# Multiple-Layer Microperforated Panels as Sound Absorbers in Buildings: A Review

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, which is real, if the MPP is to provide significant sound absorption. Hence, an additional imaginary impedance is required to counteract the reactive part of Z. The necessary reactive impedance is provided by an air cavity of thickness D. Therefore, a single-layer MPP (SL-MPP) is obtained which depends on the parameter set (d,t,ϕ,D).

## 2. SL-MPP

_{1}. When a plane wave propagating in air of characteristic impedance Z

_{0}finds this MPP system, the impedance contrast (Z

_{1}− Z

_{0}) causes a wave reflection, and as a consequence, produces sound absorption. At normal incidence, the reflection, R, and absorption, α

_{0}, coefficients are as follows:

- The visco-thermal dissipation within the holes, Z
_{hole}. - The distortion of flow in the perforation edges, Z
_{edge}. - The resonances in the air cavity, Z
_{c}. - The structural vibrations of the panel, Z
_{vib}.

_{hole}and Z

_{edge}is the MPP impedance Z

_{MPP}. The impedance of the air cavity is as follows:

_{vib}, can be obtained from the elastic properties of the panel [38]. Therefore, the input impedance to the MPP system is as follows:

_{vib}→∞, and

_{MPP}, and more specifically, for the impedances of the perforations, Z

_{hole}, and edges, Z

_{edge}, will be reviewed in the following sections.

#### 2.1. Maa Model

_{hole}, deduced from the solution of the wave equations in a cylindrical tube proposed formerly by Rayleigh and solved then by Crandall for short tubes:

_{1}is the radial coordinate in the tube, η is the air viscosity, and ∆p is the pressure difference at both sides of the tube. Solving for u, and averaging on the tube surface, the following equation is obtained:

_{0}being the air density, η the air viscosity, ω the angular frequency, r = d/2 the perforation radius, and J

_{0}and J

_{1}the Bessel functions of first class and orders 0 and 1, respectively. To extrapolate this solution to that of an MPP, it is necessary to take into account the relationship between the particle velocity inside and outside the perforations, as shown in Figure 2 [39,40].

_{hole}valid in the range 1 < s < 10. Nevertheless, the exact version of Equation (10) will be used in this article.

_{edge}term composed of two terms—one resistive, due to the friction of the air flow in the edges of the holes, and other reactive, due to the piston-like radiation of the air at both edges. The resistive term is also called surface resistance, R

_{s}. The reactive term is called mass reactance, X

_{m}. Thus,

_{m}, as follows:

- For each combination of (t,ϕ,D), there exists a value of d providing maximum absorption (Figure 6). Furthermore, the absorption bandwidth increases as d decreases.
- For each combination of (d,ϕ,D), there is a value of t yielding maximum absorption (Figure 7). The absorption curve moves towards higher (lower) frequencies as t decreases (increases).
- Keeping constant the combination of parameters (d,t,D), there is a value of ϕ affording maximum absorption (Figure 8). The absorption curve moves towards higher frequencies, and the absorption bandwidth increases as ϕ increases.
- Keeping constant the combination of parameters (d,t,ϕ), the effect of D is to move the absorption curve towards lower frequencies as D increases (Figure 9).

_{i},f

_{s}), where f

_{i}and f

_{s}are the frequencies at half absorption at each side of the peak. The number of octaves spanned is

#### 2.2. EF Model

#### 2.3. Comparison Between Maa and EF Models

#### 2.4. Microslotted Panels (MSP)

_{hole}, depends on the geometry of holes. The Maa formulation [2,3], valid for circular holes, Figure 14a, includes Bessel functions. The equation of Z

_{hole}for slits, Figure 14b, contains the hyperbolic tangent function [8,9,10]. The edge impedance also changes for slotted perforations [8,9]. The impedance of an SL-MSP of thickness t and hydraulic diameter d, is [9,10]

#### 2.5. Microperforated Insertion Units (MIUs)

_{m}

_{1}and Z

_{m}

_{2}are the impedances of each of the panels (the carrying plate and the micrometric mesh) and Z

_{c}is the air cavity impedance. Equation (35) allows obtaining the absorption curve of the MIU, once Z

_{m}

_{1}and Z

_{m}

_{2}are fixed. For the Maa model, the SL-MIU impedance is [11]

_{1},t

_{1},ϕ

_{1},d

_{2},t

_{2},ϕ

_{2},D).

_{1}and t

_{2}be the thicknesses of both plates perforated with holes of diameters d

_{1}and d

_{2}and porosities ϕ

_{1}and ϕ

_{2}, respectively. In the EF model, the edge effects are introduced through the tortuosities of both panels. For an SL-MPP, the geometrical tortuosity was given by Equation (22), ${\alpha}_{\infty}=1+\left(2{\u03f5}_{e}/t\right)$, ${\u03f5}_{e}$ being the excess of length of the vibrating air mass at each side of the holes. In this case, assuming continuity across the interface between the two plates, Figure 18, the tortuosities at both sides of the panels should be

_{1,2}= d

_{1,2}/2.

_{1},t

_{1},ϕ

_{1},d

_{2},t

_{2},ϕ

_{2},D) = (3 mm,1 mm,10%,41 μm,50 μm,31%,3 cm). As can be seen, the high frequency branch of the absorption curve provided by the EF model is slightly displaced towards higher frequencies, as compared with the Maa model curve. An SL-MIU provides an absorption curve similar to that of an SL-MPP with a bandwidth of one-to-two octaves.

#### 2.6. MPP Manufactured by Infiltration Technique

_{r},a

_{i}) real numbers close to 1.

#### 2.7. Absorption of an SL-MPP at Random Incidence

_{1}(θ), depends on the incidence angle and the type of the panel reaction [51]

- The random incidence absorption coefficient of a locally reacting SL-MPP has an absorption bandwidth rather similar to the normal incidence absorption coefficient, with a slight reduction of the peak absorption.
- The random incidence absorption coefficient of a bulk reacting SL-MPP has an absorption curve quite displaced towards higher frequencies as compared to the normal incidence absorption coefficient, with a more reduced absorption peak.

## 3. Multiple-Layer MPP (ML-MPP)

#### 3.1. Double-Layer MPP (DL-MPP)

_{m}

_{1}and Z

_{m}

_{2}with two air cavities of impedances Z

_{c}

_{1}and Z

_{c}

_{2}. The input impedances to the two interfaces are Z

_{1}and Z

_{2}. In addition, the characteristic impedance of the air is Z

_{0}. Sound waves attain the DL-MPP system from the left at normal incidence.

_{1},d

_{1},ϕ

_{1},D

_{1},t

_{2},d

_{2},ϕ

_{2},D

_{2}). As an example, Figure 25 shows the absorption coefficient of a DL-MPP for the combination of parameters (t

_{1},d

_{1},ϕ

_{1},D

_{1},t

_{2},d

_{2},ϕ

_{2},D

_{2}) = (0.15 mm,1 mm,10%,2 cm,0.15 mm,1 mm,15%,2 cm). The absorption bandwidth, obtained by Equation (17), is 3.24 octaves.

#### 3.2. Triple-Layer MPP (TL-MPP)

_{m}

_{1}, Z

_{m}

_{2}, and Z

_{m}

_{3}with three air cavities of impedances Z

_{c}

_{1}, Z

_{c}

_{2}, and Z

_{c}

_{3}. The input impedances to the three interfaces are Z

_{m}

_{1}, Z

_{m}

_{2}, and Z

_{m}

_{3}. In addition, the characteristic impedance of the air is Z

_{0}. Sound waves reach the TL-MPP system from the left at normal incidence.

_{1},t

_{1},ϕ

_{1},D

_{1},d

_{2},t

_{2},ϕ

_{2},D

_{2},d

_{3},t

_{3},ϕ

_{3},D

_{3}). To illustrate the capability of TL-MPP to provide broadband absorption, Figure 27 shows the normal incidence absorption coefficient of such absorber for the combination of parameters (d

_{1},t

_{1},ϕ

_{1},D

_{1},d

_{2},t

_{2},ϕ

_{2},D

_{2},d

_{3},t

_{3},ϕ

_{3},D

_{3}) = (0.15 mm,1.35,15%,0.89 cm,0.15 mm,1.4 mm,10%,1.3 cm,0.15 mm,1.1 mm,5%,2 cm). Notice that this TL-MPP provides absorption in a band of 4.25 octaves with a total thickness D

_{1}+ D

_{2}+ D

_{3}= 4.6 cm.

## 4. Machining MPPs

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Relationship between the particle velocity inside, u, and outside, u’, the perforations of an MPP.

**Figure 7.**Absorption coefficient of an SL-MPP as a function of (f,t) for (d,ϕ,D) = (0.4 mm,1%,2 cm).

**Figure 8.**Absorption coefficient of an SL-MPP as a function of (f,ϕ) for (d,t,D) = (0.4 mm,1 mm,2 cm).

**Figure 9.**Absorption coefficient of an SL-MPP as a function of (f,D) for (d,t,ϕ) = (0.4 mm,1 mm,1%).

**Figure 10.**Absorption coefficient of an SL-MPP as a function of (d,t) for (f,ϕ,D) = (1500 Hz,1%,2 cm).

**Figure 11.**Absorption coefficient of an SL-MPP with t = 0.5 mm, D = 2 cm and different values of d and ϕ.

**Figure 13.**Absorption curves provided by the Maa (solid line) and EF (dotted line) models for D = 3 cm and different combinations of the parameters (d,t,ϕ).

**Figure 15.**Comparison of the absorption curves of an SL-MSP and an equivalent SL-MPP for (d,t,ϕ,D) = (0.15 mm,1 mm,5%,5 cm).

**Figure 16.**Absorption curves of an SL-MSP (red lines) and the equivalent SL-MPP (blue lines) for D = 5 cm and (

**a**) (d,t,ϕ) = (0.5 mm,0.5 mm,0.5%), (

**b**) (d,t,ϕ) = (0.25 mm,0.75 mm,1%), (

**c**) (d,t,ϕ) = (0.15 mm,1 mm,5%), and (

**d**) (d,t,ϕ) = (0.1 mm,1.13 mm,10%).

**Figure 17.**Sketch of the MIU design. Two single MPPs, the carrying plate and the micrometric mesh, are used to form the MIU. The horizontal arrow above the micrometric mesh means that it is moved to bond the carrying plate.

**Figure 19.**Absorption curves of an SL-MIU with hole perforations, according to the Maa and EF models for (d

_{1},t

_{1},ϕ

_{1},d

_{2},t

_{2},ϕ

_{2},D) = (3 mm,1 mm,10%,41 μm,50 μm,31%,3 cm).

**Figure 20.**Sketch of an MPP with (

**a**) evenly distributed regular size holes and (

**b**) irregular holes unevenly distributed.

**Figure 21.**Measured absorption curves of the SL-MPP absorber manufactured using infiltration in comparison with those provided by the modified Maa and EF models.

**Figure 22.**Absorption coefficients at normal $({\alpha}_{0}$) and random incidence for local (${\alpha}_{d1}$) and bulk (${\alpha}_{2}$) reaction of an SL-MPP with parameters (d,t,ϕ,D) = (0.1 mm,0.1 mm,1.8%,4 cm).

**Figure 23.**Normal incidence absorption coefficient of an SL-MPP with d = 0.12 mm, t = 1 mm, ϕ= 10%, and D = 2 cm.

**Figure 25.**Normal incidence absorption coefficient of a DL-MPP with (d

_{1},t

_{1},ϕ

_{1},D

_{1},d

_{2}t

_{2},ϕ

_{2},D

_{2}) = (0.15 mm,1 mm,10%,2 cm,0.15 mm,1 mm,15%,2 cm).

**Figure 27.**Normal incidence absorption coefficient of a TL-MPP with parameters (d

_{1},t

_{1},ϕ

_{1},D

_{1},d

_{2},t

_{2},ϕ

_{2},D

_{2},d

_{3},t

_{3},ϕ

_{3},D

_{3}) = (0.15 mm,1.35,15%,0.89 cm,0.15 mm,1.4 mm,10%,1.3 cm,0.15 mm,1.1 mm,5%,2 cm).

**Figure 28.**Several MPPs of diameter 28.5 mm, manufactured using distinct techniques. (

**a**) Laser technology, (

**b**) MIU, (

**c**) infiltration technique, (

**d**) 3D printing, and (

**e**) drilling an epoxy laminate.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Cobo, P.; Simón, F. Multiple-Layer Microperforated Panels as Sound Absorbers in Buildings: A Review. *Buildings* **2019**, *9*, 53.
https://doi.org/10.3390/buildings9020053

**AMA Style**

Cobo P, Simón F. Multiple-Layer Microperforated Panels as Sound Absorbers in Buildings: A Review. *Buildings*. 2019; 9(2):53.
https://doi.org/10.3390/buildings9020053

**Chicago/Turabian Style**

Cobo, Pedro, and Francisco Simón. 2019. "Multiple-Layer Microperforated Panels as Sound Absorbers in Buildings: A Review" *Buildings* 9, no. 2: 53.
https://doi.org/10.3390/buildings9020053