# Broadband-omnidirectional absorption using inclined wiremesh gratings

Juan Pablo Escudero, Jean-Philippe Groby, Vincent Pagneux

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Juan Pablo Escudero, Jean-Philippe Groby, Vincent Pagneux. Broadband-omnidirectional absorption using inclined wiremesh gratings. JASA Express Letters, 2025, 5 (8), 10.1121/10.0038993 . hal-05384400

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# Broadband-omnidirectional absorption using inclined wiremesh gratings<sup>a)</sup>

Juan Pablo Escudero,<sup>b)</sup> Jean-Philippe Groby,<sup>id</sup> and Vincent Pagneux<sup>id</sup>

Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS),
CNRS, Le Mans Université, 72085 Le Mans, France

Juan_Pablo.Escudero_Lopez@univ-lemans.fr, Jean-Philippe.Groby@univ-lemans.fr, Vincent.Pagneux@univ-lemans.fr

**Abstract:** With the aim of achieving strong absorption over a wide range of frequencies and angles of incidence, the reflection of an acoustic wave by a rigidly-backed periodic layer of inclined ultrathin resistive sheet is studied theoretically and numerically. It appears that the structure exhibits either almost omnidirectional large absorption over a subwavelength frequency band for large resistivity of the wiremesh or large absorption over a broad frequency band for medium resistivity. Both optimal values are obtained for large inclination angles. These findings provide promising perspectives for the design of broadband omnidirectional subwavelength sound-absorbing devices. © 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

[Editor: Ning Xiang]

https://doi.org/10.1121/10.0038993

Received: 6 June 2025

Accepted: 5 August 2025

Published Online: 19 August 2025

## 1. Introduction

Absorbing low-frequency noise often requires bulky and heavy passive noise control treatments (Cox and d'Antonio, 2016). Metamaterials have therefore emerged to overcome this drawback, as their acoustic properties are generally better than those of conventional treatments for subwavelength dimensions (Groby et al., 2011; Romero-García et al., 2016; Li and Assouar, 2016; Yang and Sheng, 2017; Carbajo et al., 2020). Nevertheless, they are mainly designed to absorb acoustic energy efficiently at normal incidence, and new solutions are thus needed to achieve omnidirectional absorption (Cui et al., 2012; Jiménez et al., 2016, 2017; Cavalieri et al., 2020).

Designing an efficient absorbing structure requires impedance matching between the absorbing and incident media and attenuation of the incoming energy, i.e., a loss mechanism. Achieving impedance matching over a wide range of incidence angles is a difficult task. For the particular case of the interface between a homogeneous half-space and a half-space made up of periodic array of inclined thin plates, the reflection coefficient of an incident plane wave parallel to the plates is obviously zero; however, more intriguingly and interestingly, due to reciprocity, the reflection is also zero for an incident plane wave mirror-symmetric with respect to the normal to the interface (Norris and Su, 2015; Porter, 2021; Kucher et al., 2023). Since only implied by reciprocity, this property remains valid if the plates are replaced by ultra-thin resistive sheets and opens up new possibilities for the design of absorbing structures. For instance, transmission of a plane incident wave through a layer of inclined periodic resistive sheets is unity when the angle of incidence is the same as that of inclination of the sheets, whereas it is almost zero when the angle of incidence is mirror-symmetric with respect to the normal to the first interface of the layer. Consequently, the absorption coefficient of this mirror-symmetric incident wave is close to unity, since its reflection coefficient is also zero. It is worth noting that a similar idea has recently been applied to design a metainterface with good absorption properties using the Brewster angle concept (Luo et al., 2021; Fan et al., 2021) combined with the reciprocity principle. When the inclined resistive sheet array is rigidly-backed, impedance matching to the incident medium allows the wave to be freely absorbed within the attenuating layer.

In this work, we numerically and theoretically study the absorption of a plane incident acoustic wave by a rigidly-backed architecture layer of resistive lossy wiremesh. We use numerical simulations to study the structural ability to absorb low-frequency noise from different angles of incidence. The analysis is completed by an effective medium approximation. We show that the structure exhibits either (i) almost omnidirectional large absorption (98% of the acoustic energy) over a subwavelength frequency band for large resistivity of the wiremesh, or (ii) large absorption (85% of the acoustic energy) over a broad frequency band for medium resistivity.

<figure><img src="image_3.png" /></figure>

<footnote>a) This paper is part of a special issue on Sound Absorption and Diffusion: Modeling, Measurement and Application.</footnote>

<footnote>b) Author to whom correspondence should be addressed.</footnote>
## 2. Reflection by a rigidly-backed wiremesh grating

Figure 1(a) depicts the setup. We assume an implicit time dependence $e^{-i\omega t}$. The acoustic pressure satisfies the Helmholtz equation in the entire gray region, $(\Delta + k^2)p = 0$, where $k = \omega/c$ is the wavenumber with $c$ the sound speed. A plane incident wave impinges a rigidly-backed layer at an angle $\theta$. This $h$-thick layer is composed of a $d$-periodic arrangement of ultrathin wiremesh inclined at an angle of $\phi$, such that the distance between two adjacent wiremeshes is $d' = d \cos \phi$. A wiremesh is an ultrathin purely resistive sheet, typically with a millimetric thickness that is much smaller than the wavelength of the sound waves it interacts with, commonly composed of woven stainless steel wires. The pore dimensions are sufficiently small for linear viscous effects to dominate, allowing the mesh to react as a purely resistive element and resulting in significant attenuation of the acoustic wave. Its effectiveness covers a wide frequency range, without a lower frequency limit, provided the viscous boundary layer remains large compared to the pore size. The wiremesh is assumed not to vibrate. Under this condition, due to viscous friction of the creeping flow inside the pores, a pressure drop develops across the screen when an acoustic wave propagates through it: this pressure drop is directly related to the particle velocity through the mesh and governed by its acoustic resistance. The normal particle velocities and pressures on either side of each wiremesh satisfy the following conditions (Ingard, 2009; Farooqui et al., 2022; Kuznetsova et al., 2024, 2023; Coutant et al., 2020):

$$ [[\partial_n p]] = 0 \text{ and } [[p]] = \frac{i\zeta}{k} \partial_n p, \qquad (1) $$

where $\zeta > 0$ is the purely resistive wiremesh normalized impedance. Typical resistance value of commercially available wiremeshes ranges from near zero to several units. Additionally, the total resistance of closely packed wiremeshes is just the addition of each individual resistance.

The present study focuses on the low-frequency range, i.e., $kh/\pi \le 1$, where only the specularly reflected wave is propagating, i.e., $kd/\pi \le 1$ (Maystre, 2013; Bonod and Neauport, 2016; Popov, 2012), since the setup considered will always satisfy $d \le h$. In what follows, we will use $kh/\pi = 2h/\lambda$ as the frequency dimensionless parameter, where $\lambda$ is the wavelength. The scattering problem thus reduces to solving for the reflection coefficient $R$, assuming a pressure field in the upper half-space of the form,

$$ p(x, y \ge h) = e^{ik_x x} (e^{-ik_y(y-h)} + \text{Re}^{ik_y(y-h)}), \qquad (2) $$

where $k_x = k \sin \theta$ and $k_y = k \cos \theta$. Note that the reflection coefficient depends on five dimensionless parameters, thus $R = R(kh, \zeta, h/d, \theta, \phi)$.

## 3. Effective medium

To further analyze this reflection coefficient, the inclined wiremesh grating layer is approximated by an effective medium layer as depicted in Fig. 1(b). In the low-frequency approximation ($kd' \ll 1$), the $d'$-periodic arrangement of inclined ultra-thin wiremesh medium can be correctly approximated by a homogeneous anisotropic medium (Oleönik et al., 1992; Norris and Su, 2015) with principal directions $(X, Y)$, see Fig. 1(b). These two principal directions are respectively parallel and perpendicular to the orientation of the wiremeshes. The pressure field satisfies the following wave equation in this effective anisotropic medium:

$$ \nabla \cdot [\mathbf{a}\nabla p(x, y)] + k^2 p(x, y) = 0, \qquad (3) $$

where $\mathbf{a}$ is a $2 \times 2$ normalized effective density matrix. In the principal directions $(X, Y)$ (Norris and Su, 2015), this normalized effective density matrix is a diagonal matrix, which reads as

<figure><img src="image_4.png" /><figcaption>Fig. 1. (a) Reflection of an acoustic wave by a rigidly backed grating composed of inclined wiremeshes (red dashed line). (b) Effective medium modelling of the setup with effective parameters $\mathbf{a}$[as in Eq. (3)].</figcaption></figure>
$$ \mathbf{a}_0 = \begin{pmatrix} 1/\rho_{eff} & 0 \\ 0 & 1 \end{pmatrix}, \qquad (4) $$

where the normalized effective density $\rho_{eff}$ perpendicular to the wiremesh orientation, i.e., along the X-direction, is (Groby et al., 2021)

$$ \rho_{eff} = 1 + \frac{i}{kh} \left( \frac{\zeta h}{d'} \right). \qquad (5) $$

Please note that $kh$ is introduced here for convenience in view of the final use of the layer treatment. In the $(x, y)$ coordinate system, the normalized effective density matrix becomes $\mathbf{a} = \mathbf{R}^T \mathbf{a}_0 \mathbf{R}$, where $\mathbf{R}$ is a rotation matrix with angle $\varphi$, i.e.,
$$ \mathbf{R} = \begin{pmatrix} \cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{pmatrix}. $$
Finally, the matrix $\mathbf{a}$ is symmetric and reads as

$$ \mathbf{a} = \begin{pmatrix} a_x & a_{xy} \\ a_{xy} & a_y \end{pmatrix}, \qquad (6) $$

with

$$ a_x = \frac{kh \cos \varphi + i(\zeta h/d) \sin^2 \varphi}{kh \cos \varphi + i(\zeta h/d)}, \qquad (7a) $$

$$ a_{xy} = \frac{-i(\zeta h/d) \cos \varphi \sin \varphi}{kh \cos \varphi + i(\zeta h/d)}, \qquad (7b) $$

$$ a_y = \frac{\cos \varphi(kh + i(\zeta h/d) \cos \varphi)}{kh \cos \varphi + i(\zeta h/d)}. \qquad (7c) $$

Returning to the reflection problem of an acoustic plane wave by the rigidly-backed effective medium layer, the boundary conditions at the air/anisotropic media interface and the rigid-backing location are, respectively,

$$ \llbracket p \rrbracket = 0 \quad \text{and} \quad \llbracket \mathbf{n} \cdot (\mathbf{a} \nabla p) \rrbracket = 0 \quad \text{at} \quad y = h, \qquad (8a) $$

$$ \mathbf{n} \cdot (\mathbf{a} \nabla p) = 0 \quad \text{at} \quad y = 0. \qquad (8b) $$

In the layer, the pressure field is sought in the form $p(x, 0 \le y \le h) = e^{ik_x x}(Ae^{ik^+ y} + Be^{ik^- y})$, where $\kappa^{\pm}$ is the wavenumber in the anisotropic effective fluid along the y-direction

$$ \kappa^{\pm} = - \frac{a_{xy} k_x}{a_y} \pm \frac{k}{a_y} m', \qquad (9a) $$

$$ m' = \sqrt{a_y - (a_x a_y - a_{xy}^2) \sin^2 \theta}. \qquad (9b) $$

Applying the boundary conditions Eqs. (8), together with Eq. (2), the reflection coefficient $R$ finally reads as

$$ R = \frac{\cos \theta + im' \tan \frac{kh}{a_y} m'}{\cos \theta - im' \tan \frac{kh}{a_y} m'}. \qquad (10) $$

From the Eqs. (7) and (9), it becomes clear that this reflection coefficient and, consequently, the absorption coefficient only depend on the four dimensionless parameters $kh$ and $\zeta h/d$ and on the angles $\varphi$ and $\theta$ as

$$ \alpha(\zeta h/d, kh, \varphi, \theta) \equiv 1 - |R|^2. \qquad (11) $$

## 4. Results

The absorption coefficient is calculated with two methods: numerically by solving the Helmholtz equation with Eq. (1) using COMSOL Multiphysics® and, theoretically, the analysis is driven by the reflection coefficient formula [Eq. (10)] derived through the effective medium approximation. This means that these two calculation methods must first be compared to identify possible sources of discrepancy. In particular, from numerical computation, $R$ depends separately on $\zeta$ and $h/d$ (see Sec. 2), but it depends only on the product $\zeta h/d$ in the effective medium approximation (Sec. 3). Several simulations were run (not shown in this Letter) to check this point: it appears that, for a given $\zeta h/d$, the smaller $\zeta$ and the larger $h/d$, the closer the absorption coefficient calculated with the effective medium approximation to that calculated numerically. Thus, in order to ensure close agreement between theory and numerics, all the numerical simulations with COMSOL Multiphysics® presented in this Letter are performed with $h/d = 8$. This parameter $\zeta h/d$ can be written as
$\zeta(L/d) \cos \varphi$ (where L is the total length of wiremeshes in a unit cell), such that it can be interpreted as the resistance linear density weighted by $\cos \varphi$.

Let us first investigate the effect of the wiremesh inclination ($\varphi$) on the absorption coefficient. Here, we note that reciprocity implies $R(-\theta) = R(\theta)$ (Norris and Su, 2015), and we take advantage of this property to represent the comparison between numerics ($\theta < 0$) and theory ($\theta > 0$). Figure 2 depicts the absorption coefficient for $\zeta h/d = 2$ (in the numerical simulations, $\zeta = 0.25$ and $h/d = 8$) with different wiremesh inclinations $\varphi = 30^{\circ}$ (a), $45^{\circ}$ (b), $60^{\circ}$ (c), and $75^{\circ}$ (d). The absorption coefficients as simulated and as calculated with the effective medium theory are in good agreement, although small differences appear for large $kh/\pi$, i.e., at high frequencies. It appears that increasing the inclination angles results in a general increase in the absorption coefficient, the maxima decreasing to low frequency to sort of encapsulate a lower absorption coefficient zone for normal incidence, see Figs. 2(a)-2(c). For a large inclination angle, impedance matching and a large absorption coefficient are achieved when $|\theta| \approx \varphi$ [see Fig. 2(d)]. In general, these results show that increasing the angle of inclination of the layer structure results in larger absorption, which is more broadband and omnidirectional.

To examine the structure's ability to achieve omnidirectional absorption, the absorption coefficient calculated with the effective medium approximation is averaged along the angle of incidence $\theta$ using

$$ \bar{x}(\zeta h/d, kh/\pi, \varphi) = \frac{1}{\theta_{\max}} \int_{0}^{\theta_{\max}} \alpha(\zeta h/d, kh/\pi, \varphi, \theta) d\theta. \quad (12) $$

This mean absorption coefficient is computed up to $\theta_{\max} = 80^{\circ}$, because of limitations of numerical calculations at incidence angles close to $\pi/2$. It is depicted in Figs. 3(a), 3(b), and 3(c) for discrete values of $\zeta h/d$. These results show a shift in the mean absorption coefficient towards lower frequencies as $\zeta h/d$ and $\varphi$ increase. A large omnidirectional subwavelength absorption is achieved for large wiremesh inclination, resulting both from impedance matching at large incident angle and from subwavelength resonance at normal incidence, producing a zero of Eq. (10). A maximum mean absorption coefficient $\bar{x}_{\max} = 0.98$ is reached for $\varphi_{\max} = 78^{\circ}$ and $(\zeta h/d)_{\max} = 5.47$. This value is pointed out by a circle in Fig. 3(c). Note that, at normal incidence $\theta = 0^{\circ}$, the total absorption resonance is approximately obtained for $kha_y^{-1/2}$ close to $\pi/2$, yielding to solving for $kh$ the equation $kha_y^{-1/2} - (ia_y^{1/2} + \pi/2) = 0$, eventually providing $kh \approx \cos \varphi \pi/2$ (subwavelength resonance) for $\zeta h/d \approx (\pi^2/8 \cos \varphi)(1 - \cos^2 \varphi)[1 + (8 \cos^2 \varphi)/\pi^2]$ (for $\varphi = 78^{\circ}$ that gives $kh = 0.104\pi$ and $\zeta h/d = 5.87$).

The corresponding absorption coefficient, i.e., $\alpha(\zeta h/d = 5.47, kh, \varphi = 78^{\circ}, \theta)$ as a function of the frequency $kh$, is depicted in Fig. 3(d). A quasi-omnidirectional absorption coefficient is clearly visible with a narrow frequency band centered at $kh/\pi = 0.11$. Large broadband absorption is also reached for frequencies above 0.2 at the layer's inclination angle. Again, the absorption coefficients as simulated ($\zeta = 0.684$ and $h = 8$) and as calculated with the effective medium approximation are in good agreement. The quasi-perfect and quasi-omnidirectional absorption is highlighted in Figs. 3(e) and 3(f), where the real part of the total pressure fields is represented when the structure is excited by a plane wave at $\theta = 70^{\circ}$ and $\theta = 0^{\circ}$ for $kh/\pi = 0.11$ [highlighted by a rhombus and a star in Fig. 3(d)], are shown. Both fields are essentially

<figure><img src="image_5.png" /><figcaption>Fig. 2. Absorption as a function of the angle of incidence and reduced frequency; effect of the inclination angle ($\varphi$) and comparison between numerical computation (using $\zeta = 0.25$ and $h = 8$) and effective medium results for a constant $\zeta h/d = 2$. (a) $\varphi = 30^{\circ}$, (b) $\varphi = 45^{\circ}$, (c) $\varphi = 60^{\circ}$, and (d) $\varphi = 75^{\circ}$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 3. Comparison of the omnidirectionality evolution of the absorption coefficient integrated along θ for different inclination angles as ζh/d increases. (a) ζh/d = 1, (b) ζh/d = 2, and (c) ζh/dO = 5.47. The mean absorption coefficient shifts towards low frequency as ζh/d and φ increase. The circle in Fig. 3(c) represents the maximum mean absorption at the lowest frequency αO = 0.98. (d) Comparison of the maximum omnidirectional absorption over a subwavelength frequency band (φO = 78°, kh/π = 0.11 and ¯x = 0.98) between direct numerics (ζ = 0.684 and h = 8) and effective medium (ζh/dO = 5.47). (e) Real part of the total pressure field produced by an incident wave arriving over the structure with an angle θ = 70° for kh/π = 0.11 indicated with a rhombus in Fig. 3(d). (f) Real part of the total pressure field produced by an incident wave arriving over the structure with an angle θ = 0 for kh/π = 0.11 indicated with a star in Fig. 3(d).</figcaption></figure>

without reflection, composed of only the incident wave, which proves quasi-total absorption of the acoustic energy by the rigidly-backed inclined wiremesh grating.

To assess the broadband and omnidirectional absorbing performance, we define the double average absorption coefficient $\bar{\bar{\alpha}}(\zeta h/d, \varphi)$, averaged over the angle of incidence $\theta$ and frequency $kh/\pi$,

$$ \bar{\bar{\alpha}}(\zeta h/d, \varphi) = \int_{0}^{1} \bar{\bar{\alpha}}(\zeta h/d, kh/\pi, \varphi) d\left(\frac{kh}{\pi}\right). \quad (13) $$

Figure 4(a) depicts $\bar{\bar{\alpha}}(\zeta h/d, \varphi)$ for 30° < $\varphi$ < 80° and 0.1 < $\zeta h/d$ < 7. Strong broadband-omnidirectional absorption ($\bar{\bar{\alpha}} > 0.8$) is encountered for 0.65 < $\zeta h/d$ < 3.48 and 58° < $\varphi$ < 79°. The maximum value $\bar{\bar{\alpha}}_{MAX} = 0.85$ is highlighted with a star in Fig. 4(a) and is obtained for $\varphi_{MAX} = 70°$ and ($\zeta h/d$)$_MAX = 1.32$. The corresponding absorption coefficient as simulated ($\zeta = 0.165$ and $h = 8$) and as calculated with the effective medium approximation are depicted in Fig. 4(b). Here, again, the results of the two calculation methods are in good agreement. A strong omnidirectional absorption ($\alpha > 0.9$) appears for frequencies $kh/\pi$ above 0.24. Moreover, broadband absorption is achieved when the angle of incidence ranges from the inclination angle of the wiremesh (highlighted by a rhombus at $kh/\pi = 0.2$) to grazing incidence. The real part of the total pressure fields is represented Fig. 4(c) for the setup highlighted by the rhombus. The strong-absorption structure produces an undisturbed total pressure field, making the reflected wave barely perceptible.

## 5. Conclusion

In this work, we studied the reflection by a rigidly-backed architecture layer composed of inclined periodic resistive wiremeshes. The analysis of the structure was performed using numerical computations and was supplemented by an anisotropic effective medium approximation. From the obtained results, depending on the targeted absorption characteristics, two applications can be envisaged: (i) a resistive configuration capable of strongly absorbing acoustic energy at all angles of
<figure><img src="image_7.png" /><figcaption>Fig. 4. (a) Broadband-omnidirectional absorption performance of the structure from Eq. (13). The star indicate the maximum broadband-omnidirectional absorption located at $\phi_{OB} = 70^\circ$ and $\zeta h/d_{OB} = 1.32$ with a max($\bar{x}$) = 0.85. (b) Comparison of the maximum broadband-omnidirectional absorption between direct numerics ($\zeta = 0.165$ and $h = 8$) and effective medium ($\zeta h/d_{OB} = 1.32$). (c) Real part of the total pressure field produced by an incident wave arriving over the structure with an angle $\theta = 70^\circ$ for $kh/\pi = 0.2$ indicated with a rhombus in Fig. 4(b).</figcaption></figure>

incidence for a specific subwavelength thickness, and (ii) a less resistive configuration capable of strongly absorbing acoustic energy over a wide omnidirectional broadband absorption. In both cases, the angle of inclination of the wiremesh is large. The studied structure opens up new possibilities for the design of perfect omnidirectional broadband absorbers and anechoic environments.

### Acknowledgment

The European Commission is gratefully acknowledged for their support of the Horizon Europe DN METAVISION project (GA 101072415). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union. The European Union cannot be held responsible for them.

### Author Declarations

#### Conflict of Interest

The authors have no conflicts of interest.

### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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