Effect of Hilbert Fractal Acoustic Metamaterials on Ventilation Noise Control

The growth of the human population and the continuous expansion of urban areas have contributed significantly to the increasing levels of environmental noise worldwide [1]. This has led to the emergence of noise pollution as a critical urban sustainability issue that substantially impacts health and quality of life [2,3]. Ventilated windows are vital in maintaining optimal thermal comfort, air quality, and natural lighting in domestic buildings. Besides, they are essential to facilitate heat exchange in heavy machinery in industrial environments. However, these ventilated windows are the potential gateways for the noise to enter, which is a potential drawback. Thus, a well-ventilated system with higher noise control is always challenging as there is a trade-off between ventilation and noise cancellation. Therefore, the development of these well-ventilated systems with proper noise control is utmost needed.

In urban buildings, various traditional methods have been suggested in the literature to address this challenge. These methods include different types of facades, protrusions, plenum windows, active noise control windows, balconies, arrays of quarter-wavelength resonators, and Helmholtz resonators [4–10]. Additionally, acoustic enclosures, barriers, and curtains are utilized to mitigate machinery noise while allowing ventilation [11]. These techniques are effective for noise control, but they can be bulky for low-frequency noise control, as dictated by the mass frequency law. Furthermore, their acoustic performance is significantly influenced by the incident angle.

Acoustic metamaterial has significantly influenced low-frequency noise control applications in recent years. Acoustic metamaterial is an artificial structure exhibiting unique properties, such as negative effective bulk modulus and negative effective density. Liu et al. [12] experimentally proposed the first metamaterial, which unit cell consists of a lead ball covered with silicone rubber. The arrangement of these unit cells in a 3D lattice structure provides the spectral gaps on two orders smaller than the corresponding wavelength of the structure. Further continuation of research leads to broadening the application of acoustic metamaterials such as low-frequency sound absorption [13,14], noise cancellation [15], acoustic superscatterers [16,17], acoustic holography [18], acoustic black holes [19], and acoustic cloaking [20,21].

In recent years, the sonic crystal-based plenum window performance has been investigated by Lee et al. [22]. They found it is capable of traffic noise attenuation of 4.2 dBA at 1000 Hz. Similar to the sonic crystal arrangement, Shen et al. [23] experimentally demonstrated the flow noise control using a circular arrangement of a shunted Helmholtz resonator. This study helps to achieve omnidirectional acoustic shielding without disturbing the airflow. Melnikov et al. [24] proposed the arrangement of C-shaped meta-atoms for machinery noise control. The C-shaped local resonance structure is equivalent to the Helmholtz resonator, allowing the airflow between the meta-atoms and attenuating the noise simultaneously.

The literature discussed above regarding local resonance-based metamaterials plays an essential role in noise attenuation. This type of metamaterials is primarily based on the Helmholtz or quarter-wavelength resonator. The usual bottle-type Helmholtz resonator does not allow airflow. To address this, Kim and Lee [25] proposed the diffraction-type resonator. This design features an airflow passage, a diameter lesser than the wavelength of the acoustic wave to attain strong diffraction. Moreover, they followed another noise control condition by achieving a negative effective bulk modulus. Wang et al. [10] effectively demonstrated the combined effect of utilizing the quarter-wavelength resonator with absorption material in ventilated windows. Due to this combined effect, they achieved a transmission loss of 10–22 dB in the 500–4000 Hz. Similarly, Yu et al. [26] conducted a theoretical and numerical study of the ventilated duct with a periodic array of subwavelength resonators. The results show a 30 dB noise reduction from 600 to 1600 Hz. Kumar et al. [27] achieved low to mid-frequency noise attenuation by applying Helmholtz resonator behavior to the ventilated metamaterial. Wang et al. [28] investigated the perforated and constrained metamaterial for low-frequency (<500 Hz) noise control. Recently, Xiao et al. [29] studied the effect of ventilated openings on sound transmission loss and found that changing the inclination causes broadband transmission loss.

In the sonic crystal arrangement of the local resonators, a major constraint is their large size requirement for low-frequency noise control. Space-coiling acoustic metamaterials have evolved to overcome this disadvantage of the local resonance phenomenon in sonic crystal arrangement. This structure helps slow down the propagation speed within the structure itself. Due to this advantage, research on space-coiling acoustic metamaterials has grown significantly. Additionally, self-similar fractal structures are proposed over arbitrary space-coiling structures for better sound attenuation [30,31]. Hilbert curve is one of the best examples of a self-similar fractal structure. The use of self-similar fractal structures and Hilbert curves have been applied in various areas of sound manipulation, such as wave filtering [32], bandgap formation [33–35], acoustic lens [36,37], sound cloaking, tunneling [38,39] and sound absorption [40]. To improve ventilation with noise attenuation, the study of space-coiling ventilated acoustic metamaterials has been undertaken by providing additional ventilation space. However, there is limited literature on this topic, and existing studies do not compare with local resonance or diffraction-based ventilated metamaterial [41–43]. Additionally, a detailed summary of recent acoustic ventilated metamaterials is shown in Table 1. Zhao et al. [44], Man et al. [38], and Comandini et al. [40] focused on different applications of Hilbert fractal structures such as transformer noise control, noise control, quarter bending, sound cloaking, sound tunneling, and sound absorption. Still, there is a lack of literature on the theoretical and real-time experimental application of Hilbert fractal structures in ventilated metamaterials, which this study needs to address.

However, these studies primarily focused on 2D lattice configurations and did not address low-frequency broadband noise reduction in ventilated systems. In contrast, this research highlights the use of a 1D periodic duct arrangement for reducing noise in ventilated systems, particularly focusing on the low-frequency range, specifically below 1600 Hz. This frequency range is crucial for controlling outdoor urban noise in buildings, especially traffic noise. Studies [45–48] have demonstrated that traffic noise, a significant contributor to urban sound pollution, has peak sound pressure levels and high spectral energy content within this range.

The present study addresses these gaps by investigating the sound attenuation performance of an innovative Hilbert curve–based acoustic ventilated metamaterials. The evaluation encompasses a comprehensive approach using the transfer matrix method, rigorous computational analysis, and experimental validation with an impedance tube. The impact of various geometric parameters on sound transmission loss is explored by analyzing first-, second-, and third-order Hilbert curves, as well as a base model. Furthermore, the finite element method is employed to construct band diagrams and eigenmodes, elucidating the fundamental mechanisms driving sound transmission loss. Effective properties are examined through phase plots to enhance understanding. The study is grounded in controlled impedance tube experiments and realistic assessments using twin reverberation chamber tests. This multifaceted approach advances the theoretical understanding and paves the way for practical applications of ventilated metamaterials in sound attenuation.

This section provides a detailed approach for developing and analyzing Hilbert curve ventilation metamaterials. It starts by explaining the design principles and construction process of the metamaterial structure. Then, it describes the methods used to assess its acoustic performance, including the transfer matrix method, experimental setup and procedures, and computational studies. Finally, the section concludes with an analysis of the effective properties, focusing on its bulk modulus and density to offer insight into its acoustic behavior.

The base case geometry features a simple expansion chamber arrangement, which aligns with prior literature [25,26,29,51], highlighting the importance of a simple expansion chamber design in acoustic applications. This study introduces the Hilbert structure in the base case cavity space. The Hilbert curve ventilated metamaterials are designed by dividing the base case cavity into four sections. Further, each section is divided into Here, N = 1, 2, and 3 for first, second, and third orders. The center of $2N×2N$ section is connected from lower left to lower right clockwise with the Hilbert channel width K, as shown in Fig. 1(a). Additionally, the sound wave propagation length in the 1/4th of the Hilbert structure space is $Lp=L(2N−1−1/2N+1)$⁠, N = 1, 2, and 3. Here, L is the length of the unit cell.

Figure 1(b) illustrates the 3D unit cell of the first-order (O1), second-order (O2), and third-order (O3) Hilbert curve ventilated metamaterial and base case. In the Hilbert configurations, the blue area depicted in Figs. 1(a) and 1(b) represents the air domain through which acoustic waves propagate, with the Hilbert channel width K ranging from 3 to 6 mm. The overall length of this structure is denoted as L and measures 70 mm. Additionally, H is the total thickness of the unit cell varying by 24, 26, and 28 mm, and $l×l$ defines the spatial dimension for airflow entry varying between 15 × 15, 20 × 20, and 25 × 25 mm². For the O3 configuration, the K value is fixed at 3 mm due to geometry constraints and no involvement of K in the base case. Therefore, the total combinations of O1, O2, O3, and base case are 270 (108 + 108 + 27 + 27). To simplify, Table 2 shows a comprehensive overview of the various parameters utilized in this study, based on which the acoustic characteristics of the Hilbert structure are studied. Throughout the study, each sample is named by using the parameter values; for example, the sample O2_K3_l20_H24_N2 represents the second-order Hilbert curve, 3-mm Hilbert channel width, 20-mm ventilated space opening, 24 mm is the thickness of the unit cell, and 2 number of the unit cells, respectively.

The proposed transfer matrix model helps to analyze the acoustic characteristics of the Hilbert fractal structure without considering the thermoviscous effect. This approach will be used in Sec. 3.1 to predict the acoustic performance of the base and Hilbert curve ventilated metamaterials. For a more robust analysis, conducting impedance tube experiments and including the thermoviscous effect in the computational model would be beneficial. These will be studied in the following sections.

This section provides a detailed explanation of the methodologies and equipment utilized in the acoustic experiments. It begins with an overview of the sample preparation process, including the materials used and their specifications. Subsequently, it describes the impedance tube setup and the method for measuring sound transmission loss. Finally, the section concludes with an outline of the reverberation test procedure, covering the test chamber setup, measurement equipment, and the process for conducting the experiments.

To realize the experimental setup, the sample is fabricated using Fused Deposition Modeling (FDM) 3D printing technology. The material chosen for this construction is poly-lactic acid (PLA), with a density of $1250kg/m3$⁠, a Poisson's ratio of 0.35, and Young's modulus of 3.5 GPa. The 3D-printed samples, as displayed in Figs. 2(a) and 2(b), were manufactured with a diameter of 100 mm and included a face cover. A unit cell size of 70 mm is embedded in the 100-mm diameter structure to support the sample, facilitating attachment to an impedance tube. The face cover is carefully attached to the Hilbert structure model using cyanoacrylate glue to ensure airtightness and prevent potential leakage. The 3D printing process utilized the Flash Forge Creator 3 Pro 3D printing machine, noted for its printing precision of ±0.2 mm, a layer height of 0.1 mm, and a print speed of 90 mm/s. These specific printing parameters are crucial in achieving the desired metamaterial structure, ensuring precision and accuracy throughout the manufacturing process.

The acoustics performance of the Hilbert curve–based metamaterials is evaluated through experimental tests conducted in the BSWA SW 422 impedance tube system at the Acoustics and Dynamics Lab in IIT Madras. The test is performed based on the transfer matrix method as per ASTM E1050-08 and ASTM E2611-09 Standards. The experiments are performed in the frequency range of 250–1600 Hz. As shown in Fig. 2(c), this experimental test setup incorporated additional hardware components such as a 50 W power amplifier, four 1/4″ microphones, a four-channel data acquisition card—MC3242, and VA-LAB 4 software—for data processing.

A calibration process is performed before commencing the experiments. The four microphones are calibrated using a pistonphone calibrator to ensure accurate measurements. The sample is meticulously secured with Teflon tape along its external diameter to avoid sound leakage and provide a tight fit with an impedance tube. Each sample is tested five times to ensure repeatability and consistency, and the average of five measurements is used to calculate the final results. The impedance tube test results will be used to validate the computational study that incorporates thermoviscous effects, as presented in Sec. 3.1.

where $L1andL2$ is the average sound pressure level in the source and receiver room, S is the area of the test sample partition $(m2)$⁠, $A=0.612V/T$⁠, V is the volume of the sound-receiving reverberation chamber $(m3)$⁠, and T is the reverberation time $(s)$ of the receiver chamber.

The reverberation chamber test method described here will be used to evaluate the acoustic performance of Hilbert-based structures in a more realistic, multi-directional sound field. This approach complements the plane wave conditions of the impedance tube experiments by simulating reflection-rich environments. The results from these tests, presented in a subsequent section, will provide insights into the materials' behavior under diffuse sound field conditions, offering a clearer understanding of their acoustic characteristics in practical applications.

Determining the acoustic properties of a heterogeneous medium is a challenging task. To address this issue, the effective medium theory can approximate a medium as homogeneous and isotropic [59]. Furthermore, negative effective parameters play a crucial role in acoustic metamaterial and can help to explain negative refraction, acoustic cloaking, and acoustic lensing. The retrieval method is commonly used for complex metamaterial structures to determine the effective parameters using complex reflection and transmission coefficients. This method involves finding effective parameters using complex transmission and reflection coefficients. The effective impedance $ξ$ and refractive index (n) are obtained using Eqs. (12) and (13) [51,59].

The effective bulk modulus and density are calculated using computational results of reflection and transmission coefficients. The results from this method are presented in Sec. 3.6, demonstrating how these effective properties contribute to the overall acoustic performance of the structure.

This section discusses the performance of the Hilbert curve acoustic metamaterials and a base case using analytical, experimental, and computational methods. It also explores the impact of thermoviscous properties. The study delves into parametric studies to optimize the structure for broadband transmission loss. Additionally, it discusses the results of band diagrams and effective properties to understand the reasons for sound transmission loss.

Figure 5 shows the transmission loss results from the transfer matrix method (theory) and simulation (lossless) results. Three different Hilbert orders and base cases are analyzed. The geometry features are a 3 mm Hilbert channel width (K3), 15 × 15 mm² ventilated space (l15), a structure thickness of 24 mm (H24), and a single unit cell (N1). Among the results, the Hilbert-based structure performs better than the base case due to the longer effective path length leading to the higher interaction time between the acoustic waves and the structure, which enhances the sound attenuation performance in the Hilbert structure. Notably, the second-order configuration provides broadband sound transmission loss between 700 Hz and 1600 Hz. Additionally, the results show that the resonance frequencies obtained by the transfer matrix method are in acceptable agreement with the lossless computational result.

However, in real situations, thermoviscous effects play a crucial role in propagating acoustic waves in complex and narrow geometries. This effect involves the dissipation of sound energy due to thermal conduction and the viscous nature of the medium through which the acoustic waves propagate. In Fig. 6, the transmission loss of the COMSOL simulation result is compared with the impedance tube experiment. Six different samples are used for validation. The computational model is well adapted to the experimental data and agrees well with the resonance frequency of the structures.

All Hilbert configurations followed the distinct asymmetric Fano-like resonance [60,61], except for the base case. Fano-like resonance is an asymmetric profile formed by interference between the sound wave passing through the resonating and non-resonating channels. The asymmetric profile has the peak portion formed due to destructive interference. In the first-order geometry (O1), there is a maximum transmission loss of 40 and 50 dB for 3 (K3) and 6 (K6) mm Hilbert channel width, respectively. This comparison shows that the maximum Hilbert channel width (K) results in the maximum transmission loss, the physical insights of which will be discussed in Sec. 3.3. Among the tests, the O2_K3_l15_H24_N1 and O2_K6_l15_H24_N1 configurations performed in the broad range of transmission loss, which is almost one-octave band, i.e., 695–1600 Hz and 715–1600 Hz, respectively. The computational analysis of the third-order (O3) configuration indicates the presence of two distinct Fano-like resonances, one at around 400 Hz and the other at around 1100 Hz. However, the experimental observations slightly mismatch the computational study. The reason for this is the lower wall thickness of the O3 geometry, which creates unwanted sound leakage. Overall, the computational study agrees well with the experiments. Additionally, widening the (K) from 3 to 6 mm also increases peak transmission loss observed for O1 and O2 configurations.

While comparing the sound transmission loss with and without consideration of thermoviscous loss (Figs. 5, 6, and Fig. 15 in the  Appendix), it is found that considering thermoviscous loss gives a more realistic result. The lossless model predicts resonant behavior but overestimates sound transmission loss by neglecting the viscous damping effect. For example, the peak transmission loss for model O1_k3_l15_H24_N1 considering thermoviscous effects is 38.6 dB, whereas the lossless case is 53.5 dB. For model O2_k3_l15_H24_N1, the transmission loss is 28 dB when considering losses, compared to 43.2 dB in the lossless case. Similarly, for model O3_k3_l15_H24_N1, the lossless case overestimates the peak transmission loss by 15 dB at both the resonance frequencies. On the other hand, considering the thermoviscous impact in the simulation aligns well with the experimental results. However, a curved narrow channel is not involved in the base case, so the thermoviscous effect is not dominant. These results emphasize the necessity of a thermoviscous model to design the fractal narrow structure metamaterial. The effect of various parameters as mentioned in Table 2 is discussed in the following sections.

Figure 7 shows the transmission loss performance of O1, O2, O3, and base configuration for three differently sized ventilated openings $(l×l=15×15,20×20and25×25mm2)$⁠. In this analysis, other parameters are fixed, including a Hilbert channel width of 3 mm, a thickness of 24 mm, and a single unit cell. The results are derived from COMSOL transmission loss simulations using the thermoviscous model. Among the models, the O2_K3_l15_H24_N1 model provides the best average transmission loss of 22 dB in the frequency range of 695–1600 Hz. However, for O2 configuration, when the ventilated space is increased to $25×25mm2$⁠, the average transmission loss reduces to 15 dB, and the resonant frequency shifts to 800 Hz. This trend is observed in other configurations as well. It is clear that increasing the ventilated space results in a resonance frequency shift to the higher range. Besides, the maximum transmission loss decreases due to the reduction in the reflection of the acoustic wave. Despite using the Hilbert channel, there is still a trade-off between airflow and control of sound wave transmission. However, based on observations, the Hilbert-based configuration is better equipped to control sound wave propagation over a broader frequency range than the base case.

Figure 8(a) illustrates the effect of varying Hilbert channel width (K) for O2 configuration. The K value varies from 3 to 6 mm with fixed parameters of l15, H24, and N1. The results are obtained through COMSOL transmission loss simulation with the thermoviscous model. When the K value is increased in this configuration, there is an increase in transmission loss along with a frequency shift. The same trend is observed for the O1 configuration (see Fig. 16 in the  Appendix). To enhance the observed result, the inset plot in Fig. 8(a) illustrates the relationship between the real part of the impedance and the variation in K. It shows that K = 6 mm results in a higher impedance value than K = 3 mm, indicating greater energy dissipation through viscous losses in the fluid, leading to higher transmission loss. Additionally, the resonance frequency plot inset in Fig. 8(a) shows an increase in resonance frequency with respect to K, resulting in the shift of transmission loss to a higher frequency range. In summary, the increase in frequency and transmission loss with increasing K is mainly due to changes in resonance frequencies and impedance characteristics of the structure, respectively.

Figure 8(b) shows the variation of the unit-cell thickness (H) of the O2 configuration with fixed parameters of K3, l15, and N1. It is observed that the resonance frequency does not change significantly with varying H values. However, the transmission loss has an average variation of 2 dB with low to high H value, and a similar trend is observed for O1 and O3 configurations (Fig. 17 in the  Appendix).

This section demonstrates the effect of the number of unit cells in the periodic arrangement (in x-direction as inset in Fig. 9) on the transmission loss capabilities. The analysis considers the samples of O1, O2, O3, and base cases, with varying unit cell numbers ranging from 1 to 3 by using thermoviscous module in COMSOL simulation. Figure 9 reveals no significant difference in the peak frequency when increasing the number of unit cells. However, there is a slight decrease in the broadband range due to an unsmooth transition of impedance mismatch. Additionally, the results show that as the number of unit cells increases, the transmission loss spectrum tends to align more closely with the bandgap in the band diagram (Fig. 18 in the  Appendix). This trend is consistent with the work of Xiao et al. [29], which shows that increasing the number of unit cells leads to alignment with the bandgap in the band diagram.

The band diagram is studied in the following section to comprehend the broadband range of sound transmission loss. Moreover, the amplitude of sound transmission loss increases due to enhanced wave reflection within the periodic structure. The results from various geometric configurations suggest that the parameters K and l are more important than H. Therefore, for further analysis, H and l are set at their lower limit of 24 mm and 15 mm, respectively, while K at 3 and 6 mm.

Figure 10, known as a dispersion curve or band diagram, depicts the relationship between the normalized wavenumber and frequency. This study defines the Brillouin zone [62] $−π/Htoπ/H$ from the unit cell. The bandgaps are in band diagram highlights in cyan color. It is observed that increasing the order of the Hilbert structure results in the bandgap shift to a lower frequency. The flat bands observed in O1, O2, and O3 are responsible for the fundamental resonance of their structures. Additionally, increasing the Hilbert channel width (K) shifted the fundamental frequency to a slightly higher range. For instance, in the second-order configuration (O2), increasing K shifted the fundamental frequency from 695 to 732 Hz. Moreover, the transmission loss spectra in Fig. 6 align well with the corresponding bandgap.

As the tortuosity increases, the fundamental frequency shifts toward the lower frequency, leading to increased bands. In the base case, no flat band is observed due to the absence of a fractal structure. The presence of a flat band is closely connected to the asymmetrical Fano profile in the transmission loss spectra. Fano-like resonance is an asymmetrical profile caused by the interaction of a sound wave moving through resonating and non-resonating channels. The Hilbert structure introduces localized discrete forms of resonant modes at a specific frequency (resonant channel). This resonant mode interacts with the continuous propagating wave mode in the periodic structure (non-resonant channel), leading to Fano resonance. This phenomenon means that the horizontal line represents the resonant frequency, showing that these resonant modes are localized and prevent the acoustic wave from propagating through the structure.

Furthermore, in Fig. 10, the eigenfield or pressure distribution of the fundamental resonance frequency is also included. The bandgap exhibits dipolar characteristics attributed to Fano resonance in all cases except the base case. Besides, the third-order (O3) geometry has an additional bandgap in the higher frequency range due to the increased wave propagation length and belongs to higher order modes (1118 Hz). Among all the configurations, the O2 exhibits a wide bandgap range primarily for higher Hilbert channel width (K = 6 mm), ranging from approximately 700 to 1400 Hz.

Figure 11 illustrates the relationship between bandgap width and the variation of K, H, and l values, as determined from the band diagram using the eigenfrequency module in COMSOL [63]. In Fig. 11(a), a clear trend shows for constant H and l; the band gap width expands with an increase in K, such as in the O2 configuration, the band gap width increases from 525 Hz (695–1220 Hz) at K = 3 mm to 668 Hz (732–1400 Hz) at K = 6 mm. Notably, the base case does not involve K, and there is no variation of K for O3 due to geometry constraints. Figure 11(b) reveals that for the Hilbert configuration, there is no significant variation in band gap width as H increases. However, the base case band gap widens from 296 Hz (1304–1600 Hz) to 434 Hz (1166–1600 Hz) with increasing H. This band gap variation aligns with the transmission loss variation observed in Figs. 8(a) and 8(b). In contrast, Fig. 11(c) demonstrates a decreasing trend in band gap width as l increases, consistent with the results presented in Fig. 7. Overall, these findings suggest that the K value has the most pronounced positive impact on band gap width, while l exhibits an inverse trend. The H value primarily influences the base case configuration, with minimal effect on the Hilbert structure.

The results of the effective properties obtained through the retrieval method (Sec. 2.5) are displayed in Fig. 13. It is observed that only the effective bulk modulus has a negative value for all configurations, which is highlighted in cyan color. The negative effective bulk modulus is due to the inward and outward movement of the structure's diagonal (insert in Fig. 13), resulting in the external force reversing the rate of volumetric change. These negative effective properties are responsible for bending the sound wave and leading to high acoustic wave reflection, ultimately causing transmission loss.

The effective bulk modulus for O1_K3_l15_H24_N1 and O1_K6_l15_H24_N1 is negative and ranges from 1385 to 1600 Hz and 1411 to 1600 Hz, respectively. In the negative bulk modulus regime, the transmission loss and bandgap are well-aligned due to the decrease in acoustic wave propagation velocity. A similar trend is observed for O2_K3_l15_H24_N1 and O2_K6_l15_H24_N1 configurations with a broader range of transmission loss. For O3, the negative bulk modulus occurs within two specific frequency ranges (372–950 Hz and 1118–1600 Hz), as observed in the bandgap and sound transmission loss study. Moreover, the base structure has the most negligible impact on negative bulk modulus among the different configurations. Evidently, for all Hilbert configurations, the resonance modes are dominated mainly by dipole resonance and, for the base case, by monopole resonance.

In the impedance tube experiment, the O2_K6_l15_H24_N1 configuration exhibited broadband transmission loss effectively. It is worth noting that the impedance tube experiment operates under plane wave conditions only, whereas real-life situations present a diffuse field environment. Consequently, the O2_K6_l15_H24_N1 and base configuration are chosen for the reverberation chamber test to achieve accurate real-time experimental results. Figure 14 compares the transmission loss of O2_K6_l15_H24_N1 with the base configuration. This O2 configuration demonstrates a significant increase in transmission loss across frequencies, ranging from approximately 700–1400 Hz. It consistently outperforms the base case with an average transmission loss of 5 dB. These results confirm the expected formation of a bandgap, which is further supported by impedance tube measurements. This improvement in sound attenuation with the O2 configuration emphasizes the use of Hilbert curve–based ventilated metamaterials to reduce sound propagation, which has implications for various noise control and acoustic engineering applications.

This study evaluated the effectiveness of using the Hilbert curved acoustic metamaterials over the simple expansion chamber for sound attenuation. The transfer matrix method, computational analysis, and experimental results are used to study the transmission loss for Hilbert and base configurations. The study delved into the impact of Hilbert orders and geometry parameters, noting that increasing the Hilbert order reduces the fundamental frequency and affects broadband transmission loss spectra due to resonance frequencies and impedance characteristics of the structure, respectively. Besides, increasing the K value raised the fundamental frequency of the Hilbert configuration. The results showed that the Hilbert-based configuration had a Fano resonance in the transmission loss spectra. Specifically, the Hilbert second-order 3-mm channel width, 15-mm ventilation space, and 24-mm thickness (O2_K3_l15_H24_N1) configuration exhibited a broadband transmission loss of 22 dB within the frequency range of 695–1600 Hz, while the Hilbert second-order 6-mm channel width, 15-mm ventilation space, and 24-mm thickness (O2_K6_l15_H24_N1) configuration achieved a 24-dB transmission loss within the frequency range of 732–1600 Hz.

Furthermore, the study also investigated the band diagram of Hilbert and base configurations to find the relation between broadband transmission loss spectra and bandgaps. The broadband transmission loss spectra matched well with the bandgap in the band diagram. The results showed that increasing the number of orders led to an increase in the number of bands and the O2_K3_l15_H24_N1 and O2_K6_l15_H24_N1 configurations had wider bandgaps in the subwavelength thickness of $λ/20$⁠. Analysis of effective parameters shows that the negative bulk modulus is due to dipole resonance in the Hilbert configuration and monopole for the base.

Real-time testing in the twin reverberation chamber demonstrated that the Hilbert structure, particularly the O2_K6_l15_H24_N1 configuration, achieved a 5-dB transmission loss compared to the base case within the frequency range of approximately 700–1400 Hz. These results highlight the promising role of Hilbert curve–based ventilated metamaterials in mitigating sound propagation and have implications for various noise control and acoustic engineering applications.

The authors would like to thank Professor Chandramouli Padmanabhan, Dr. Jhalu Gorain, and Mr. K. Suresh for their kind help with the impedance tube and twin reverberation chamber facility. The authors are also thankful to the central skill training and fabrication facility at IIT Madras and Mr. R Vasantha Kumar for fabricating 3D printing samples.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Additional supporting data of the cases presented in the article are given in Figs. 15–18.