# Experiment-driven Simplification of Johnson-Champoux-Allard-Lafarge Model for Fibrous Materials

Tao Yang, Martin Eser, Xiaoman Xiong, Jean-Philippe Groby, Jonas Schmid, Marcus Maeder, Yu-Hao Chang, Steffen Marburg

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Tao Yang, Martin Eser, Xiaoman Xiong, Jean-Philippe Groby, Jonas Schmid, et al. Experiment-driven Simplification of Johnson-Champoux-Allard-Lafarge Model for Fibrous Materials. Journal of the Acoustical Society of America, 2025, 157 (5), pp.3497-3507. 10.1121/10.0036644. hal-05406498

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Experiment-driven Simplification of Johnson-Champoux-Allard-Lafarge Model for
Fibrous Materials

Tao Yang,¹ Martin Eser,¹ Xiaoman Xiong,² Jean-Philippe Groby,³ Jonas M. Schmid,¹
Marcus Maeder,¹ Yu-Hao Chang,⁴ and Steffen Marburg¹

¹Chair of Vibro-Acoustics of Vehicles and Machines, Department of
Engineering Physics and Computation, TUM School of Engineering and Design,
Technical University of Munich, Boltzmannstrasse 15, 85748 Garching,
Germany

²State Key Laboratory of New Textile Materials and Advanced
Processing Technologies, Wuhan Textile University, Wuhan, 430200,
China

³Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613,
Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université,
France

⁴School of Aerospace Engineering, Technical University of Munich Asia,
510 Dover Road #05-01, 139660, Singapore

(Dated: 14 April 2025)
Widely used empirical and semi-phenomenological models lack practicality due to
unstable predictions, which exhibit poor reproducibility, or reliance on numerous pa-
rameters that are not directly measurable. This paper presents an experiment-driven
method to simplify the Johnson-Champoux-Allard-Lafarge (JCAL) model using
a two-microphone impedance tube. Bayesian inference is applied to determine the
non-acoustical parameters (i.e., tortuosity, airflow resistivity, viscous and thermal
characteristic lengths, and static thermal permeability), and the regression method
is used to correlate porosity with these non-acoustical parameters. The simplified
JCAL model is validated against the original model and experimental results for
acrylic, silk, and wool fibers. The simplified JCAL model shows good agreement
with the original model for acrylic and wool fibers. Both models perform reliably
for porosities between 92% - 98%, but show a mismatch with experimental data at
99% porosity, likely due to frame vibration effects. The poroelastic behavior leads to
the original and simplified JCAL models failure for silk fiber. This experiment-driven
simplification method reduces the JCAL model to a single parameter, porosity,
which significantly enhances computational efficiency and enables faster predictions.
It also enhances the model's applicability to optimization strategies, such as tuning
design parameters for optimal sound absorption while minimizing thickness and mass.

**Keywords:** Semi-phenomenological model; Acoustic material characterization; Bayesian Inference; Model simplification; Fibrous material
## 22 I. INTRODUCTION

Semi-phenomenological acoustical models for porous materials have been developed based on the geometry of pores since the mid-20th century¹⁻³. They combine empirical data with theoretical frameworks to describe how sound interacts with materials having complex internal structures. These semi-phenomenological models employ a set of four to eight non-acoustic parameters to approximate the dynamic density and the dynamic bulk modulus⁴.

Among semi-phenomenological models, the Johnson-Champoux-Allard-Lafarge¹⁻³ (JCAL) model is now extensively utilized due to its robustness and accuracy in describing sound propagation in porous media across a broad frequency range⁵. The JCAL model employs six non-acoustic parameters, namely porosity, tortuosity, viscous and thermal characteristic lengths, and viscous (related to airflow resistivity) and thermal permeability. The primary limitation of the JCAL model lies in the challenges associated with precisely measuring and estimating the non-acoustical parameters it depends on due to few of these parameters being directly measurable⁶.

Several scholars have proposed simplifying semi-phenomenological models to increase their practicality while maintaining accuracy. The typical approach is to correlate microstructural characteristics with non-acoustic parameters using two-scale homogenization⁷⁻⁹. Consequently, directly measurable microstructural characteristics, which can be obtained without the need for inversion characterization, are used in semi-phenomenological models instead of those non-acoustic parameters. Horoshenkov et al.¹⁰ have simplified the Johnson-Champoux-Allard (JCA) model, which has five non-acoustical parameters, to a
four-parameter model by relating the pore size distribution with viscous and thermal char-
acteristic lengths for the media with log-normal pore size distribution. Subsequently, they
investigated how utilizing porosity, median pore size, and standard deviation of pore size in
porous media with a near log-normal pore size distribution (including granular, fibrous, and
foam materials) can reduce the number of non-acoustical parameters in the JCAL model by
half¹¹. Doutres et al.¹²,¹³ have linked the microstructural characteristics (i.e., cell size, strut
length, and thickness) with non-acoustical parameters, also called macroscopic parameters,
including porosity, airflow resistivity, tortuosity, and viscous and thermal characteristic
lengths, to simplify the JCA model for polyurethane foam.

Throughout numerous studies, fibrous material takes an essential role in sound absorp-
tion and reverberation time control<sup>14–18</sup>. The acoustic properties of fibrous materials have
been comprehensively studied based on the geometrical characteristics of their microstruc-
ture<sup>19–21</sup>. Nevertheless, it is easier to use empirical models, such as Delany–Bazley<sup>22</sup>, Miki<sup>23</sup>,
and Garai–Pompoli<sup>24</sup> models, to predict the acoustic properties of fibrous materials since
only airflow resistivity is required<sup>25,26</sup>, compared to semi-phenomenological models. How-
ever, the prediction accuracy of empirical models can be unstable, exhibiting poor repro-
ducibility when applied to new fibrous materials. This instability arises because these models
are based on regression methods, developed using extensive experimental data, such as char-
acteristic impedance and airflow resistivity measurements, for specific fibers (e.g., glass fiber
and mineral wool in Delany *et al*.<sup>22</sup>). In addition, empirical models may fail to accurately
predict the acoustic behavior of fibrous materials in all directions, leading to instability, char-
acterized by inconsistent or unreliable predictions, or inaccuracy when applied to anisotropic
fibrous materials<sup>27,28</sup>. Researchers have attempted to modify the eight coefficients in the De-
lany-Bazley or Miki models to enhance the accuracy for specific fibrous materials, including
some natural fibers<sup>29-32</sup>. Only a few studies have shown acceptable agreement between ex-
periments and predicted values based on original empirical models for natural fibers<sup>31,32</sup>.

The unreliable prediction of the acoustic properties of fibrous materials is mainly attributed
to their large microstructure diversity, especially in the case of natural fibers. Natural
fibers' size/diameter varies widely, with a broad and random distribution, often assumed
to be normal or log-normal<sup>33-35</sup>. In addition, the significant difference in surface roughness
among these natural fibers directly affects tortuosity, which in turn influences their acoustic
properties<sup>36</sup>. Such differences in the microstructure of pores directly affect the shape factor,
consequently influencing the acoustic properties<sup>37</sup>. Hence, widely used empirical models are
usually inaccurate for fibers, especially natural fibers. Moreover, note that the airflow re-
sistivity represents the interaction of low-velocity airflow (down to 0.5 mm/s<sup>38</sup>) with porous
media, making this parameter more challenging to control compared to porosity or density.
This difficulty implies that preparing fibrous media with a precise airflow resistivity value
is not straightforward, even though empirical models may be accurate for certain specific
fibers. In contrast, porosity and density are relatively easy to control.

Overall, commonly used semi-phenomenological and empirical models each have their
limitations. Although the semi-phenomenological models are more accurate than empirical
models, some required non-acoustical parameters are complicated to measure directly. Some
researchers have tried to simplify the former model, but other methods are needed to measure
the microstructure. So far, empirical models exhibit inaccurate predictions for fibrous mate-
rials that differ from those on which the models were initially based on. There is a significant
trend towards using natural fibers to replace synthetic fibers to enhance sustainability¹⁶,³⁹,⁴⁰.

Furthermore, predicting the properties of mixtures of different fiber materials (e.g., recycled fibers) presents another challenge for current models, as some natural materials may contain resonant elements that influence their acoustic behavior at frequencies of interest in acoustics⁴¹. Semi-phenomenological and empirical models are not ideal or practical solutions for estimating the acoustic properties of regular or sustainable fibrous materials. Hence, developing a method to overcome the limitations of semi-phenomenological and empirical models is worthwhile and meaningful.

This paper presents an experiment-driven method to simplify the JCAL model by utilizing
and relying on a two-microphone impedance tube only. The results from the simplified JCAL
model and the original JCAL model are compared with measured results on acrylic, silk, and
wool fibers. The results provide an enhanced model in terms of efficiency, which facilitates
a faster prediction, making the model applicable for optimization strategies.

Section II of this paper discusses the JCAL model, its simplification, data acquisition
from impedance tube measurements, and the application of Bayesian inference for parameter
estimation. Section III presents the inferred parameters and the accuracy of the simplified
JCAL model, followed by the conclusion in Section IV.

## II. MODELS AND METHODS

In the following, we briefly review the Johnson-Champoux-Allard-Lafarge (JCAL) model
to provide a theoretical foundation for the reader. The simplification applied in this study
108 is based on the Bayesian inference of key parameters derived from the measurements, con-
109 cluding this section.

110
A. The Johnson-Champoux-Allard-Lafarge model

111
Fibrous media's complex propagation coefficients and characteristic impedance are calcu-
112
lated using the Johnson-Champoux-Allard-Lafarge (JCAL) model. In this work, we assume
113
the fibers are isotropic. Here, the equivalent dynamic density $\tilde{\rho}(\omega)$ and the dynamic bulk
114
modulus $\tilde{K}(\omega)$ follow as$^{1-3,42}$:

$$
\tilde{\rho}(\omega) = \frac{\alpha_{\infty} \rho_0}{\phi} \left[ 1 + \frac{\phi \eta}{j \omega} k_0 \rho_0 \alpha_{\infty} \sqrt{1 + j \omega \frac{4 k_0^2 \alpha_{\infty}^2 \rho_0}{\eta \Lambda^2 \phi^2}} \right], \quad (1)
$$

$$
\tilde{K}(\omega) = \frac{\gamma P_0 / \phi}{\gamma - (\gamma - 1) \left[ 1 - j \frac{\phi \eta}{\omega} k'_0 P_r \rho_0 \sqrt{1 + j \omega \frac{4 k'_0^2 P_r \rho_0}{\eta \Lambda'^2 \phi^2}} \right]^{-1}}, \quad (2)
$$

115 where j² = -1 denotes the imaginary unit, ω = 2πf is the angular frequency with the
116 corresponding frequency, f, ρ₀ denotes the density of air, P₀ is the atmospheric pressure,
117 η is the dynamic viscosity of air, γ is the specific heat ratio, and Pr is the Prandtl number.
118 Furthermore, φ represents the open porosity, α∞ is the high-frequency limit of the tortuosity,
119 k₀ = η/σ, where σ is the airflow resistivity, and k'₀ are respectively the static viscous and
120 thermal permeabilities, Λ and Λ' denote the viscous and thermal characteristic lengths,
121 respectively. Subsequently, the characteristic impedance, Zc, and the complex propagation
122 coefficient, k, result using the following equations¹³:
$$Z_c = \sqrt{\tilde{\rho}(\omega) \tilde{K}(\omega)}, \qquad (3)$$

$$k = \omega \sqrt{\frac{\tilde{\rho}(\omega)}{\tilde{K}(\omega)}}. \qquad (4)$$

In the JCAL model (Eqs. (1-4)), the equivalent dynamic density accounts for viscous losses, while the equivalent dynamic bulk modulus addresses thermal losses¹⁶. Once the characteristic impedance and complex propagation coefficient are obtained, the surface characteristic impedance $Z_s$ and the normal sound absorption $\alpha$ of a rigidly backed layer of porous material with a finite thickness $l$ are subsequently determined as:

$$Z_s = Z_c \coth(kl), \qquad (5)$$

$$\alpha = 1 - |R|^2 = 1 - \left| \frac{\frac{Z_s}{\rho_0 c_0} - 1}{\frac{Z_s}{\rho_0 c_0} + 1} \right|^2, \qquad (6)$$

where $R$ is the reflection coefficient, and $c_0$ is the sound speed in air.

## B. Model simplification

The key microstructural parameters that influence acoustic properties, such as fiber diameter, cross-sectional shape, and surface roughness, remain unchanged even when the porosity, density, or thickness of the fiber assembly varies, as long as the fibers are identical and randomly distributed. Only the pore size changes with the change in porosity, which means the cross-sectional shape factor of the pore remains the same. Consequently, the potential
exists to streamline the non-acoustical parameters, condensing them into expressions solely governed by the open porosity. This approach simplifies the JCAL model by circumventing the necessity for numerous experimental measurements, including omitting the need to measure fiber diameter. Table I provides the original and simplified expressions of the five non-acoustical parameters. We adopt the numerical model for airflow resistivity from the literature<sup>44,45</sup>. While the models for characteristic lengths, which combine the work of John-son et al.<sup>1</sup> and Eichhorn et al.<sup>46</sup>, offer greater precision than the original models for polyester fibers<sup>47</sup>, they are also more complex to simplify. Therefore, only the original expressions of characteristic lengths<sup>48</sup> are used for simplification. Some studies have investigated static thermal permeability<sup>3,49</sup>, but the existing analytical models are not easily simplified. Hence, we propose a new model to establish the relationship between porosity and static thermal permeability.

Impedance tube measurements are conducted for each fiber type across an open porosity range of 92% to 99%, which are 92, 93, ..., 99%, making it straightforward to derive coefficients for the simplified models. The porosity is obtained according to $\phi = 1 - \rho/\rho_f$, where $\rho$ is fiber assembly density, and $\rho_f$ is the fiber density which are 1170 kg/m³, 1361 kg/m³, and 1314 kg/m³ for acrylic, silk, and wool fibers, respectively<sup>50-52</sup>. The fiber assembly density is obtained according to $\rho = w/v$, where w is the fiber weight, and v is the volume of space occupied by the fibers and the saturated fluid, which is air in this work, in the impedance tube. To ensure the accuracy of porosity, the weight of fibers is measured on a weight scale with reproducibility of 0.003 g (Pocket balance TGD 50-3CS05, KERN & SOHN GmbH, Ebingen, Germany). The relationship between porosity and the five remaining non-acoustical
<table><caption>TABLE I: Original and simplified models for the non-acoustical parameters in JCAL model.</caption><thead><tr><th>Non-acoustical parameters</th><th>Original forms</th><th>Simplified forms</th></tr></thead><tbody><tr><td>Airflow resistivity (σ)<sup>44,45</sup></td><td>σ = <sup>16η(1-φ)</sup>/<sub>r<sup>2</sup>(-0.640 ln(1-φ)-φ+0.263)</sub></td><td>σ = a <sup>(1-φ)</sup>/<sub>(-0.640 ln(1-φ)-φ+0.263)</sub></td></tr><tr><td>Tortuosity (α<sub>∞</sub>)<sup>53</sup></td><td>α<sub>∞</sub> = 1 - q ln φ</td><td>α<sub>∞</sub> = 1 - b ln φ</td></tr><tr><td>Viscous characteristic length (Λ)<sup>48</sup></td><td>Λ = √<sup>8ηα<sub>∞</sub></sup>/<sub>σφ</sub></td><td>Λ = c√<sup>(1-b ln φ)(-0.64 ln(1-φ)-φ+0.263)</sup>/<sub>(1-φ)φ</sub></td></tr><tr><td>Thermal characteristic length (Λ')<sup>48</sup></td><td>Λ' = 2√<sup>8ηα<sub>∞</sub></sup>/<sub>σφ</sub></td><td>Λ' = d√<sup>(1-b ln φ)(-0.64 ln(1-φ)-φ+0.263)</sup>/<sub>(1-φ)φ</sub></td></tr><tr><td>Static thermal permeability(k'<sub>0</sub>)<sup>49</sup></td><td>k'<sub>0</sub> ≥ η/σ</td><td>k'<sub>0</sub> = e<sup>φ<sup>2</sup></sup>/<sub>(1-φ)</sub></td></tr></tbody></table>

Note: *q* is a constant parameter in the original form of tortuosity, *a*, *b*, *c*, *d*, and *e* are the coefficients for the simplified expressions of non-acoustical parameters in the simplified JCAL model, and *r* is the fiber diameter. The references following the non-acoustical parameters correspond to their original forms.

parameters, obtained through impedance tube measurements and Bayesian inference, is subsequently established using regression methods for each fiber type. This approach accurately determines the coefficients required for the simplified models, which helps to facilitate the optimization and simulation process for fibrous materials. Similar optimization techniques
have been applied using the original Delany-Bazley and JCAL models for acoustic material design, demonstrating the practical benefits of such simplifications⁵⁴⁻⁵⁶.

### C. Data acquisition

Here, the reflection coefficient, and normalized surface impedance of three types of loose fibers (i.e., acrylic, silk, and wool) with varying porosity ranging from 92% to 99% are measured using a two-microphone impedance tube (Gesellschaft für Akustikforschung Dresden mbH, Dresden, Germany), see Eser et al.⁵⁷ for more details. Each specimen is measured three times. After each measurement, the specimen holder is removed from the impedance tube, the fibers are taken out, and then randomly reinserted into the holder for the next measurement. The final result is obtained by averaging the measured reflection coefficients over three measurements. A central moving average with a window size of two is then applied separately to the real and imaginary parts of the averaged reflection coefficients to smooth the data. Finally, the sound absorption coefficient is calculated from the smoothed reflection coefficient values. The moving average is given by:

$$ \Re(R')_i = \frac{\Re(R_i) + \Re(R_{i+1})}{2}, \quad \Im(R')_i = \frac{\Im(R_i) + \Im(R_{i+1})}{2} \qquad (7) $$

where $\Re(R')_i$ and $\Im(R')_i$ are the smoothed real and imaginary parts of the reflection coefficient at index $i$, and $\Re(R_i)$, $\Im(R_i)$ are the original real and imaginary parts.

The reflection coefficient of loose fibers has been used for the parameter estimation via Bayesian inference. The measurements are carried out using a small tube (40 mm inner diameter) covering a frequency range between 100 Hz and 4.95 kHz. Figure 1 illustrates the
<figure><img src="image_2.png" /><figcaption>FIG. 1: The (a) schematic and (b) setup of the 40 mm inner diameter impedance tube.</figcaption></figure>

schematic and the experimental setup of the impedance tube. The position at $x = 0$ indicates the rigid termination, whereas $x_1$ and $x_2$ are the distances between the specimen surface and the two microphones, Mic 1 and Mic 2, see Figure 1(a), respectively. A loudspeaker, positioned at one end of the impedance tube, excites the tested sample acoustically. More concretely, the loudspeaker generates broadband, stationary random sound waves, which propagate as incident plane waves in the tube and interact with the sample surface. The reflected sound wave signals are then captured and compared to the incident sound waves. Two separate measurements are carried out by changing the microphone position from ‘Mic 1’ to ‘Mic 1~’ to account for the full frequency range. The data acquisition is realized by an
Apollo light frontend (SINUS Messtechnik GmbH, Leipzig, Germany) and a PA601 single channel power amplifier (AtlasIED, Phoenix, United States of America), processed with the AED 1001 - AcoustiTube analysis software (Gesellschaft für Akustikforschung Dresden mbH, Dresden, Germany). Prior to the measurements, both microphones have been calibrated at 250 Hz using a pistonphone Type 4228 (Hottinger Brüel & Kjær, Sound & Vibration Measurement A/S, Virum, Denmark).

## D. Bayesian inference

Bayesian inference has become a key tool in acoustics, enhancing model accuracy and robustness$^{58-61}$. In this study, we employ Bayesian inference to estimate non-acoustical parameters of fibrous materials, moving beyond traditional deterministic approaches. The Bayesian framework offers probabilistic estimation by incorporating prior knowledge and quantifying uncertainty, leading to more robust parameter identification.

Probabilistic inverse methods estimate the parameters using Bayes' theorem, which is formulated as:

$$p(\theta|D, I) \propto p(D|\theta, I) p(\theta|I) \qquad (8)$$

where $\theta$ is the vector of unknown model parameters, $D$ is the measured reflection coefficient data, and $I$ represents prior assumptions. The Bayesian framework can be interpreted as updating one's prior knowledge $p(\theta|I)$ about the model parameters after incorporating the measured reflection coefficient data $D$ through the likelihood function $p(D|\theta, I)$. This re-
sults in the posterior probability distribution $p(\theta|D, I)$, which represents the updated belief about the parameters given the data and the background information $I$.

In this work, a uniform prior is assigned for each parameter, defined as:

$$ \theta_i \sim \mathcal{U}(\theta_{i,\min}, \theta_{i,\max}) \qquad (9) $$

The prior bounds are set by the lower and upper bounds for each parameter (Table II), ensuring no bias within the defined physical range.

The likelihood is assigned as Gaussian likelihood according to the principle of maximum entropy$^{59}$. It can be formulated as:

$$ p(D|\theta, I) = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi\tau^2}} \exp \left( -\frac{(D_i - \hat{D}_i(\theta))^2}{2\tau^2} \right) \qquad (10) $$

where $\hat{D}_i(\theta)$ is the predicted reflection coefficient from the forward model given parameters $\theta$, $n$ is the number of measurement points, and $\tau$ is the standard deviation capturing measurement noise.

Markov Chain Monte Carlo (MCMC) sampling, specifically the Metropolis-Hastings algorithm, was used to sample from the posterior distribution. We generated 10$^6$ samples for each of three parallel chains, discarded the first 30% of each chain as burn-in, and combined the remaining 700,000 samples to estimate parameters and quantify uncertainty. The mean and variance of the posterior distribution provide parameter estimates and credible intervals for uncertainty.

To ensure the physical relevance of the inferred parameters, and promote uniqueness in the parameter estimation, constraints based on established models and fiber diameter ranges
are applied to the five parameters (airflow resistivity, tortuosity, viscous and thermal characteristic lengths, and static thermal permeability) in the Bayesian framework, as detailed in Table II. The constraints of tortuosity are based on Allard et al.⁴³ and Fotsing et al.⁶², which suggest a range from 0 to 2 - $\phi$. To allow for broader exploration during inference, we extend the upper bound to 3 - $\phi$. The upper and lower bounds of airflow resistivity and viscous and thermal characteristics lengths are defined with models from the literature and the fiber diameter ranges¹⁵,¹⁸, while the range of static thermal permeability (10⁻¹⁰ to 10⁻⁸ m²) is directly taken from the literature⁶³. The fiber diameter ranges are 10–40 µm⁶⁴, 4–15 µm⁵², and 16–40 µm⁶⁵ for acrylic, silk and wool fibers, respectively.

<table><caption>TABLE II: Constraints of non-acoustical parameters.</caption><thead><tr><th>Non-acoustical parameters</th><th>Lower bounds</th><th>Upper bounds</th></tr></thead><tbody><tr><td>Airflow resistivity ($\sigma$)⁴⁵</td><td>$\sigma_1 = \frac{16\eta(1-\phi)}{r_2^2(-0.640 \ln(1-\phi)-\phi+0.263)}$</td><td>$\sigma_2 = \frac{16\eta(1-\phi)}{r_1^2(-0.640 \ln(1-\phi)-\phi+0.263)}$</td></tr><tr><td>Tortuosity ($\alpha_\infty$)⁶²</td><td>$\alpha_\infty_1 = 1$</td><td>$\alpha_\infty_2 = 3 - \phi$</td></tr><tr><td>Viscous characteristic length ($\Lambda$)⁴⁸</td><td>$\Lambda_1 = \sqrt{\frac{8\eta\alpha_\infty_1}{\sigma_2\phi}}$</td><td>$\Lambda_1 = \sqrt{\frac{8\eta\alpha_\infty_2}{\sigma_1\phi}}$</td></tr><tr><td>Thermal characteristic length ($\Lambda'$)⁴⁸</td><td>$\Lambda'_1 = 2\sqrt{\frac{8\eta\alpha_\infty_1}{\sigma_2\phi}}$</td><td>$\Lambda'_2 = 2\sqrt{\frac{8\eta\alpha_\infty_2}{\sigma_1\phi}}$</td></tr><tr><td>Static thermal permeability ($k'_0$)⁶³</td><td>$k'_0 = 10^{-10}$</td><td>$k'_0 = 10^{-8}$</td></tr></tbody></table>

Note: non-acoustical parameters and fiber diameter with subscripts 1 and 2 indicate the lower and upper bounds or values, respectively. The references following the non-acoustical parameters correspond to their models or constraints, which are determined by porosity and fiber diameter.
## 234 III. RESULTS AND DISCUSSION

Mounting loose fibers in an impedance tube for accurate measurements can be challenging due to the difficulty in maintaining the fiber's stability and uniformity. Custom-designed specimen holders are created using SolidWorks (SolidWorks Corp., Waltham, USA) and fabricated utilizing advanced additive manufacturing technology on the Bambu Lab X1-Carbon 3D printer (Shenzhen Tuozhu Technology Co., Ltd., Shenzhen, China). The effect of the specimen holder on the normalized surface impedance and the sound absorption coefficient is thoroughly investigated to ensure that the holder itself introduces no significant measurement errors. Comparisons are made between measurements with the specimen holder and those taken with an empty tube of the same thickness, as Figures 2(a), 2(b) and 2(c) illustrate. This comparison is crucial to validate the neutrality of the holder in acoustic measurements. Specifically, the specimen holder designed for a 30 mm thick single-layer specimen is provided in Figure 2(d). This holder features a wall thickness of 2 mm and an inner depth of 30 mm to accommodate the specimen securely. The grille mesh, which is essential for stabilizing the fibers and facing the speaker, has a thickness of 2 mm and a width of 1 mm, ensuring minimal interference with the acoustic measurements. Additionally, the eight sidebars provide structural support and are strategically positioned at angles of 15° from the central point, with their width aligned at this angle to enhance stability without obstructing the acoustic path. Figures 2(e-g) demonstrate the specimen holders with acrylic, silk, and wool fibers, respectively. Although some fibers may extend beyond the specimen holder, the inner surface of the impedance tube helps contain the fibers within a defined volume
<figure><img src="image_3.png" /><figcaption>FIG. 2: Comparison of (a) resistance, (b) reactance and (c) sound absorption coefficient between the empty specimen holder and the empty tube, (d) CAD model of the specimen holder, (e)-(g) different fiber types inside the specimen holder, and (h) specimen holder mounted in the impedance tube with acrylic fibers.</figcaption></figure>
(see Figure 2(h)). Figures 2(a), 2(b) and 2(c), show that the impact of the specimen holder on the normalized surface impedance and the sound absorption coefficient is insignificant. This indicates that the specimen holder does not alter the acoustic properties of the samples, and its effect can be considered negligible. Consequently, the designed holder provides a reliable solution for mounting loose fibers in an impedance tube, facilitating accurate and reproducible measurements.

The inferred non-acoustical parameters, which include airflow resistivity, tortuosity, viscous characteristic length, thermal characteristic length, and static thermal permeability, are depicted as functions of the porosity in Figure 3. Their values and the standard deviation are shown in Table 1 in **Supplementary Material**. The correlation coefficient $R^2$ between the inferred values and their corresponding fitted values has been calculated for each of these inferred parameters. Figure 3 also presents the fitted curves along with their corresponding $R^2$ values, regression equations, and shaded areas indicating the 95% confidence intervals. These regression equations are derived from the simplified forms presented in Table I, which provide a streamlined approach to modeling these parameters. The derived coefficients in the simplified forms are listed in Table 2 in **Supplementary Material**. Additionally, each subgraph includes the constraints applied to the non-acoustical parameters during the inversion process, as detailed in Table II. It shows that the inferred parameters fall within reasonable ranges, demonstrating the robustness of the inversion process. The fitted curves, together with the 95% confidence intervals, follow the pattern of the inferred data points well for most parameters, showing that the simplified Johnson-Champoux-Allard-Lafarge (JCAL) model works effectively. The high $R^2$ values
<figure><img src="image_4.png" /><figcaption>FIG. 3: The inversely characterized non-acoustical parameters in the JCAL model, their correlations with porosity, and correlation coefficient. $\sigma$ is the airflow resistivity, $\alpha_{\infty}$ is the tortuosity, $\Lambda$ is the viscous characteristic length, $\Lambda'$ is the thermal characteristic length, $k'_{0}$ is the static thermal permeability, and $R^2$ is the coefficient of determination.</figcaption></figure>
277 indicate a strong correlation for most non-acoustical parameters across the three fiber types.

278 However, there is an exception in the case of the tortuosity of silk fiber, which exhibits a
279 relatively low correlation coefficient of 0.57. In general, the inferred parameters for acrylic
280 and wool fibers exhibit good agreement with the regression curves, indicating reliable fits
281 across most porosity levels. A slight exception is observed in the case of acrylic fiber, where
282 the viscous characteristic length tends to be overestimated at lower porosities. Furthermore,
283 an interesting observation from the study is that the thermal characteristic lengths of the
284 three types of fibers are approximately 1.4 to 9.8 times the viscous characteristic lengths
285 rather than the previously assumed factor of two<sup>66-68</sup>. This deviation highlights the unique
286 properties of the materials under study and suggests that further investigation is warranted
287 into the relationships between these characteristic lengths.

288 To validate the original and simplified JCAL models, the inversely characterized non-
289 acoustical parameters are employed in the original JCAL model, while the single porosity
290 parameter is used in the simplified JCAL model. Figure 4 exemplarily demonstrates the
291 validation of the simplified JCAL model against the original JCAL model for 95% porosity
292 acrylic, silk, and wool fibers. This figure provides a clear comparison, showcasing how the
293 simplified model performs relative to the original. The complete validation results, which
294 include reflection coefficient and sound absorption coefficient based on the original and
295 simplified JCAL models, are provided in Figures 1-6 in **Supplementary Material**. These
296 results cover 30 mm thick fibrous materials with porosities ranging from 92% to 99%. Both
297 models show relatively good predictions for the real and the imaginary parts of the reflection
298 coefficient for acrylic and wool fibers. However, the prediction accuracy of both the original
and simplified JCAL models for the reflection coefficient and sound absorption of silk fibers with porosities ranging is less satisfactory. The silk fiber specimens with porosities ranging from 92% to 96% exhibit significant poroelastic behavior within the frequency range of approximately 1 kHz to 1.6 kHz, while a noticeable enhancement in sound absorption in this range occurs. This unique acoustic behavior may be due to the Biot resonance$^{69,70}$ and the high displacement sensitivity in this frequency range$^{71}$ that results in transverse vibration of fibers$^{72,73}$. Since silk fibers are much finer than acrylic and wool fibers and are similar in size to the Kevlar fibers in Lambert *et al.*$^{72}$, this may explain why acrylic and wool fibers exhibit less resonance across the frequency range. Moreover, the JCAL model assumes a motionless porous frame, which does not hold for silk fibers with such poroelastic behavior. Regarding the variation in the resonance behavior of silk fibers with changing porosity, the tension along the fiber direction affects displacement, while different porosities correspond to varying levels of compression and tension. This variation affects the acoustics behavior, highlighting the importance of considering mechanical properties and fiber interactions in acoustic modeling. Nevertheless, it is necessary to note that silk fibers with 95% and 96% porosities exhibit good sound absorption around 1 kHz, despite having a thickness of only 30 mm. Regarding the sound absorption coefficient of acrylic and wool fibers, the original and simplified JCAL models provide acceptable predictions for fibers with lower porosities (e.g., 92% to 98%). But both models **show a mismatch** for fibers with 99% porosity. This **mismatch** can be attributed to the frame vibration caused by the increased elasticity of the fibers when more air cavities are present in the medium due to the lack of mechanical connections between the fibers$^{74}$. By incorporating these detailed validations and observations, the effectiveness
321

322

323

and limitations of both the original and simplified JCAL models are clearly illustrated,
providing valuable insights for future improvements and applications in acoustical material
characterization.

<figure><img src="image_5.png" /><figcaption>FIG. 4: Validation of original JCAL model based on the inferred parameters and the simplified JCAL model for 95 % porosity and 30 mm thickness acrylic, silk, and wool fibers' reflection coefficient (upper) and sound absorption coefficient (bottom).</figcaption></figure>
324 IV. CONCLUSION

325 This study investigates the feasibility of simplifying the Johnson-Champoux-Allard-Lafarge (JCAL) model for fibrous materials. Three types of fibers—acrylic, silk, and wool—are analyzed. Two-microphone impedance tube measurements are conducted for fibrous specimens with porosities ranging from 92% to 99% for each fiber type with the help of a custom-designed specimen holder. Subsequently, Bayesian inference is employed to characterize the non-acoustical parameters in the JCAL model, with porosity treated as a known input. 330
331 The simplified forms of the non-acoustical parameters and their constraints are well-defined.
332 A regression method is utilized to establish correlations between porosity and five other non-acoustical parameters, including tortuosity, airflow resistivity, viscous and thermal characteristic lengths, and static thermal permeability, which are necessary for the JCAL model. Simplified JCAL models, relying solely on porosity, are developed using these correlations, and their accuracy is validated against measurement results. 333
334

337 The fitted values exhibit a strong correlation with the inversely characterized non-acoustical parameters, with coefficients of determination exceeding 0.89, except for the correlation between inferred tortuosity and fitted values for silk fiber. The original and simplified JCAL models are found to work well on acrylic and wool fibers with various porosities (92%–98%). Due to the Biot resonance and the displacement sensitivity of silk fibers, the measured results show a significant difference for porosity ranging from 92%–96%. The resonance occurs in the frequency range between 1 kHz and 1.6 kHz, which leads to the original and simplified JCAL models failure for silk fibers. However, further investigation 340
341
342
343
344
of the acoustic behavior of silk fibers is necessary, given their superior sound absorption
around 1 kHz despite having a thickness of only 30 mm. The simplified JCAL model mis-
matches the sound absorption of fibers with high porosity (i.e., 99%) due to the frame
vibration effects in acrylic and wool fibers. This study exclusively relies on two-microphone
impedance tube measurements, eliminating the need for complex characterizations, such
as the time-consuming microstructure determination of fibers, three- or four-microphone
measurements, and corresponding assessment of dynamic mass density and dynamic bulk
modulus. This experiment-driven simplification approach enhances the practicality of the
JCAL model by simplifying its dependency from six parameters to one, thereby facilitating
its broader application while preserving fundamental physical principles. Furthermore, the
simplification method significantly reduces computational complexity, enhancing their suit-
ability for real-time applications and large-scale simulations. Importantly, this methodology
is adaptable to various types of fibers and requires only the availability of two-microphone
impedance tubes commonly found in acoustic laboratories.

359 SUPPLEMENTARY MATERIAL

See the supplementary material at [URL will be inserted by AIP] for a comprehensive com-
parison between the measurements, the original Johnson-Champoux-Allard-Lafarge (JCAL)
model, and the simplified JCAL model.
363 ACKNOWLEDGMENTS

364 This project has received funding from the European Union's Horizon Europe research and innovation program under the Marie Skłodowska-Curie with the associated grant agreement No. 101063867. X.X. acknowledges support from the Youth Talent Project of Scientific Research Program (Q20231704) funded by the Hubei Provincial Department of Education. The authors would like to thank Dr. Mohanapriya Venkataraman and Prof. Hideaki Morikawa for their help with material acquisition. The authors would like to thank the anonymous reviewers for their valuable comments.

371 AUTHOR DECLARATIONS

372 The authors declare no conflicts of interest regarding this manuscript. All authors contributed to the research and writing of this paper.

374 DATA AVAILABILITY

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