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Abstract

The in situ characterization of materials is a crucial challenge in room acoustics, as laboratories measurement cannot always be applied in consultancy practices. In particular, there is a lack of method to characterize in situ systems with perforated facings, which are commonly encountered systems in room acoustics. In this paper, the in situ characterization of a rigidly-backed porous material behind a rigid perforated facing by means of pressure–velocity measurements is presented. The method includes an inverse impedance model fitting based on measurement in a limited frequency range. The applicability of this method was studied by measuring a variety of perforated facings, whether in front of an air cavity or backed by a porous layer, and comparing the obtained impedance model parameters to reference values. Good agreement was observed between the retrieved parameters and the references, with the errors in all retrieved parameters moving mass, facing thickness, cavity depth, porous layer thickness and porous layer flow resistivity not exceeding 15%.

Keywords

In situ measurements acoustic characterization model fitting PU probe surface impedance absorption coefficient

Introduction

The acoustic characterization of materials is one of the challenges in room acoustics, as the properties of the laid–out materials have a significant impact on the acoustics of an indoor space. In some cases, it is necessary to measure the material in situ, because the mounting conditions and deterioration over time can result in material properties different from those measured in a laboratory. Moreover, measuring data in situ avoids the problem of bringing a material sample to the laboratory for a standardized impedance tube measurement.¹

An in situ characterization method was introduced in a previous paper.² This method combines a pressure–velocity (PU) measurement for a sound wave with normal incidence to the surface to be characterized with an impedance model fitting procedure. In the same paper, it was shown that this approach allowed an accurate characterization of hard-backed porous layers. However, in most applications, the surfaces of the acoustic materials installed in a room are not left bare. Their visible surface is usually covered for mechanical and aesthetic purposes. While these facings can have a negligible effect on the acoustics, in some cases they are designed to be an integral part of the acoustic system, usually adding a resonant sound absorption mechanism to the porous absorption. One of the most common types of acoustic covering is a perforated wooden plate.³

The influence of such facings can be predicted using various modeling approaches. One such approach uses a Helmholtz resonator to characterize the facing as an acoustic mass and an acoustic resistance, to account for the movement of air within the perforations and the associated losses. This model assumes that the wavelength is much larger than the dimension of the perforation. A separate model has been developed by Maa⁴ for micro-perforated panels (MPPs), which have facing perforations with sub-millimetric dimensions. A more recently developed approach is the equivalent Johnson–Champoux–Allard (JCA) model, which considers the facing as a rigidly framed porous material whose properties are described with parameters related to the geometry of the facing. The JCA model was found to correctly predict the acoustic performance of both perforated facings and MPPs.⁵ Other approaches to predict the performances of more complex perforated facing systems such as heterogeneous perforations or grooved facings have also been presented.^6,7 These more complex geometries, as well as MPPs, are out of the scope of this paper.

This paper focuses on the inverse in situ characterization of systems with perforated facings either on an air cavity or a porous layer, in an in situ context. In a study from 2011.⁸ such estimation of the acoustic parameters of two perforated facings was achieved by combining impedance tube measurements and comparisons of the results to the model of Atalla and Sgard.⁵ This study was however based on laboratory measurements. In another publication,⁹ an in situ measurement of a slotted panel absorber was done with the help of a small pressure–velocity, and yielded estimations of the complex surface impedance and absorption coefficient of the panel from measurements realized without an impedance tube and a special environment like an anechoic room. However, it presented no attempts to compare the results to an impedance model and obtain an inverse characterization of the panel. The same pressure–velocity apparatus was used in a third publication¹⁰ to measure the absorption coefficient and the transmission loss of a micro-perforated panel in an anechoic room environment and the results compared to an impedance model, but no acoustic parameters were extracted through an inverse characterization. However, to the knowledge of the authors, there is no existing literature on the investigation of PU-based, in situ inverse characterization of perforated systems in an in situ context.

The goal of this study is to evaluate the applicability of an in situ inverse PU-characterization method for multilayered acoustic systems with rigid perforated facings. To assess their strengths and weaknesses with respect to the applicability and accuracy of in situ inverse characterization, two approaches to model the rigid perforated facings have been investigated: the Helmholtz resonator model and the equivalent JCA model. Various combinations of perforated panels and backings (air cavity and hard-backed porous layers) have been measured experimentally and the retrieved parameters obtained from the inverse characterization were compared to reference values. The measurements presented in this study are limited to perforated facings with backing layers consisting of either a porous layer (two types) backed by a hard wall or by an air cavity without material. Furthermore, this study is limited to square samples of perforated panels with dimension 0.6 m made of medium density fiberboard (MDF). Finally, the only perforation type measured was with circular holes, homogeneously spread in lattice patterns.

This paper is structured as follows. The method used to characterize the system in situ is first presented in section “Methods” and introduces the impedance models used for the fittings. Next the measurement setup employed to verify the validity of the approach is described in section “Practical implementation,” after which the results are presented and discussed in section “Results.” Finally, the conclusions of this work are given in section “Conclusion.”

Methods

Measurement procedure

The measurement procedure revolves around the acquisition of four impulse responses (IRs). The acquisition of IRs has an advantage over steady-state transfer functions in that it allows the removal of parasitic reflections via time-windowing. First, pressure and particle velocity in the direction of the source-probe axis ( $p_{f f}$ and $u_{f f}$ , respectively) have to be acquired in free field, that is, with the measurement device located away from any surfaces. These IRs are used to estimate the free-field impedance, $Z_{f f} = p_{f f} / u_{f f}$ , which serves as a normalization value. Subsequently, the normal incidence pressure and velocity IRs ( $p$ and $u$ , respectively) are captured at a point M just above the surface of the porous material in a similar way to yield an estimation of the impedance near the material, $Z_{m} = p / u$ . The measurement point $M$ is placed above the surface of the material at a distance $h$ . The sound source is vertically aligned with $M$ with distance $d$ and oriented to ensure a normal incident sound wave on the surface of the material. This setup is shown in Figure 1. The four IRs obtained from the pressure and velocity signals are then time-windowed using an Adrienne window.¹¹ The window is applied to the IRs such that the starting point of the flat portion of the window is set 0.2 ms before the initial peak. The length of the time window $t_{w i n}$ , must be slightly shorter than the approximate time delay between the incident sound peak and the first parasitic reflections, so as to exclude parasitic reflections from the recorded IRs.

Figure 1.

Arrangement of sound source and PU probe above the perforated facing system.

Subsequently, a Fourier transform is applied to the windowed signals and the complex reflection coefficient is calculated using the image-source model presented in Li et al.¹² and Alvarez and Jacobsen.¹³ With the speaker-probe distance, $d$ , and probe-sample distance, $h$ , both known, the complex reflection coefficient (using the $e^{+ j ω t}$ convention) can be written as

R_{i n s i t u} (f, h) = \frac{\frac{Z_{m}}{Z_{f f}} - 1}{\frac{Z_{m}}{Z_{f f}} (\frac{d}{d + 2 h}) (\frac{j k (d + 2 h)}{j k d}) + 1} (\frac{d + 2 h}{d}) e^{2 j k h},

(1)

where $Z_{m}$ and $Z_{f f}$ are, respectively, the impedances (ratio of pressure over normal velocity spectra) captured near the surface of the material and in the pseudo-free-field situation, $k = 2 π f / c_{0}$ is the wavenumber in air, $f$ is the frequency, and $c_{0}$ the speed of sound in air. This formula relies on the assumption that both incident and reflected waves present a plane wave front but includes a correction for the amplitude decay of spherical propagation.¹³

Model fitting procedure and error estimation

The reflection coefficient obtained from equation (1) depends on the probe-sample distance, $h$ . However, this parameter is hard to measure accurately in practice because the measurement point needs to be close to the measured surface (<20 mm),¹⁴ and the exact location of the acoustic center of the sensor can be subject to small deviations compared to its geometrical location, a problem reported in previous research.¹⁵ Thus, instead of treating the probe-sample distance as a fixed value obtained from a geometrical measurement, it was treated as an input variable of the model fitting procedure presented in this section.

The complex reflection coefficients acquired from the in situ measurements are fitted to those predicted by impedance models via an optimization routine. The plane normal-incidence complex reflection coefficient at the surface of the system is related to the modeled surface impedance, $Z_{s} (\bar{x})$ , via the following equation:

R_{m o d e l} = \frac{Z_{s} (\bar{x}) - ρ_{0} c_{0}}{Z_{s} (\bar{x}) + ρ_{0} c_{0}},

(2)

where $ρ_{0} = 1.2$ kg/m³ is the mass density of air, $c_{0} = 343$ m/s is the speed of sound in air (at 20°C), and $\bar{x}$ is the vector of the system’s parameters to be retrieved. The expression of the surface impedance depends on the model used.

Two separate models are studied to characterize the perforated panel: the Helmholtz resonator approach and the equivalent JCA model approach. The various expressions of $Z_{s} (\bar{x})$ for these two models are presented in subsection “Impedance models” along with the parameters they contain. The next section also presents the models used for the backing of the systems (air cavity and porous layers). The parameters vector can thus be decomposed into the vector of parameters relative to the facing, ${\bar{x}}_{f}$ , and the vector of parameters relative to the backing, ${\bar{x}}_{b}$ , such that ${\bar{x} =(\bar{x}}_{f} {, \bar{x}}_{b})$ . The reflection coefficient obtained by the model, together with the reflection coefficient measured in situ (equation (1)), is used to find optimal values for the probe-sample distance and parameters of the impedance models.

By using equations (1) and (2), a cost function, $F_{o p t}$ , is defined to minimize the absolute value of the difference between the frequency vectors of both the imaginary and real parts of $R_{i n s i t u} (f, h)$ and $R_{m o d e l} (f, \bar{x})$ , which becomes

F_{o p t} (\bar{x}, h) = \sum_{f_{l}}^{f_{u}} | | ℜ (R_{i n s i t u} (f, h) - R_{m o d e l} (f, \bar{x})) | | + \sum_{f_{l}}^{f_{u}} | | ℑ (R_{i n s i t u} (f, h) - R_{m o d e l} (f, \bar{x})) | |,

(3)

where $h$ is the probe-to-sample distance, $\bar{x}$ is the vector of the parameters of the impedance models, and $f_{l}$ and $f_{u}$ are the lower and upper bounds of the frequency range of the fitting $(Δ_{f} = [f_{l}, f_{u}])$ , respectively. Because $R_{i n s i t u}$ has a dependency in $h$ , it does not truly serve as a reference value in the cost function. However, it will be shown in section “Results” that the values retrieved for $h$ are in agreement with the measurement setup used.

The reflection coefficient was chosen as the basis of the cost function because the absolute values of both the real and imaginary parts of reflection coefficients are bounded by 1, which provides a natural normalization of deviations between the fittings of the real and imaginary parts during the optimization process. The lower bound of the frequency range is determined by limitations related to the plane wave approximation used in equation (1), the type of source used in the setup, and the source-to-probe distance. For the experimental setup used in this study (see section “Practical implementation”), the assumption is valid down to around 500 Hz; the wavelength below this frequency becomes large compared to the source-to-probe distance used (260 mm) and thus invalidates the plane-wave assumption.

Based on this model fitting procedure, the final estimates of the measured and fitted reflection coefficients are extracted as

R_{i n s i t u, o p t} = R_{i n s i t u} (h_{o p t}),

(4)

R_{m o d e l, o p t} = R_{m o d e l} ({\bar{x}}_{opt}).

(5)

Physically, $R_{m o d e l, o p t}$ is the complex reflection coefficient obtained from the impedance model parameters retrieved by the model fitting, and $R_{i n s i t u, o p t}$ is the measured normal-incidence complex reflection coefficient corrected with the probe-sample distance retrieved by model fitting. Furthermore, $h_{o p t}$ and ${\bar{x}}_{opt}$ are respectively the retrieved sample-to-probe distance and model parameter vector that minimize the cost function $F_{o p t}$ . The comparison of the retrieved parameters in ${\bar{x}}_{o p t}$ to reference values is then the approach chosen used to estimate the precision of the material characterization performed. Thus, for a retrieved parameter ${\bar{x}}_{o p t} (i)$ for which the known true value is $x_{r e f}$ , the error $e$ is computed as

e = \frac{{\bar{x}}_{o p t} (i) - x_{r e f}}{x_{r e f}} .

(6)

Impedance models

Helmholtz resonator model

A rigid perforated facing can be modeled as the combination of a generic acoustic resistance term, $r_{h}$ , and a generic acoustic mass, $m_{h}$ :

Z_{s} (m_{h}, r_{h}, {\bar{x}}_{b}) = r_{h} + j ω m_{h} + Z_{b} ({\bar{x}}_{b})

(7)

where $Z_{b}$ is the surface impedance of the backing (air cavity or porous layer) immediately behind the facing. In this case, the acoustic mass corresponds to the mass of air plugs moving in and out of the perforations, and the resistance accounts for all the losses resulting from the friction of the moving air against the walls of the perforations. With this model, the parameters of the facing to be retrieved are ${\bar{x}}_{f} = (m_{h}, r_{h})$ .

For reference, the moving mass within the perforations can be estimated in the case of circular perforations with:

m_{h, r e f} = \frac{ρ_{0}}{ϵ} [t + 2 δ a]

(8)

where $ϵ$ is the perforation ratio of the facing, $t$ is the thickness, $a$ is the perforation radius, and $δ$ is the end correction factor. The value of the correction factor $δ$ is usually expressed as a polynomial of the open ratio of the panel. The polynomial coefficients depend on the nature and dimensions of the perforations, inter-perforation distance and pattern. In this study (as shown in Figure 2), the perforated panels feature circular holes in lattice patterns, a configuration for which an expression of the end correction was established by Jaouen and Bécot⁸:

δ = 0.85 (1 - 1.13 ϵ^{1 / 2} - 0.09 ϵ + 0.27 ϵ^{3 / 2}).

(9)

Figure 2.

Picture of a perforated facing (Perf II) measured with Material A as backing.

Further expressions for more perforation shapes and patterns can be found in Chapter 7 of Cox and D’ Antonio.³

A limitation of the Helmholtz resonator model (equation (8)) is that it is only valid as long as the dimensions of the perforations are very small compared to the wavelength.

A theoretical prediction for the loss within the perforated facing has been derived by Ingard¹⁶:

r_{h, r e f} = \frac{\sqrt{4 ρ_{0} η π f}}{2 ϵ}

(10)

where $η = 1.84 \times 10^{- 5}$ N·s/m² is the dynamic viscosity of air. However, in the experimental work described in this paper, this expression was found to greatly underestimate the losses within the measured systems. A generic, frequency-independent resistance term, $r_{h}$ , was preferred to model the loss of the system. It will be shown in “Results” section that this simple expression of the losses allows a good fitting of the experimental data.

Equivalent JCA model

Another way to model a perforated facing was introduced by Atalla and Sgard.⁵ Their method models the perforated facing with an equivalent propagation constant and characteristic impedance, based on the JCA theory. In this framework, the parameters of the equivalent JCA fluid (resistivity, $σ$ , porosity, $ϕ$ , tortuosity, $α_{\infty}$ , viscous length, $Λ$ and thermal length, $Λ^{'}$ ) are derived from the geometry of the facing: the perforation radius, $a$ , the open ratio, $ϵ$ and the tube length, $t$ . For a facing with circular perforations, the equivalent JCA parameters are estimated from

ϕ_{e q} = ϵ,

(11)

Λ_{e q} = {Λ^{'}}_{e q} = a,

(12)

σ_{e q} = \frac{8 η}{ϵ a^{2}},

(13)

α_{\infty, e q} = 1 + \frac{2 δ a}{t} .

(14)

Following these formulas, the characteristic impedance, $Z_{J C A, e q}$ , and propagation constant, $k_{J C A, e q}$ , of the equivalent fluid are then calculated using the JCA model formulas presented in the Appendix.

This model of a facing depends on the perforation radius, $a$ , and open ratio, $ϵ$ , of the perforated facing, which can be evaluated by direct geometrical measurement and the thickness of the facing (or tube length) $t$ . The thickness remains the only parameter to retrieve to characterize the perforated facing because it is generally not possible to directly measure it without taking off the facing. In this work, a generic resistance, $r_{c}$ , is added in series to the surface impedance of the system to account for the fact that the material from which perforated facing is made (in this case, MDF) is not perfectly reflective and thus induces additional losses. In this approach, the facing vector to be retrieved is thus ${\bar{x}}_{f} = (t, r_{c})$ . From these considerations, the surface impedance of the system is computed with a regular transfer matrix calculation:

Z_{s} (t, r_{c}, {\bar{x}}_{b}) = r_{c} + \frac{- j Z_{b} ({\bar{x}}_{b}) Z_{J C A, e q} \cot (k_{J C A, e q} t) + Z_{J C A, e q}^{2}}{Z_{b} ({\bar{x}}_{b}) - j Z_{J C A, e q} \cot (k_{J C A, e q} t)} .

(15)

Air cavity backing

Where the perforated facing is backed by an air cavity, the surface impedance of the backing layer is computed with

Z_{b} ({\bar{x}}_{b}) = Z_{b} (D) = - j ρ_{0} c_{0} \cot (k D),

(16)

where $D$ is the cavity depth. This is valid for normal incidence.

Porous layer backing

Where the perforated facing is backed by a porous layer, the surface impedance of the backing layer becomes:

Z_{b} ({\bar{x}}_{b}) = Z_{b} (σ, l) = - j Z_{c} \cot (k_{c} l),

(17)

where $l$ is the thickness of the layer, and $Z_{c}$ and $k_{c}$ are the characteristic impedance and propagation constant of the porous material, respectively. In this work these metrics are calculated from the flow resistivity, $σ$ , of the layer via the expressions provided by the Delany–Bazley–Miki (DBM) model.¹⁷ These expressions are reported in Appendix.

Practical implementation

In situ measurements

Materials

Measurements were carried out to characterize a set of systems featuring perforated facings. The three perforated facings investigated were made of 16 mm thick MDF and featured straight circular perforations in lattice patterns, with varying perforation radii: 3, 4, and 5 mm. The geometrical properties of the three facings, as well as the predicted acoustic mass calculated from the Helmholtz resonator model from equation (8), are given in Table 1. The lateral dimensions of the samples were 0.6 × 0.6 m. Each of the panels was coupled with three rigidly backed conditions:

• a 45 mm air cavity, which was created by putting the facing on top of rubbers bars;

• a 40 mm glass wool layer (porous material A);

• a 60 mm rock wool layer (porous material B).

Table 1.

Reference radii, open ratios, thicknesses and moving masses (via Equation (8)) of the perforated facings measured.

Sample	Perf. I	Perf. II	Perf. III
$a_{r e f}$ (mm)	5.0	4.0	3.0
$ϵ_{r e f}$ (-)	0.31	0.20	0.11
$t_{r e f}$ (mm)	16	16	16
$m_{h, r e f}$ (kg/m²)	0.076	0.119	0.208

Thus, there were nine acoustic systems to characterize. The reference thicknesses and flow resistivities of the two porous materials are shown in Table 2. The reference flow resistivities were obtained from the low-frequency limit of the imaginary part of the dynamic mass density,¹⁸ extracted from impedance tube measurements via the two-cavity method.¹⁹

Table 2.

Reference flow resistivities, $σ_{r e f}$ , and thicknesses, $l_{r e f}$ , of porous materials A and B.

Sample	$σ_{r e f}$ (kPa s/m²)	$l_{r e f}$ (mm)
Material A	30	40
Material B	13	60

Procedure

The measurements took place in the workshop space of the Echo building at the Eindhoven University of Technology. The room was not empty, as measurement equipment and various objects were present, but a clearance radius of approximately 1.5 m around the measurement location was arranged. The systems to be characterized were mounted on the floor of the room, a 300 mm-thick smooth concrete slab, which served as rigid backing. Compression of the porous layers by the rigid facings was avoided by supporting the two opposite edges of the facing with piles of books of equal heights so that the facings were supported at a level less than 3 mm above the surfaces of the porous layers. This small air gap was neglected in the fitted models. The measurement conditions are shown in Figure 2.

The measurement device used was the “In Situ Absorption setup” by Microflown Technologies, which was mounted on a lightweight tripod. This device consisted of a PU probe attached to a small spherical loudspeaker via a structure that decouples vibrations between the loudspeaker and sensors. The PU probe comprises of an omnidirectional free-field microphone and a particle velocity sensor capturing the velocity in one direction. The two sensors are positioned close to each other in the probe such that they are considered to be in the same position. The decoupling structure sets the distance from the loudspeaker diaphragm to the probe, $d$ , to 260 mm. This distance was measured with a ruler with a ±3 mm accuracy.

For all recordings, the probe is oriented so that the velocity is recorded along the direction orthogonal to the panel’s surface. For the measurement of $Z_{m}$ , the device is placed so that the source is vertically aligned with the sensor and the sound wave reaches the measured point on the surface of the system to be characterized with normal incidence. Additionally, the distance from the surface to the probe $(h)$ is set to 10 mm, less than 20 mm as recommended in a previous publication.¹⁴ This was measured with a ruler, with a ±3 mm accuracy. Finally, the horizontal position of the probe is set within the “confidence region” defined in Brandão et al.,²⁰ where it is shown that such positioning of the probe mitigates the potential error resulting from sound wave diffraction on the panel’s side edges.

The output of the PU probe was connected to the Microflown signal conditioner, which separated the pressure and velocity signals into two channels. The outputs of the conditioner, as well as the loudspeaker, were connected to a 2-in/2-out USB audio interface (Triton V2 Standard, Acoustics Engineering), connected to a laptop running the DIRAC 6 room acoustics acquisition software (Acoustics Engineering). The software was used to perform the de-convolution of sent and recorded signals to derive the impulse responses for both pressure and velocity channels. The source signal, a 5-s-long exponential sine sweep (ESS), was sent from DIRAC 6 to the speaker via an intermediate amplifier.

The distance from the source and receiver to the closest reflective surface of the measurement environment was measured at around 1.5 m. The approximate time delay between the incident sound peak and the first parasitic reflections was thus computed as $Δ t_{r e f l} = 7.4$ ms, and was then used as the time-window length for the Adrienne window applied to the IRs.

Model fitting

The cost function, $F$ , is minimized in MATLAB with the $f m i n c o n$ function used as the core solver to a $M u l t i S t a r t$ object, which runs the solver for a given number of starting points distributed uniformly across the search space and yields the lowest minimum. The search space, which is defined by the lower and upper boundaries ( $x_{l}$ and $x_{u}$ , respectively) for each parameter, is presented in Table 3.

Table 3.

Lower $(x_{l})$ and upper $(x_{u})$ limits of search space for the optimization variables in thi study.

Parameter	$x_{l}$	$x_{u}$
$m_{h}$ (kg/m²)	0	10
$r_{h}$ (Pa·s/m)	0	$1 \times 10^{4}$
$t$ (mm)	0	100
$σ$ (Pa·s/m²)	$1 \times 10^{3}$	$1 \times 10^{6}$
$l$ (mm)	0	200
$D$ (mm)	0	200
$h$ (mm)	0	30

The number of starting points fed to the MultiStart procedure, $N_{x}$ , was chosen as the lowest number ensuring no variation of the output parameters ( ${\bar{x}}_{o p t}$ and $h_{o p t}$ ) over seven fitting iterations based on the same experimental data. This number was empirically set as $N_{x} = 30$ . For the fitting process, reflection coefficients at 300 logarithmically spaced frequencies in the optimization frequency range were used.

The frequency ranges of the model fitting were different for the two impedance models because the Helmholtz resonator model becomes invalid above 5 kHz, as the ratio between the wavelength $(\approx 68 mm)$ and the dimension of the perforations becomes smaller than 5. The frequency ranges chosen were $Δ f_{h e l m} = [500, 4000]$ Hz and $Δ f_{J C A, e q} = [500, 10000]$ Hz for the Helmholtz model and the JCA model, respectively.

Results

In this section the results are presented. For each configuration, figures are shown comparing both real and imaginary parts of $R_{m o d e l, o p t}$ and $R_{i n s i t u, o p t}$ , and a table of retrieved parameters is presented. In these tables, the errors $e$ in retrieved parameters for which reference values are available (and found in Tables 1 and 2) are shown in parenthesis next to the retrieved values.

Helmholtz resonator model characterization

This section presents the characterization results of the perforated facings with a Helmholtz resonator model fitting. Figure 3 shows the complex reflection coefficient curve fittings obtained for the three perforated facings backed by the air cavity. The model and measurement are in good agreement from 500 Hz to around 4 kHz. At higher frequencies, the measurements and fitted models deviate significantly from each other because the Helmholtz resonator model is valid only when the perforation size is small compared to the wavelength, a condition that begins to be violated at higher frequencies. The curves start to deviate from 4 kHz upwards, which corresponds to a wavelength of 86 mm, that is, more than 15 times the radii of the perforations.

Figure 3.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II and (c) Perf. III backed with a 45 mm air cavity, using the Helmholtz resonator model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Vertical Red dashed lines: limits of model fitting frequency range.

Table 4 shows the results obtained from the in situ measurements of the facings backed by a 45 mm cavity. Slight inaccuracies (smaller than 5 mm) are observed for the cavity depth retrieved in the case of Perf I and II, which are the facings with greater open areas. This could be due to small differences in height in the rubber bars used to create the cavities. It could also be because the greater perforation radii in Perfs I and II result in plugs of air of increased volume, thereby altering the effective depth of the cavities. The error in the retrieved moving mass varies depending on the facing: 20% for Perf. III and only 7% for Perf. II. The reason for these errors is not clear. The underestimates for Perfs. I and II could be due to a boundary layer effect in the perforations partially preventing the air from moving because of friction forces. Additionally, as seen in all the result tables presented in this section, the values retrieved for the probe-to-sample distance, $h$ , hover within the range from 8 to 13 mm, less than 3 mm from the targeted value of 10 mm.

Table 4.

Characterization results of the perforated facings backed by a 45 mm cavity from in situ measurement.

Parameter	Perf. I	Perf. II	Perf. III
$m_{h}$ * (kg/m²)	0.065 (−14%)	0.127 (−7%)	0.249 (+20%)
$r_{h}$ * (Pa·s/m)	224	317	324
$D$ * (mm)	49 (+9%)	48 (+7%)	45 (0%)
$h$ * (mm)	10	12	8

The characterization results of the facings when backed by porous layers are given in Table 5 and the curve fittings in Figures 4 (Material A) and 5 (Material B). Comparing these new plots with Figure 3 shows that the replacement of the air cavity with a porous layer changes greatly the resulting reflection coefficient. Two main differences can be observed: first, the multiple sharp cavity resonances above 3 kHz have disappeared with the introduction of a porous layer, as the layer prevents the formation of modes in the cavity. Moreover, the sharpness (or quality factor) of the Helmholtz resonance (lowest first dip on the plots) is decreased in the porous layer configuration because of the additional damping produced by that layer.

Table 5.

Characterization results of the perforated facings backed with porous layers with the Helmholtz resonator model.

Parameter	Perf. I	Perf. II	Perf. III
Material A
$m_{h}$ * (kg/m²)	0.052 (−31%)	0.101 (−14%)	0.183 (−12%)
$r_{h}$ * (Pa·s/m)	0	71	155
$σ$ * (kPa·s/m²)	24 (−18%)	27 (−10%)	28 (−6%)
$l$ * (mm)	44 (+10%)	45 (+12%)	47 (+17%)
$h$ * (mm)	13	11	11
Material B
$m_{h}$ * (kg/m²)	0.056 (−25%)	0.106 (−11%)	0.190 (−9%)
$r_{h}$ * (Pa·s/m)	30	114	200
$σ$ * (kPa·s/m²)	12 (−9%)	11 (−14%)	14 (+8%)
$l$ * (mm)	61 (+2%)	63 (+5%)	64 (+7%)
$h$ * (mm)	11	10	12

Figure 4.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II, and (c) Perf. III backed with Material A, using the Helmholtz resonator model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Vertical Red dashed lines: limits of model fitting frequency range.

Table 5 shows that the moving masses retrieved are about 30% smaller than those retrieved in the air cavity configuration. This could be caused by the presence of the porous layer behind the perforations, which reduces the effective length of the plug of air moving in the perforations, and hence the total moving mass. This is consistent with the fact that the masses retrieved when Material B is used as backing are slightly higher than those retrieved when Material A is used as backing. As material A is a “stiffer” material with a higher flow resistivity, it is expected to offer a greater resistance to the movement of the air out of the perforations. The acoustic resistances retrieved are also lower than those for the air cavity configuration, and a sharp increase in acoustic resistance is observed with the reduction of the perforation radius. The fact that the resistance values are lower than for the cavity configuration could be due to the inclusion of the porous layer in the model, which allows the attribution of a part of the damping to the porous layer instead of a generic resistance.

The retrieved porous layer parameters are in fair agreement with the references. However, the match is much better for Material B, whereas overestimates in thickness of about 10–17% (4–7 mm) and underestimates in resistivity of about 6–18% (2–6 kPa·s/m²) are found for Material A. This might be because the reflection coefficient is not very sensitive to the thickness of the porous layer. The curve fitting itself was in good agreement with the measurement from 500 Hz to 5 kHz for both porous backings, as can be seen from Figures 4 and 5.

Figure 5.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II, and (c) Perf. III backed with Material B, using the Helmholtz resonator model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Vertical Red dashed lines: limits of model fitting frequency range.

Equivalent JCA characterization

The results obtained by using an equivalent JCA model for the facings are presented in this section. To limit the number of variables within the optimization process, and because these values can be easily retrieved by visual inspection, the open ratio, $ϵ$ , and the perforation radii, $a$ , of the perforated panels are considered to be given data. As a result, the JCA model fitting retrieves only the thickness of the perforated panel (or tube length), $t$ , the generic resistance, $r_{c}$ , and the parameters of the backing layer (the air cavity depth, $D$ , or the resistivity $σ$ and thickness, $l$ , of the porous material).

The results obtained for a 45 mm cavity backing are shown in Table 6. The retrieved values of the cavity depth and facing thickness are retrieved with very good accuracy, with maximum errors of 4% and 6%, respectively. The values of the generic resistances retrieved increase as the perforation radius decreases, as is the case for the resistances retrieved in the Helmholtz resonator modeling.

Table 6.

JCA characterization results of the perforated facings backed by a 45 mm cavity from in situ measurement. The perforation ratio and radii are treated as input data.

Parameter	Perf. I	Perf. II	Perf. III
$t$ * (mm)	17 (+6%)	16 (0%)	16 (0%)
$r_{c}$ * (Pa·s/m)	210	302	342
$D$ * (mm)	46 (+2%)	47 (+4%)	46 (+2%)
$h$ * (mm)	11	8	8

Looking at the curve fitting obtained from the JCA modeling of the facings in Figure 6, a very good agreement is obtained up to 10 kHz, which was not the case when using the Helmholtz resonator model. This is because the JCA model does not assume that the wavelength is large compared to the perforation size.

Figure 6.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II, and (c) Perf. III backed with the 45 mm air cavity, using the equivalent JCA model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Vertical Red dashed lines: limits of model fitting frequency range.

The characterization results obtained for measurements of the facings with porous layer backings are presented in Table 7. The error values are very low. The error for the facing thickness compared to the reference value is always less than 0.5 mm, which results in a relative error $< 0.5 %$ . The relative error values for the flow resistivity and the thickness of the porous layers are below 8% and 5%, respectively. Similarly to the case of the air cavity, the retrieved buffer resistance, $r_{c}$ , increases when the radius of perforation decreases. However, the retrieved resistance values are lowered by an offset of around 150–250 Pa·s/m. This offset value can thus be interpreted to represent the losses within the 45 mm air cavity (which is now filled with a porous layer), while the values retrieved here account for the losses in the perforations due to the porosity of the MDF, which are not accounted for in the JCA model.

Table 7.

JCA characterization results of the perforated facings backed by porous layers. The perforation radius and open ratio of each perforated panel are fixed for the fitting.

Parameter	Perf. I	Perf. II	Perf. III
Material A
$t$ * (mm)	16 (0%)	16 (0%)	16 (0%)
$r_{c}$ * (Pa·s/m)	0	60	143
$σ$ * (kPa·s/m²)	31 (+3%)	30 (0%)	28 (−6%)
$l$ * (mm)	39 (−3%)	40 (0%)	39 (−3%)
$h$ * (mm)	9	9	10
Material B
$t$ * (mm)	16 (0%)	16 (0%)	16 (0%)
$r_{c}$ * (Pa·s/m)	9	40	170
$σ$ * (kPa·s/m²)	14 (+8%)	14 (+8%)	13 (0%)
$l$ * (mm)	57 (−5%)	60 (0%)	57 (−5%)
$h$ * (mm)	7	8	11

Figures 7 and 8 show that the model matches the measurement very well up to 10 kHz, demonstrating that the JCA equivalent model for the perforated facing allows for the prediction of the properties at higher frequencies than the Helmholtz resonator model when the facing is backed by a porous layer.

Figure 7.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II, and (c) Perf. III backed with Material A, using the equivalent JCA model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Red dashed lines: limits of model fitting frequency range.

Figure 8.

Measured and model fitted complex reflection coefficient of (a) Perf. I, (b) Perf. II, and (c) Perf. III backed with Material B, using the equivalent JCA model for the facing. Gray line: $R_{i n s i t u, o p t}$ ; Black line: $R_{m o d e l, o p t}$ ; Red dashed lines: limits of model fitting frequency range.

Conclusion

In this paper, the in situ characterization of perforated panels when backed by an air cavity and two porous layers were experimentally investigated using a method combining a PU-based in situ measurement and an impedance model fitting procedure. Two models were investigated for the characterization of the perforated panels: the Helmholtz resonator model and the equivalent JCA model.

The characterizations performed with the Helmholtz resonator model provided a good agreement between the measurement and the fitted model between 500 Hz and 5 kHz, as well as a good estimation of the backing layer properties (air cavity or porous layers). However, the moving mass retrieved showed some deviation from the theoretical prediction. The retrieved acoustic damping (in the form of a generic acoustic resistance of the resonator model) accounting for the combined acoustic losses within the perforation and the cavity was significantly higher for an air cavity backing than for a porous layer backing, suggesting that non-negligible losses is occurring in the cavity. This may be because the air cavity allows the plug of air to move more freely, which increases the friction losses in the perforations. However, a close match between the measured data and the fitted model was observed in all cases.

The equivalent JCA model of the perforation allowed a much closer curve fitting at higher frequencies (up to 10 kHz). It also retrieved with excellent accuracy the thickness of the facing and the properties of the backing layers. The generic resistance term retrieved was non-zero in most cases, showing that besides the losses in the air cavity, not all losses in the facing can be predicted by the equivalent JCA model, because the MDF facings have a non-zero porosity value.

Overall, the equivalent JCA model allowed a better fitting and better retrieval of the parameters of the backing layer, especially in the case of a porous backing layer, and should thus be preferred. However, it should be noted that the geometry of the perforations (except for the tube length, $t$ ) was assumed to be known for the JCA model fitting. If the geometry is concealed or more complex (e.g., perforations with varying diameters or grooved panels), using a generic Helmholtz resonator model could prove more useful as fewer details are needed.

One potential drawback of the proposed method is that only the data in a certain frequency range is analyzed. This suggests that for systems where the defining absorption characteristic lies at beyond this range, the proposed methods will be less precise. For example, the characterization of a resonant absorber whose resonance lies below 500 Hz will most likely be inaccurate following the proposed method, as this is a frequency too low to be measured precisely enough with the proposed setup.

Regarding future works, it was noted that several manufactured perforated panel samples (not presented in this paper) had a thin fleece layer glued at their back. This additional thin layer was not modeled in this research. Future work could be done to describe the influence of the fleece on the overall performance of the perforated facing and to derive an effective impedance model for it. Also the rigid facing study addressed only the case of circular macro-perforations (i.e., radii of several millimeters). Investigation of the slotted, grooved and micro-perforated facings constitutes a potential future work of interest. Finally, studying the influence of the panel size on the results, and the influence of the room in which the measurement is carried are other future works of interest.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement number 721536.

ORCID iDs

Baltazar Briere de La Hosseraye

Jieun Yang

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In situ acoustic characterization of a perforated panel on a cavity by means of PU measurement and model fitting

Abstract

Keywords

Introduction

Methods

Measurement procedure

Model fitting procedure and error estimation

Impedance models

Helmholtz resonator model

Equivalent JCA model

Air cavity backing

Porous layer backing

Practical implementation

In situ measurements

Materials

Procedure

Model fitting

Results

Helmholtz resonator model characterization

Equivalent JCA characterization

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iDs

References