the relevanteigenvector.
If the scatterer Σ is mirror symmetric, R+ = R− ≡ R and the problem can be reduced to two uncoupled subproblems by choosing incident waves that are symmetric or antisymmetric with the reflection coefficients Rs =R + T and Ra = R − T . In particular, the reflection and transmission coefficients of the initial problem can beexpressed as R = (Rs + Ra )/2, and T = (Rs − Ra )/2 while the eigenvalues of the S-matrix can be written as
λ1 = Rs and λ2 = −Ra . For an one-sided incident wave, the absorption coefficient defined as α = 1 − |R|2 − |T |2becomes α = (αs + αa )/2, where αs ≡ 1 − |Rs |2 and αa ≡ 1 − |Ra |2 . Achieving α( fmax ) = 1 at a frequency fmax ,is equivalent to getting simultaneously the minima of the reflection coefficients of the two sub-problems, i.e.,
Ra ( fmax ) = Rs ( fmax ) = 0 [αs ( fmax ) = αa ( fmax ) = 1]. This has been achieved in Ref. [?] for a mirror symmetricslab through intensive numerical calculations.
In this talk we analyze the perfect absorption in ventilation problems by using mirror symmetric scatterers madeby a panel with a periodic distribution of slits, the upper wall of which are loaded by Helmholtz Resonators.
The propagation in the slit is highly dispersive due to the presence of the resonators, producing a slow soundpropagation before the resonance, and down shifting it to low frequencies. By controlling the geometry of theresonator, the visco-thermal losses can be tuned in order to compensate the leakage of the system and fulfill thecritical coupling condition so activating the perfect absorption of sound. In the case of transmission, a double
slit is needed, one acting as symmetric and the other one as antisymmetric resonators.

ACKNOWLEDGEMENTS
The authors gratefully acknowledge the ANR-RGC META- Room (ANR-18-CE08-0021) project and the project
HYPERMETA funded under the program E?toiles Montantes of the Région Pays de la Loire. This article isbased upon work from COST (European Cooperation in Science and Technology) Action DENORMS CA15125,
supported by COST (European Cooperation in Science and Technology).

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