Toward identifying efficient sound absorbers, we have formulated and analyzed the micro-perforated panels (MPPs) comprising cylindrical pores of arbitrary axial profiles for out-of-plane wave propagation. First, the forward problem was solved wherein an exact analytical expression for the absorption coefficient $\alpha $ was developed for these MPPs. The dependence of $\alpha $ was analyzed for various profiles comprising convex and concave-shaped sections including divergent and convergent linear and sinusoidal profiles, conic sections, Kilroy, and chirp shapes. The MPPs with pores having a diverging shape at the entrance were found to have higher sound absorption. The non-symmetric fluctuations in the profile led to fluctuations in the spectrum of $\alpha $. Aiming to address the long-standing challenge of low-frequency sound absorption, we solved the inverse problem to identify the axial profile of the cylindrical pore for maximum sound absorption for frequencies up to 2500 Hz. Given the advances in additive manufacturing, the results of this comprehensive work help in designing MPPs comprising complex-shaped pores suitable for a particular spectral regime.

## I. INTRODUCTION

The absorption of sound has been of interest to the acoustic (sonic) and ultrasonic communities for attenuation of sound waves in fluids and solids.^{1} The traditional way to absorb sound is by the use of porous viscoelastic materials^{2} and the recent focus emphasizes biodegradability in these materials.^{3} Over the past three decades, structured materials were developed to control the propagation of sound and elastic waves in solids;^{4,5} these materials were referred to as acoustic metamaterials (AMs). Phononic crystals of metals^{6} and polymers,^{7} micro-perforated panels,^{8} topological insulators,^{9} etc., at various length scales are some examples of acoustic metamaterials, which have had applications as waveguides,^{10} isolators,^{11} sensors,^{12} and controlling the flow of sound.^{13} The characteristic size of AMs often dictates the spectral regime of where the “meta” effect is observed. For example, the central wavelength of the bandgap of a phononic crystal is approximately the size of the period of the crystal. The spectral regime of special characteristics is often a narrow band. Micro-perforated panels (MPPs) are the only type of AMs, similar to the traditional acoustic materials,^{14} where (a) the attenuation of the sound is over a broad spectral regime^{15} and (b) the mechanism is by sound absorption. Furthermore, the micrometer-sized perforations lead to absorption in the audible regime. The effectiveness of MPPs is measured in terms of the sound absorption coefficient $\alpha $, which represents the fraction of the sound absorbed.

MPPs are thin structured composite panels comprising one panel perforated with a large number of sub-millimeter-sized pores, followed by an air cavity, and then a rigid panel.^{16} MPPs are used for sound absorption in the sonic regime. The development of the understanding of MPPs began in the year 1975 with the work by Maa^{17} and subsequent developments by him.^{18,19} The full thickness of an MPP lies between 50 and 100 mm, with the thickness of the perforated panel typically ranging from 0.5 to 3 mm, the pore diameter typically in the range 50–500 $\mu $m, and the air cavity of at least 50 mm for better performance. The shapes of the pores are often upright cylinders,^{19} and in rare cases, slanted^{20} or tapered cylindrical.^{20,21}

Additive manufacturing technologies have aided in the rapid prototyping of AMs,^{22} including MPPs comprising pores of tapered^{21} and petal-shaped^{23} for experimental verification. Furthermore, MPPs comprising multiple walls,^{24} honeycomb backing,^{25,26} and coiled-up channels backing^{27,28} have also been developed due to the advancements in manufacturing as well as the high efficiency in sound absorption over a broadband spectral regime.

Wave propagation through MPPs is often analyzed in terms of the equivalent electrical circuit rather than solving the boundary value problem.^{16} However, these circuit-based analyses cannot be extended when the profile of the pore is complex. Furthermore, in a large part of the literature on MPPs, the analysis is made for normally impinged waves. Efforts for waves with wavevector lying in the plane of the panel have only begun recently.^{29} In this work, we focused our efforts on the propagation of waves with wavevectors perpendicular to the plane of the MPP. We refer to this as out-of-plane wave propagation.

In a project dedicated to the development of sound-absorbing materials, we explored two types of acoustic metamaterials: locally resonant materials^{30} and MPPs, in addition to biodegradable sound-absorbing materials.^{31} In the work focused on MPPs, we explored the study of waves propagating in-plane and out-of-plane. The study was also extended for MPPs comprising multiple shells in each pore for both kinds of wave propagation. In this work, we present the analysis for wavevectors perpendicular to the plane of MPP, i.e., out-of-plane wave propagation. In existing MPPs, the shape of the pore is upright cylindrical. In a more advanced MPP, the pore is replaced by a multi-shelled-pore.^{24} As additive manufacturing techniques can rapidly prototype complex shapes,^{32,33} first, we chose to investigate MPPs comprising symmetric pores, but with the radius of the profile varying arbitrarily with depth. Starting from the first principles, exact analytical expressions were determined for the generic profile of radius, and various easily fabricated cases were analyzed. Numerical cases of standard concave and convex geometries of pores including conic sections, Kilroy, and chirp shapes were analyzed to establish the effect of shape on the absorption spectrum.

The need to optimize MPPs arose due to the long-standing challenge in attenuating the sound for frequencies between 20 and 2000 Hz of the audible regime.^{34,35} This noise control or minimization is crucial not only for indoor environments such as offices and houses but also for engineering applications such as ground and aerial vehicles, and industrial machinery.^{36} Hence, in tandem with the developments of the forward analysis, where the architecture of the MPP or AM is altered, efforts were also made to maximize the desired outcome. Some examples are maximization of the bandgap in phononic crystals by varying the geometry by topology optimization^{37} or genetic algorithms,^{38} increasing the sound absorption of Helmholtz resonator-based absorbers using deep neural network efficiency,^{39} or more recently by the use of machine learning models for AMs.^{40} While inverse problems are solved by a variety of numerical methods,^{41} the traditional method involves the use of heuristic,^{42} non-heuristic optimization^{43} algorithms, regularization methods,^{44} and the machine and deep learning models.^{40,45} Among the many methods, traditional optimization methods fail to achieve the desired output due to the complex nature of the equations. Genetic algorithms (GAs) operate by simulating the process of natural selection, where candidate solutions undergo successive generations of reproduction, mutation, and selection to evolve toward optimal or near-optimal solutions to a given problem. In essence, GAs are particularly well-suited for exploring large solution spaces, especially in cases where the objective function is complex or difficult to evaluate analytically. A survey of literature also established the success of using GAs for various AMs such as phononic crystals,^{38} locally resonant materials,^{46} and MPPs.^{47} Hence, we attempted to maximize the sound absorption over a broad spectral regime by GA, rather than aiming for a maximum over a local spectral regime. In the next part of the work, we solved the inverse problem for out-of-plane propagation in an MPP and identified the axial profile of the symmetric pore to have maximum broadband soundabsorption for frequencies below 2500 Hz.

The plan for the remaining paper is as follows: Starting with the theoretical preliminaries, we formulate and determine the velocity fields for a pore of the arbitrary axial profile in Sec. II A. The impedance and absorption coefficient are determined in Sec. II B. The results of various numerical cases of the forward model are discussed in Sec. III. The inverse problem to identify the shape of the pore for maximum absorption is detailed in Sec. IV. We end with the conclusions in Sec. V.

## II. FORWARD PROBLEM: THEORY

### A. Velocity profiles in arbitrarily shaped pores

The schematic of a typical MPP is shown in Fig. 1(a). The pore diameter is $d$, the thickness of the panel is $\tau $, and the spacing between the panel and wall is $D$. These geometric parameters ${d,\tau ,D}$ represented in the figure can be tuned to achieve the desired bandwidth of absorption. As the MPP is a parallel arrangement of a large number of small pores, the following assumptions are made in line with the literature: (a) The distance between the pores is greater than their diameter so as to neglect the effect of a pore on another and (b) the distance between pores must be small compared to the wavelength of sound, to neglect the reflection of sound by the solid part of the panel.

^{17–19}

*Ansatz*$ u h= \u2211 j = 0 \u221e a j r j + n$ and equating coefficients of the polynomial to zero gives $ a 1=0$ and

^{th}order. The particular solution $ u p$ is obtained by setting all the partial derivatives in Eq. (5) to zero, i.e.,

### B. Impedance and absorption coefficient

## III. FORWARD PROBLEM: RESULTS AND DISCUSSION

We first validated the theory developed for MPPs of arbitrarily shaped profiles by reconstructing the absorption coefficient spectrums for two cases from the literature: (a) MPPs comprising upright cylindrical pores of different thicknesses and pore diameters^{20} and (b) MPPs comprising tapered pores with 0.8 mm diameter on the exposed side and 0.5 mm diameter on the other side.^{20} In all the profiles of the pores presented, a constraint $0.1537<r<0.3074$ mm for the radius of the pore was enforced, based on the literature.^{48} All computations were implemented in MATLAB.^{49}

### A. Pores of conic sections

In exploring alternative geometries that may offer unique acoustic characteristics, we considered pore profiles of conic sections such as parabolas and ellipses along with circular ones. The profiles are shown in the inset of Fig. 2(a). The lower and upper limits of 0.15 and 0.3 mm radii for the profiles were set. The circular profile was constructed by pulling apart two identical semi-circular sections at a certain distance. The ellipse-based profiles are formed by carving out the middle portion of a full ellipse. Unlike circular profiles, joining the two sections of the ellipse-based profiles would not result in a complete ellipse as the middle portion of the full ellipse was carved out. Hence, we see a good overlap in the circular and elliptical pore profiles.

A comparison of sound absorption for MPPs with pores of different geometric shapes circle, parabola, ellipse, and hyperbola is shown in Fig. 2(a). We note that for a pore with fixed minimum and maximum radial distance, the value of sound absorption decreases for pores of ellipse, circle, parabola, and then hyperbola, especially in the low-frequency regime. We further considered MPPs comprising circular pores, but the pore separation is more in one case. The schematics for these pore profiles along with the corresponding absorption coefficients are shown in Fig. 2(b). For comparison, the spectrum of $\alpha $ of an MPP comprising cylindrical pores is also overlaid. It can be seen that the absorption is better for $ circle 1$ profile, then $ circle 2$ and the least for the cylinder. As cavities enhance resonance and mixing, the absorption is higher for the pore with $ circle 1$ profile than that for the pore with $ circle 2$ profile, despite the latter having a part of the profile lower than the cylinder.

### B. Divergent–convergent, convergent–divergent, and sinusoidal pores

The sound absorption coefficients of MPPs comprising two units of convergent-divergent (CD) and divergent-convergent (DC) pores are shown in Fig. 3(a). The results of an MPP with cylindrical pores of equivalent radius are also superposed for comparison. Even with an increase in the number of cycles, we find that the $\alpha $ spectrum of MPPs with cylindrical pores lies between that of MPPs comprising DC or CD pore profiles. Further comparing with Fig. 4, we find that $\alpha $ is independent of the number of repetitions of the profile. The conclusions differ slightly for the MPPs comprising pores of CD- and DC-styled sinusoidal profiles, shown in Fig. 3(b). We find that neither the phase of the profile nor the number of repetitions of the CD- or DC-styled sinusoidal profiles altered the spectrum of the absorption coefficient. The difference between $\alpha $’s of sinusoidal CD and DC pores from that of the corresponding triangular pores can be attributed to the smooth transition between the converging and diverging parts of the pores in the former.

Sound absorption coefficients of MPPs with DC of cycles $n=1$ with either a fixed minimum or maximum radius are shown in Figs. 4(a) and 4(b), respectively. The same for $n=1$ cycles MPPs of CD pores are shown in Figs. 4(c) and 4(d). We note that for DC pores with fixed minimum radius, shown in Fig. 4(a), an increase in maximum radius led to an increase in $\alpha $ and an increase in the peak-frequency of $\alpha $. For DC pores with a fixed maximum radius, shown in Fig. 4(b), an increase in minimum radius led to a decrease in $\alpha $ as well as the frequency at which peak in $\alpha $ was observed. The increase in absorption can be attributed to the expansion of air due to the widening of the pore, leading to more relaxation times or equivalently more absorption. For CD pores with a fixed maximum radius, shown in Fig. 4(c), an increase in minimum radius led to increase in $\alpha $ and an increase in the peak-frequency of $\alpha $. For CD pores with fixed minimum radius, shown in Fig. 4(d), an increase in maximum radius led to a decrease in $\alpha $ and decrease in the frequency at which peak in $\alpha $ was observed. As the mean radius increases for CD pores, the pore becomes more inclined, and hence a part of the incoming wave gets reflected and goes out, thereby decreasing the absorption characteristics. The values of $\alpha $ increase with the mean radius as the pore becomes widened and incoming waves have a larger part to enter. From these figures shown in Fig. 4, we see that the absorption characteristics of MPPs with cylindrical pores lie midway in between that of MPPs comprising DC or CD pore profiles. Furthermore, we conclude that an increase in the curvature of the pore either (concave or convex) enhances sound absorption.

### C. Kilroy and chirp profiles

Kilroy was a meme that became popular during World War II. We designed four types of curves based on the Kilroy profile. The first curve was designed using the equation $y=log\u2061(|sin( k 1x)/ k 2x|),$ where $ k 1$ and $ k 2$ are constants. Considering $ k 1= 10 4$ and $ k 2= 10 3$, curve 1 was obtained and further scaled to fit the radius of the pore in the range of 0.15–0.3 mm. By flipping curve 1, curve 2 was obtained. By appending curve 1 with curve 2 and fitting into the same range, curve 3 was obtained. By appending curve 2 with curve 1, curve 4 was obtained. The schematics for these four variations of Kilroy-curve based pores are shown in Fig. 5(a) and Fig. 5(b) along with their sound absorption spectrums of the corresponding MPPs. As the pressure wave propagates within the pore of curve 1 or 3, shown in Fig. 5(a) or (b), despite a slightly converging entry, the increasing number of cavities helps in having larger values of $\alpha $ than that of the profile with curve 2 or 4, respectively. For all four variants of the Kilroy profiles, we find that the broad spectrum of spatial frequencies has induced a broad fluctuation in the spectrums of $\alpha $. The spatial fluctuations being more for the profiles of curves 3 and 4 as compared to that from pores made of curves 1 and 2, the same was also reflected in the corresponding spectrums of $\alpha $. The spatial frequencies of curves 1 and 2 are identical and hence, the corresponding frequencies at which fluctuations occur in the $\alpha $s of these curves are also identical. The same conclusion holds for the cases discussed in Fig. 5(b). The spatial frequency of the sinusoidal pores did not induce any fluctuations in the spectrum of $\alpha $ due to the symmetry of the profile. Furthermore, we found that the spatial fluctuation in the pore profiles also locally altered the frequency at which a maximum exists in $\alpha $.

A cosine-based linear chirp profile was generated using the function $y= chirp(t, f 0, t 1, f 1)$ in MATLAB. The instantaneous frequency at time $t=0$ is $ f 0$ and the instantaneous frequency at time $ t 1$ is $ f 1$. We set $ f 0=0$, $ t 1=0.005$, and $ f 1=1000$ and obtained four different variations of the chirp curve, similar to that of the Kilroy pore. The schematics are shown in Figs. 5(c) and 5(d) along with their sound absorption spectrums. The results and reasons remain the same as that of the Kilroy profiles. Despite the converging profile (chirp curve $ 1$), shown in Fig. 5(c), the $\alpha $ of chirp curve $ 1$ is marginally higher than that of chirp curve $ 2$ due to the increase in the number of cavities. The same reason can be attributed to the $\alpha $ of chirp curve $ 3$ as compared to that of chirp curve $ 4$, Fig. 5(d). For all the spectrums shown in Figs. 5(c) and 5(d), as the fluctuations are more closely spaced along the length, very few fluctuations are seen in $\alpha $ of chirp curve $ 1$ and chirp curve $ 3$ as compared to that of chirp curve $ 2$ and chirp curve $ 4$.

## IV. INVERSE PROBLEM

### A. Formulation and numerics

### B. Results

The obtained optimized pore profile (in red) constructed from $U$ is shown in Fig. 6(a). Further, the reverse of the optimized profile (in black) along with the minimum and maximum radius are also compared. Since the symmetry of the pore was not imposed along $ x ^$, we intend to compare the corresponding the spectrum of $\alpha $ for the optimized profile along with that of the reversed optimum profile. The reverse of the optimized profile (in black) along with the minimum and maximum radius are also compared. The corresponding sound absorption coefficients for these four pore profiles are shown in Fig. 6(b). It is clear that the obtained spectrum has maximum absorption not just around the a local peak, but all through the spectrum. The spectrum of the reversed profile has a spectrum intermediate to that of minimum and maximum profiles. This observation is in line with our reasoning from forward models, that a diverging entry to the pore leads to more absorption.

While the initial calculations indeed indicated the effectiveness of circular and elliptical cross sections in achieving higher absorption coefficients, it is important to note that the inversion process considers the optimization of the entire pore profile, rather than individual sections in isolation. The inclusion of a linear variation in the profile likely serves to introduce additional acoustic impedance variation, thereby enhancing sound absorption properties within the desired frequency range. Furthermore, we note that the GA algorithm has not led to an arc-shaped divergent section for the following reasons: (a) the GA in pursuit of the global optimum may prioritize certain configurations that offer a more balanced trade-off between complexity and performance and (b) the optimization process may have converged toward a solution that exhibits greater efficacy in achieving the desired acoustic objective of maximum overall absorption coefficient across a broad spectral range.

## V. CONCLUSIONS

The main advancement of this work is the establishment of analytical expressions for absorption coefficient $\alpha $ of microperforated panels (MPPs) comprising cylindrical pores of arbitrary axial profiles. After developing the generalized theory, we identified $\alpha $ spectrum of MPPs comprising pores of conic sections, divergent and convergent pores, sinusoidal pores, Kilroy pores, and chirp pores. Given the progress of additive manufacturing techniques, the second advancement from this work is on identifying the optimized axial profile of the pore for maximum sound absorption. The novelty of the work is in developing the theory from first principles. The immediate extension to this work is to determine the absorption coefficient of MPPs comprising pores of arbitrary non-symmetric three-dimensional pores. Such analysis is of use to understand MPPs comprising slanted and zig-zag-shaped cylindrical pores. The forward model could also be used to design MPPs with a desired peak in the spectrum of $\alpha $.

## ACKNOWLEDGMENTS

This work was performed with partial funds through funds from Initiation Grant (No. IITK/ME/2017445) from the Indian Institute of Technology Kanpur and SERB Early Career Research Award (No. ECR/2018/02341) to C.C.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**O. V. Vigneswar:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (supporting); Software (equal); Validation (lead); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). **C. Chandraprakash:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Resources (lead); Software (equal); Supervision (lead); Validation (supporting); Visualization (supporting); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

### APPENDIX: END CORRECTION

^{17–19}

^{,}

- $ k r 0<1$$ z pore= 4 3i\omega \rho 0\tau + 32 \mu \tau d 2.$
- $ k r 0>10$$ z pore=i\omega \rho 0\tau + 4 \mu \tau d \omega \rho 0 2 \mu (1+i).$
- $1< k r 0<10$ (in this work)$ z pore= 32 \mu \tau d 2 ( 1 + k r 0 2 32 ) 1 / 2+i\omega \rho 0\tau ( 1 + ( 9 + k r 0 2 2 ) \u2212 1 / 2 ).$

^{19,50}The apparent increase in panel thickness is caused by the radiation of a cylindrical mass of air via the opening. This increase in thickness is accounted by using a correction term in the imaginary part of impedance.

^{50}Taking into account these end corrections, the final expression for impedance is

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*Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials*

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*2019 Third International Conference on Inventive Systems and Control (ICISC)*

*Theory of Vibrating Systems and Sound*