Acoustic metamaterials, sometimes referred to as sonic or phononic crystals, are architected materials designed to manipulate sound waves or phonons in gases, liquids, and solids. By tailoring effective parameters such as bulk modulus (β), density (ρ), and in some cases chirality, they can be engineered to transmit, trap, or attenuate waves at selected frequencies, functioning as acoustic resonators when local resonances dominate. Within the broader field of mechanical metamaterials, acoustic metamaterials represent the dynamic branch where wave control is the primary goal. They have been applied to model large-scale phenomena such as seismic waves and earthquake mitigation, as well as small-scale phenomena such as phonon behavior in crystals through band-gap engineering. This band-gap behavior mirrors the electronic band gaps in solids, enabling analogies between acoustic and quantum systems and supporting research in optomechanics and quantum technologies. In mechanics, acoustic metamaterials are particularly relevant for designing structures that mitigate vibrations, shield against blasts, or manipulate wave propagation in civil and aerospace systems.
History
Acoustic metamaterials trace their origins to the broader field of metamaterials. The concept of artificial media with unusual effective properties was first proposed by Victor Veselago in 1967 and later advanced by John Pendry in the late 1990s, leading to the first realization of negative-index electromagnetic materials in 2000. Building on these developments, the acoustic counterpart emerged the same year, when Liu and colleagues demonstrated locally resonant sonic materials composed of heavy inclusions in a soft matrix, showing band gaps at subwavelength scales. This work is widely regarded as the foundation of acoustic metamaterials. Subsequent studies expanded the field by adapting ideas from electromagnetic metamaterials, including analogs of split-ring resonators, and by achieving double-negative parameters (simultaneously negative bulk modulus βeff and density ρeff). This was followed by transposing the behavior of the split-ring resonator to research in acoustic metamaterials. Then a group of researchers presented the design and test results of an ultrasonic metamaterial lens for focusing 60 kHz. These progressions enabled applications such as ultrasonic metamaterial lenses, vibration control, and seismic shielding, firmly establishing acoustic metamaterials as a dynamic branch of mechanical metamaterials.Research in acoustic metamaterials has the same goal of broader material responses with sound waves. A facility of the NSF provides added material to the original paper - The Work of Jagadis Chandra Bose: 100 Years of MM-Wave Research.
Acoustical engineering is typically concerned with noise control, medical ultrasound, sonar, sound reproduction, and how to measure some other physical properties using sound. With acoustic metamaterials the direction of sound through the medium can be controlled by manipulating the acoustic refractive index. Therefore, the capabilities of traditional acoustic technologies are extended, for example, eventually being able to cloak certain objects from acoustic detection.
The first successful industrial applications of acoustic metamaterials were tested for aircraft insulation.
Basic principles Properties of acoustic metamaterials usually arise from structure rather than composition, with techniques such as the controlled fabrication of small inhomogeneities to enact effective macroscopic behavior.
Bulk modulus and mass density
The bulk modulus β is a measure of a substance's resistance to uniform compression. It is defined as the ratio of pressure increase needed to cause a given relative decrease in volume.
The mass density (or just density) of a material is defined as mass per unit volume and is expressed in grams per cubic centimeter (g/cm3). In all three classic states of matter—gas, liquid, or solid—the density varies with a change in temperature or pressure, with gases being the most susceptible to those changes. The spectrum of densities is wide-ranging: from 1015 g/cm3 for neutron stars, 1.00 g/cm3 for water, to 1.2×10−3 g/cm3 for air. Other relevant parameters are area density which is mass over a (two-dimensional) area, linear density - mass over a one-dimensional line, and relative density, which is a density divided by the density of a reference material, such as water.
For acoustic materials and acoustic metamaterials, both bulk modulus and density are component parameters, which define their refractive index. The acoustic refractive index is similar to the concept used in optics, but it concerns pressure or shear waves, instead of electromagnetic waves.
Theoretical model Acoustic metamaterials or phononic crystals can be understood as the acoustic analog of photonic crystals: instead of electromagnetic waves (photons) propagating through a material with a periodically modified optical refractive index (resulting in a modified speed of light), the phononic crystal comprises pressure waves (phonons) propagating through a material with a periodically modified acoustic refractive index, resulting in a modified speed of sound.
In addition to the parallel concepts of refractive index and crystal structure, electromagnetic waves and acoustic waves are both mathematically described by the wave equation.
The simplest realization of an acoustic metamaterial would constitute the propagation of a pressure wave through a slab with a periodically modified refractive index in one dimension. In that case, the behavior of the wave through the slab or 'stack' can be predicted and analyzed using transfer matrices. This method is ubiquitous in optics, where it is used for the description of light waves propagating through a distributed Bragg reflector.
Negative refractive index acoustic metamaterials In certain frequency bands, the effective mass density and bulk modulus may become negative. This results in a negative refractive index. Flat slab focusing, which can result in super resolution, is similar to electromagnetic metamaterials. The double negative parameters are a result of low-frequency resonances. In combination with a well-defined polarization during wave propagation; k = |n|ω, is an equation for refractive index as sound waves interact with acoustic metamaterials (below):
The inherent parameters of the medium are the mass density ρ, bulk modulus β, and chirality k. Chirality, or handedness, determines the polarity of wave propagation (wave vector). Hence within the last equation, Veselago-type solutions (n2 = u*ε) are possible for wave propagation as the negative or positive state of ρ and β determine the forward or backward wave propagation.
In electromagnetic metamaterials negative permittivity can be found in natural materials. However, negative permeability has to be intentionally created in the artificial transmission medium. For acoustic materials neither negative ρ nor negative β are found in naturally occurring materials; they are derived from the resonant frequencies of an artificially fabricated transmission medium, and such negative values are an anomalous response. Negative ρ or β means that at certain frequencies the medium expands when experiencing compression (negative modulus), and accelerates to the left when being pushed to the right (negative density).
Electromagnetic field vs acoustic field The electromagnetic spectrum extends from low frequencies used for modern radio to gamma radiation at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. In comparison, infrasonic frequencies range from 20 Hz down to 0.001 Hz, audible frequencies are 20 Hz to 20 kHz and the ultrasonic range is above 20 kHz.
While electromagnetic waves can travel in vacuum, acoustic wave propagation requires a medium.
Mechanics of lattice waves
In a rigid lattice structure, atoms exert force on each other, maintaining equilibrium. Most of these atomic forces, such as covalent or ionic bonds, are of electric nature. The magnetic force, and the force of gravity are negligible. Because of the bonding between them, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength λ is marked.
There is a minimum possible wavelength, given by the equilibrium separation a between atoms. Any wavelength shorter than this can be mapped onto a long wavelength, due to effects similar to aliasing.
Research and applications Applications of acoustic metamaterial research include seismic wave reflection and vibration control technologies related to earthquakes, as well as precision sensing. Phononic crystals can be engineered to exhibit band gaps for phonons, similar to the existence of band gaps for electrons in solids and to the existence of electron orbitals in atoms. However, unlike atoms and natural materials, the properties of metamaterials can be fine-tuned (for example through microfabrication). For that reason, they constitute a potential testbed for fundamental physics and quantum technologies. They also have a variety of engineering applications, for example they are widely used as a mechanical component in optomechanical systems.Eichenfield, M., Chan, J., Camacho, R. et al. Optomechanical crystals . Nature 462, 78–82 (2009). https://doi.org/10.1038/nature08524
Sonic crystals In 2000, the research of Liu et al. paved the way to acoustic metamaterials through sonic crystals, which exhibit spectral gaps two orders of magnitude smaller than the wavelength of sound. The spectral gaps prevent the transmission of waves at prescribed frequencies. The frequency can be tuned to desired parameters by varying the size and geometry.
The fabricated material consisted of high-density solid lead balls as the core, one centimeter in size and coated with a 2.5-mm layer of rubber silicone. These were arranged in an 8 × 8 × 8 cube crystal lattice structure. The balls were cemented into the cubic structure with an epoxy. Transmission was measured as a function of frequency from 250 to 1600 Hz for a four-layer sonic crystal. A two-centimeter slab absorbed sound that normally would require a much thicker material, at 400 Hz. A drop in amplitude was observed at 400 and 1100 Hz.
The amplitudes of the sound waves entering the surface were compared with the sound waves at the center of the structure. The oscillations of the coated spheres absorbed sonic energy, which created the frequency gap; the sound energy was absorbed exponentially as the thickness of the material increased. The key result was the negative elastic constant created from resonant frequencies of the material.
Projected applications of sonic crystals are seismic wave reflection and ultrasonics.
Split-ring resonators for acoustic metamaterials
In 2004 split-ring resonators (SRR) became the object of acoustic metamaterial research. An analysis of the frequency band gap characteristics, derived from the inherent limiting properties of artificially created SRRs, paralleled an analysis of sonic crystals. The band gap properties of SRRs were related to sonic crystal band gap properties. Inherent in this inquiry is a description of mechanical properties and problems of continuum mechanics for sonic crystals, as a macroscopically homogeneous substance.