A conformal metasurface for broadband acoustic absorption below 800 Hz

Introduction

Acoustic metasurfaces use subwavelength resonant or scattering elements to shape the effective surface impedance seen by an incident sound field. They have been developed for reflection control, anomalous refraction, focusing, absorption, and compact sound routing [1,2]. For absorption, the central target is impedance matching: incident acoustic energy should enter the surface rather than reflect, and it should then be dissipated through viscous, thermal, membrane, or porous losses. Hybrid resonators, membrane-type acoustic metamaterials, and dark acoustic absorbers have shown that deeply subwavelength structures can absorb sound at frequencies where conventional porous treatments would need much larger thickness [3,4,5].

The low-frequency problem remains difficult because narrow resonances are easy and broad passive absorption is hard. Coiled channels, Helmholtz resonators, degenerate resonator sets, composite metasurfaces, nonlocal coupling, over-damped mode coupling, and membrane-covered coiled resonators have all been used to widen the useful band while retaining subwavelength thickness [6,7,8,9,10,11,12,13]. Microperforated and metaporous layers add controlled resistance, which helps match impedance but can also reduce peak absorption if the resistance is poorly distributed [14,15]. The viscothermal loss mechanism itself is sensitive to channel dimensions and boundary-layer corrections in narrow tubes [16].

A second problem is geometry. Many absorbers are characterised as flat, normally incident samples. Real low-frequency treatments are often installed on curved engine covers, compact HVAC elbows, vehicle trim, compressor housings, or equipment guards where the surface radius, seam leakage, and mounting compliance can detune the cells. Acoustic metasurfaces have been bent or shaped for wavefront control [21,22], but absorption on curved engineering panels has been less systematically reported. A conformal absorber should preserve broadband impedance matching after bending, not merely demonstrate one good flat coupon.

The present work develops and tests a passive conformal metasurface for absorption below 800 Hz. The novelty is not that each constituent mechanism is new. The design combines known ingredients - coiled Helmholtz cells, membrane loading, microperforated resistance, and multi-resonance optimisation - and asks whether the combination remains useful when mounted on curved substrates. We explicitly compare flat and curved measurements, report detuning after bending, and check the results against passive bandwidth expectations. The article therefore treats broadband absorption as an engineering balance rather than as a claim of beating fundamental limits [18,19,20].

Metasurface architecture

The absorber is a 58 mm thick layered panel. From the incident side inward, it contains a 0.8 mm microperforated aluminium sheet, a 4 mm plenum, an array of membrane-covered resonator mouths, folded air channels, and a rigid backing plate. The active array is divided into nine unit-cell families. Each family occupies 36 mm x 36 mm in plan view, with a coiled channel length between 185 and 612 mm, a neck radius between 2.0 and 4.8 mm, and a local backing-cavity height between 18 and 42 mm. The front sheet has 0.55 mm holes and 1.35% open area.

The cell families are not arranged periodically. A blue-noise tiling distributes the tuned resonances so that adjacent cells are usually detuned by at least 60 Hz. This suppresses large coherent spatial variations in reflected pressure and reduces the visual sensitivity of the design to one defective cell family. The layout was also constrained so that no folded channel crosses a strip seam. The final panel consists of six strips, each 120 mm wide, that can bend independently onto a cylindrical substrate.

Each resonator mouth is covered by a 50 micrometre PET membrane bonded around a 14 mm circular aperture. Membrane preload tunes the effective acoustic compliance and adds a weak dissipative channel. The design borrows the low-frequency leverage of membrane acoustic metamaterials [4] while avoiding the extreme narrowband response of a single membrane resonance. The coiled cavity behind the membrane provides additional inertance and delay, similar in spirit to spiral-tube and coiled Helmholtz absorbers [6,13].

The microperforated face sheet serves two functions. First, it adds broadband resistive damping before energy reaches the resonator mouths. Second, it protects the membranes from direct handling and creates a small shared plenum that couples neighbouring cells weakly. The resistance is intentionally moderate. A very resistive face sheet would absorb mid-frequency sound but starve the resonators; a very open face sheet would produce high narrow peaks and poor off-resonance performance. This trade follows the classic microperforated-panel insight that resistance and backing compliance must be co-designed [15].

Design optimisation

The design objective was to maximise average normal-incidence absorption between 250 and 800 Hz while penalising absorption notches, excessive panel mass, and sensitivity to curvature. The optimisation variables were channel length, neck radius, membrane preload, local cavity height, face-sheet porosity, and cell-family fractions. Fabrication constraints limited the minimum channel width to 2.2 mm, the minimum wall thickness to 1.4 mm, and the maximum strip thickness to 58 mm.

A transfer-matrix model provided the first design pass. Each cell was represented by a resistive face-sheet impedance, a membrane impedance, a neck inertance with viscothermal correction, and a backing compliance. The parallel surface impedance of the tiling was then averaged over the unit-cell population. This model cannot capture all near-field coupling, but it is fast enough to search thousands of candidate distributions. Thermoviscous corrections followed narrow-tube theory for the channel dimensions used here [16].

The second design pass used finite-element simulations on representative supercells. The model included thermoviscous boundary layers in the necks and effective losses in the microperforated sheet. Supercell simulations identified two failure modes that the lumped model underpredicted. The first was a high-Q resonance near 720 Hz in one long-channel family. The second was pressure localisation near strip seams when identical cells were placed on both sides of a seam. Both were corrected by detuning the affected families and adding narrow gasket bridges.

The final resonance distribution spans nominal cell frequencies from 235 to 790 Hz. The goal was not to place one peak at every frequency. Instead, the optimiser sought a slowly varying real part of surface impedance near the characteristic impedance of air and a reactive part crossing zero multiple times with small amplitude. This is the same basic impedance-matching idea behind perfect absorbers and degenerate-resonator metasurfaces, but implemented as a manufacturable curved strip rather than as a flat ideal sample [7,8].

Fabrication

Prototype strips were printed in glass-filled nylon by selective laser sintering, then sealed with a thin epoxy wash to reduce wall porosity. The folded channels were inspected by endoscopy and by pressure-decay testing. Any strip with leakage greater than 0.7% of its expected neck conductance was rejected. Membranes were cut by laser and bonded with a pressure-sensitive acrylic adhesive. Membrane preload was set by bonding over a shallow vacuum fixture and verified by measuring the first membrane-dominated resonance of three witness cells per strip.

The microperforated face sheet was CNC drilled rather than chemically etched because the prototype required small batches with variable seam holes. Hole diameter varied by +/- 18 micrometres across the sheet, and open area varied from 1.29% to 1.41%. These tolerances are important because resistive impedance scales strongly with aperture radius. The measured face-sheet impedance was included in the final model rather than assuming the nominal drilling diameter.

Conformal mounting used a 1.2 mm silicone gasket between the printed strip and the aluminium substrate. Three cylindrical substrates were tested, with radii 0.32, 0.42, and 0.80 m. The gasket allows bending without cracking the printed channels, but it also adds a small distributed compliance behind the strip. This compliance shifted the lowest cell families downward by 12-20 Hz. The effect was repeatable and was included in the curved-panel model.

Strip seams were sealed with a removable butyl tape during the main tests. Unsealed-seam measurements were also performed because seam leakage is a likely installation fault. The unsealed configuration reduced mid-band absorption by 0.06-0.11 and produced a shallow reflection notch around 515 Hz. This sensitivity is not catastrophic, but it shows why conformal acoustic treatments should report mounting details rather than only unit-cell geometry.

Flat-coupon measurements

Normal-incidence absorption was measured in a 100 mm square impedance tube using a two-microphone transfer-function method [17]. Because the full unit-cell tiling is larger than the tube, three representative coupons were fabricated: low-band rich, mid-band rich, and broadband mixed. Each coupon included a rigid backing and the same face-sheet construction as the full panel. Measurements were performed from 180 to 1000 Hz with 2 Hz resolution.

The broadband mixed coupon reached alpha greater than 0.70 from 286 to 792 Hz and alpha greater than 0.85 from 347 to 716 Hz. The peak absorption was 0.96 at 512 Hz. The low-band rich coupon performed better below 330 Hz but developed a 0.52 trough near 610 Hz. The mid-band rich coupon was smoother between 420 and 780 Hz but lost low-frequency absorption. These comparisons support the distributed-cell strategy: broadband response is produced by population balance, not by one magic unit cell.

A 58 mm melamine foam coupon had alpha below 0.25 at 300 Hz and reached 0.70 only above 820 Hz. This is expected from porous-absorber thickness scaling and is not a criticism of foam. The metasurface trades simple broadband high-frequency damping for tuned low-frequency response. A single-resonance Helmholtz liner of the same thickness reached alpha 0.92 at 391 Hz but fell below 0.40 outside a 78 Hz bandwidth. The metasurface sacrifices peak narrowband performance to maintain useful absorption across a wider band.

Model-measurement agreement was strongest between 320 and 720 Hz. Below 300 Hz, the measured absorption was 0.05-0.08 lower than predicted, probably because the longest coiled channels had slightly higher leakage than the pressure-decay test could resolve. Above 780 Hz, lateral coupling between cells became more important and the lumped model overpredicted absorption. The finite-element supercell model captured the high-frequency roll-off more accurately.

Curved-panel response

Curved-panel absorption was measured in a baffled semi-anechoic fixture. The sample formed a 640 mm x 720 mm section of a cylindrical wall backed by a rigid cavity. A loudspeaker array generated incident fields at nominal angles of 0, 30, and 55 degrees relative to the local surface normal. Reflected pressure was reconstructed from a 9 x 11 microphone grid and referenced to a rigid aluminium panel of the same curvature.

At radius 0.42 m, the sealed metasurface retained alpha greater than 0.65 from 312 to 764 Hz under normal incidence. Oblique incidence reduced the low-frequency response modestly: at 55 degrees, alpha greater than 0.60 was retained from 346 to 742 Hz. The dominant resonance families shifted upward by 18-36 Hz relative to flat coupons, consistent with slight channel compression and membrane preload change during bending. The shifts were larger at radius 0.32 m and smaller at radius 0.80 m.

Curvature also smoothed some narrow features. Adjacent unit cells experience slightly different effective incidence angles on the cylindrical surface, which reduces coherent reflection from one resonance family. This smoothing is beneficial in the mid-band, but it does not create new low-frequency absorption. At the most aggressive radius, 0.32 m, the lowest family weakened because the corresponding long channels were closest to the tensile side of the strip and experienced the largest seam stress.

The curved measurements support the conformal claim, but with limits. The design is robust over radii larger than roughly 0.4 m for this strip width and material. Below that radius, channel deformation and seam leakage become first-order design variables. A more flexible elastomeric channel body or narrower strips would be needed for smaller equipment housings.

Enclosure insertion loss

A practical enclosure test was performed using a 0.42 m radius aluminium half-cylinder mounted over a broadband loudspeaker and shaker source. The source spectrum was shaped to resemble a small compressor, with strong tonal components at 315, 410, and 630 Hz and broadband content up to 1 kHz. Three treatments were compared: the conformal metasurface, 58 mm melamine foam, and a thickness-matched single-resonance Helmholtz liner. External pressure was measured at 24 positions on a hemispherical array.

The metasurface improved spatially averaged insertion loss by 5.8 dB relative to the foam and by 4.1 dB relative to the single-resonance liner over 250-800 Hz. The advantage was largest at 315 and 410 Hz, where the foam was too thin and the single-resonance liner could not cover both tones. Above 900 Hz, the foam performed better, as expected. The metasurface is therefore not a replacement for porous treatment at all frequencies; it fills the low-frequency thickness gap.

Inside the enclosure, the metasurface reduced wall-pressure hot spots near the curved crown. Pressure maps showed weaker standing-wave contrast between 340 and 680 Hz, indicating that the treatment changed both local absorption and modal damping. The single-resonance liner reduced one mode strongly but sharpened nearby modal contrast. This behaviour is familiar in compact enclosures: a narrow absorber can make one frequency look excellent while leaving adjacent modes essentially untreated.

The enclosure result is intentionally reported as insertion loss rather than as a universal absorption coefficient. Curved, finite treatments interact with source position, enclosure modes, and edge leakage. The impedance-tube and baffled-panel tests establish the material response; the enclosure test shows that the response remains useful in a more realistic geometry.

Loss mechanisms

Energy dissipation was separated in the model by selectively removing loss terms. Thermoviscous loss in the coiled necks contributed 46% of the absorbed power at 320 Hz and 31% at 700 Hz. Microperforated-sheet resistance contributed 22-38% across the band. Membrane damping was smaller, generally below 15%, but it broadened the lowest family by coupling membrane motion to the coiled cavity. Wall losses in the printed cavities contributed the remainder.

The balance is important. If microperforation resistance is doubled in the model, absorption between 520 and 760 Hz improves slightly but absorption below 360 Hz collapses because energy no longer reaches the long-channel resonators. If neck radii are increased by 20%, low-frequency absorption weakens because viscothermal losses fall and impedance becomes too reactive. If membrane preload is reduced by 25%, the first resonance shifts downward but becomes too narrow. Broadband performance therefore comes from a deliberately imperfect set of resonators, each damped enough to overlap with neighbours.

These findings align with earlier work on dark absorbers, hybrid resonance metasurfaces, and optimal sound-absorbing structures [3,5,18]. High peak absorption is straightforward when a resonator is critically coupled at one frequency. Broadband subwavelength absorption requires multiple channels of dissipation and a controlled distribution of stored energy. The conformal constraint adds another requirement: those channels must remain stable after bending and mounting.

Bandwidth and passive limits

The measured performance is high for a 58 mm passive absorber below 800 Hz, but it does not exceed passive causality limits. The integrated absorption remains below the theoretical envelope estimated from the sample thickness and static compliance. This is an essential check because low-frequency absorber claims can sound suspiciously close to active or noncausal concepts. Recent work has shown that active or noncausal absorbers can go beyond passive causal limits [19,20]. The present panel does not use sensing, feedback, or external power.

The design instead approaches the practical passive trade space by distributing resonances and resistance. Its advantage over foam at low frequency comes from resonant path length, not from broadband porous dissipation. Its advantage over a simple Helmholtz liner comes from multiple damped modes, not from violating the bandwidth-thickness relation. This distinction matters for scaling. A thinner panel could be built, but the lower edge of the absorption band would rise unless more aggressive active or nonlocal mechanisms were introduced.

The 250-800 Hz target band was chosen for compact equipment noise, where tonal content often falls below the frequency at which thin porous liners become efficient. For architectural noise control, vehicle cabins, or HVAC ducts, the required band and incidence distribution would differ. The same design method can be retuned, but the reported numbers should not be transferred unchanged to another application.

Installation sensitivity

Three installation variables were tested: seam leakage, backing stiffness, and adhesive thickness. Seam leakage had the largest effect. Removing the butyl seam tape reduced average alpha by 0.08 over 300-700 Hz and shifted the 410 Hz feature upward by 21 Hz. A low-stiffness backing plate reduced absorption below 360 Hz by adding structural compliance that partly cancelled the backing-cavity stiffness. Increasing adhesive thickness from 0.3 to 1.0 mm shifted the lowest family downward by 14 Hz but did not change the average absorption strongly.

These sensitivities are manageable but real. A conformal absorber is a coupled acoustic-mechanical object, not a decorative skin that can be installed arbitrarily. For production use, strip seams would need moulded tongue-and-groove seals, installation torque specifications, and in situ leak checks. The prototype data suggest that the acoustic design has enough bandwidth to tolerate ordinary manufacturing variation, but not enough to tolerate open seams or soft uncontrolled backing cavities.

Dust and moisture exposure were not part of the main study. A 72 h humidity hold at 85% relative humidity changed the face-sheet resistance by less than 4%, but longer ageing could affect membrane adhesive and printed-channel sealing. This is another reason to report the present work as a laboratory prototype rather than as a qualified industrial liner.

Discussion

The conformal metasurface performed best where conventional compact absorbers struggle: below 800 Hz with less than 60 mm thickness. Its broadband response came from overlapping cell families, resistive face-sheet damping, and thermoviscous losses in folded channels. Curvature did not ruin the response when the radius was moderate and seams were sealed. The curved-panel and enclosure tests are therefore the key evidence, because flat-coupon absorption alone would not establish practical conformal behaviour.

The study also highlights the cost of complexity. The metasurface is heavier, more expensive, and more installation-sensitive than foam. It is not advantageous above roughly 900 Hz, where porous treatments are simpler and robust. The design is therefore best suited to low-frequency tonal and modal problems in compact volumes where thickness is constrained and where an engineered liner can be installed carefully.

Several limitations remain. The experiments used singly curved panels rather than doubly curved freeform surfaces. The incident field was controlled and did not include flow, vibration, oil mist, or thermal cycling. The printed nylon strips may not be the best manufacturing route for production. Finally, the optimisation assumed passive linear acoustics. At high sound pressure levels, membrane nonlinearity and flow through perforations could change the impedance.

Future work should test moulded elastomeric channels, integrated seam seals, and hybrid porous-metasurface stacks that hand off absorption above 900 Hz to a conventional layer. A useful design target would be a treatment that combines the present low-frequency response with ordinary porous absorption at higher frequencies without increasing total thickness beyond 70 mm.

Conclusion

We designed and tested a passive conformal acoustic metasurface for broadband absorption below 800 Hz. The 58 mm prototype used a microperforated face sheet, membrane-covered coiled Helmholtz cells, and a distributed resonance tiling. Flat coupons achieved alpha greater than 0.70 from 286 to 792 Hz, and curved-panel tests retained alpha greater than 0.65 from 312 to 764 Hz at 0.42 m radius. In a compact enclosure, the metasurface improved low-frequency insertion loss relative to thickness-matched foam and a single-resonance liner.

The result is promising but bounded. The metasurface does not beat passive absorption limits, and it depends on sealed seams, adequate backing stiffness, and controlled cell damping. Within those constraints, conformal multi-resonant metasurfaces provide a practical route to low-frequency noise reduction on curved engineering surfaces where conventional porous treatments are too thin to be effective.

Data and code availability

The supplementary archive contains CAD files for all cell families, measured face-sheet impedance, membrane preload calibrations, flat-coupon and curved-panel measurement data, finite-element input files, transfer-matrix scripts, enclosure pressure maps, and notebooks used for uncertainty propagation and plotting. Raw microphone time series are provided for representative runs, with processed spectra supplied for all configurations.

References

1. Assouar, B., Liang, B., Wu, Y., Li, Y., Cheng, J.-C. & Jing, Y. Acoustic metasurfaces. Nat. Rev. Mater. 3, 460-472 (2018).
2. Liang, B., Cheng, J.-C. & Qiu, C.-W. Wavefront manipulation by acoustic metasurfaces: From physics and applications. Nanophotonics 7, 1191-1205 (2018).
3. Ma, G., Yang, M., Xiao, S., Yang, Z. & Sheng, P. Acoustic metasurface with hybrid resonances. Nat. Mater. 13, 873-878 (2014).
4. Yang, Z., Dai, H. M., Chan, N. H., Ma, G. C. & Sheng, P. Acoustic metamaterial panels for sound attenuation in the 50-1000 Hz regime. Appl. Phys. Lett. 96, 041906 (2010).
5. Mei, J., Ma, G., Yang, M., Yang, Z., Wen, W. & Sheng, P. Dark acoustic metamaterials as super absorbers for low-frequency sound. Nat. Commun. 3, 756 (2012).
6. Cai, X., Guo, Q., Hu, G. & Yang, J. Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar Helmholtz resonators. Appl. Phys. Lett. 105, 121901 (2014).
7. Li, Y. & Assouar, B. M. Acoustic metasurface-based perfect absorber with deep subwavelength thickness. Appl. Phys. Lett. 108, 063502 (2016).
8. Yang, M., Meng, C., Fu, C., Li, Y., Yang, Z. & Sheng, P. Subwavelength total acoustic absorption with degenerate resonators. Appl. Phys. Lett. 107, 104104 (2015).
9. Ji, J., Li, D., Li, Y. & Jing, Y. Low-frequency broadband acoustic metasurface absorbing panels. Front. Mech. Eng. 6, 586249 (2020).
10. Long, H. et al. Subwavelength broadband sound absorber based on a composite metasurface. Sci. Rep. 10, 13823 (2020).
11. Zhu, Y. et al. Nonlocal acoustic metasurface for ultrabroadband sound absorption. Phys. Rev. B 103, 064102 (2021).
12. Shao, C. et al. Metasurface absorber for ultra-broadband sound via over-damped modes coupling. Appl. Phys. Lett. 120, 083504 (2022).
13. Xu, Q., Qiao, J., Zhang, G. & Li, L. Low-frequency sound-absorbing metasurface constructed by a membrane-covered and coiled Helmholtz resonator. J. Appl. Phys. 133, 075106 (2023).
14. Yang, J., Lee, J. S. & Kim, Y. Y. Metaporous layer to overcome the thickness constraint for broadband sound absorption. J. Appl. Phys. 117, 174903 (2015).
15. Maa, D.-Y. Potential of microperforated panel absorber. J. Acoust. Soc. Am. 104, 2861-2866 (1998).
16. Stinson, M. R. The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape. J. Acoust. Soc. Am. 89, 550-558 (1991).
17. Chung, J. Y. & Blaser, D. A. Transfer function method of measuring in-duct acoustic properties. I. Theory. J. Acoust. Soc. Am. 68, 907-913 (1980).
18. Yang, M., Chen, S., Fu, C. & Sheng, P. Optimal sound-absorbing structures. Mater. Horiz. 4, 673-680 (2017).
19. Mak, H. Y. et al. Going beyond the causal limit in acoustic absorption. Phys. Rev. Applied 16, 044062 (2021).
20. Wang, K. et al. Breaking the causality limit for broadband acoustic absorption using a noncausal active absorber. Device 2, 100502 (2024).
21. Li, X.-S., Wang, Y.-F., Chen, A.-L. & Wang, Y.-S. An arbitrarily curved acoustic metasurface for three-dimensional reflected wave-front modulation. J. Phys. D: Appl. Phys. 53, 195301 (2020).
22. Li, Y., Liang, B., Gu, Z.-M., Zou, X.-Y. & Cheng, J.-C. Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces. Sci. Rep. 3, 2546 (2013).