Architected acoustic metamaterials: An integrated design perspective Open Access

Acoustic metamaterials (AMs) provide an innovative approach to interacting with acoustic waves, surpassing the constraints of conventional porous materials and presenting new opportunities for sound and noise management.

This review concentrates on acoustic metamaterials in fluids, distinguishing them from elastic metamaterials (EMs) that affect mechanical waves in solids.

Acoustic metamaterials are specifically engineered to control and manipulate the propagation of sound waves. They can be classified geometrically into several distinct categories, as shown in Fig. 1. Each type of metamaterial has unique or shared characteristics and applications. This geometrical classification provides a comprehensive framework for a better understanding of their design and offers insights into their potential applications in various fields.

Acoustic metamaterials have gathered considerable attention in the field of materials science and technological design, due to their various applications in areas such as noise mitigation,¹ energy harvesting,² and medical imaging.³ 

AMs are characterized by their extraordinary properties, such as unusual bulk modulus and density,⁴ which can interact with acoustic waves innovatively; however, it is important to note that these characteristics are only applicable when equivalent fluid models are used to describe these metamaterials, a condition that does not universally apply.

These unusual or artificially generated properties have led to various real-world applications, like minimizing noise in hospital rooms during the recent pandemic,⁵ or making the external chiller unit of air conditioning systems more silent.⁶ 

Among all the possible geometries, the fractal [Fig. 1(a)] is one of the most commonly and effectively used.^1,8–19 The advantage of using fractal geometries relies on the adoption of self-repeating patterns, which exhibit multi-scale structures.²⁰ Among all the space-filling fractal geometries, the Hilbert fractal appears to be one of the most robust and thoroughly investigated in terms of acoustic wave modulation.

Another architecture widely used in acoustic metamaterials is the coiled or space-coiling one [Fig. 1(b)]. These geometries control the propagation of sound waves through the number of zigzag units inside the periodic unit area. Coiled structures can also be designed with tapered channels²¹ and openings, allowing the further modulation of their acoustic properties for wider broadband effects.

Architected acoustic metamaterials can provide, in particular, tailored transmissibility^12,22–26 and focused acoustic wave projection capability.^1,27–31 AMs can also provide multifunctional capabilities, like structures with soundproofing and air passage capabilities to help reduce noise pollution without blocking ventilation,^32–40 generating a more comfortable and sustainable built environment.

Their utility can span various industries, addressing heating, ventilation, and air conditioning (HVAC) noise⁴¹ in buildings.^42–45 In terms of multifunctionality, some acoustic metamaterials can be used to enable sound insulation at extreme temperatures,^46–49 opening up new possibilities for aeronautics and high-temperature machining. Acoustic metamaterials have been applied to combustion engine mufflers^50–53 or turbojets noise passing through an airplane fuselage.⁵⁴ 

Noise barriers⁵⁵ represent one of the most promising applications of acoustic metamaterials. Those barriers function by attenuating noise pollution, thereby fostering a quieter environment (Fig. 2). The design of AMs barriers often involves the creation of materials that can absorb a wide spectrum of noise frequencies.

Beyond these, AMs seem to have found applications in everyday life, like reducing toilet flushing noise⁵⁶ in condominium buildings, where the turbulent water noise can interfere with other people's sleep and peace of mind. Acoustic meta-absorber prototypes have also shown their efficiency in reducing the noise of stage machinery in theaters and opera houses,³⁵ thus improving the acoustic quality of performances and enhancing the overall experience in those leisure activities.

Architected acoustic metamaterials have also been used to exploit the tunneling effect²² in nondestructive testing^57–63 of structural and nonstructural parts for Structural Health Monitoring (SHM)⁶⁴ in a nondestructive way. Acoustic metamaterials also demonstrate some promising energy harvesting capacity, transforming seemingly disruptive noise into a renewable energy source.^2,65–70 They can convert ambient acoustics into electrical energy through resonance processes that exploit the acoustic modes of the metamaterials, paving the way for sustainable and renewable energy sources (Fig. 3).

Quite interestingly, the exploration of the design and performance space of AMs is not solely limited to noise mitigation and energy usage. Biomedical applications have also been developed; examples are the improvement of the efficiency and effectiveness of hearing aids technologies^71–76 or more effective noise-canceling headphones^17,77,78 for the mass market. By manipulating the acoustic signals, acoustic metamaterials provide improved sound quality and clarity.

Similarly, an artificial cochlea⁷¹ has been designed with a selective frequency picking along its architecture and modeled after the mammalian hearing organ [Fig. 1(i)], further advancing the field of audiology. Another biomedical application of AMs is related to the improvement of imaging technologies,^{45,58,60,79–87} for which the presence of a higher accuracy of image diagnostic would improve patient care.⁸⁸ 

One of the metamaterial effects, responsible for this enhanced image accuracy, is the tunneling effect (Fig. 4) that can improve the visible features of interior structures.⁸⁹ This meta behavior lets acoustic waves travel through a metamaterial, mimicking the diode effect.⁹⁰ Along an acoustic tunnel, it is possible to determine which one of the two potential orientations is restricted for the acoustic wave, using the reflection modulated by acoustic tunnel walls.⁹⁰ Thus, as a consequence, a decreased wave scattering²² is obtained, thanks to its quasi-diode behavior. The tunneling effect (Fig. 4) can also increase image resolution for biological applications, such as body scans through ultrasounds, by reducing the presence of undesired reflection and refraction of the waves, augmenting, as a consequence, the image resolution quality. Moreover, the tunneling effect could also be helpful in acoustic underwater communications,⁹¹ for which traditional radio techniques do not work properly.

The use of acoustic metamaterials can improve the quality of water, due to ultrasonic treatments^92–95 that enable more efficient and sustainable solutions to decontaminate waste and drain waters from pollutants. This application employs high-frequency sound waves to eliminate impurities from water, improving its overall quality with fewer chemical agents added to the water to purify it. Furthermore, acoustic metamaterials have been used for acoustic cloaking^96–106 to effectively manipulate sound waves around an object and making it acoustically invisible because of the minimum amount of reflected or refracted waves¹⁰⁷ (Fig. 5). This feature has promising applications in defense applications: an acoustic metamaterial skin could be applied to a stealth submarine¹⁰⁸ to make it acoustically invisible to enemy sonar.

Surprisingly, AMs can also be designed to act as transistors^109–112 to perform math operations with acoustic signals for acoustic computing applications.¹¹² This latter will form the basis of an acoustic calculator that can solve ordinary differential equations and more in environments where, due to harsh and strong electromagnetic fields, a traditional computational system would not work.

Further exciting applications of acoustic metamaterials include acoustic levitation (Fig. 6), with ultrasound^82,113–120 to manipulate extremely hot, reactive, or toxic chemical compounds in a contactless manner.

Architectural acoustics is another domain exploiting the benefits of acoustic metamaterials, leveraging their soundproofing abilities to create quieter spaces. Urban planning applications constitute another possible field that would benefit from adopting acoustic metamaterials. Applying fractal geometries to architectural forms^121,122 can lead to a higher noise diffusion within cities, with a consequent drop in noise pollution and improved life quality of all the urban residents.

Another unique attribute facilitates their use in sensors that detect⁸⁷ and respond¹²³ to sound. The use of AMs as antennae^{80,87,123–131} allows for the enhancement of weak acoustic signals within random noise, which could help with diagnostics in rotating and complex mechanisms to avoid imminent catastrophic failures. Broadband acoustic sensing (Fig. 7), for instance, allows for capturing both low-level and high-level features of audio signals, offering opportunities in various sectors such as surveillance¹²⁷ and healthcare.¹³² Acoustic sensors can be classified into two types: directional⁸⁷ and non-directional⁸⁰ acoustic antennas. Directional antennas are useful to spot specific frequencies in a noisy environment. For instance, the diagnostics of a different frequency from the one generated by an imminent failure can help avoid stopping production in an assembly line. On the other hand, the non-directional antenna can spot the noise direction in a three-dimensional (3D) environment. This feature can be useful in robotics to detect a generic noise source position (Fig. 8). In the field of wave propagation, acoustic metamaterials with chiral Archimedean spirals¹³³ or other coiled geometries can modify the wavefront,¹³⁴ allowing for precise control of propagation with limited or null waves refracted or reflected. This application of metamaterials could lead to the creation of acoustic barriers,¹³⁵ which would direct sound waves into designated areas where noise is not a concern.

A significant innovation within this domain is the study of topological acoustic metamaterials. These systems utilize concepts from topological physics to generate edge states that exhibit resistance to defects and irregularities. These topologically protected modes allow for directional sound propagation without backscattering, offering substantial benefits for the development of durable acoustic waveguides and devices.¹³⁶ For example, research on adjustable acoustic acceleration beams illustrates the application of topological ideas to achieve precise manipulation of the propagation of sound waves.¹³⁷ Incorporating topological principles into metamaterials not only strengthens their durability and efficiency but also sets the stage for creating devices capable of maintaining functionality despite structural flaws. This research highlights the transformative potential of topological acoustic metamaterials to advance sound manipulation technologies, providing innovative solutions to intricate acoustic problems.

Non-reciprocal acoustic metamaterials signify a major breakthrough in controlling sound, facilitating directional sound propagation, and fostering groundbreaking uses such as acoustic diodes and isolators. These systems leverage non-reciprocity, a phenomenon in which sound waves propagate in a single direction while being restricted in reverse, thus optimizing sound control in critical settings such as recording studios and concert halls.^138–140 Recent advances have highlighted diverse strategies for performing non-reciprocal acoustic propagation, including spatiotemporal modulation and magnetoelastic effects, which successfully disrupt the time-reversal symmetry.^139,141 Furthermore, the creation of acoustic metamaterials that feature extensive band gaps has proven effective in isolating sound over broad frequency ranges, making them ideal for scenarios that require meticulous acoustic control.¹⁴² Incorporating these innovations not only improves sound isolation but also sets the stage for the design of functional acoustic devices specifically adapted to unique acoustic settings.^140,143,144

The field of multifunctional acoustic metamaterials has gained prominence, focusing on the design of materials capable of performing several tasks at once. Examples include merging sound-damping properties with structural support or combining acoustic wave control with heat regulation.^145,146 The achievement of multiple functionalities in a single-material system enhances performance and minimizes the requirement for separate components, resulting in more streamlined and cohesive designs. Alongside advances in active and programmable metamaterials, interest has grown in hybrid metamaterials that integrate diverse functionalities. Recent investigations have highlighted the feasibility of acoustic metamaterials acting as both noise barriers and energy generators. By embedding piezoelectric elements within the metamaterial framework, researchers have engineered systems capable of transforming sound energy into electrical energy, enabling dual-purpose noise mitigation and energy production.^68,147,148 This innovation is especially applicable in urban areas where noise pollution is a significant issue, presenting an eco-friendly solution for both noise reduction and energy harvesting. Such developments underscore the movement toward the creation of materials with versatile applications, increasing their practicality in real-world scenarios.

The incorporation of artificial intelligence (AI) and machine learning (ML) techniques has experienced a remarkable growth in advancing the creation of innovative metamaterials. These computational approaches reveal groundbreaking design possibilities that may surpass human intuition, thus expediting the development of state-of-the-art acoustic technologies.¹⁴⁹ Using generative models within machine learning aids in exploring expansive design spaces, uncovering unique metamaterial configurations that human designers may overlook.^150,151

For example, machine learning tools can process experimental data to uncover relationships between structural characteristics and acoustic properties, refining design specifications through data-driven insights.^152,153 Furthermore, AI-powered optimization methods can be utilized to boost the effectiveness of acoustic metamaterials in practical applications, such as adaptive noise control systems that modify their properties dynamically in response to variable acoustic conditions.^154,155

Additionally, combining AI and ML with acoustic metamaterials has enabled the creation of multifunctional devices capable of tackling diverse acoustic challenges at the same time. For example, recent studies have highlighted the potential for acoustic metamaterials to integrate both soundproofing and energy harvesting functions, uniting two traditionally separate functions into a material system.^147,156 This multifunctionality offers significant benefits in fields such as architectural acoustics, where materials must operate efficiently over a wide range of frequencies and environmental factors.^157,158

Advancements in acoustic metamaterials encompass not only their functional aspects but also the approaches used for their production. Developments in additive manufacturing and 3D printing have allowed the creation of intricate metamaterial designs with tailored acoustic features.^159,160 These innovations allow for the realization of complex configurations that were previously challenging to achieve through conventional fabrication methods. Among the most promising techniques is two-photon lithography, which facilitates the production of microacoustic metagratings designed to function at extremely high frequencies. This method provides precise control over the geometry and spatial arrangement of the structure, which is crucial to attaining the desired acoustic effects such as sound focusing and negative refraction.¹⁶¹ The capability of manipulating microstructures with such precision facilitates the creation of advanced acoustic lenses and devices that can guide sound waves in unconventional ways. Additive manufacturing, along with two-photon lithography, has become a critical method for producing acoustic metamaterials. This technology supports the fabrication of intricate structures that are often unachievable with standard manufacturing techniques. For example, 3D printing has been used to create acoustic metamaterials based on tetrakaidecahedron cells, which exhibit notable sound absorption properties.¹⁶² The versatility of additive manufacturing not only enables the construction of complex shapes but also allows the tailoring of material properties to address specific acoustic demands.¹⁶³ Furthermore, the use of soft metamaterials, which can be manufactured by 3D printing, has revealed exceptional acoustic characteristics, including the ability to exhibit negative acoustic indices. This proves especially advantageous for applications such as underwater acoustics, where the distinct features of soft metamaterials enhance sound transmission and absorption.¹⁵⁶ 

The domain of acoustic metamaterials has seen remarkable progress in recent years, especially with innovations in tunable and active acoustic metamaterials. Tunable acoustic metamaterials enable modifications of acoustic responses through external factors such as mechanical strain, temperature changes, or magnetic influences, expanding their applicability in scenarios that require adaptable acoustic filtering or focus.¹⁶⁴ Incorporating origami-based structures into metamaterials, for example, offers substantial design flexibility to create programmable and responsive systems, underlining the critical role of tunability in contemporary acoustic technologies.^165–167 Active acoustic metamaterials leverage active components, including actuators and smart materials, to dynamically adjust their acoustic characteristics in real time under external stimuli. This real-time adaptability supports uses such as adaptive noise suppression and reconfigurable acoustic devices, which are becoming increasingly important in technological applications.¹⁶⁸ The programmable capabilities of these materials provide advanced functionalities that are unattainable with passive designs.¹⁶⁹ For example, the use of dynamic actuators to regulate the effective surface mass density of membrane-based acoustic metamaterials illustrates how active features achieve specific acoustic results.^170,171 This strategy facilitates the creation of materials that adapt to changing acoustic conditions, addressing needs in areas such as noise mitigation and soundproofing.

The advent of programmable metamaterials has facilitated the modulation of their properties. By altering geometric structures or employing materials with, for instance, electronically tunable attributes, these metamaterials can be customized to produce specific acoustic outcomes as required. This capability is particularly beneficial in dynamic acoustic environments characterized by rapid fluctuations.^172,173 This flexibility is vital for applications in noise mitigation, sensing, and communication systems.^173,174

Recent progress has highlighted the capabilities of active acoustic metamaterials, which can dynamically modify their sound-dampening properties via intrinsic tuning mechanisms.¹⁷⁵ For example, incorporating piezoelectric components allows real-time adjustments to acoustic characteristics, thereby broadening their applicability in various settings.¹⁷⁶ Furthermore, the development of modular acoustic metamaterials has exhibited the potential to achieve precise noise control, resulting in notable advances in design effectiveness and overall performance.¹⁷⁷ 

Acoustic metamaterials constitute a step-up in the application and understanding of already well-known materials for acoustic/sound management^178–182 and inspire scientific and industrial stakeholders to reconsider the use and functionality of conventional materials.^183–187 Fueled by the need for lighter and more efficient acoustic devices, the potential applications of AMs extend well beyond the confines of traditional noise control techniques.

From solving complex mathematical problems to their role in aerodynamic noise reduction and acoustic imaging, acoustic metamaterials represent truly multifunctional platforms of materials systems. Within the present review, we will examine the most frequently encountered metamaterial architectures that make up these unique constructs. We will then delve into these individual and combined configurations. The synergy between traditional acoustic components like resonators,^188–192 foams,^193–197 and porous panels,^198–202 with the aforementioned AMs geometries will also be described.

This study will also cover materials commonly used in manufacturing AMs and provide an outline of the cutting-edge technologies that facilitate their production.

Our journey through the acoustic metamaterial landscape will continue by describing the main physical mechanisms behind the performance of AMs and also discussing various measurement tools capable of capturing their effects.

The final paragraph contains the Authors' perspective on the main obstacles and complexities associated with the development and use of acoustic metamaterials.

Acoustic metamaterials (AMs) and elastic metamaterials (EMs) are designed for specific wave manipulation purposes, targeting sound waves and elastic waves, respectively. AMs are specialized for the control of sound waves in fluids, using methods such as resonance and impedance tuning to improve noise reduction and sound absorption.^203–205 Their diverse applications include architectural sound management and medical imaging, showcasing their ability to handle airborne pressure waves effectively.²⁰⁶ On the other hand, EMs devices are designed to influence elastic waves within solids, addressing longitudinal and transverse waves that are critical for uses, such as vibration isolation and seismic shielding.^207,208 Although these two types of metamaterial focus on different waveforms, they are based on shared principles such as periodic structures and localized resonances, resulting in effects such as negative effective mass density.^204,209 A clear grasp of these differences is essential to delineate their applications and the core physical principles that drive their functionality.^203,206

Acoustic metamaterials have become a key research focus due to their distinctive ability to control sound waves through in particular designed structures that possess characteristics absent in traditional materials. The configurations used in acoustic metamaterials, including fractals, coiled designs, chiral and antichiral formations, auxetic structures, honeycomb patterns, helicoidal shapes, spirals, cochlea-inspired forms, spider web configurations, Kelvin cells, and labyrinthine designs, each provide unique physical principles and acoustic functionalities suitable for diverse applications.

Fractal geometries [Fig. 1(a)], known for their self-similar patterns and intricate structures, repeat on varying scales. These features enhance wave manipulation and sound absorption by increasing the surface area-to-volume ratio. Studies show that fractal metamaterials enable broadband focus, essential for applications that require precise acoustic control, such as acoustic sensing and medical imaging.²¹⁰ The distinctive characteristics of fractals facilitate the creation of acoustic lenses capable of focusing sound waves with greater efficiency than conventional designs, thereby boosting the performance of devices such as ultrasonic transducers.²¹⁰ 

Coiled configurations [Fig. 1(b)], such as space coil designs, employ spiral structures to control sound waves by adjusting their phase and amplitude. These metamaterials enable concurrent modulation of various acoustic properties, making them effective for tasks such as sound filtration and wavefront alteration.⁵⁸ The crafting of coiled designs at scales smaller than the wavelength enhances their acoustic capabilities, especially in underwater scenarios where conventional materials often fail.²¹¹ The remarkable ability of coiled geometries to manipulate waves has been shown to substantially increase the performance of acoustic technologies.⁵⁸ 

Chiral metamaterials [Fig. 1(c)] exhibit asymmetrical configurations that interact distinctively with sound waves, resulting in effects such as acoustic rotation and amplified sound transmission. By tailoring chiral characteristics, it becomes possible to create devices capable of directing sound propagation, offering valuable applications in fields such as acoustic imaging and sensing.²¹² Research has recently shown that chiral designs can be customized to produce negative refractive indices, facilitating the innovation of advanced acoustic lenses and cloaking systems.²¹² This potential smooths the way for breakthroughs in stealth technologies and sound control solutions.

Antichiral configurations [Fig. 1(d)], characterized by their mirrored symmetry relative to chiral structures, are also capable of effectively manipulating sound waves. These designs can be customized to achieve distinctive acoustic characteristics, such as improved sound absorption and transmission, making them highly applicable for noise mitigation purpose.²¹² The dynamic interaction between chiral and antichiral attributes facilitates the creation of cutting-edge acoustic devices that innovatively control sound and opens the door to advancements in acoustic cloaking and sound filtration technologies.²¹² 

Auxetic materials [Fig. 1(e)], exhibit a negative Poisson's ratio, which means that they expand laterally when pulled. This distinctive attribute is used in acoustic metamaterials to increase sound absorption and transmission.²¹² Auxetic configurations can be customized to construct structures that efficiently capture and dissipate sound energy, making them well suited for noise reduction and soundproofing applications.²¹² The adjustable properties of auxetic materials enable the creation of adaptive acoustic devices capable of responding to changes in sound frequencies and intensities, thus enhancing their overall functionality.²¹² 

Honeycomb configurations [Fig. 1(f)] are renowned for their lightweight nature and the exceptional specific mechanical attributes, which make them advantageous for acoustic purposes. These patterns can be customized to make acoustic metamaterials with improved properties for sound absorption and transmission.²¹³ The hexagonal layout supports efficient wave propagation and can be adjusted to target particular frequency ranges, making it ideal for use in architectural acoustics and noise mitigation.²¹³ The adaptability of honeycomb structures allows their integration into various acoustic technologies, including sound barriers, acoustic liners, and sophisticated acoustic lenses.²¹³ 

Helicoidal designs [Fig. 1(g)], featuring spiral or helical configurations, influence sound waves by leveraging their distinct geometric characteristics. These metamaterials are tailored to produce targeted acoustic effects, such as focusing or scattering of sound waves.²¹³ The helicoidal configuration facilitates the development of devices capable of managing sound transmission in three-dimensional space, making them ideal for use in medical imaging and ultrasound therapy.²¹³ The capacity to fabricate helicoidal structures at the microscale increases their efficiency in directing sound waves across various applications.²¹³ 

Spiral configurations [Fig. 1(h)] are employed to fabricate acoustic devices capable of distinctive wave manipulation properties. These designs focus on sound waves and improve sound transmission through their specific geometric arrangements.²¹⁴ Incorporating spiral patterns into acoustic metamaterials facilitates the creation of devices with exceptional sound control, which is critical for use in telecommunications and audio systems.²¹⁴ The adjustable nature of spiral geometries supports the development of adaptable acoustic devices that can accommodate changes in sound frequencies and intensities.²¹⁴ 

Cochlea-inspired designs [Fig. 1(i)] replicate the intricate architecture of the human ear, facilitating effective sound manipulation and processing. These structures can be customized to improve sound absorption and transmission, making them highly applicable for use in acoustic sensors and hearing aids.²¹³ The biomimetic approach of cochlea-inspired geometries enables the creation of sophisticated acoustic devices that emulate biological systems, enhancing their capability for sound detection and processing²¹³ (Fig. 8). Incorporating cochlea-inspired configurations into acoustic devices has the potential to drive remarkable progress in auditory technologies.²¹³ 

Spider web formations [Fig. 1(j)] possess remarkable mechanical and acoustic characteristics, which can be utilized in the design of metamaterials. The complex architecture of spider webs facilitates efficient sound absorption and propagation, making them ideal for noise mitigation and soundproofing applications.²¹³ The customization of spider web geometries on a microscale enhances their ability to influence sound waves in various functionalities.²¹³ The lightweight and adaptable properties of spider web structures support their integration into various acoustic technologies, ranging from soundproof barriers to sophisticated acoustic lenses.²¹³ 

Kelvin cell structures [Fig. 1(k)] exhibit a distinctive geometric configuration that can be tailored to achieve superior acoustic functionality. These designs can deliver targeted acoustic results, including improved sound absorption and transmission properties.²¹⁵ Incorporating Kelvin cells into acoustic metamaterials facilitates the creation of systems adept at capturing and dissipating sound energy, making them highly suitable for applications such as noise mitigation and sound insulation.²¹⁵ The adjustable nature of Kelvin cell geometries supports the development of adaptive acoustic systems capable of responding to different frequencies and intensities of sound, thus improving their efficiency.²¹⁵ 

Labyrinthine designs [Fig. 1(l)] offer innovative solutions to build acoustic devices with exceptional wave control properties. These configurations are designed to improve sound absorption and propagation through their complex layouts.²¹³ Incorporating maze-like patterns into acoustic metamaterials facilitates the creation of devices capable of capturing and dissipating sound energy effectively, making them ideal for noise reduction and soundproofing applications.²¹³ The adaptability of labyrinthine structures makes them applicable in a range of acoustic technologies, from noise barriers to sophisticated acoustic lenses.²¹³ 

Fifteen main physical mechanisms underpinning the behavior of acoustic metamaterials can be defined. Acoustic impedance match/mismatch between two media is essential for controlling the reflection and transmission of sound waves. Bragg scattering blocks specific frequency bands, shaping the sound wave propagation. Mie and Fabry–Pérot resonances affect wave interactions and specific resonance effects. Fano-like interference offers some unique acoustic profiles through asymmetric responses. Double open-ended and open-closed resonators fine-tune the resonance behavior within the material. Helmholtz resonators control sound by manipulating the vibration of air, while thermoviscous losses affect sound propagation by converting acoustic energy into heat. Together, all these physical mechanisms contribute to creating the metamaterials' unique acoustic properties.

Moreover, essential physical concepts like band theory, effective medium theory, generalized Snell's law, topological phonon edge states, parity-time (PT) symmetry, and non-Hermitian acoustics are crucial in shaping the design and analysis of acoustic metamaterials. These frameworks facilitate novel control over sound waves, unlocking applications that range from noise isolation to cutting-edge acoustic technologies.

The acoustic impedance is related to the resistance offered by a material against impinging sound waves [Fig. 9(a)]. For a given sound wave, the impedance is calculated as the ratio derived from the sound pressure about the particle velocity at a specific point within the material. Factors that determine the acoustic impedance include the geometry of the metamaterial and its boundary conditions.

The principle of impedance matching⁴⁹ facilitates the efficient transfer of acoustic energy from the medium to the metamaterial. Impedance matching can be beneficial in the case of sound cloaking since a good impedance match will generate a limited amount of reflected acoustic energy.

A different scenario arises with the presence of impedance mismatch²¹⁶ when the material and the medium differ significantly in terms of acoustic properties. This situation leads to the reflection of the acoustic waves until no acoustic waves can successfully pass through the acoustic metamaterial, resulting in an entirely reflective effect. Impedance mismatch, however, is intentionally utilized in designing acoustic metamaterials. For instance, impedance mismatch can create a sonic barrier, which can obstruct specific frequencies, such as those generated by HVAC systems, thereby making them less noisy.²¹⁷ 

Bragg scattering [Fig. 9(b)] is a crucial occurrence in acoustics and happens when sound waves encounter a periodic structure. This interaction heavily affects the wave propagation. The core of this interaction is the “Bragg condition,”²²⁰ represented mathematically as $ΔL=λ/2$⁠, which represents a particular case of the Laue equations.²²¹ In the Bragg equation, $ΔL$ stands for the typical length scale of the periodic structure, while $λ$ refers to the wavelength of the sound wave. The condition is usually fulfilled when the wavelength is double the distance between the scattering elements. When this scenario occurs, the scattering of the waves from the periodic structure leads to destructive interference that creates a bandgap. This bandgap represents a particular frequency range in which waves are prevented from propagating through the material, effectively blocking sound at those frequencies from penetrating the acoustic metamaterial. Figure 9(b) represents the mutual effect of a series of scatters at a distance $ΔL$ on the incident acoustic wave. As a consequence, a bandgap is generated on the opposite side of the metamaterial.

An acoustic wave infiltrating a metamaterial can encounter scattering elements present within the metamaterial architecture. When the frequency of the incoming sound wave coincides with the scatterers' natural frequency, Mie resonance⁶³ may be initiated due to interference between the resonant frequency of the material and the medium's monopolar, or multipolar resonance.

The arrangement of scatterers can facilitate diverse and useful effects. A notable consequence is the formation of multiple band gaps within the frequency spectrum.

The rise of Mie resonances can be triggered when the size of the metamaterial's components is comparable to the wavelength of the incoming sound wave as illustrated in Fig. 9(c), in which the dimension of the scatter and the acoustic wavelength are the same.

Mie resonances in acoustic metamaterials result from the interaction between acoustic waves and sub-wavelength scatterers, which resembles Mie scattering observed in electromagnetic contexts. The resonance condition for a spherical scatterer is given by $ka=ωac$⁠, where k denotes the wave number, $ω$ represents the angular frequency, c is the speed of sound and a refers to the radius of the scatterer. This equation is crucial to comprehending how acoustic waves interact with objects smaller than their wavelength, producing distinctive resonant effects that can be harnessed in various applications, such as sound absorption and cloaking.^224,225 The theoretical framework used to examine these resonances mirrors electromagnetic Mie theory, as both involve resolving wave equations constrained by the geometrical and material properties of scatterers. This parallel enables the application of scattering theories to anticipate and control acoustic wave interactions within engineered materials.^226,227 The significance of Mie resonance is far-reaching, introducing novel approaches to influence sound and light, particularly in the fields of optics and materials science.^228–230 

A Fabry–Pérot resonator^231,232 operates by implementing two reflective planes positioned in a parallel configuration. An incoming wave is subjected to repeated reflections between these surfaces, thereby producing a series of increasingly reflected waves. Constructive interference is observed in the transmitted waves when the width of the resonator aligns with an integer multiple of half the wavelength. This type of interference maximizes the transmission coefficient.²³³ The phase synchronization among the reflections generates an improved transmitted wave.

A Fabry–Pérot resonator can function as a frequency filter. It facilitates the efficient transmission of waves at particular frequencies while causing at the same time significant attenuation of waves in other parts of the spectrum.²³⁴ This selective frequency transmission serves many applications, including the design of acoustic devices, sensors, and systems dedicated to noise management. As shown in Fig. 9(d), the parallel walls of the acoustic duct generate constructive interference. A similar phenomenon is exploited in the design of laser systems.

In this expression, $T(ω)$ signifies transmission at angular frequency $ω$⁠, $T0$ is the peak transmission, F refers to the finesse coefficient, and $δ=2ωLv$ represents the phase shift per round trip, where v is the speed of sound. This resonance mechanism facilitates the selective amplification or attenuation of specific frequencies, which is useful in devices such as filters and sensors.^235,236

Fano-like interference⁵⁵ happens when acoustic waves impact a multifaceted configuration that supports an array of oscillation modes. A single mode can be narrow and well-defined, like the resonance of a tuning fork, for instance. The remaining modes generate however a broad and almost continuous spectrum of resonances. The simultaneous excitation of the discrete resonance and the continuum by the sound waves leads to interference, culminating in a highly irregular scattering profile [Fig. 9(e)]. This effect shows up as asymmetric transmission peaks, which occur when resonant and non-resonant wave modes interact.²³⁷ 

Fano-like interference has particular relevance in the field of acoustic metamaterials to design open or ultra-open sound insulators³⁴ for HVAC systems, for instance.

Locally resonant materials are AMs featuring resonant structures with unique acoustic properties.²⁴³ These materials have lattice constants much smaller than the sound wavelengths they interact with, allowing them to block specific frequencies of sound waves. This effect can also be created by embedding dense core materials, like lead, inside soft coatings such as silicone rubber. When sound waves hit these composites, the localized resonances can cause the material to reflect almost all the sound in certain frequency ranges.²⁴⁴ 

Double open-ended resonators^{8,27,40,252,253} could be represented as vacant passages or conduits permeable at both extremities and with a constant cross section [Fig. 9(f)]. The inlet of acoustic waves into these resonators triggers the air or any other entrapped medium to pulsate at certain distinct frequencies, which are the resonant frequencies associated with the geometrical characteristics of the resonator.

This latter phenomenon seems to be responsible for the position of the absorption coefficient peaks in acoustic metamaterials built with Hilbert or coiled geometry.⁸ 

The potential applications for this type of resonator are broad and mainly around noise management. It is possible to modulate and tune frequencies maximizing/minimizing the acoustic absorption of a metamaterial by a careful design of the resonators' length and shapes.

Open-closed resonators^{10,58,252,256–260} provide a particular case of Fabry–Pérot resonance.^{68,198,233,234,252,253,261–263} Open-closed resonators represent another variation of constant cross section resonators used for the manipulation of acoustic waves. These resonators can be described as tubular constructs that bear a resemblance to their double open-ended resonators counterparts [Fig. 9(f)]. There is, however, one critical distinction: one of their ends remains sealed.

The permeable end of the resonator is at the ingress of the sound waves, whilst the closed extremity establishes a fixed condition. This distinct arrangement alters the modalities of the oscillation of the air cavity or any other entrapped medium within the resonator, in response to the incoming sound waves.

These open-closed resonators display different resonant frequencies and diverse oscillation patterns when contrasted with double open-ended resonators. They are responsible for the position of the transmission loss peaks in specific parts of the frequency spectrum for Hilbert and coiled geometries.^10,264

Quarter-wavelength resonators work through standing waves: they are tubes in which one end is closed and the other is open, while the length equals one-quarter of the resonant frequency wavelength for the operation it is designed to carry out. The consequence is a standing wave pattern of minimum air pressure at the open end, an antinode of maximum particle velocity at the open back, and vice versa at the closed end.²⁶⁷ The resonant frequency, set by the tube length L, is $f=c4L$⁠, where c is the speed of sound.¹⁷⁸ Such resonators find wide use in musical instruments like clarinets and flutes in producing some definite pitch.²⁶⁸ They are undoubtedly important in noise control, for instance, by making mufflers and silencers, achieving this end by matching the resonator's frequency to an unwanted frequency.²⁶⁹ The same can apply in architectural acoustics to manage the acoustics of a room by the same principle: to cut down on reverberation and take charge of a particular frequency band.²⁷⁰ 

The incorporation of Helmholtz resonators^67,271–276 in the architectural design of acoustic metamaterials allows the exploitation of their well-known narrow-band effect.

Each of these resonators can be calibrated to a particular frequency to absorb acoustic energy, effectively acting as a “trap” for that specific frequency within the sound wave.²⁷⁷ 

A system comprising multiple such resonators, with each resonator tuned to a distinct frequency, can provide an efficient noise reduction across a broad frequency bandwidth. The versatility of this solution is made possible by adjusting the volume, length, and section of the Helmholtz resonator neck.

The positioning of coiled or fractal geometries inside the resonators also introduces an additional element of space efficiency. This design feature decreases the volume required for the spring component of the acoustic mechanism, thereby requiring less cavity volume to be used. A classic example of a Helmholtz resonator is the classic glass bottle, where the mass of air in the neck oscillates according to the spring action of the internal volume [Fig. 9(g)].

Thermoviscous losses represent a special energy dissipation mechanism occurring within fluid systems, such as air or water. These losses occur near the boundary or borders with the material substrate, affecting the transmission of acoustic waves. As sound waves traverse a medium, the pressure and particle velocity oscillates, triggering viscothermal effects due to the near-zero particle velocity on the surface of the channel [Fig. 9(h)]. The interplay between these effects culminates in energy dissipation.

When used in acoustic metamaterials, thermoviscous losses acquire significant importance in the design process of systems at scales where such losses become substantial.

Thermoviscous losses are especially critical in three main cases; perforated and microperforated panels,²⁸⁵ acoustic foam,²⁸⁶ and narrow channels inside acoustic metamaterial geometries.⁸ 

Typically, thermoviscous losses²⁸⁷ take place at the microscale, where dimensional features of the metamaterials are comparable with the thickness of the boundary layer in which the thermoviscous effects occur. This becomes particularly critical when designing metamaterials for applications in ultrasound or other high-frequency domains, where these losses must be accounted for to achieve optimum sound energy transmission.

As the sound wave advances, these losses become evident as attenuation or a reduction of the acoustic energy of the wave, thus impeding the effectiveness of the acoustic metamaterial. This behavior is highly significant if the objective of the metamaterial is to optimize the sonic energy transmission. However, the transmission of the acoustic wave, for applications like acoustic absorbers can be enhanced by thermoviscous losses. In such cases, the losses can be engineered to augment the absorption of sound energy, thereby mitigating noise and increase the magnitude of the transmission loss provided by the metamaterial.

Membrane-type acoustic metamaterials (MAMs) are innovative materials that use lightweight,²⁸⁹ flexible membranes, sometimes enhanced with strategically placed rigid masses.^290–292 These enhancements help the membranes interact with the acoustic field, making them effective in influencing sound waves.^293–295 The design of these masses—considering their size, weight, and position—is crucial in creating specific resonances that can block or reflect sound waves.

The core idea behind MAMs is to utilize the resonance created by these masses to interfere with and significantly reduce sound transmission.²⁹³ When sound waves hit the membrane, the attached masses cause it to vibrate in a manner that cancels out certain frequencies.¹⁸³ This is particularly effective for low-frequency sounds, which are typically harder to manage. By tweaking the properties of the masses and the tension of the membrane, specific frequencies can be targeted.^292,296

There are two primary ways to position these membranes relative to incoming sound waves: normal incidence,²⁹³ where the membrane faces the wave head-on, and grazing incidence,^297,298 where the membrane is parallel to the wave.

For instance, increasing the tension in the membrane can shift its resonance frequency, allowing it to target different sound frequencies.²⁹³ Theoretical models have been developed to predict the behavior of these metamaterials,²⁹⁵ and experiments have consistently shown that MAMs can achieve significant transmission loss at specific frequencies.

A different approach involves active membranes,¹⁷⁰ such as in the case of loudspeakers.^299,300 Here, the membrane's coil can be paired with either a passive or active shunt to enhance control.

Shunts play an important role in manipulating sound waves to achieve desired acoustic effects. By adjusting the impedance characteristics of the shunt,³⁰¹ the interaction between the loudspeaker and the metamaterial can be fine-tuned, ensuring efficient energy transfer and improved acoustic properties like resonance and bandwidth. Shunts can also be designed to shape the frequency response of the loudspeaker-metamaterial system,³⁰² creating bandgaps where sound is not transmitted or enhancing specific frequency ranges.

Moreover, active shunts enable dynamic adaptation of the system's acoustic properties in real-time.¹⁴³ This feature allows for the development of reconfigurable metamaterials that can adjust their acoustic characteristics based on the environment or specific requirements, providing a high level of versatility.

One of the most notable advantages of MAMs is their ability to provide high sound insulation without adding significant weight or thickness, unlike traditional soundproofing materials. This quality makes them particularly attractive for use in the aerospace and automotive industries, where reducing weight is crucial. Integrating shunts with MAMs further enhances their capabilities, offering more precise control over sound attenuation and the potential for real-time adjustments to the acoustic environment.³⁰³ 

Parity-time (PT) symmetry in acoustics pertains to systems where a balance between gain and loss results in extraordinary phenomena like exceptional points (EPs), where eigenvalues merge.^310,311 The scattering characteristics of PT-symmetric acoustic systems exhibit notable effects, such as unidirectional transparency and asymmetric reflection, made possible by precisely engineered materials with complex parameters. However, achieving such materials often necessitates active devices, as passive materials alone struggle to provide the required gain and loss profiles. A PT-symmetric medium is defined by $κ(r)=κ*(−r)$ and $ρ(r)=ρ(−r)$⁠, where $ρ$ represents the mass density, and $κ$ denotes the complex bulk modulus.³¹² Recent progress in PT-symmetric acoustics has revealed applications such as asymmetric cloaking and improved sensor capabilities.^310,313 Research into these systems has expanded to diverse configurations, including piezoelectric composites and metamaterials, often integrating active components to achieve PT symmetry. This versatility highlights the potential of PT symmetry in controlling acoustic waves.^314,315 These discoveries have significant implications, proposing innovative approaches for the design of acoustic devices that take advantage of the distinctive characteristics of non-Hermitian systems, although their implementation often depends on active devices.³¹⁶ 

The incorporation of advanced techniques like machine learning (ML) and genetic algorithms in the development of acoustic metamaterials marks a notable improvement over conventional strategies, which often depend on human intuition combined with iterative simulations. These advanced methods allow the exploration of intricate design spaces, aiding in the discovery of unconventional configurations that can achieve enhanced acoustic performance. For example, ML algorithms expedite the design process by optimizing unit cell geometries based on insights from predefined simulations, thereby fostering the creation of innovative structures.^324,325 Genetic algorithms similarly contribute by emulating natural selection to identify ideal design parameters, offering a versatile alternative to traditional trial-and-error techniques.^326,327 Furthermore, these intelligent methods are particularly advantageous in the creation of adaptive and multifunctional acoustic metamaterials that can dynamically respond to changing environmental conditions.^328,329 By generating effective initial configurations, these techniques complement conventional design methods, streamlined the process, and promoted innovation.^325,330 The transition to intelligent design methodologies not only improves the capabilities of acoustic metamaterials but also expands their potential applications, setting the stage for future progress in the field.^324,331

Investigating acoustic phenomena uncovers a nuanced interplay between mechanisms, properties, and applications that range from basic to advanced principles. Core effects such as mismatches in acoustic impedance and Helmholtz resonators play crucial roles in sound insulation and noise suppression.^332–334 Cutting-edge topics, including topological acoustics and parity-time (PT) symmetry in non-Hermitian systems, extend the boundaries of acoustic control, facilitating directional wave travel and bandgap optimization organized.^335,336

Table I describes the fundamental acoustic phenomena associated with wave propagation, resonance, impedance, and bandgap creation. It provides an organized overview of the mechanisms, properties, applications, geometries, and materials, serving as a foundational reference for principles of noise suppression and sound management. Expanding on these basics, Table II presents advanced acoustic control phenomena made possible through metamaterials. It showcases intricate designs and engineered interactions, allowing precise wave control for innovations such as acoustic cloaking, sound concentration, and advanced sensing applications.

In the open literature, terms like “zigzag,” “coiled,” “maze,” and “labyrinthine,” are frequently applied to acoustic metamaterial geometries, often creating ambiguity due to their interchangeable use (Table III). We propose here a possible classification based on the internal geometrical configuration of these metamaterials. The classification is based on how the acoustic wave navigates within the metamaterial basic periodic unit. The multichannel configuration [Fig. 11(a)] exemplifies structures that allow multiple paths of the propagating wave due to the intricacy of the internal metamaterial architecture. Maze or labyrinthine geometries [Fig. 1(l)] are excellent examples of these multichannel configurations because they enable both constructive and destructive wave interference. The single-channel configuration [Fig. 11(b)] represents a category of metamaterial in which the acoustic wave travels along a forced path inside the meta atom. Typical structures of this type are zigzag or coiled [Fig. 1(b)], which offer a unique trajectory for wave propagation. To further refine the classification, we also identify an additional “dead-end” category [Fig. 11(c)], applicable to both single and multichannel configurations. This subclass describes configurations where the acoustic paths terminate prematurely, as observed in the coiled channels of acoustic antennas. The proposed classification (Fig. 11) offers a simplified approach to categorize acoustic metamaterials, providing a clear distinction between metamaterial architectures based on the trajectory that an acoustic wave undertakes within each configuration.

The proposed classification approach, which organizes acoustic metamaterials based on the number of internal channels, provides a clear and straightforward method for understanding and categorizing structural designs. This method is particularly advantageous for visualizing the internal architecture of metamaterials and quickly identifying design patterns that might be suitable for specific sound manipulation applications. It effectively captures the diversity of geometries, ranging from simple coiled channels to complex, labyrinthine structures, that contribute to sound control through mechanisms like space-coiling, resonance enhancement, or wave redirection.⁴⁴⁶ 

• Simplicity and accessibility: This classification is easy to understand and does not require extensive analysis of complex physical properties. It allows for a quick assessment of structural designs, making it particularly useful during the early phases of metamaterial development.

• Versatility in design: The approach accommodates a wide variety of structural configurations, including fractal, coiled, and hierarchical designs, thereby offering flexibility for exploring innovative configurations that may not fit neatly into more restrictive classification schemes.¹⁸⁴ 

• Limited predictive capability: The method primarily focuses on geometric characteristics and does not inherently provide insights into how these structures will interact with acoustic waves across different frequencies. This lack of a direct link between structural design and acoustic performance can limit its utility for predicting the precise behavior of metamaterials, such as sound absorption peaks or bandwidths.^402,446

• Oversimplification: By concentrating solely on the internal channel count, the method may overlook other critical aspects, such as material composition, resonance behaviors, or dynamic tunability, which are essential for understanding and optimizing acoustic properties in specific applications.^184,399

To address these limitations, it is beneficial to consider integrating the proposed geometric classification with existing frameworks that focus on physical and functional properties. Such integration can enhance the utility of the classification in the following ways:

Resonance-based classifications categorize acoustic metamaterials according to their interaction with sound waves at specific frequencies, often focusing on local resonance phenomena that create unique sound manipulation effects, such as negative mass density or impedance matching.^184,446 By integrating our geometric approach, we can first use the internal channel count to identify potential designs and then refine these designs by analyzing their resonance behaviors. This combined method allows researchers to predict how internal structures will affect sound propagation, leading to more precise control over acoustic properties.

Example: Coiled structures identified through geometric classification can be further analyzed to determine their resonance frequencies. This additional step can help optimize these designs to achieve effective sound absorption at specific low frequencies, which is crucial for applications like noise reduction in industrial environments.³⁹⁹ 

Topological methods classify metamaterials based on unique waveguiding features and topologically protected states that prevent backscattering and ensure robust sound transmission, even in the presence of defects.⁴⁰² By combining our classification, which focuses on internal channel architecture, with a topological approach, it is possible to identify and design metamaterials that not only fit a certain geometric category but also exhibit desirable topological behaviors, such as unidirectional waveguiding or tunability.

Example: A metamaterial initially categorized by its geometric design can be assessed for topological properties to see if its structure supports edge states. If so, the design can be further modified to enhance these properties, leading to robust, multifunctional devices useful for applications like acoustic sensing or waveguiding.⁴⁴⁷ 

Hybrid classification systems combine structural design with physical parameters, such as effective bulk modulus, mass density, and impedance. Integrating our geometric classification into this framework can offer a multi-layered approach where internal structures serve as the baseline for categorization, and physical parameters refine this categorization based on performance requirements.^399,447 This can lead to a more holistic understanding of how structural features influence acoustic performance, enabling designers to target specific acoustic outcomes, such as broadband absorption or focused sound wave manipulation.

Acoustic metamaterials (AMs) are engineered to regulate the propagation of sound waves using principles such as resonance, Bragg diffraction, and impedance discontinuity. These materials serve a wide range of purpose, including reducing noise in building designs and altering sound in consumer products such as hearing aids and noise canceling devices.^354,488 Their ability to generate bandgap-specific frequency ranges where sound waves are heavily dampened facilitates efficient sound control, making them indispensable in sectors such as medical imaging and noise pollution mitigation.^354,362 Recent innovations emphasize the enhancement of the functionality of acoustic metamaterials by incorporating features such as adjustable resonances and auxetic geometries, which exhibit unique mechanical characteristics.³⁷⁵ For example, the creation of broadband sound absorbers and tunable filters underscores the adaptability of these materials to customized acoustic settings, thus enhancing their utility in the real world.^253,433 Furthermore, investigations into locally resonant structures have prepared the ground for effective low-frequency sound absorption, a critical aspect for applications in urban noise control and industrial environments.^325,355

Figure 1 shows the baseline geometrical shapes employed in acoustic metamaterials. Most of these configurations can be described as two-dimensional structures, which can be extruded into a third dimension without alteration to their initial 2D shape. This property holds true for all configurations, excluding the helicoidal [Fig. 1(g)], the cochlear [Fig. 1(i)], and the Kelvin cell [Fig. 1(k)].

Another useful pattern is the one that utilizes coiled and tapered fingers-like [Fig. 1(b)] channels.^40,489,490 This configuration is also used for acoustic sensors, where the peculiar geometry can be tailored to pick and amplify specific frequencies in a noisy environment.

Maze geometries [Fig. 1(l)] of acoustic metamaterials involve the use of a labyrinth-like pattern that creates pathways and barriers to manipulate sound waves. This design principle allows for the creation of complex acoustic landscapes, which can be used to shape the propagation of sound waves^{106,124,479,483,491–493} with designed reflective and refractive angles, focusing,^{1,27,45,106,469,478} (Fig. 9) or tunneling.^{106,494–496}

Bio-inspired⁴⁵⁵ hierarchical assembly geometries of acoustic metamaterials imitate the structure found in biological systems. Examples include the spider web [Fig. 1(j)] geometry^{69,305,461,497–502} and honeycomb [Fig. 1(f)] structures^503–513 that use hexagonal cells arranged in repeating patterns. These bio-inspired designs offer unique acoustic properties and can be used to create a wide range of AMs with tailored properties, like low-frequency sound and vibration control for the spider geometry and broadband hierarchical sound absorption at high temperatures for honeycomb.

Chiral lattice [Fig. 1(c)] geometry^514–521 in AMs utilizes a lattice structure with chirality, resulting in asymmetric arrangements of unit cells. This design principle allows for creating of complex structures that can be used, such as sound-diffusing metasurfaces that allow non-specular scattering.

Antichiral lattices [Fig. 1(d)] geometry^522,523 in AMs are structures with an opposite chirality compared to their mirror image. For instance, the use of I-shaped anti-chiral units⁵²⁴ to build AMs has been reported. This design principle allows for the creation of complex, multi-scale structures for enhanced broadband sound insulation in the lower part of the acoustic spectrum.

Auxetic lattice [Fig. 1(e)] geometry^{522,525–528} in acoustic metamaterials make use of structures that exhibit a negative Poisson ratio behavior. Some researchers have constructed curved shells with 3D reentrant auxetic cellular metamaterials.⁷⁷ In contrast, other researchers have exploited the auxetic behavior of these structures by applying external loads or other inputs to modify their acoustic performances.^525,527 This unique property allows the design of acoustic metamaterials with potentially high tunable characteristics.

The increasing focus on bio-inspired and hierarchical acoustic metamaterials is largely due to their ability to mimic intricate natural geometries, thus improving the wave control capabilities. Natural structures such as tree branches and moth wings^529–531 present hierarchical configurations that allow better management of wave propagation, as highlighted in recent research.^49,532 These structures frequently incorporate self-similar or varied unit cell arrangements, which facilitate adjustable wave interaction properties, demonstrating improved sound absorption and filtering over diverse frequency ranges.^533–535 Hierarchical metamaterials embed smaller components within larger ones, enhancing energy dissipation while simultaneously managing both low- and high-frequency waves.^502,536 This nesting strategy has been shown to significantly enhance the functionality of phononic crystals, leading to the creation of versatile metamaterials that address both structural and acoustic requirements.^49,537 Incorporating bio-inspired elements into metamaterials not only strengthens their mechanical characteristics but also establishes the foundation for novel applications in noise control and energy dissipation.^539,541

One of the most captivating categories among these configurations involves the use of self-similar structures at various scales. Fractal geometries can be subdivided into two macro-categories: space-filling (SF) and not-space-filling (nSF). SF tend to fill a confined space completely, with a polygonal line that theoretically can reach every point in the area or volume. At the same time, nSF cannot be contained in a 2D or 3D enclosed space if the number of iterations increases.⁵⁴⁰ These geometries offer unique acoustic properties in terms of acoustic absorption and transmission loss, for instance, making them useful in various applications, like barriers in transformer noise reduction.⁵⁴¹ 

Fractal-inspired^{18,106,456,468,542–544} or recursive structures, also known as pseudo-fractal geometry, are used in AMs to create patterns that mimic fractal properties. These structures exhibit self-similarity at different scales and include designs such as hexagonal or triangular lattices,^{456,498,542,545,546} hierarchical lattices,^498,542,547 and self-similar beam lattices.^456,542 Moreover, it has been experimentally shown that the number of 90° angles in a 2D fractal or coiled structure does not influence the acoustic output in terms of acoustic absorption and transmission loss if the total length of the cavity patterns is the same. Thus, Figs. 1(a) and 1(b) contain metamaterials architectures that have the same absorption and transmission performance if the length and section of the polygonal shape is the same.^8,264 A notable example is the utilization of Hilbert fractals,^{1,8,10,12,18,210,261,541,548,549} which are characterized by intricate self-similarity patterns inspired by the Hilbert curve [Fig. 1(a)]. The hierarchical nature of the Hilbert fractal design offers flexibility in tuning the acoustic response across a wide range of frequencies, thereby enhancing acoustic properties such as sound absorption⁸ and transmission loss.¹⁰ 

In addition to the Hilbert fractal [Fig. 12(a)], the Menger fractal,^13,17,22,483 also known as the Menger sponge [Fig. 12(b)], has been employed in the design of acoustic metamaterials. The unique structural properties of the Menger fractal, particularly its self-similarity at different scales, are exploited to control, manipulate, and block sound waves. The Menger fractal geometry provides a lightweight solution, allowing air passage due to the high porosity. It has also shown a remarkable tunneling effect.

The Sierpinski fractal [Fig. 12(c)], also referred to as the Sierpinski triangle or pyramid, is a self-similar structure widely implemented in acoustic metamaterials.^{9,15,16,548,550–554} This fractal is characterized by its recursive triangular patterns, which significantly influence sound wave propagation due to its repeating geometry. Its distinctive properties allow for the formation of band gaps; in addition, this geometry is used to enhance sound scattering in architectural acoustics, both inside buildings and on facades, to optimize sound wave diffraction in urban environments. Moreover, using this pattern in barriers between streets and residential areas can be an effective noise mitigation tool. The Koch fractal, also referred to as the Koch snowflake [Fig. 12(d)], is a mathematical curve that can be applied to a segment or a polygonal shape perimeter.^19,510,555 It is defined by a recursive process that incrementally enhances the number of sides, and its sophisticated design is leveraged in the construction of acoustic metamaterials to boost properties such as wave scattering.¹⁹ In addition to promoting wave scattering, this fractal can also modulate acoustic wave speed, create bandgaps, and steer the wavefront in a waveguide plate by modifying the fractal order.

The T-fractal [Fig. 12(e)], represents a potential fractal pattern that manifests the letter T at different scales and orientations within a 2D space.⁵⁵⁶ It could also be constructed using other alphabet letters or shapes, thereby providing bandgap capabilities and a spectrum of acoustic effects that remain largely unexplored.

Finally, the polyadic Cantor fractal [Fig. 12(f)], a variant of the conventional Cantor set, has been utilized in the design of acoustic metamaterials.^14,557,558 This fractal is derived from a process that iteratively excises portions from a line segment, mirroring the construction methodology of the Cantor set. Its primary applications are in ultrasonic imaging and, lately, low-frequency porosity absorbers.

Apart from fractal and fractal-inspired curves,^{18,456,468,542} recursive geometries have been shown to induce a phenomenon referred to as negative density,⁵⁵⁹ bulk modulus,⁵⁶⁰ or both,⁵⁶⁰ which is rare in the natural world.⁵²⁹ This unnatural set of physical properties extends their utility to various applications encompassing noise reduction,^253,484,497 sound focusing,^27,28 and even acoustic cloaking.^{100,102,103,272,561,562} The application of materials in recursive geometries can be illustrated using cylinders^563,564 or other extruded 2D shapes, like the c-shape;⁵⁶⁵ arranged in a well-defined square, hexagonal, or Kagome⁴⁵⁶ patterns. The effect of such geometric configurations will offer an overall density and bulk modulus different from the single scatter, forming the series [Fig. 10(b)].

Another interesting example entails the deployment of membranes along a 1D dimension, at a specified fixed interval from each other, with or without an additional mass incorporated within the membrane in different positions.^294,566 This configuration further enhances the versatility of acoustic metamaterials, even with just a monodimensional repeated pattern that can generate zero reflection.⁵⁶⁶ 

Recent scholarly endeavors have begun exploring the potential application of Kelvin cells^215,567 in developing acoustic metamaterials, yielding promising results. A Kelvin cell is a tetrakaidecahedron [Fig. 1(k)], essentially a polyhedron with 14 faces, comprising six squares and eight hexagons. This distinctive property of the Kelvin cell enables it to create a 3D space-filling tessellation, which makes it helpful in obtaining band gaps at designed frequencies.

Other geometrical configurations used to make AMs are the spiral^{33,104,253,515} and the helix,^28,66,568 as shown in Figs. 1(h) and 1(g). AMs that use spiral geometries can mainly develop their shape in a 2D plane. Spiral structures [Fig. 1(h)] can be arranged in a variety of configurations. The possibility of incorporating patterns that exemplify this versatility as Archimedean spirals^57,569 or logarithmic spirals⁵⁷⁰ to make waveguides, each with unique properties, like slowing down, guiding, and confining the wave propagation.

Helix AMs add another dimension to the design possibilities in this rapidly advancing field. These metamaterials are characterized by incorporating helical structures [Fig. 1(g)] into their design, arranged in a periodic or variable pattern of the helix coils. The uniqueness of the helix, with its continuous curve winding about a central axis, can generate, for instance, phase shifting⁵⁶⁸ of the acoustic wave, and consequent passive noise cancelation.

The construction of acoustic metamaterials entails a strategic integration of specific geometries and acoustic technologies. The breadth and diversity of design approaches employed in developing these materials provide a wide spectrum of potential combinations. An illustrative strategy for the development of AMs can be concentrated into six primary elements, each reflecting the combination of different geometries and technologies. The first approach involves using mixed internal geometries or a singular one in combination with perforated or micro-perforated panels [Fig. 13(a)] internally or on top^{49,198,460,473,489,497,548,571–579} (the perforations can be usually circular or rectangular). This method enhances the material's ability to damp certain frequencies through thermoviscous losses.

In general, microperforations [Fig. 13(a)] are used to damp high frequencies; meanwhile, the internal geometries [coiled, zigzag, maze, labyrinthine or fractal, as exemplified in Fig. 13(d)] act toward the spectrum's low-frequency range. This specific union can produce a broadband effect regarding acoustic losses if compared to the effect of the single “pieces” alone.

The second approach incorporates mixed geometries or a single one, with one or more Helmholtz resonators.^{39,56,273–275,475,485,580–583} Helmholtz resonators [Fig. 13(c)] are structures designed to resonate at specific frequencies, thereby absorbing sound at these frequencies and reducing noise. Sound absorption can be maximized by integrating more than one resonator within the AMs design, enabling a more broadband absorption of the acoustic energy.

The next approach mixes membranes [Fig. 13(f)] and Helmholtz resonators [Fig. 13(c)], providing exceptional control over sound waves. By combining Helmholtz resonators with membranes—sometimes weighted with added mass—these materials can significantly shift the frequency range at which they operate.^584–587 This combination is particularly effective at lowering the frequency range, allowing the metamaterial to trap lower-frequency sounds more effectively. Additionally, this design approach broadens the range of frequencies the metamaterial can manage for a variety of applications, like mufflers^{586,588–590} or ventilated structures.⁵⁹¹ It is possible to fine-tune these materials to suit specific needs, such as noise reduction and sound isolation by tweaking the properties of the membranes and the size of the resonators.⁵⁹² 

In the fourth approach, porous materials, such as acoustic foam like melamine [Fig. 13(b)], are combined with a specific geometry that takes care of the low-frequency range.^{286,449,454,459,583} In this combination, the porous material dampens sound wave propagation, reducing reflected sound and facilitating noise control through their double action separated by the Biot limit.⁴⁵⁴ Under the Biot limit, the solid part of the foam will contribute to the metamaterials effect in the middle-frequency range. On the other hand, over the Biot limit, the air inside the pores, with their thermoviscous action, will contrast the middle-high frequency range.

The fifth approach merges one or more of our geometries with one or more piezoelectric patches.^65,66,68,69 Piezoelectric materials generate an electric charge in response to applied mechanical strain stress. By embedding piezoelectric patches [Fig. 13(e)] within the design of AMs, it becomes possible to control sound wave propagation and harvest electric energy from the sound waves. This approach creates multifunctional acoustic metamaterials that can serve dual roles; sound control and energy harvesting.

The last approach combines some of the aforementioned four methods,^593,594 offering a comprehensive solution for constructing acoustic metamaterials. This strategy provides higher flexibility regarding covered frequency but with a consequent securely higher complexity and manufacturing cost.

Elastic metamaterials are engineered to control mechanical waves, including stress waves and vibrations, within solid media. These distinctive characteristics enable applications ranging from seismic protection to vibration mitigation, which are vital for bolstering structural durability during phenomena such as earthquakes and dynamic loads in machinery and frameworks. Advances in fabrication technologies, especially 3D printing, have allowed the development of intricate elastic metamaterials with customized mechanical properties, improving their utility in real-world scenarios.^595–597 For example, research has illustrated the utility of such materials in forming seismic metamaterials capable of transforming harmful seismic waves into less destructive forms, offering groundbreaking approaches to earthquake resilience.^598,599 Moreover, incorporating locally resonant structures within elastic metamaterials has demonstrated significant potential to control low-frequency waves, a critical aspect of effective vibration isolation.^600,601 In conclusion, ongoing advances in elastic metamaterials continue to broaden their applications in diverse engineering domains.

The fabrication of acoustic metamaterials involves a range of technologies, each with its own unique strengths and capabilities. This section offers a detailed, though not exhaustive, overview of the main methods used to create acoustic metamaterials (AMs). We also look at the types of machinery involved and their specifications, particularly regarding precision and accuracy. This is especially important for the geometry of surfaces, such as rugosity, which can play a major role in energy dissipation through thermoviscous losses. Additionally, the methods and their tolerances are closely tied to the frequencies being studied with the manufactured samples. For example, at higher frequencies, it is crucial to minimize imperfections in the manufacturing process due to the shorter wavelengths involved.

For example, if the targeted frequency range is 1 kHz, using photolithography to produce metamaterials with operational wavelengths close to 34 cm would be excessive, given the precision requirements of the machining process. Conversely, small imperfections at reduced wavelengths for ultrasound applications can significantly affect the response of the metamaterials. The techniques considered in this work are classified as additive and subtractive manufacturing methods and are ordered from the least to the most precise.

Fused deposition modeling (FDM) is often used in fabricating acoustic metamaterials. FDM operates by extruding a thermoplastic filament layer by layer to construct a tridimensional object, with the advantage of a wide range of filament materials with varying properties, including PLA, one of the cheapest but reliable options for relatively fast prototyping. For example, the machine “FORTUS 400mc” has a building volume of 406.4 × 355.6 × 406.4 mm³, with a printing tolerance of ±0.127 mm, and a possible layer height between 0.127 and 0.3302 mm.⁶⁰² 

PolyJet⁷⁰ makes use of inkjet printing principles to create three-dimensional objects, making it a versatile 3D printing process for fabricating acoustic metamaterials with complex geometries. PolyJet printing works similarly to 2D inkjet printing. The process, deposits layers of curable photopolymer resin onto the printer's building area rather than propelling ink droplets into a paper surface. According to the manufacturer Stratasys, the machine Objet500,⁶⁰³ has a building volume of 490 × 390 × 200 mm³, with an accuracy of ±0.1 mm if the highest dimension of the printed object is below 100 mm. Otherwise, if the highest dimension of the printed object is bigger than 100 mm, the accuracy drops from ±0.1 to ±0.2 mm.

Powder bed fusion (PBF) technology is an alternate method of additive manufacturing. To begin, the powder bed fusion printing process, a thin (around 0.1 mm) layer of it, is applied to the construction platform. Following that, a laser merges the primary stratum, or the design sliced sectional design. A roller is then used to distribute successive layers of dust over the previous layer, incorporating additional layers. This sequence is repeated until the entire prototype is formed. The loose and non-merged powder remains in place throughout the process but is removed during the subsequent cleaning phase. For instance, according to the manufacturer “Wematter,” the machine Wematter Gravity SLS 3D printing system,⁶⁰⁴ has a building volume of 300 × 300 × 300 mm³, with a precision of ±0.1 mm.

Direct Ink Writing (DIW) is a relatively new layer-by-layer 3D printing technology that uses a small nozzle to distribute a pressure-driven viscoelastic ink, resulting in detailed structures. This process normally consists of three steps: first, creating 3D models with the CAD software, then designing a 3D route for the nozzle with slicing software to create a G-code, and finally, a controlled ink release. DIW is distinguished by its extensive flexibility for customizing the inks, allowing for the exact creation of 3D objects from the microscopic scale to the bigger ones. The precision and resolution of the printed object are heavily influenced by critical determinants such as nozzle diameter and printing speed, and it can print soft materials such as hydrogel to thermoset polymers. The x-y plane layer resolution ranges from ±0.1 to ±1.2 mm, and the z-axis resolution ranges from ±0.1 to ±0.4 mm, with the smallest potential feature size being around 0.5 mm.⁶⁰⁵ 

Digital Light Processing (DLP) is typically considered to be faster than Stereolithography. This is because SLA uses a technology in which the laser cures the resin point by point; meanwhile, DLP uses a projector screen to expose a complete horizontal slice (layer) at once, allowing for simultaneous curing of all locations inside the slice. Thus, the time necessary to harden a single layer is lower than SLA. Compared to stereolithography, the DLP printing quality is slightly worse but faster than SLA; this is due to the possibility of specular or reflection effects that can make the polymerization of the layers less precise. According to the manufacturer FlashForge, the machine Hunter 3D Printer,⁶⁰⁶ has a building volume of 120 × 67.5 × 150 mm³, with a layer resolution between 0.025 and 0.02 mm, and a print accuracy of ±0.05 mm.

Stereolithography (SLA),⁴⁷⁵ an additive manufacturing technique, has been instrumental in creating three-dimensional objects through a process of photopolymerization. This technique involves selectively solidifying a liquid photopolymer resin point by point using a light source, such as a laser, to form the desired 3D shape. For instance, according to the manufacturer “3D Systems,” the machine Projet 7000 HD,⁶⁰⁷ has a building volume of 380 × 380 × 250 mm³, with accuracy between ±0.025 and 0.05 mm for every 25.4 mm of printed element.

Due to the dimensions of the typical wavelengths in acoustic metamaterials, techniques like electron beam lithography (EBL), ion beam lithography, and x-ray lithography are not considered. Those techniques, due to their high cost, nanometric precision, and accuracy, are rather used for optical metamaterials.

Multimaterial printing opens up exciting possibilities to design metamaterials that make use of coupled structural vibration effects due to the use of different materials. Techniques like PolyJet printing, which works like an inkjet but with materials, can blend different photopolymers, giving parts with different mechanical and acoustic properties.

Directed energy deposition (DED) methods,⁶⁰⁸ such as laser engineering net shaping⁶⁰⁹ (LENS), allow for the simultaneous use of multiple powders, creating structures with potentially customized acoustic traits. advanced fused deposition modeling^437,610 (FDM) printers with one,⁶¹¹ or multiple extruders can produce parts with different stiffness and density,⁶¹² crucial for acoustic performance. While Binder Jetting has typically been a single-material process,⁶¹³ new advancements allow it to use multiple powders or add secondary materials afterward, resulting in complex acoustic behaviors.^614,615 Finally, technologies like digital light processing (DLP) and multi-material stereolithography (SLA) can switch between or combine different resins, crafting detailed structures like piezoelectric microphones,⁶¹⁶ or an ultrasonic sensor equipped with three-dimensional printed holographic elements.⁶¹⁷ These multimaterial methods make it possible to fine-tune how parts handle vibrations, leading to better performance in many applications.⁶¹⁸ 

Subtractive manufacturing processes, such as laser cutting and water jet cutting, or classical machining, have also found applications in producing AMs. Water jet cutting employs a high-pressure jet of water (around 4000 bars, for instance), mixed with abrasive particles, to cut through various materials. The accuracy for a straight cut of 1 m is, for instance, ±0.1 mm for the machine Mach 100⁶⁰³ of the company Flow,⁶¹⁹ until ±0.02 mm for a straight cut of 30 cm for the NanoJet machine from the same company.⁶²⁰ 

On the other hand, laser cutting makes use of a high-power laser beam to cut or engrave materials, creating intricate patterns and structures. One downside is the possible difficulty in assembling precisely 3D AMs structures using laser-cut 2D slices.²⁶⁴ It is worth noticing that if the pattern to laser-cut has relatively small recursive geometries, like in the case of fractals, particular care needs to be paid toward overheating of the material during the cutting session.²⁶⁴ Moreover, it is helpful to consider that the X-Y resolution is a function of the material, material thickness, and cutting speed.⁶²¹ For instance, the cut accuracy is ±0.015 mm for the machine Speedy 400 flexx⁶⁰³ of the company Trotec.⁶²² 

CNC milling machines can produce AMs by machining solid blocks or sheets of materials with desired patterns or structures.⁶²³ A five-axis CNC machine, for instance, has five planes of motion: sideways (X-axis), vertically (Y-axis), back and forth (Z-axis), and two additional axes for worktable rotation and tilt. Its primary advantage is enhanced productivity and efficiency when producing complex forms using continuous milling. The five-axis movement produces more precise and smooth parts, minimizing manual tool changes and saving time. The versatility of this machine enables a vast range of components and forms, highlighting its adaptability across diverse metamaterial complex shapes. According to the manufacturer Bridgeport, the machine VMC 600,⁶²⁴ has a building volume of 600 × 410 × 520 mm³, with an accuracy of ±0.005 mm.⁶²⁵ 

Electrical discharge machining (EDM) uses heat energy to carve accurate, complicated shapes in electrically conductive materials, which standard processes such as CNC milling may not be able to accomplish because of the material toughness. This distortion-free technology, which does not make physical contact with the workpiece, enables the production of complex shapes, produces a high-quality finish, and ensures remarkable precision, making it particularly useful for small-scale manufacturing. Even though materials like inconels and tungsten carbide can be used for EDM. However, the slower material removal rate increases production time and costs, and the extreme heat utilized can change the material's metallurgy. According to the manufacturer Agie, the machine Agiecut classic 2,⁶²⁴ has a building volume of 750 × 550 × 250 mm³, with an accuracy of ±0.006 mm.

Photolithography is a micromanufacturing technique used to create intricate structures on a 2D substrate. Photolithography is an intricate micromanufacturing procedure, it involves subjecting a polymer surface to ultraviolet (UV) light to eliminate the areas that are not covered by the design mask. This masking process on the two-dimensional polymer surface is akin to the function of a photo negative. Areas covered by the mask are protected from UV light exposure. As a result, there is no curving or deformation of the geometric pattern beneath these masked regions. For instance, this technique has been used in AMs to enable active broadband tunability⁶²⁶ and create ultrasonic transducers.⁶²⁷ For instance, the machine SLX 3 e1 from the company Micronic⁶²⁸ can ensure a minimum linewidth of 400 nm, with an accuracy of ±30 nm for an area of 150 × 150 mm².

The advancement of acoustic metamaterials (AMs) depends significantly on sophisticated fabrication methods that offer precise control over microstructural features while remaining scalable for practical use. Techniques at ultrafine scales, such as electron-beam lithography (EBL) and focused ion beam lithography (FIB), enable detailed patterning with nanometer accuracy, which is vital for influencing sound waves at subwavelength dimensions.⁶²⁹ However, these approaches often face challenges in scalability and expense, restricting their application mainly to experimental environments.⁶³⁰ On the other hand, macroscale production methods, including large-scale additive manufacturing processes such as fused deposition modeling (FDM) and powder bed fusion (PBF), are crucial in creating AMs intended for practical use, such as noise barriers and construction materials.²¹¹ These techniques allow the incorporation of sound-diffusing and sound-absorbing components into larger systems, improving their functional applicability.⁶³¹ The combination of multiscale fabrication strategies, such as Direct Ink Writing (DIW) and Multi-Material Jetting, offers a promising pathway to develop hierarchical acoustic metamaterials that merge nanoscale accuracy with macroscale designs. This multiscale capability can tackle technical issues in the field, especially for applications requiring wide frequency attenuation and durability under environmental conditions.⁶³² Despite these progressions, the scaling of AMs remains challenging due to material and process limitations, highlighting the need for further exploration into the integration of top-down lithographic techniques with scalable 3D printing methods.¹⁶³ 

Advancements in modern manufacturing techniques, especially multi-material additive manufacturing (AM), have markedly improved the acoustic properties of metamaterials. This approach enables the precise arrangement of materials with distinct mechanical properties, facilitating the development of customized material gradients that enhance both vibration damping and sound absorption.^211,633,634 These capabilities allow for the design of metamaterials optimized for targeted acoustic uses, enhancing structural integrity and enabling greater control over acoustic attributes.⁶³⁵ Topology optimization has become a vital technique in designing intricate geometries that maximize acoustic efficiency while reducing material consumption. By strategically distributing materials, engineers can create lightweight structures capable of accurately manipulating sound waves, leading to improved impedance matching and sound attenuation.^325,636 Moreover, integrating machine learning into the design process accelerates the iteration and optimization of metamaterials, enabling predictions of acoustic behaviors across extensive frequency ranges.⁶³⁷ Advanced manufacturing techniques featuring real-time monitoring and adaptive controls significantly enhance the precision and reproducibility of acoustic metamaterials. Materials with intelligent properties that adapt to environmental changes smooth the way for novel applications in adaptive noise mitigation and energy harvesting.^326,638,639 The choice of fabrication technique for acoustic metamaterials is pivotal in optimizing their functional characteristics. Additive manufacturing, particularly multi-material printing, enables the fabrication of complex gradients that enhance sound absorption over broader frequency ranges, expanding their utility.^640,641 On the other hand, subtractive techniques such as CNC milling and electrical discharge machining (EDM) deliver the accuracy required for high-frequency applications, where slight imperfections can significantly alter acoustic behavior.⁶³⁸ Hybrid manufacturing approaches, which combine additive and subtractive techniques, provide a balanced solution, achieving both scalability and the precise microstructural details necessary for particular acoustic properties.^211,642 Additionally, integrating machine learning into these processes enables real-time adjustments and predictive modeling, boosting the adaptability and reproducibility of acoustic metamaterials.^394,643 This integration supports the creation of multifunctional metamaterials designed for noise control and energy absorption, facilitating their implementation across various settings.^639,644,645

Additive manufacturing methods, including 3D printing and multi-material printing,^646,647 enable the production of complex structures that effectively minimize unwanted resonances and enhance broadband absorption. These techniques offer precise control over geometry and material distribution, which is vital for optimizing acoustic performance in devices such as transducers and metamaterials.^648,649 For instance, polymeric cellular structures created through additive manufacturing have shown significant advantages in acoustic absorption due to their custom-designed microstructural attributes.⁶⁴⁸ Conversely, subtractive methods like CNC milling enhance surface quality and dimensional precision, which is critical for reducing wave scattering and achieving optimal impedance matching. The high accuracy of CNC milling ensures smooth, uniform surfaces on acoustic devices, improving sound wave transmission and minimizing reflections at material interfaces.^650,651 Nanoscale fabrication methods, such as nanoscale lithography, refine microstructural features, improving frequency selectivity and resonance performance. These techniques are essential for applications like noise suppression, where controlling specific frequencies is critical.^164,652 The integration of various materials and manufacturing methods further enhances performance; for example, polymers used in additive manufacturing allow for the flexible tuning of metamaterial properties, making them ideal for lightweight acoustic control applications.^653,654

Techniques for measuring acoustics are fundamental in defining the properties of acoustic metamaterials within controlled settings. Commonly used configurations include reverberation chambers, impedance tubes, and anechoic chambers. Reverberation chambers generate a diffuse sound field, allowing evaluation of the sound absorption and scattering properties of materials in a controlled environment, which is critical for analyzing their acoustic performance.⁶⁵⁵ In contrast, impedance tubes are used to determine the acoustic impedance and absorption coefficients of materials, shedding light on sound-surface interactions.⁶⁵⁶ Anechoic chambers are uniquely designed to suppress reflections, allowing precise measurements of sound propagation and material characteristics without interference from echoes or external noise.⁶⁵⁷ These approaches are indispensable for advancing the research of acoustic metamaterials, ensuring accurate and consistent measurements that support the creation of materials with custom acoustic properties.

In contrast, impedance tubes offer a quicker and more economical means to assess acoustic characteristics, although they may lack the precision necessary to capture intricate wave behaviors in some metamaterials.⁶⁵⁸ This contrast emphasizes the need to carefully select measurement methods based on the particular demands of the research or application, striking a balance between the desire for accuracy and the constraints of time and budget.

Recent advances in optimization are shaping acoustic measurement methods. Automated systems and real-time data processing are becoming standard in experimental setups, speeding up data analysis, and minimizing the costs associated with repetitive testing.⁶⁵⁹ Machine learning algorithms are also being used to improve the interpretation of complex datasets, enabling faster insight derivation.⁶⁶⁰ In addition, adaptive systems that adjust measurement frequency ranges based on initial results are gaining traction, significantly increasing measurement efficiency in industrial contexts.⁶⁶¹ These advances streamline the testing process and enhance the reliability and precision of the results, driving forward the practical implementation of acoustic metamaterials.

Emerging research highlights how innovative measurement methods address practical limitations. For example, 3D printed prototypes have proven effective in assessing acoustic properties quickly, allowing quicker iterations in design and performance analysis.⁶⁶² This technique accelerates development while offering crucial insight into the acoustic behaviors of novel materials and structures. Furthermore, combining diverse measurement methods has provided a holistic understanding of acoustic properties, helping researchers choose techniques that optimize precision, scalability, and resource efficiency.⁶⁶³ Such examples underscore the importance of flexibility in measurement approaches as researchers tackle the challenges posed by acoustic metamaterials and their wide-ranging applications.

The section provides an overview of the materials used to produce acoustic metamaterials, with an emphasis on those most frequently employed in research-based applications. The selection of these materials varies significantly based on the intended application; specifically, the materials can range from rigid to flexible, particularly in scenarios involving structural–acoustic coupling.

Photosensitive resins, or photopolymers,¹ are liquid polymers that solidify when exposed to a specific wavelength of light, typically ultraviolet light. They are commonly used in stereolithography⁴⁷⁵ or digital light processing 3D printing technologies. These resins offer sufficient precision for acoustic applications and can produce intricate structures with fine details, making them ideal for creating complex geometries.

Metals such as copper,⁶⁶⁴ brass,¹⁴ and aluminum⁵⁶⁴ have also been used to create AM if we consider multifunctionality. Copper's excellent electrical conductivity and high thermal conductivity make it desirable for AMs applications. Brass and aluminum, due to their relatively high density (compared to polymers), stiffness, and stability over time, are valuable for manipulating sound waves. Thin composite plates, using steel and rubber,⁶⁶⁵ have also been used due to their insulation power against a wide range of frequencies, exploiting their resistance to substantial underwater pressure.

Polylactic acid²⁶⁴ (PLA), a biodegradable thermoplastic polymer, is commonly used in 3D printing⁶⁶⁶ and various applications. PLA is one of the most typical materials used for fast prototyping, because of its relatively low cost and good printability. Similarly, other thermoplastic polymers such as polyethylene terephthalate⁶⁶⁷ (PET), acrylonitrile butadiene styrene⁶⁶⁸ (ABS), acrylonitrile styrene acrylate⁵⁷⁵ (ASA), and polymethyl methacrylate¹¹⁴ (PMMA) have been used in certain scenarios due to their unique properties, for instance, PMMA can be used to build waveguides, meanwhile, the others polymers can be selected for their respective mechanical properties like stiffness, impact, and heat resistance.

Another material widely used in AMs prototyping is epoxy resin.⁴⁸² This type of thermosetting polymer, undergoes a chemical reaction when combined with a hardener to form a strong and rigid material. This material has been explored for use in acoustic metamaterials due to its strength and rigidity.

Polydimethylsiloxane⁶⁶⁹ (PDMS) and thermoplastic polyurethane⁶⁷⁰ (TPU) are flexible and versatile materials that can be utilized to create deformable structures. Their flexibility and elasticity enable the creation of AMs that can adapt to different acoustic environments.

Composite structures can be created by combining metal honeycomb frameworks with ethylene vinyl acetate (EVA) copolymer films.⁵⁰⁴ The metal honeycomb frameworks provide structural support, a high strength-to-weight ratio, and sound transmission control, while EVA copolymer films⁶⁷¹ offer flexibility, damping properties, and additional acoustic properties.

Materials such as aluminum nitride⁶⁷² (AlN), silicon dioxide⁶⁷³ (⁠ $SiO2$⁠), barium titanate⁶⁷⁴ (⁠ $BaTiO3$⁠), and lithium niobate⁶⁷⁵ (⁠ $LiNbO3$⁠) have been used for applications that require converting mechanical energy to electrical.

Natural materials like wood have been tested as well to produce AMs. Due to the low density and workability, cylindrical wood⁶⁷⁶ scatterers have been used in 2D waveguide prototypes without finding substantial differences in terms of acoustic output compared to other materials like iron or aluminum, but are interesting in terms of manufacturing costs due to the relatively easy workability.

Melamine foam,²⁸⁶ known for its high porosity and interconnected open-cell structures, is another material used in acoustic metamaterials. It contributes to sound attenuation and absorption through multiple reflections, diffusion, and conversions of sound energy into thermal energy.

Metal fibers,⁴⁶ made of various materials such as steel, aluminum, or copper, provide stiffness, structural integrity, and heat resistance; meanwhile, at the same time, the porous matrix contributes to sound control and can be tailored to specific acoustic requirements and applications, like inside mufflers.

Polyurethane foam,¹⁹³ with an open-cell structure, can be molded, cut, or shaped into various forms, allowing flexibility in the design and manufacturing process, especially used in anechoic chambers, perimeters of 2D waveguides, and impedance tubes.

Flexibility can be granted by nonwoven materials⁴⁵⁴ made by entangling fibers together (without weaving or knitting), resulting in a porous and flexible structure. These can be manufactured from various fibers, such as polyester and polypropylene, or natural fibers, like cotton or wool. Their porous structure makes them particularly suitable for sound absorption applications.

Finally, polyvinyl chloride⁶⁷⁷ (PVC) foam can be used as a material to create AMs, providing sound absorption and transmission control properties. PVC offers advantages such as flexibility and lightweight, making it a suitable material for AMs prototyping.

In conclusion, the choice of the material depends on the specific requirements for the application (if prototype or not, for instance), including the desired acoustic properties, fabrication method, and environmental and operational considerations. The materials discussed in this review offer a wide range of possibilities for designing and fabricating acoustic metamaterials, each with unique and sometimes similar advantages and challenges.

The exploration of acoustic metamaterials encompasses a wide array of frequencies, each with distinct properties and potential applications. At the lower end of the spectrum, we find infrasound frequencies, typically below 20 Hz, which are not detectable by the human ear. Despite being imperceptible, these frequencies play a significant role in both natural events, such as earthquakes, and human-induced activities, including specific industrial operations.

Noise manifesting at low frequencies conventionally bracketed within 20–600 Hz, originates from various sources. These may encompass the deep grumble produced by lorry engines, auditory manifestations of distant meteorological phenomena such as thunderstorms, or the persistent hum generated by sizable air conditioning apparatus. Normally, these sound waves with long wavelengths can traverse physical barriers like walls in a relatively easier manner, if compared to their high-frequency counterparts. This exacerbates potential noise pollution issues. Due to the low-frequency spectra involved, stopping them with a thin layer of traditional acoustic materials is difficult. This is one of the most researched subtopics of AMs for sound insulation.^{1,253,261,449,453,471,497,524,574,576,678} The goal is to induce sound blocking with a reasonable thin layer of material.^{63,198,234,274,459,461,463,473,484,574,577,668,679,680}

Sounds falling within the medium frequency range between approximately 600 and 2 kHz have different sources. These can include conversational exchanges conducted within typical vocal ranges, the operational noise emanating from domestic electrical devices like a vacuum cleaner, or the audible hustle and bustle associated with heavy traffic on populous thoroughfares. Moreover, middle frequencies play a crucial role in facilitating intelligible speech. Thus, the contribution of AMs can potentially be linked to the combined use of foams plus coiled structures to mitigate and modulate sound energy.^{49,55,57,217,257,460,485,548,579,582}

High-frequency noise encompasses the spectrum from 2 kHz upward to 20 kHz. Representative sources include the high-pitched hiss discharged by a pressure cooker under strain, the piercing emission from a blown whistle or the acute shriek of a distressed child. Owing to their comparatively shorter wavelength, these types of sound are often more effectively attenuated or absorbed by conventional acoustic materials. For this frequency range, traditional acoustic technologies like acoustic foams having open cells and Helmholtz resonators are also efficient.^449,469,579

Finally, ultrasounds are sound waves with frequencies exceeding the upper limit of human audibility, typically above 20 kHz. Previous studies have discussed the generation of ultrasound within the 20–240 kHz frequency interval. Ultrasounds are widely used in medical imaging and industrial testing because they can penetrate materials and tissues, exposing their internal features or defects.^{14,45,57,70,114,681}

Laboratory tests are essential to evaluate the effects of acoustic metamaterials before developing real-world applications. We report here on the fundamental techniques used in experimental acoustics, categorized by the measured quantities and specific applications of the metamaterials.

Interestingly, most acoustic metamaterials proposed in the literature have not been validated experimentally or are tested only in very controlled settings, like 1D ducts with normal incidence. A few studies explore grazing incidence or flow conditions, but these are not frequent. Additionally, most validations are performed under monomodal configurations—i.e., scenarios that only consider simple geometries and specific angles like 0° and 90°. This means very few acoustic metamaterials have been tested under complex, real-world setups that resemble industrial applications.

There is a clear need for experimental setups that can handle more complicated configurations. This includes the 2D apparatus mentioned later in this study, which is designed to test AMs in conditions that are much closer to what you would find in real-life industrial settings. By using such advanced testing devices, we can ensure that AMs are not just theoretically sound but also practically viable in a variety of real-world applications.

The reverberation chamber,⁶⁸² or reverberation room, aims to generate a homogeneous sound field. It is designed so that sound waves can reflect uniformly off the encompassing surfaces, including walls, ceilings, and floors.⁶⁸³ This construct's primary objective is to facilitate quantifying the absorption coefficient and the reverberation time of various AMs from several angles of incidence of the acoustic waves.⁴⁸⁵ For instance, in architectural acoustics, metamaterials are tested to measure the absorbing acoustic capability to limit the echo inside structures. Another significant measure that can be obtained is the diffusion coefficient, which assesses the evenness of sound distribution within an enclosure. This is particularly crucial in designing spaces such as concert halls and recording studios, where a balanced sound field is paramount for an optimal acoustic experience.

An experimental apparatus commonly used to determine the characteristics and effectiveness of acoustic metamaterials is the Kundt tube equipped with generally two⁶⁸⁴ or four microphones.⁶⁸⁵ A speaker is placed on one side of the tube. Recently, the three-microphone technique has been gaining popularity due to its advantages in complexity and cost reduction.⁶⁸⁶ 

The two-microphone configuration can extract the acoustic impedance, which is a measure of resistance to sound wave propagation in a medium, and the absorption coefficient, which indicates a percentage of acoustic energy absorbed when the wave hits the acoustic metamaterial sample. Moreover, the reflection coefficient indicates how strong the impedance mismatch is between air and AMs.

The three-microphone method, specifically the three-microphone two-load method, as highlighted in the report “Complement to standard method for measuring normal incidence sound transmission loss with three microphones,” offers a significant modification to the ASTM E2611-09 standard. It proposes a setup that involves fewer transfer function measurements and one less microphone, thereby simplifying the experimental procedure and potentially reducing costs. This method has been validated on both symmetrical homogeneous and nonsymmetrical non-homogeneous specimens. For symmetrical specimens, the process can be further simplified to a three-microphone one-load method.

The four-microphone setup allows us to estimate mainly the transmission loss and the transmission coefficient. With the transmission loss, we quantify the number of decibels that specific AMs can cut. On the other hand, the transmission coefficient specifies the portion of acoustic energy passing through the generic sample, in a specific frequency interval. The measurements taken with four microphone configurations are generally more time-consuming than the two microphone ones since it is necessary to perform a double set of measurements for each sample to get transmission loss and transmission coefficient.

A two-dimensional waveguide²⁸⁶ is a structure that extends primarily across two planes and consists of an anechoic foam encircling a two-dimensional plate, a wave generator, and one or more microphones. Several values are typically measured in these setups, including transmission and reflection coefficients, which provide information about the amount of incident acoustic waves that are either reflected or transmitted by the metamaterial. Stop bands are also important to consider as they represent frequency ranges where sound propagation is heavily reduced or completely blocked. Other values like sound pressure level and pressure field can give data on sound intensity after interacting with the metamaterial. At the same time, measurements of the phase and amplitude of the acoustic wave can provide insights into the wavefront shaping capabilities of the metamaterial. 2D waveguides are especially important to measure effects related to the bending (Fig. 14) of the wavefront,¹³⁴ which can happen with different angles, and cloaking effects, where a total or partial lack of reflection of the acoustic waves can be observed.

Anechoic chambers,⁶⁸⁷ or semi-anechoic chambers, are in particular designed rooms that minimize sound wave reflection, resulting in an environment with minimal echo and reverberation. These chambers are often used for various acoustic measurements, including those related to acoustic metamaterials. In an anechoic chamber, we can perform several measurements. For instance, we can measure sound pressure and sound intensity,⁶⁸⁸ and moreover, the transmission loss, as shown in the work of Wang et al.⁶⁸⁹ and Wu et al.⁶⁹⁰ Plus, it can also obtain near and far field measurements of ultrasonic wave beams, generated by AMs.⁶⁸¹ 

Acoustic metamaterials can revolutionize the way we build anechoic rooms. Normally, these rooms require a lot of space filled with foam wedges to absorb sound. However, metamaterials can do the same job without taking up as much room. They can absorb sound effectively, with an absorption coefficient higher than 0.85 for frequencies above 100 Hz. This makes them a great alternative to traditional methods, as shown in a study by Yang et al.⁶⁹¹ 

Electroacoustic resonators, comprising a combination of loudspeakers and microphones, are employed for noise reduction in enclosed spaces. Traditionally, these resonators utilize a linear control model, resulting in predictable responses to sound variations. However, recent research indicates the potential to shift to a nonlinear control approach by employing a model-inversion technique. This technique adjusts the loudspeaker's electrical current based on sound pressure measurements, allowing for a more dynamic response.⁶⁹² A state-space representation model and a control algorithm to achieve a Duffing-like nonlinear response, which has proven to be particularly effective at noise reduction.¹⁶⁷ The findings suggest that this advanced nonlinear control technique could lead to more efficient noise reduction devices with broader applications across various domains, including buildings, vehicles, and industrial environments.

Moreover, it is possible to create new cutting-edge acoustic devices such as high-performance lenses, cloaks, and absorbers. These innovations adopt designs from electromagnetic metamaterials, but with the added benefit of the slower propagation speed of acoustic waves, enabling more precise control and enhanced stability.¹⁸⁶ For example, the dynamic tunability of AMs makes them ideal for applications like unidirectional sound transmission and acoustic diodes, thanks to their mechanical robustness and the ability to adjust properties electronically.¹⁴⁴ Moreover, the capability to change the properties of these materials on the fly, without altering their physical structure, allows for versatile functions such as sharper imaging through nonlinear harmonic responses.¹⁷⁶ 

The potential applications of these materials are extensive. They include sound insulation, acoustic cloaking, wavefront engineering, and noise reduction across various fields like underwater acoustics, medical ultrasound, and architectural acoustics.¹⁶⁶ Despite the progress, practical challenges such as achieving real-time control and maintaining stability across a wide frequency range persist. Nevertheless, research has shown that active acoustic metamaterial cells, which can be controlled in real-time, offer significant advantages over their passive counterparts, setting the stage for next-generation acoustic devices.⁶⁹³ 

Additionally, the development of non-reciprocal acoustic metamaterials using nonlinear electronic circuits offers a compact and efficient solution for applications that require unidirectional sound propagation. This includes noise mitigation in aircraft fuselages and underwater vehicles, where controlling the direction of sound is crucial.¹⁴³ Experimental prototypes and theoretical models continue to validate these designs' robustness and efficiency, suggesting a promising future for active control strategies in advanced acoustic applications.⁶⁹⁴ 

Recent advancements in acoustics have underscored the potential of innovative plasma-based technologies for sound control, as evidenced by two pivotal studies. The first study, titled “Ultrabroadband sound control with deep-subwavelength plasmacoustic metalayers,” explores the utilization of plasmacoustic metalayers for comprehensive sound control across a wide frequency range.⁶⁹⁵ These metalayers, consisting of small layers of air plasma, achieve perfect sound absorption and tunable acoustic reflection from several hertz to the kilohertz range. This approach effectively overcomes the limitations of traditional noise absorption techniques, such as porous materials and acoustic resonators, which typically underperform below 1 kHz and are inherently narrowband. The technology offers promising applications in noise control, audio engineering, room acoustics, imaging, and metamaterial design.

Complementing this, a second study introduces the creation of a plasma electroacoustic actuator by using corona discharge to generate sound.⁶⁹⁶ This actuator provides a compact and efficient solution for active noise control. The design features two metallic electrodes separated by an air gap, with one electrode comprising thin wires and the other a coarse grid. Upon the application of high voltage, the system produces positive ions that interact with air particles, functioning as a dipolar acoustic source with minimal inertia. The theoretical and experimental analysis demonstrates the actuator's capability to significantly reduce noise across a wide frequency spectrum. Together, these studies underscore the transformative potential of plasma-based technologies in advancing sound control methodologies

Integrating active control mechanisms in acoustic metamaterials marks a significant advancement in terms of acoustic engineering. By overcoming the limitations of passive designs, these tunable metamaterials provide a versatile and effective means to manipulate acoustic waves, potentially transforming various acoustic technologies and applications.⁶⁹⁷ However, further research is necessary to address current challenges and fully harness these innovative materials' potential for practical use.⁶⁹⁸ 

Active acoustic metamaterials offer enhanced control and tunability over passive counterparts but present significant disadvantages that must be considered for specific applications. The primary drawbacks include increased complexity and cost due to the integration of external control mechanisms, sensors, actuators, and power supplies.⁶⁹⁹ These elements lead to higher production and maintenance expenses and increased energy consumption, necessitating a continuous power supply, which can be problematic in power-constrained environments. Additionally, these materials face issues with battery life, reliability, and stability under varying conditions, as well as susceptibility to wear and tear, demanding regular maintenance.³⁹⁹ The complexity of control systems, involving sophisticated algorithms and real-time processing, adds further challenges, including potential latency and response time delays. The risk of malfunction due to numerous electronic and mechanical components, along with environmental constraints and electromagnetic interference, further limits their applicability. Finally, the design and fabrication of active elements require advanced techniques, posing additional hurdles. These factors collectively present substantial challenges that need careful consideration against the benefits of active acoustic metamaterials for their intended applications. However, it is essential to recognize the significant advantages that active acoustic metamaterials bring to the table. Unlike passive systems, they can achieve low-frequency absorption in limited spaces and offer real-time tunability. This adaptive capability means they can adjust to changing conditions, which passive systems simply cannot match. Yes, they are more complex and expensive, which explains why they are not widespread, but in some specialized cases, they can be game-changers.

Another important issue to address is their frequency range performance. Active acoustic metamaterials typically perform well only over a limited frequency range, especially at low frequencies. When they do manage to cover low frequencies, it is often at the cost of a very narrow bandwidth due to resonance or requiring much space. This is a significant limitation that needs to be discussed more prominently.

It's also worth noting that the benefits of active control are not limited to piezoelectric materials. Electroactive transducers, like loudspeakers, can achieve similar effects. In practical applications, energy from the system is often dissipated into an electric circuit to optimize target impedance, similar to what is done with distributed cells. This highlights the versatility and potential of active acoustic metamaterials beyond just piezoelectric systems.

Active acoustic metamaterials mark a substantial improvement over their passive counterparts, offering dynamic capabilities for sound control, including noise suppression, adaptable acoustic devices, and on-the-fly sound adjustment. These materials incorporate active elements to respond to environmental variations, improving their functionality in fields such as medical technology and the automotive sector.^700,701

Modern design strategies emphasize the integration of microactuators and piezoelectric components to instantly modify acoustic properties.^212,701 Advanced materials such as shape memory alloys and magnetorheological materials further enhance adjustability through external stimuli.²¹² Additive manufacturing methods, especially multimaterial 3D printing, enable the construction of intricate designs that were previously unattainable by conventional approaches.⁷⁰¹ 

In engineering contexts, active metamaterials surpass passive solutions in soundproofing and adaptive noise barriers by actively tuning acoustic properties for optimal performance. However, issues such as energy consumption and the complexity of embedding active components persist.^212,701 Future advancements might use machine learning tools to refine design settings for greater adaptability.⁷⁰¹ 

Traditional electromechanical devices, including loudspeakers, play a key role in active acoustic metamaterials by efficiently transforming electrical inputs into mechanical vibrations, allowing precise control of sound waves for applications such as noise reduction and adaptive acoustic environments.⁷⁰² Cutting-edge innovations in lightweight designs, such as membrane-based loudspeakers, leverage thin membranes to reduce weight and size, improving portability.²⁷⁶ Furthermore, plasma-based acoustic sources provide ultralight and efficient options that could revolutionize optimized active metamaterial configurations.⁷⁰³ Incorporating multiple distributed sensors and actuators poses considerable challenges in managing signal processing. The handling of associated data requires rapid and adaptive algorithms capable of real-time adjustments to fluctuating environmental conditions.^704,705 Using sophisticated signal processing methods, particularly those enhanced by machine learning, can greatly enhance computational performance and flexibility, critical for the advancement of active and tunable acoustic metamaterials.¹⁵⁰ 

Passive acoustic metamaterials are an emerging field that promises to revolutionize acoustics and noise control technologies. However, several complex challenges in this exciting and dynamic field of study should be addressed to enable the real-world implementation of AMs. These challenges span multiple domains, ranging from design constraints to manufacturing complications and inherent physical limitations. A significant concern in the domain of acoustic metamaterials is the broadband behavior and tunability issue. Most AMs are designed to operate efficiently within a targeted frequency range, which essentially limits their application in scenarios requiring broadband noise reduction from the low to the high-frequency range. Moreover, some applications require acoustic tailorability, like in the case of biomedical applications where ultrasonic beam precision is fundamental. Thus, designing acoustic metamaterials with adaptable tunability and broadband behavior may require both theoretical advances and innovative material design.

The potential for energy harvesting in acoustic metamaterials is a tantalizing prospect but poses unique challenges. The amount of energy that can be harvested from sound waves is relatively small compared to the amount of sonic energy hitting a generic piezoelectric metamaterial. This requires the development of metamaterials with the ability to harvest a higher amount of energy efficiently, exploiting, for instance, resonance phenomena. Currently, the integration of AMs, with piezoelectric patches is a good starting point, exhibiting promising performances for the future in terms of acoustic energy exploitation.

The directionality problem also poses a significant obstacle in developing acoustic metamaterials. Unlike electromagnetic waves, sound waves spread out in all directions when they encounter a medium. This omnidirectional characteristic of sound poses challenges in designing metamaterials that can precisely guide sound propagation in a specific direction. The need for directionality is imperative in many applications, such as medical imaging and acoustic antennas.

Another challenge faced is the lack of ready-to-use design equations for acoustic metamaterials, which allow engineers to design and use this relatively new technology more efficiently. Furthermore, the manufacturing difficulties and associated costs also pose substantial challenges in developing acoustic metamaterials. These materials often involve the use of unconventional materials and intricate geometric designs that are not easy to produce using standard techniques. Furthermore, the processes involved are often time-consuming and resource-intensive, making the production of AMs potentially too expensive. High production costs and manufacturing complications may limit the widespread adoption of these materials unless new and more effective solutions are found to streamline production on an industrial scale.

Multifunctionality, or lack thereof, is another pressing issue in acoustic metamaterials. While these materials demonstrate remarkable properties such as a negative refractive index or superabsorption, their single-function nature may limit their applicability. There is a potential growing demand for metamaterials with multiple functionalities, such as simultaneous sound absorption and energy harvesting or sound absorption with airflow and/or heat resistance. The challenge lies in the design of such multifunctional materials, which must strike a balance between competing design requirements and material properties.

Finally, the thickness of acoustic metamaterials also constitutes a significant impediment to their applicability. The thickness of these materials often needs to be proportional to the wavelength of sound for effective operation, which results in a bulky apparatus for low-frequency sound waves. These size constraints may limit the use of acoustic metamaterials in space-sensitive applications. However, the attempt to miniaturize acoustic metamaterials is complex, requiring a careful balance between reducing size and maintaining performance. It involves theoretical considerations and experimental campaigns. These structures can significantly reduce the thickness of the material while maintaining its ability to control sound propagation. Figure 15, accompanied by Table IV, which provides explanatory details, and Table V, showcase a variety of representative works published on the subject of layer or metamaterial thickness. In Fig. 15, on the y-axis, a series of coefficients C, when multiplied by the wavelength, determine the necessary thickness of the AM for the geometry used. In Table V, we have the AMs layer thickness, independent from the wavelength, as in Fig. 15. Here, the thickness layer is displayed directly in millimeters (Table V). Interestingly, it is worth noting that the five examples involving the use of fractal geometries,^{1,453,548,577,582} which are discussed in the context of acoustic performance, require a comparatively thinner metamaterial layer compared to the other ones. The other internal geometries reported include the coiled, helicoidal, maze, and honeycomb configurations. Among these, fractal geometry appears to be one of the most effective solutions for the internal structure of the metamaterial layer. However, searching for multifunctional acoustic metamaterials inspires and challenges interdisciplinary collaborations, combining acoustics, materials science and engineering. Because of that, multidisciplinary knowledge is needed to address multiphysics challenges.

Acoustic metamaterials encounter various structural difficulties that influence their multifunctional capabilities, geometric constraints, and fabrication challenges. A primary concern is directional control, where the omnidirectional nature of sound waves complicates their use in domains such as medical imaging and spatial audio systems.⁷¹⁰ Furthermore, these materials are often optimized for specific frequency ranges, limiting their effectiveness in low-frequency applications such as underwater acoustics, as their design typically sacrifices bandwidth to achieve reduced spatial dimensions.^211,538

In addition, elaborate structures needed for precise sound manipulation pose significant challenges for manufacturing and scalability. The reliance on advanced fabrication methods, such as additive manufacturing, to achieve the necessary structural accuracy makes production complex and expensive, limiting widespread application.⁶⁴⁵ However, advances in multifunctional metamaterials are promising, especially in the integration of sound absorption, energy harvesting, and thermal insulation, which have great potential for use in aerospace and industrial machinery.⁷¹¹ 

Future work in the field of advanced material innovation should emphasize utilizing the development of intelligent and adaptive materials capable of adjusting their properties in real-time to adapt to changing environmental conditions. These materials can utilize machine learning for customizable designs, expanding their use in various domains, including acoustic metamaterials, which benefit from precise production methods such as multi-material 3D printing and nanoscale lithography.^163,712 Furthermore, studying topological and nonreciprocal metamaterials is imperative, as these materials enable reliable sound manipulation and directional wave propagation, critical for use in complex settings.^713,714

Collaboration between disciplines is essential to progress in these areas, merging knowledge from materials science, engineering, and artificial intelligence to improve material capabilities and utility.^{324,715–717} The promising role of smart materials in cutting-edge applications, such as monitoring of structural integrity and energy-efficient systems, highlights the importance of continued investigation and innovation within this rapidly evolving field.^718,719

This comprehensive review has summarized various discoveries and insights into acoustic metamaterials, providing an overview of the materials and manufacturing methodologies underpinning this field. Moreover, the smart blending of various geometric configurations in the creation of hybrid metamaterials further illustrates the complexity and potential of this area.

The synthesis of this literature review points out that the performance of acoustic metamaterials is greatly enhanced by employing intricate fractal formations and different geometric configurations. This recognized variety has necessitated a new classification system, complementary to the existing ones, focused on the quantity and variety of channels inherent in the geometry of metamaterials. Further exploration of combining various geometric configurations to synthesize more intricate or hybrid acoustic metamaterials revealed interesting design intersections, such as combining geometries with and without acoustic apparatuses, such as acoustic foams or resonators. Furthermore, this study also describes the materials and machines used in manufacturing, underlining achievable precision and applications.

The principal test rigs and testing rooms used in research with the relative measurable quantities to get their characterization and performance evaluation are outlined for the acoustic measurements.

However, acoustic metamaterials present many opportunities but are not without challenges. This review identified complexities that may hinder their successful application, including their broadband behavior, tunability, and directionality for the wave entering and leaving the metamaterial cell, plus energy harvesting efficiency and multi-functionality. A critical challenge is that acoustic metamaterials often exhibit effects in a relatively narrow bandwidth spectrum, which poses difficulties in fine-tuning or translating the same wanted metamaterial effects to other frequencies. Furthermore, the directionality angle of an acoustic wave entering the metamaterial cell can significantly alter the effectiveness of the AMs. Achieving accuracy in directing the acoustic wavefront at a predetermined angle can also be problematic. Furthermore, the thickness of the metamaterial layer can have a significant impact on phenomena such as sound attenuation, particularly within thin metamaterial layers. This effect arises from the direct correlation between the volume occupied by the metamaterial and the frequency spectra that are affected. Although this challenge might be addressed with the use of active acoustic metamaterials, precise tuning remains critical for broadband performance.

Another challenge lies in the limited energy harvesting of acoustic metamaterials, even when resonance phenomena are strategically employed to enhance the collected vibrations. A further consideration is a move toward multifunctionality in AMs, to solve more than one issue within the same metamaterial layer, for instance, vibroacoustic solutions for industrial machinery.

Finally, clarifying the differences between acoustic and elastic metamaterials is essential for advancing research and development in wave manipulation technologies, ensuring that each type is effectively utilized in its specific application domain.

G.C. acknowledges the support of UK EPSRC through the EPSRC Centre for Doctoral Training in Advanced Composites for Innovation and Science (No. EP/L016028/1) and Charles P. Macleod for the technical assistance. F.S. and G.C. acknowledge the support of ERC-2020-AdG-NEUROMETA (No. 101020715). M.O. acknowledges the support of EIPHI (No. ANR-17-EURE-0002). V.T. acknowledges funding from the EPSRC (No. EP/R01650X/1).

The authors declare no competing interest.

G. Comandini: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Visualization (lead); Writing - original draft (lead); Review & editing (equal). M. Ouisse: Review & editing (equal); Supervision (supporting); Conceptualization (supporting). V. P. Ting: Review & editing (equal); Supervision (supporting); Conceptualization (supporting). F. Scarpa: Funding acquisition (lead); Review & editing (lead); Conceptualization (equal); Supervision (lead).

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