A Review of Acoustic Metamaterials and Phononic Crystals

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Review A Review of Acoustic Metamaterials and Phononic Crystals

Junyi Liu 1, Hanbei Guo 2 and Ting Wang 3,4,5,* 1 College of Engineering, Huazhong Agricultural University, Wuhan 430070, China; [email protected] 2 Wuhan Second Ship Design and Research Institute, Wuhan 430064, China; [email protected] 3 School of Naval Architecture & Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China 4 Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, China 5 Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamic (HUST), Wuhan 430074, China * Correspondence: [email protected]

Received: 20 March 2020; Accepted: 10 April 2020; Published: 15 April 2020

Abstract: As a new kind of artificial material developed in recent decades, metamaterials exhibit novel performance and the promising application potentials in the field of practical engineering compared with the natural materials. Acoustic metamaterials and phononic crystals have some extraordinary physical properties, effective negative parameters, band gaps, negative refraction, etc., extending the acoustic properties of existing materials. The special physical properties have attracted the attention of researchers, and great progress has been made in engineering applications. This article summarizes the research on acoustic metamaterials and phononic crystals in recent decades, briefly introduces some representative studies, including equivalent acoustic parameters and extraordinary characteristics of metamaterials, explains acoustic metamaterial design methods, and summarizes the technical bottlenecks and application prospects.

Keywords: acoustic metamaterial; phononic crystal; effective negative parameters; band gaps; research overview

1. Research Background Noise is one of the four major public hazards in the world which seriously affect people’s daily life and work. Noise is caused by the random vibration of objects, mainly mechanical vibrations. Mechanical vibration itself is also harmful to mechanical structures and engineering equipment, and even more serious for precision machine tools and instruments. Mechanical vibrations not only cause fatigue and wear of mechanical equipment and affects its performance, but also greatly shorten the service life of the equipment. At the same time, noise generated by mechanical equipment will reduce people’s quality of life, threatening their well-being. The most direct harm to the human body caused by noise is damage to hearing, and damage to the central nervous system of a person. Low frequency vibrations and noise are able to travel further, so if captured by the enemy sonar, they can expose and endanger underwater vehicles. Therefore, in order to improve people’s living environment and survival capability, it is necessary to develop vibration suppression and noise reduction techniques [1]. There are three main ways to control vibration and noise [2]. The ﬁrst is to control the source, design and manufacture products with small noise, low vibration or silent; the second is to control the propagation path, mainly via isolation, absorption and attenuation, etc.; the third is to protect the receiver. As people’s requirements for product functions become higher and higher, product structures

Crystals 2020, 10, 305; doi:10.3390/cryst10040305 www.mdpi.com/journal/crystals Crystals 2020, 10, 305 2 of 26 become more and more complex, and it has become increasingly difficult to reduce noise from the source. Moreover, the receivers are often changing their locations, so techniques to protect the receiver are not reasonable. Therefore, it is a more sensible choice to try to reduce vibration and noise in the propagation path. The ways to control noise are divided into active controls and passive controls. Here we mainly discuss passive control methods. In passive control, various vibration suppression and sound insulation materials have been developed rapidly in recent years, and people have also designed various vibration isolators and sound absorbers. There are some limitations in traditional vibration reduction techniques [3], such as vibration insulation, dynamic vibration absorption and damping vibration absorption. The equipment must be larger for vibration insulation technology. What’s more, it shows a narrow frequency band and is greatly affected by the environment. For dynamic vibration absorption, the disadvantages are that it focuses on a large mass and has a narrow frequency band too. However, the damping vibration absorption only has an impact on the intermediate and the high frequencies. In the low frequencies, the effect is poor. Traditional sound insulation materials generally use rigid plates, in order to isolate low-frequency sound waves of hundreds of Hertz [4]. A concrete partition with a thickness of 1 m needs to be designed. Traditional sound-absorbing materials are mostly porous structures, such as organic fiber materials, rock wool, mineral wool, etc. These materials mainly rely on small holes from the surface to the inside to attenuate sound waves [5]. According to the frequency range, sound waves can be divided into the following types: low frequency, intermediate frequency and high frequency. Noise in the low-frequency range has strong penetrating power and dissipates slowly during propagation, making it difficult to control the sound waves; low-frequency sound waves can also resonate with certain important organs of the human body, affecting health. Therefore, it is necessary to strengthen the research on low frequency noise and vibration control. No matter whether the above-mentioned vibration-isolating materials or sound-absorbing materials are considered, they basically absorb medium and high frequency sound waves, and the effect on isolating low-frequency noise is not significant. In recent decades, research on artificial composite materials has become the focus of many scholars, and acoustic metamaterials and phononic crystals have been discussed in this context.

2. Basic Concept of Acoustic Metamaterials and Phononic Crystals

2.1. Proposals for Acoustic Metamaterials and Phononic Crystals “Metamaterials” refer to a class of materials that are artificially synthesized and have properties different from those of conventional materials without violating objective physical laws. The concept of metamaterials first came from electromagnetics. In 1968, Veselago [6], a theoretical physicist in the former Soviet Union, discovered that the dielectric properties and the permeability of electromagnetic materials are both negative, which is different from conventional materials with positive material parameters, and thus theoretically proposed electromagnetic metamaterials. This kind of optical material with negative dielectric constant and magnetic permeability has a negative refractive index. The incident and outgoing streams are on the same side of the normal line instead of the opposite side. The wave vector K, the electric field intensity E, and the magnetic field intensity H constitute a left-handed relationship. This material is called a left-handed material, also known as a double negative material. An analog optical metamaterial is designed using a structure much smaller than the sub-wavelength size to obtain an acoustic metamaterial [7]. The effective parameters of the acoustic metamaterial become negative within specific frequency ranges, called band gaps (see Figures1 and2). For natural materials, the material parameters such as the mass density, Young’s modulus, Poisson’s ratio, are positive in natural circumstances, while with artificial construction, the effective material parameters may turn negative within specific frequency ranges. When mechanical waves propagate in this type of structure, due to the modulation of the structure, the propagation of mechanical waves is hindered, thereby achieving a sound absorption and vibration isolation effect [8]. Crystals 2020, 10Crystals, x FOR2020 ,PEER10, 305 REVIEW 3 of 26 3 of 26 Crystals 2020, 10, x FOR PEER REVIEW 3 of 26

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(a) (b) (a) (b) Figure 1. A unit cell of a typical acoustic metamaterial plate: (a) front view; (b) perspective view [7]. Figure 1. A unit cell of a typical acoustic metamaterial plate: (a) front view; (b) perspective view [7]. Figure 1. A unit cell of a typical acoustic(a) metamaterial plate: ((b)a) front view; (b) perspective view [7].

Figure 1. A unit cell of a typical acoustic metamaterial plate: (a) front view; (b) perspective view [7].

(a) (b) (a) (b) Figure 2. Dispersion(a) surfaces and band gap of the above metamaterial(b): (a) dispersion surfaces; (b) Figure 2. DispersionFigurestopband 2. Dispersion(gray surfaces rectangle) surfaces and [7] . band and band gap gap of ofthe the above above metamaterial:metamaterial (a): dispersion(a) dispersion surfaces; surfaces; (b) (b) stopband (gray rectangle) [7]. Figurestopband 2. Dispersion(gray rectangle) surfaces [7]. and band gap of the above metamaterial: (a) dispersion surfaces; (b) In the 1990s, the concept of a phononic crystal was modeled after the concept of a photonic stopband (grayIn the rectangle) 1990s, the concept [7]. of a phononic crystal was modeled after the concept of a photonic crystal. crystal. By artificially designing the structure and defects of the photonic crystal, the propagation of By artificially designing the structure and defects of the photonic crystal, the propagation of light waves In the 1990s,light waves the can concept be controlled of a phononic[9–11]. Later, crystal it was discovered was modeled that elastic after waves the propagate concept inof a a photonic can be controlled [9–11]. Later, it was discovered that elastic waves propagate in a periodic elastic In the 1990s,periodic theelastic concept composite of medium, a phononic which also crystal produces was a similar modeled band gap after phenomenon the concept (see Figure of a photonic crystal. By artificiallycomposite medium, designing which alsothe producesstructure a similar and banddefects gap phenomenonof the photonic (see Figure crystal,3)[ 12 ].the In 1992,propagation of crystal. By artificially3) [12]. In 1992, designing Sigalas et theal. [13] structure buried spherical and defects objects ofin athe matrix photonic material crystal,in a periodic the latticepropagation of light wavesSigalas can et be al. controlled[13] buried spherical [9–11] objects. Later, in a matrix it was material discovered in a periodic that lattice elastic structure waves arrangement. propagate in a structure arrangement. Under the action of external sound waves, the composite produced an light wavesUnder can the be action controlled of external [9 sound–11] waves,. Later, the compositeit was discovered produced an acoustic that elastic band gap waves phenomenon, propagate in a periodic elasticacoustic composite band gap medium, phenomenon, which and thus also phononic produces crystals a similar were theoretically band gap synthesized phenomenon for the (see Figure periodic elasticand thuscomposite phononic medium, crystals were which theoretically also produces synthesized a forsimilar the first band time. gap Kushwaha phenomenon et al. [14] (see Figure 3) [12]. In 1992,first time. Sigalas Kushwaha et al. et [13] al. [14] buried formally spherical proposed theobjects concept in of aphononic matrix crystals material in 1993, in thata periodic is, a lattice formally proposed the concept of phononic crystals in 1993, that is, a material with a periodic elastic 3) [12]. In 1992,material Sigalas with aet periodic al. [13] elastic buried medium spherical structure objectswith a phononic in a matrix forbidden material band. Similarly, in a periodic by lattice structure arrangement.medium structure Under with a phononic the acti forbiddenon of band. external Similarly, sound by changing waves, the the internal composite structure and produced an changing the internal structure and defects of the phononic crystal, the propagation path of the elastic structure arrangement.defects of the phononic Under crystal, the the acti propagationon of external path of the sound elastic wave waves, can be the artificially composite adjusted, produced an acoustic bandwave gap can phenomenon, be artificially adjusted, and thus thereby phononic achieving crystals the purpose were of vibration theoretically isolation synthesized and noise for the acoustic bandthereby gap achieving phenomenon, the purpose and of vibration thus phononic isolation and crystals noise reduction were [theoretically15]. synthesized for the first time. Kushwahareduction [15] et. al. [14] formally proposed the concept of phononic crystals in 1993, that is, a first time. Kushwaha et al. [14] formally proposed the concept of phononic crystals in 1993, that is, a material with a periodic elastic medium structure with a phononic forbidden band. Similarly, by machangingterial with the internal a periodic structure elastic and medium defects structure of the phononic with a crystal,phononic the forbiddenpropagation band. path Similarly, of the ela sticby changingwave can the be internal artificially structure adjusted, and therebydefects of achieving the phononic the purposecrystal, the of propagation vibration isolation path of andthe ela noisestic wave can be artificially adjusted, thereby achieving the purpose of vibration isolation and noise reduction [15]. reduction [15].

(a) (b)

FigureFigure 3. 3.Proposed Proposed metamaterial metamaterial with with defect defect states states [ 12[12]]. .

The biggest difference between a phononic crystal and an acoustic metamaterial is that its resolution is limited by anisotropy and the lattice constant, but the acoustic metamaterial is basically unrestricted [16]. Phononic crystals require the lattice size to be the same order of magnitude as the wavelength in the direction of sound wave propagation, while the acoustic metamaterial is able to restrain the waves with much larger wavelength than the lattice size. The acoustic metamaterial is composed of many tiny units(a) with mechanical resonators , and each(b) unit can avoid interference with (a) (b) Figure 3. Proposed metamaterial with defect states [12]. Figure 3. Proposed metamaterial with defect states [12]. The biggest difference between a phononic crystal and an acoustic metamaterial is that its The biggest difference between a phononic crystal and an acoustic metamaterial is that its resolution is limited by anisotropy and the lattice constant, but the acoustic metamaterial is basically resolutionunrestricted is [16]limited. Phononic by anisotropy crystal sand require the lattice the lattice constant, size tobut be the the acoustic same order metamaterial of magnitude is basically as the unrestrictedwavelength in[16] the. Phononic direction crystal of sounds require wave thepropagation lattice siz,e while to be the sameacoustic order metamaterial of magnitude is able as the to wavelength in the direction of sound wave propagation, while the acoustic metamaterial is able to restrain the waves with much larger wavelength than the lattice size. The acoustic metamaterial is restraincomposed the of waves many withtiny unitsmuch with larger mechanical wavelength resonators than the, andlattice each size unit. The can acoustic avoid interference metamaterial with is composed of many tiny units with mechanical resonators, and each unit can avoid interference with

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2.2. Band Structure Calculation Method of Phononic Crystal and Acoustic Metamaterials The research on forbidden bands in phononic crystals and acoustic metamaterials mainly focuses on the calculation of elastic wave forbidden bands. At present, there are several mature methods for calculating forbidden bands, such as the transfer matrix method, the plane wave expansion method, the finite difference time domain method and the multiple scattering method, etc. The transfer matrix method [17,18] is a method of converting the wave equation into the form of a transfer matrix equation and solving the eigenvalues. This method starts with the basic equations including state parameters (stress, particle velocity, etc.) to obtain the transfer equation of a flat solid medium, then obtains the boundary conditions according to the characteristics of the surrounding medium and the intermediate medium, and finally obtains the solution of the system. This method is mainly used for the calculation of band gaps of phononic crystal and acoustic metamaterial in a one-dimensional model. Because the transfer matrix is generally small and is an analytical solution, its calculation effort is small. The plane wave expansion method [14,19–22] is the earliest and most widely used method in the theoretical analysis of phononic crystals and acoustic metamaterials. In this method, the physical quantities such as displacements and elastic constants in the wave equation are expanded by a Fourier transform in the form of superposition of plane waves in the inverse lattice space, and then the wave equation is solved to obtain the dispersion relationship between the characteristic frequency and the wave vector, that is, the band structure. The plane wave expansion method can be used for two- dimensional and three-dimensional band structures. In the calculation of solid [14,19,20], liquid- liquid [21,23,24], and solid-gas [25–28] systems, great success was achieved with the band gaps of phononic crystals and acoustic metamaterials. However, this method has certain limitations, as its convergence is slow, and it may even diverge during expansion, which makes it impossible to solve. The finite-difference time-domain method [29,30] discretizes partial differential equations by discretizing time and space, transforms the partial differential equations into differential equations, and then uses numerical calculation methods to solve all vibrations at three points during wave propagation. Parameters are a function of time. This method can simulate a variety of complex periodic structures, and can accurately simulate the non-uniformity, anisotropy, nonlinear problems and dispersion characteristics of the medium. At the same time, the method can calculate the characteristics of the band structure in the periodic structure and the transmission and reflection in the finite structure. When using this method to calculate the instantaneous nonlinear response of a phononic crystal or an acoustic metamaterial, attention should be paid to the numerical stability and convergence. Because the plane wave expansion method cannot solve the problem of solid scatterers in a liquid matrix, scholars have proposed a multiple scattering method based on the Korringa-Kohn-Rostoker (KKR) theory in the calculation of the electronic band structure [31]. The theory of multiple scattering of elastic waves holds that the frequency band structure of a crystal depends on the Mie scattering between spheres. By calculating the scattering of sound waves from other spheres incident on the Crystals 2020, 10, x FOR PEER REVIEW 5 of 26 element method. Every method has its own limitations. When analyzing different structural forms of phononic Crystalscrystal2020 and, 10, 305 acoustic metamaterials, we should consider its structural form,5 the of 26 structural complexity, the structural dimensions and others to choose the exact one. At the same time, based on the existingsurface methods, of a single we sphere, should the also characteristic develop frequency new ones equation with fast can be calculation solved [32]. and This methodgood convergence is to lay the veryfoundation suitable for for the the calculation follow-up of phononic research. crystals with special structures. For this kind of band structure, general plane waves cannot give accurate solutions. The multiple scattering method also has disadvantages. It is complicated to calculate and difficult to use. 3. EquivalentIn Acoustic addition to Parameters the above methods, of Acoustic there are Metamaterials some other methods to calculate the band structures of phononic crystals and acoustic metamaterials, such as the finite element method and the spectral Mass density ρ and bulk elastic modulus K are two key parameters of acoustic materials. They element method. Every method has its own limitations. When analyzing different structural forms of determinephononic the propagation crystal and acousticcharacteristics metamaterials, of acoust we shouldic waves consider in the its structural medium. form, Both the the structural sound velocity and the characteristiccomplexity, the structuralimpedance dimensions of a medium and others are to chooseexpressed the exact by one.these At thetwo same parameters: time, based onthe speed of sound 𝑐=the existing𝐾/𝜌and methods, the characteristic we should also impedance develop new onesof the with medium fast calculation𝑍= and𝐾𝜌. good In conventional convergence to materials, lay the foundation for the follow-up research. both the mass density ρ and the bulk elastic modulus K are positive and depend on the material composition3. Equivalent and microstructure Acoustic Parameters of the of media. Acoustic Howeve Metamaterialsr, if local resonance units are introduced into the material toMass enhance density ρtheand acoustic-matter bulk elastic modulus interactK are twoion, key it parameters is possible of acoustic that the materials. equivalent They acoustic parametersdetermine can become the propagation negative characteristics within a specific of acoustic frequency waves in the range medium. not Both found the sound in natural velocity materials. and the characteristic impedance of a medium are expressed by these two parameters: the speed Based on the resonancep mechanism, we can achieve negative equivalent pmass density and equivalent of sound c = K/ρ and the characteristic impedance of the medium Z = Kρ. In conventional bulk modulus,materials, as well both theas extraordinary mass density ρ and acoustic the bulk parameters elastic modulus suchK are as positive negative and refractive depend on theindex [33]. material composition and microstructure of the media. However, if local resonance units are introduced 4. Typicalinto Types the material of Acoustic to enhance Metamaterials the acoustic-matter and interaction, Phononic it is Crystals possible that the equivalent acoustic parameters can become negative within a specific frequency range not found in natural materials. In 2000,Based Liu on theet resonanceal. [32] mechanism,used the we"core-shell can achieve structure" negative equivalent for the mass first density time and to equivalentmake an acoustic metamaterial.bulk modulus,Based ason well the as extraordinary"core-shell acousticstructure" parameters concept such asscholars negative have refractive further index [33developed]. the following4. other Typical structures. Types of Acoustic Metamaterials and Phononic Crystals In 2000, Liu et al. [32] used the “core-shell structure” for the first time to make an acoustic metamaterial. 4.1. Structures Composed of Functional Units with Certain Mechanical Characteristics Based on the "core-shell structure" concept scholars have further developed the following other structures.

The spring-mass4.1. Structures Composed model ofis Functional periodically Units with arranged Certain Mechanical on a two-dimensional Characteristics thin plate, which is a common local resonance model. This model has a simple structure, and the resonance of the vibrator The spring-mass model is periodically arranged on a two-dimensional thin plate, which is a is also easycommon to adjust. local resonance It is frequently model. This modelused hasin areal simple life. structure, There and are the three resonance main of the components vibrator is of this structure.also Component easy to adjust. 1 is It a is mass frequently of mass used inm real, component life. There are 2 threeis a matrix main components of mass ofM this, and structure. component 3 is a spring connectingComponent 1components is a mass of mass 1 andm, component 2. 2 is a matrix of mass M, and component 3 is a spring In 2011,connecting Huang components et al. [34] 1 and proposed 2. two types of mass spring structures with local resonators In 2011, Huang et al. [34] proposed two types of mass spring structures with local resonators and and performedperformed equivalent equivalent simulations, simulations, and and found found that theythat have they negative have negative effective elastic effective modulus elastic and modulus and selectiveselective filtering. filtering. In In 2016, 2016, Zhu Zhu etet al. al. [35 [35]] periodically periodically arranged arranged a spring-vibrator a spring-vibrator resonance unit resonance on unit on a stiffeneda stiff enedplate plate to obtain to obtain a alocal local resonanceresonance sti ffstiffenedened plate plate structure structure (see Figure (see4) and Figure simulated 4) and its simulated its performance.performance. They They found found that that the the structurestructure can ca generaten generate a lower a frequencylower frequency local resonance local band resonance gap band and widen the high frequency Bragg band gap compared to ordinary stiffened plates. gap and widen the high frequency Bragg band gap compared to ordinary stiffened plates.

Figure 4. Schematic diagram of the unit cells of a locally resonant stiﬀened plate [35]. Figure 4. Schematic diagram of the unit cells of a locally resonant stiffened plate [35].

The same year, Wu et al. [36] obtained a multi-frequency local resonance type phononic crystal plate by periodically attaching multiple double cantilever structures on a thin plate (see Figure 5 and Figure 6). Through their theoretical research, it is found that the structure can generate multiple low- frequency bending wave forbidden bands, and the bending vibration of the plate is significantly reduced in the band gap range.

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The same year, Wu et al. [36] obtained a multi-frequency local resonance type phononic crystal plate by periodically attaching multiple double cantilever structures on a thin plate (see Figures5 and6). Through their theoretical research, it is found that the structure can generate multiple low-frequency bending wave forbidden bands, and the bending vibration of the plate is signiﬁcantly reduced in the CrystalsCrystalsband 2020 20 gap, 2010,, 10 range.x FOR, x FOR PEER PEER REVIEW REVIEW 6 of6 26of 26

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(a)(a) (b)(b)

FigureFigureFigure 5. 5.Schematic 5. SchematicSchematic diagram (a) diagram diagram of of of a a multi a multi-frequency multi-frequency-frequency locally locally resonant resonant resonant phononic phononic phononic(b)plate: plate: plate: (a) Schematic(a )( aSchematic) Schematic diagramdiagram of the of thephononic phononic plate plate with with several several periodic periodic unitunit cells;cells;( (bb)) schematic schematic diagram diagram of theof the attached attached diagramFigure of 5.the Schematic phononic diagram plate with of a several multi-frequency periodic locallyunit cells;( resonantb) schematic phononic diagram plate: (a) ofSchematic the attached doubledouble -cantilever -cantilever beams beams [36] [36. ]. doublediagram -cantilever of the phononic beams [36] plate. with several periodic unit cells;(b) schematic diagram of the attached double -cantilever beams [36].

(a) (b) (a)(a) (b) (b) FigureFigure 6. S 6.implifiedSimplified model model of of the the proposed proposed multi-frequency multi-frequency locally resonant locally phononic resonant plate: phononic (a) Schematic plate: (a) FigureFigure 6. S 6.implified Simplified model model of of the the proposed proposed multi multi--frequencyfrequency locally locally resonant resonant phononic phononic plate: plate: (a) (a) diagram of the simpliﬁed model with several periodic unit cells; (b) The low-frequency equivalent SchematicSchematicSchematic diagram diagram diagram of ofthe ofthe simplifiedthe simplified simplified model modelmodel with with several severalseveral periodic periodic periodic unitunit unit cells; cells; ((bb) ) ( ThebThe) The low low -lowfrequency-frequency-frequency “mass-spring-mass” system of the attached double-cantilever beams [36]. equivalentequivalentequivalent “mass “mass “mass-spring-spring-spring-mass”-mass”-mass” system system system of theofof the attached attached double double double-cantilever-cantilever-cantilever beams beams [36] [36] [36]. . . In 2019, Ma et al. [37] proposed a damping structure with multi-band gap characteristics based on In 2019, Ma et al. [37] proposed a damping structure with multi-band gap characteristics based theIn spring-vibrator 2019,In 2019, Ma Ma et al.et resonance al. [37] [37] p roposedp unitroposed (see a Figurea damping damping7), and structurestructure found that with with the multi structuremulti-band-band has gap thegap characteristics characteristics characteristics based of based onon thelow theon spring frequency the spring spring-vibrator-vibrator band-vibrator and resonance lowresonance resonance frequency unit unit unit limit. (see (see (see Figure FigureFigure 7 ), 7 7),), and and and found found found that that the the the structure structure structure has has hasthe the the characteristicscharacteristicscharacteristics of oflow lowof frequencylow frequency frequency band band band and and and low lowlow frequency frequency limit. limit. limit.

FigureFigure 7. 7.Schematic Schematic diagram diagram of of the the suppressors suppressors periodically periodically arranged arranged on on the th basee base beam beam [37[37]]. . FigureFigure 7. Schematic7. Schematic diagram diagram of theof the suppressors suppressors periodically periodically arranged arranged on onth eth basee base beam beam[37][37]. . AtAt present, present, referring referring to to the the idea idea “local “local resonance resonance phononic phononic crystals”, crystals”, a a kind kind of of local local artiﬁcial artificial periodicperiodic structure structure can can be be constructed constructed by by periodically periodically attaching attaching spring-mass spring-mass units units to to traditional traditional At present, referring to the idea “local resonance phononic crystals”, a kind of local artificial engineeringengineeringAt present, structures structures referring such such to as the as rods, rods,idea beams beams“local and aresonancend plates. plates. The The phononic structures structures describedcrystals”, described above a abovekind do notofdo local neednot need toartificial periodic structure can be constructed by periodically attaching spring-mass units to traditional periodicdestroyto destroy structurethe uniformity the uniformity can and be integrity constructed and integrity of the bysubstrate of periodicallythe substrate structure, structure, attaching which has which springwide applicabilityhas-mass wide unitsapplicability in practical to traditional in engineeringengineeringpractical structures engineering.structures such such Afterwards, as asrods, rods, beams the beams research and and plates. will plates. focus The Theon structures the structures constitution described described of more above localabove doresonan do not not needce need to todestroy destroycommon the the uniformity engineering uniformity and structures, and integrity integrity and of study ofthe the substrate their substrate bandgap structure, structure, characteristics which which has and has wide vibration wide applicability applicability reduction in in practicalpracticalcharacteristics, engineering. engineering. to Afterwards, provide Afterwards, more the ideasthe research research for the will vibrationwill focus focus on reduction on the the constitution constitution design of of common ofmore more local engineering local resonan resonan ce ce commoncommonstructures. engineering engineering structures, structures, and and study study their their bandgap bandgap characteristics characteristics and and vibration vibration reduction reduction characteristics,characteristics, to toprovide provide more more ideas ideas for for the the vibration vibration reduction reduction design design of ofcommon common engineering engineering 4.2. Helmholtz Resonator Periodic Arrangement Structures structures.structures. Helmholtz resonance refers to the resonance phenomenon of air in a cavity. A container made 4.2.4.2. Helmholtz Helmholtzusing this R esonator phenomenonResonator Periodic Periodic is called Arrangement Arrangement a Helmholtz Struct S resonancetructureures s cavity. It comprises a rigid container with a known volume and a shape close to a sphere. The container has a narrow opening at one end and a Helmholtz resonance refers to the resonance phenomenon of air in a cavity. A container made smallHelmholtz opening resonance at the other refers end. Helmholtzto the resonance resonators phenomenon can filter sounds of air of in specific a cavity. frequencies. A container made using this phenomenon is called a Helmholtz resonance cavity. It comprises a rigid container with a using this phenomenon is called a Helmholtz resonance cavity. It comprises a rigid container with a knownknown volume volume and and a shapea shape close close to toa sphere.a sphere. The The container container has h asa narrowa narrow opening opening at atone one end end and and a a smallsmall opening opening at atthe the other other end. end. Helmholtz Helmholtz resonators resonators can can filter filter sounds sounds of ofspecific specific frequencies. frequencies.

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engineering. Afterwards, the research will focus on the constitution of more local resonance common engineering structures, and study their bandgap characteristics and vibration reduction characteristics, to provide more ideas for the vibration reduction design of common engineering structures.

4.2. Helmholtz Resonator Periodic Arrangement Structures Helmholtz resonance refers to the resonance phenomenon of air in a cavity. A container made using this phenomenon is called a Helmholtz resonance cavity. It comprises a rigid container with a Crystalsknown 2020, volume10, x FOR and PEER a shapeREVIEW close to a sphere. The container has a narrow opening at one end and a7 of 26 small opening at the other end. Helmholtz resonators can ﬁlter sounds of speciﬁc frequencies. In 2006, Fang et al. [38] firstly proposed an acoustic metamaterial composed of a periodic one-dimensional CrystalsIn 2006, 2020 Fang, 10, x FOR et al. PEER [38] REVIEW firstly proposed an acoustic metamaterial composed of a periodic7 of 26one - Helmholtz resonator cavity with acoustic inductance and capacitance (see Figure8). Experiments have dimensional Helmholtz resonator cavity with acoustic inductance and capacitance (see Figure 8). proved that this material can achieve negative elastic modulus. They research found that when the ExperimentsIn 2006, have Fang proved et al. [38] that firstly this materialproposed canan acoustic achieve metamaterial negative elastic composed modulus. of a periodic They research one- frequencydimensional of the Helmholtz acoustic resona wavetor is close cavity to the with resonance acoustic frequency, inductance the and frequency capacitance at the (see exit Figure of the 8). foundresonant that when cavity the will frequency be opposite of the to the acoustic frequency wave of theis close external to the acoustic resonance wave, frequency, thereby achieving the frequency a at theExperiments exit of the have resonant proved cavity that thiswill material be opposite can achieve to the negativefrequency elastic of the modulus. external They acoustic research wave, transmissionfound that when band the gap. frequency of the acoustic wave is close to the resonance frequency, the frequency thereby achieving a transmission band gap. at the exit of the resonant cavity will be opposite to the frequency of the external acoustic wave, thereby achieving a transmission band gap.

(a) (b) (a) (b) FigureFigure 8. The 8. The schematic schematic diagram diagram of of proposed proposed structures:structures: (a ()a One-dimensional) One-dimensional Helmholtz Helmholtz resonator; resonator; (b) Helmholtz(Figureb) Helmholtz 8. The resonator resonatorschematic composed composed diagram of of proposed periodic arrangements arrangementsstructures: (a [)38 One [38]]. -.dimensional Helmholtz resonator; (b) Helmholtz resonator composed of periodic arrangements [38]. In 2017,In 2017, Jiang Jiang et etal. al. [39] [39 ]adopted adopted aa double-openingdouble-opening internal internal and and external external cavity cavity design design to obtain to aobtain In 2017, Jiang et al. [39] adopted a double-opening internal and external cavity design to obtain a doubledouble-opening-opening Helmholtz Helmholtz periodic structure structure (see (see Figure Figure9). It9). is It found is found that that the structure the structure has good has good low-banda double-opening gap characteristics, Helmholtz periodic and its lowest structure band-gap (see Figure section 9). isIt is 86.9–138.2 found that Hz. the At structure a certain has outer good low-band gap characteristics, and its lowest band-gap section is 86.9−138.2 Hz. At a certain outer diameter,low-band thegap low-band characteristics, gap is a ﬀandected its bylowest the arc band length-gap of section the cavity, is 86.9 the−1 internal38.2 Hz and. At external a certain cavity outer diameter,spacing,diameter, the and thelow the low-band spacing-band gap gap between is isaffected affected cells. by by the the arc arc lengthlength of thethe cavity, cavity, the the internal internal and and external external cavity cavity spacing,spacing, and and the the spacing spacing between between cells. cells.

(a) (b) (a) (b) FigureFigure 9.9. TheThe schematicschematic diagramdiagram of of the the double-opening double-opening Helmholtz Helmholtz periodic periodic structure: structure: (a )(a The) The unit unit Figure 9. The schematic diagram of the double-opening Helmholtz periodic structure: (a) The unit structure’sstructure’s cross cross section section of of the the double-split double-split Helmholtz Helmholtz resonant resonant structure; structure; (b) ( Theb) The ﬁnite finite structure structure with with 5 structure’s cross section of the double-split Helmholtz resonant structure; (b) The finite structure with cells5 cells of of the the double-split double-split Helmholtz Helmholtz resonant resonant structure structure [39 ].[39]. 5 cells of the double-split Helmholtz resonant structure [39]. InIn 2018,2018, Yu Yu et et al. al. [40 [40]] changed changed the the rigidity rigidity of Helmholtz of Helmholtz resonators resonators (HRs) (HR ands designed) and designed a tube witha tube awithIn periodically 2018, a periodically Yu arrangedet al. [40] arranged elastic changed HR elastic (seethe H FigurerigidityR (see 10 ofFigure). TheirHelmholtz 10 research). Their resonators found research that (HR for founds a) singleand that designed Helmholtz for a single a tube withcavityHelmholtz a periodically unit ofcavity the sameunit arranged of size, the the same elastic resonant size, HR thefrequency (see reson Figureant of frequency the 10 elastic). Their of HR the research is elastic much lowerHR found is thanmuch that that lower for of thea than single Helmholtzthat of the cavity rigid unit HR. of Secondly, the same the size, factor the determining resonant frequency the elastic of HR the resonance elastic HR frequency is much is lower the top than that plate,of the not rigid the HR.cavity Secondly, size. For periodicthe factor pipelines, determining periodic the elastic elastic HR HR is moreresonance conducive frequency to generat is theing top plate,low not-frequency the cavity and size. wide For-band periodic gaps. pipelines, periodic elastic HR is more conducive to generating low-frequencyHelmholtz and resonator wide-bands take gaps. advantage of the resonance characteristics of air, and they have good advantageHelmholtzs in resonator light weights take and advantage low frequency of the design resonance only characteristicsby structural design of air, without and they additional have good advantagemass, whichs in light has weight been a and great low basic frequency model to design create acousticonly by metamaterialsstructural design recognized without by additional many scholars. Up to now, metamaterials made from Helmholtz resonators have been widely used in the mass, which has been a great basic model to create acoustic metamaterials recognized by many engineering field, like for pipe sound absorption. Nevertheless, in the periodic structure design of scholars. Up to now, metamaterials made from Helmholtz resonators have been widely used in the Helmholtz resonators, the local resonance region is small due to the number and the depth of engineeringopenings, field which, like results for inpipe the soundneed for absorption. a large volume Nevertheless, to obtain low in- frequencythe periodic characteristics. structure design of Helmholtz resonators, the local resonance region is small due to the number and the depth of openings, which results in the need for a large volume to obtain low-frequency characteristics.

Crystals 2020, 10, 305 8 of 26

rigid HR. Secondly, the factor determining the elastic HR resonance frequency is the top plate, not the cavity size. For periodic pipelines, periodic elastic HR is more conducive to generating low-frequency and wide-band gaps. Helmholtz resonators take advantage of the resonance characteristics of air, and they have good advantages in light weight and low frequency design only by structural design without additional mass, which has been a great basic model to create acoustic metamaterials recognized by many scholars. Up to now, metamaterials made from Helmholtz resonators have been widely used in the engineering ﬁeld, like for pipe sound absorption. Nevertheless, in the periodic structure design of Helmholtz resonators, the local resonance region is small due to the number and the depth of openings, which Crystalsresults 2020 in, 10 the, x FOR need PEER fora REVIEW large volume to obtain low-frequency characteristics. 8 of 26

(a) (b)

FigureFigure 10. 10. SchematicSchematic diagram diagram of of the the proposed proposed structure:structure: ((aa)) Sketch Sketch of of an an elastic elastic HR; HR; (b )( Sketchb) Sketch of an of an infiniteinﬁnite periodic periodic pipe pipe with with elastic elastic HRs HRs [40] [40]..

4.3. Thin Film Acoustic Metamaterials 4.3. Thin Film Acoustic Metamaterials The film-type acoustic metamaterial unit is a film that is tensioned and fixed on the peritoneal The film-type acoustic metamaterial unit is a film that is tensioned and fixed on the peritoneal ring, and then a rigid mass unit is fixed on the circular film [41,42]. The working mechanism of the thin ring,film and acoustic then a metamaterial rigid mass unit is to increaseis fixed theon the energy circular density film of low-frequency[41,42]. The working sound waves mechanism inside theof the thinmetamaterial film acoustic by metamaterial increasing the is resonance to increase mode. the Whenenergy the density metal sheetof low is- beaten,frequency the contactsound waves boundary inside thewith metamaterial the thin film by contains increasing enhanced the resonanceelastic curvature mode. energy, When and the the metal coupling sheet eff ect is bwitheaten, the the acoustic contact boundarywave is with cancelled the thin to achieve film contains Resonant enhanced cavity-like elastic sound curvature absorption energy, purposes and [43 the]. coupling effect with the acousticIn 1991, wave Hashimoto is cancelled et al.to [achieve44] first Resonant proposed cavity a membrane-like sound structure absorption with additional purposes masses,[43]. theIn so-called 1991, Hashimoto membrane et with al. [44] additional first prop weightsosed (MAW),a membrane and found structure that thiswith structure additional can improvemasses, the so-soundcalled insulationmembrane in with the low additional frequency w rangeeights through (MAW), sound and insulationfound that experiments this structure under can normal improve soundincidence insulation effects. in Inthe 2008, low Yangfrequency and Mei range of The through Hong Kongsound University insulation of experiments Science and Technologyunder normal incidenceformally effect proposeds. In the2008, concept Yang of and "thin Mei film of acoustic The Hong metamaterial", Kong University and pointed of Science out that itand is composed Technology formallyof an elastic proposed film, an the additional concept mass, of "thin and film a supporting acoustic structure metamaterial", of a fixed and film. pointed The structure out that was it is composedfound to of have an an elastic acoustic film, band an gap additional in the frequency mass, and range a ofsupporting 200–300 Hz structure [45]. Later, of it wasa fixed found film. that The the structure composed of four layers of thin film materials with different sound insulation frequency structure was found to have an acoustic band gap in the frequency range of 200–300 Hz [45]. Later, it bands and different mass masses can achieve 50–1000 Hz wide-band sound insulation, and its sound was found that the structure composed of four layers of thin film materials with different sound insulation is as high as 40 dB [46] insulation frequency bands and different mass masses can achieve 50–1000 Hz wide-band sound In 2019, He et al. [47] designed a new type of thin film acoustic metamaterial by embedding a insulation, and its sound insulation is as high as 40 dB [46] piezoelectric mass connected to an external circuit into a tensioned elastic membrane and fixed on a metalIn 2019, frame He (see et Figureal. [47] 11 designed). They found a new that type the of structure thin film has acoustic good sound metamaterial insulation by performance embedding a piezoelectricin the 20–1200 mass Hz connected frequency to band an external and there circui are twot into sound a tensioned insulation elastic peaks membrane above 50 dBand and fixed one on a metaladjustable frame (see sound Figure insulation 11.). peak.They found Secondly, that by the adjusting structure the has circuit good parameters, sound insulation the tunability performance of the in thesound 20– insulation1200 Hz frequency performance band of the and material there canare betwo achieved. sound insulation peaks above 50 dB and one adjustable sound insulation peak. Secondly, by adjusting the circuit parameters, the tunability of the sound insulation performance of the material can be achieved.

Figure 11. The structural sketch [47].

The lateral local resonance metamaterial structure proposed by Wang et al. [48,49] can also suppress low frequency noise (see Figure 12). By analyzing the energy band structure diagram of the metamaterial, they found that the metamaterial plate could generate two low-frequency band gaps by the equivalent mass. The vibration power flow of metamaterial plate was obtained by using the finite element method and the energy flow method, so as to reveal its vibration suppression mechanism.

Crystals 2020, 10, x FOR PEER REVIEW 8 of 26

(a) (b)

Figure 10. Schematic diagram of the proposed structure: (a) Sketch of an elastic HR; (b) Sketch of an infinite periodic pipe with elastic HRs [40].

4.3. Thin Film Acoustic Metamaterials The film-type acoustic metamaterial unit is a film that is tensioned and fixed on the peritoneal ring, and then a rigid mass unit is fixed on the circular film [41,42]. The working mechanism of the thin film acoustic metamaterial is to increase the energy density of low-frequency sound waves inside the metamaterial by increasing the resonance mode. When the metal sheet is beaten, the contact boundary with the thin film contains enhanced elastic curvature energy, and the coupling effect with the acoustic wave is cancelled to achieve Resonant cavity-like sound absorption purposes [43]. In 1991, Hashimoto et al. [44] first proposed a membrane structure with additional masses, the so-called membrane with additional weights (MAW), and found that this structure can improve sound insulation in the low frequency range through sound insulation experiments under normal incidence effects. In 2008, Yang and Mei of The Hong Kong University of Science and Technology formally proposed the concept of "thin film acoustic metamaterial", and pointed out that it is composed of an elastic film, an additional mass, and a supporting structure of a fixed film. The structure was found to have an acoustic band gap in the frequency range of 200–300 Hz [45]. Later, it was found that the structure composed of four layers of thin film materials with different sound insulation frequency bands and different mass masses can achieve 50–1000 Hz wide-band sound insulation, and its sound insulation is as high as 40 dB [46] In 2019, He et al. [47] designed a new type of thin film acoustic metamaterial by embedding a piezoelectric mass connected to an external circuit into a tensioned elastic membrane and fixed on a metal frame (see Figure 11.). They found that the structure has good sound insulation performance in the 20–1200 Hz frequency band and there are two sound insulation peaks above 50 dB and one adjustableCrystals 2020 sound, 10, 305 insulation peak. Secondly, by adjusting the circuit parameters, the tunability9 ofof 26 the sound insulation performance of the material can be achieved.

FigureFigure 11. TheThe structural sketchsketch [ 47[47].].

TheThe lateral lateral local local resonance resonance metamaterial metamaterial structure proposedproposed byby WangWang et et al. al. [48 [48,49],49] can can also also suppress low frequency noise (see Figure 12). By analyzing the energy band structure diagram of the suppressCrystals 20 low20, 10 frequency, x FOR PEER noise REVIEW (see Figure 12). By analyzing the energy band structure diagram9 ofof 26 the metamaterial,metamaterial, they they found found that the metamaterialmetamaterial plate plate could could generate generate two two low-frequency low-frequency band band gaps bygaps bythe the equivalent equivalent mass. mass. The The vibration vibration power power flow flow of metamaterial of metamaterial plate wasplate obtained was obtained by using by the using finite the Crystalsfiniteelement 20element20, method10, x FORmethod andPEER the REVIEWand energy the flowenergy method, flow so meth as tood, reveal so itsas vibrationto reveal suppression its vibration mechanism. suppression9 of 26 mechanism.

Figure 12. The unit cell of the metamaterial plate: (a) front view, (b) perspective view [48]

In 2020, Zhou et al. [50] proposed a polymorphic anti-resonance cooperative thin-film acoustic metamaterialFigureFigure 1(see2 12.. T heFigureThe unit unit cell13 cell). of The of the the materialmetamaterial metamaterial consists plate: plate: of ( (a afour)) front metal view,view, foils ((bb)) perspective perspective and a cross view view-shaped [48 [48]]. flexible ethylene-vinyl acetate (EVA) material swing arm attached to the surface of a polyimide (PI) film, and usesIn In2020,EVA 2020, materialZhou Zhou et etasal. al. a[50] frame. [50 proposed] proposed The study a apolymorphic polymorphicfound that the anti anti-resonance material-resonance has cooperativeachievedcooperative widening thin-film thin-film of acousticthe acoustic low metamaterialmetamaterialfrequency sound (see (see Figure Figureinsulation 1313).). ThebandThe material materialand improvement consists consists of four of sound metalfour metal foils insulation. and foils a cross-shaped and a cross flexible-shaped ethylene- flexible ethylenevinylThe acetate-vinyl advantage acetate (EVA) material (EVA)of this typematerial swing of armacoustic swing attached armmetamaterials toatt theached surface to is thethat of surface a the polyimide band of agap polyimide (PI) frequency film, and (PI) domain uses film, EVA and is usesmaterialvery EVA low, material as which a frame. asmeans a The frame. that study theyThe found studycan thatsuppress found the material thatlow -thefrequency has material achieved elastic has widening achievedwave propagation. of widening the low frequencyHowever, of the low frequencysoundthe structure insulation sound is insulation too band thin and with band improvement low and stiffness improvement of sound to bear insulation. ofweight, sound which insulation. limits its application in practical engineering.The advantage of this type of acoustic metamaterials is that the band gap frequency domain is very low, which means that they can suppress low-frequency elastic wave propagation. However, the structure is too thin with low stiffness to bear weight, which limits its application in practical engineering.

FigureFigure 13.13. A small size MAM experimental samplesample [[50]50]..

4.4. SpaceThe advantage Coiling Structure of this types of acoustic metamaterials is that the band gap frequency domain is very low, which means that they can suppress low-frequency elastic wave propagation. However, the structure Research on local resonance acoustic metamaterials has made great progress, but it has is too thin with low stiffnessFigure to 13 bear. A weight,small size which MAM limits experimental its application sample in [50] practical. engineering. disadvantages such as narrow frequency band and difficult design and manufacture. In response to the abovementioned shortcomings, scholars have introduced the coiled space type structure into 4.4.4.4. Space Space Coiling Coiling Structure Structuress acoustic metamaterials, and designed coiled space type acoustic metamaterials. The sound Research on local resonance acoustic metamaterials has made great progress, but it has disadvantages absorptionResearch mechanismon local resonanceof the curly acousticspace stru metamaterialscture is to allow hassound made waves great to propagate progress, in the but coiled it has such as narrow frequency band and difficult design and manufacture. In response to the abovementioned disadvantagesspace, increasing such theas narrowpropagation frequency time and band reducing and difficult the effective design phase and velocitymanufacture. [51,52] In. response to shortcomings, scholars have introduced the coiled space type structure into acoustic metamaterials, the abovementionedIn 2015, Li et al. shortcomings, [53] designed a scholars curly space have structure introduced composed the coiled of a Jerusalem space type cross structure groove air into and designed coiled space type acoustic metamaterials. The sound absorption mechanism of the curly acousticmatrix metamaterialperiodically arrangeds, and designedin a square coiled lattice. spaceIt is found type that acoustic the structure metamaterial can generates. The a large sound absorptionband gap mechanism in low frequency of the curlymode, space and the stru bandcture gap is to formation allow sound mechanism waves isto thepropagate resonance in themode coiled of space,the increasinginternal cavity the ofpropagation the cross groove time andstructure reducing. the effective phase velocity [51,52]. The same year, Wang et al. [54] designed an S-shaped groove structure in a square lattice (see In 2015, Li et al. [53] designed a curly space structure composed of a Jerusalem cross groove air Figure 14). They found that the structure can produce a more complete and larger band gap in the matrix periodically arranged in a square lattice. It is found that the structure can generate a large lower frequency range compared to a Jerusalem cross groove structure with the same lattice band gap in low frequency mode, and the band gap formation mechanism is the resonance mode of parameters. the internal cavity of the cross groove structure. The same year, Wang et al. [54] designed an S-shaped groove structure in a square lattice (see Figure 14). They found that the structure can produce a more complete and larger band gap in the lower frequency range compared to a Jerusalem cross groove structure with the same lattice parameters.

Crystals 2020, 10, 305 10 of 26

space structure is to allow sound waves to propagate in the coiled space, increasing the propagation time and reducing the eﬀective phase velocity [51,52]. In 2015, Li et al. [53] designed a curly space structure composed of a Jerusalem cross groove air matrix periodically arranged in a square lattice. It is found that the structure can generate a large band gap in low frequency mode, and the band gap formation mechanism is the resonance mode of the internal cavity of the cross groove structure. The same year, Wang et al. [54] designed an S-shaped groove structure in a square lattice (see Figure 14). They found that the structure can produce a more complete and larger band gap in the lower Crystalsfrequency 2020, 10 range, x FOR compared PEER REVIEW to a Jerusalem cross groove structure with the same lattice parameters. 10 of 26

Crystals 2020, 10, x FOR PEER REVIEW 10 of 26

(a) (a) (b)(b)

FigureFigure Figure14. 14. Schematic Schematic14. Schematic diagram diagram diagram of of proposed proposedof proposed structures:structure structures (: a (()aa) The) TheThe unit unit unit cellcell cell ofof ananof S-shapeSan-shape S-shape slotslot phononicslot phononic crystal(PC)crystalcrystal(PC) (PC);; (b) ( bSchematic); ( Schematicb) Schematic view view view of of theof the the cross cross-section cross-section-section of ofof the the two-dimensional twotwo-dimensional-dimensional PCPC [PC[54]54]. .[54] .

In 2017, Xia et al. [55] designed a spatially coiled acoustic metamaterial based on the Hilbert fractal In 2017,In Xia2017, et Xia al. et [55] al. [55]designed designed a spa spatiallyatially coiledcoiled acousticacoustic metamaterial metamaterial based based on the on Hilbert the Hilbert structurefractal (see structure Figure (see15). ItFigure is found 15). through It is found research through that research the high-order that the Hilbert high-order fractal Hilbert structure fractal can fractal structure (see Figure 15). It is found through research that the high-order Hilbert fractal eﬀectivelystructure realize can sub-wavelength effectively realize ﬁltering, sub-wavelength and as the filtering, fractal order and increases, as the fractal the number order increases, of band gaps the structureof thenumber metamaterial can effectivelyof band increasesgaps realizeof the and metamaterial sub the- totalwavelength band increases gap filtering, widthand the gradually total and band as becomes gap the width fractal larger. gradually order becomes increases, the numberlarger. of band gaps of the metamaterial increases and the total band gap width gradually becomes larger.

FigureFigure 15. Space 15. Space coiled coiled acoustic acoustic metamaterials metamaterial baseds based on Hilbert on Hilbert fractals: fractal ﬁrst-orders: first-order fractal, fractal, second-order second- fractal,order and fractal, third-order and third fractal-order [55 fractal]. [55]. FigureIn 2019, In15 .2019, Space Dong Dong etcoiled al. et [ 56al.acoustic][56] proposed proposed metamaterial a new a new structure sstructure based with on with broadbandHilbert broadband fractal double-negativity s:double first-order-negativity fractal, based based second on on the - cavityorderthe structure fractal, cavity andand structure third spatial- order and curling fractal spatial mechanism, [55] curling. revealing mechanism, the revealingmost useful the topological most useful characteristics topological of metamaterialscharacteristics based of on metamaterials the above structures, based and on thetwo abovetypical structures,The negative and refraction two ty andpical sub-wavelength The negative imagingIn refraction2019, of Dong the and structure et sub al.-wavelength[56] were proposed detailed imaging numericallya new of the structure structure simulated. with were The broadband detailed same year,numerically double Wu et al.- negativitysimulated. [57] proposed Tbasedhe on same year, Wu et al. [57] proposed a hybrid acoustic metamaterial composed of micro-perforated the a cavity hybrid acoustic structure metamaterial and spatial composed curling of micro-perforated mechanism, revealing plates (MPP) the and most curled useful Fabry-Perot topological plates (MPP) and curled Fabry-Perot (FP) channels (see Figure 16 and Figure 17), and found that the characteristics(FP) channels of (see metamaterials Figures 16 and 17 based), and foundon the that above the material structures, works at and a frequency two typical greater The than negative 30 timesmaterial the total works thickness. at a frequency At the resonance greater than frequency 30 times (< the500 total Hz), thickness. the sound At absorption the resonance effect frequency can reach refraction(<500 and Hz sub), the-wavelength sound absorption imaging effect of can the reach structure 99%. The were sound detailed absorption numerically performance simulated. of this The same year,materia Wul is et mainly al. [57] caused proposed by the frictional a hybrid loss acoustic of sound metamaterial wave energy in composed MPP. of micro-perforated plates (MPP)This and resonance curled Fabryfrom the-Perot folded (FP) space chann producesels (see a very Figure wide 1 band6 and refractive Figure 1index7), and with found a small that the materialabsorption works at loss a frequency and a negative greater broadband than 30 response. times the Meanwhile, total thickness. this kind At of thestructure resonance has a highfrequency (<500 Hzrequirement), the sound for manufacturing absorption effect which canmay reachmake them 99%. very The expensive sound .absorption Later, we can performance adopt a proper of this strategy to design acoustic metamaterials with new geometric structures. material is mainly caused by the frictional loss of sound wave energy in MPP. This resonance from the folded space produces a very wide band refractive index with a small absorption loss and a negative broadband response. Meanwhile, this kind of structure has a high requirement for manufacturing which may make them very expensive. Later, we can adopt a proper strategy to design acoustic metamaterials with new geometric structures.

Crystals 2020, 10, 305 11 of 26

99%. The sound absorption performance of this material is mainly caused by the frictional loss of sound wave energy in MPP. This resonance from the folded space produces a very wide band refractive index with a small absorption loss and a negative broadband response. Meanwhile, this kind of structure has a high requirement for manufacturing which may make them very expensive. Later, we can adopt a proper strategyCrystals 20 to20, design10, x FOR acoustic PEER REVIEW metamaterials with new geometric structures. 11 of 26 Crystals 2020, 10, x FOR PEER REVIEW 11 of 26

(a)(a) (b)(b) (c)(c)

FigureFigureFigure 16. 1 6 1.Schematic6 .S chematicSchematic diagram diagram diagram of proposed of of proposed structure: structure: structure: (a) The (a (a) top ) TheThe micro-perforated top top micro micro-perforated-perforated panel; (b panel; ) panel;Schematic (b ) (b) ofSchematicSchematic the hybrid of of the metamaterial the hybrid hybrid metamaterial metamaterial absorber; (c absorber;) Diagram ((cc) of) Diagram Diagram one coiled-up of of one one coiled FP coiled channel-up-up FP FP [ 57channel channel]. [57] [57]. .

Figure 17. An approximate analytical two-dimensional (2D) model of a unit cell of a space-coiled FigureFigmetamaterialure 17.17. An approximate[57]. analytical two-dimensionaltwo-dimensional (2D) model of a unit cell ofof aa space-coiledspace-coiled metamaterialmetamaterial [[57]57].. 4.5. Other Structures 4.5.4.5. Other StructuresStructures In addition to the above four structures, some other acoustic metamaterial structures have also beenInIn additionadditiondeveloped. toto theHerethe aboveabove we discuss four structures, some other somesome types. other Based acoustic on soft metamaterial technology, structuresstructures a class of havehave local alsoalso beenbeenresonance developed.developed. ultrasonic Here Here wesuper we discuss -liquids discuss some composed some other other types. of macroporous types. Based Based on softmicrosphere on technology, soft technology, concentrated a class of a local suspensions class resonance of local ultrasonicresonancewere designed. super-liquids ultrasonic The supernegat composedive-liquids index of composed macroporousof metamaterials of macroporous microsphere is derived concentrated microspherefrom the low suspensions -concentratedfrequency resonance were suspensions designed. of Theweresubwavelength negative designed. index The particles. ofnegat metamaterialsive In 2015, index Ouyang of is metamaterials derived et al. [58] from used theis derived elastic low-frequency membranes from the resonance low to -makefrequency ofhollow subwavelength resonance spheres. of particles.subwavelengthThe interior In 2015, of particles.the Ouyang sphere In etis 2015, filled al. [58 Ouyangwith] used air elastic toet obtainal. [58] membranes a used soft resonance elastic to make membranes unit. hollow The resonanceto spheres. make hollow Thefrequency interior spheres. is of theThearound sphere interior 860 is of filledHz the, and withsphere the air propagation is to filled obtain with a loss soft air of resonanceto sound obtain waves a unit. soft isresonance The 11 dB resonance. A bandunit. frequency Thegap isresonance generated is around frequency near 860 the Hz, is resonance frequency. Good sound insulation. The same year, Leroy et al. [59] studied bubble andaround the propagation860 Hz, andloss the ofpropagation sound waves loss is of 11 sound dB. A bandwaves gap is is11 generated dB. A band near gap the is resonance generated frequency. near the resonators, which are single-layer gas inclusions in soft solids. This bubble ultrafiltration membrane Goodresonance sound frequency. insulation. GoodThe same sound year, insulation. Leroy et al. The [59] studiedsame year, bubble Leroy resonators, et al. which[59] studied are single-layer bubble can be used as an ultra-thin coating to achieve acoustic super absorption over a wide frequency range. gasresonators, inclusions which in soft are solids. single This-layer bubble gas inc ultrafiltrationlusions in soft membrane solids. This can bebubble used asultrafiltration an ultra-thin membrane coating to In 2016, Wang et al. [60] proposed an acoustic metamaterial with periodic spindle-shaped achievecan be used acoustic as an super ultra- absorptionthin coating over to achieve a wide frequencyacoustic super range. absorption over a wide frequency range. inclusions embedded in a rubber matrix (see Figure 18). This structure can generate a large complete InIn 2016, 2016, Wang Wang et al.et [60 al.] proposed[60] proposed an acoustic an acoustic metamaterial metamaterial with periodic with spindle-shaped periodic spindle inclusions-shaped embeddedband gap in in a the rubber range matrix of 163 (see–222 Figure Hz, and 18 the). This generation structure of canthe generateforbidden a band large is complete mainly due band to gapthe in increcirculationlusions embedded motion inand a rubbershear motion matrix of (see the Figurematrix around18). This the structure inclusions. can generate a large complete the range of 163–222 Hz, and the generation of the forbidden band is mainly due to the recirculation band gapThe in acoustic the range black of hole 163 –phenomenon222 Hz, and isthe a promisinggeneration method of the forforbidden controlling band vibration, is mainly which due hasto the motion and shear motion of the matrix around the inclusions. recirculationattracted more motion and moreand shear attention motion of scholars of the matrixin recent around years. Thethe inclusions.black hole phenomenon originates The acoustic black hole phenomenon is a promising method for controlling vibration, which has fromThe astrophysics acoustic black and holerefers phenomenon to a space-time is a rpromisingegion of sufficient method mass for controlling that can absorb vibration, any material which has attracted more and more attention of scholars in recent years. The black hole phenomenon originates attractedoutside morethe region, and more and the attention absorbed of scholarsmaterial (includingin recent years. photons) The cannot black holeescape. phenomenon originates from astrophysics and refers to a space-time region of sufficient mass that can absorb any material outside the region, and the absorbed material (including photons) cannot escape.

Crystals 2020, 10, 305 12 of 26

from astrophysics and refers to a space-time region of suﬃcient mass that can absorb any material Crystals 2020, 10, x FOR PEER REVIEW 12 of 26 Crystalsoutside 2020 the, 10 region,, x FOR PEER and REVIEW the absorbed material (including photons) cannot escape. 12 of 26

(a)(a) (b) (b)

FigFigureure 18 18.8.. ((aa()a) T)T Thehehe unit unit unit cell cell cell with with spindle spindle-shapedspindle-shaped-shaped inclusion; inclusion; inclusion; (b (b) ( )Sbchematic Schematic) Schematic view view view of of proposed proposedof proposed PC PC [60] PC [60. ]. [60].

InIn 1988, 1988, Mironov Mironov found found a a similar similar phenomenon phenomenon in in a awedge wedge-shaped-shaped structure. structure. When When the the thickness thickness In 1988, Mironov found a similar phenomenon in a wedge-shaped structure. When the thickness ofof the the thin thin plate plate structure structure is is reduced reduced as as a a power power function function ( (h((x) = AxAxmm, ,mm ≥ 2)2),, the the wave wave velocity velocity of of the the of the thin plate structure is reduced as a power function (h(x) = Axm, m≥ ≥ 2), the wave velocity of the bendingbending wave wave will gradually decreasedecrease with with the the decrease decrease of theof the thickness. thickness. Under Under ideal ideal conditions, conditions, the wave the bending wave will gradually decrease with the decrease of the thickness. Under ideal conditions, the wavevelocity velocity can be can reduced. be reduced. Wave Wave reflections reflections as small as small as zero as are zero achieved are achieved [61,62 ].[61,62] In 2019,. In Liu2019, et Liu al. [ 63et ] wave velocity can be reduced. Wave reflections as small as zero are achieved [61,62]. In 2019, Liu et al.designed [63] designed an acoustic an acoustic superstructure superstructure by embedding by embedding multiple multiple acoustic acoustic black hole black units hole in an units array in onan a al.array10 [63] mm on designed thin a 10 plate mm thin (seean acoustic Figureplate (see 19 superstructure). Figure It was 1 found9). It was throughby foundembedding experiments through multiple experiments that the acoustic average that blackthe sound average hole insulation units sound ofin an arrayinsulationthe structure on a of10 themm in the structure thin frequency plate in the(see band frequency Figure of 50–160 19 band). Hz It was can of 50 reachfound–160 nearly Hz through can 30 reach dB, experiments and nearly the average 30 dBthat, and soundthe the average insulation average sound insulationsoundin the insulation frequency of the structure bandin the of frequency 100–600 in the frequency Hz band can of reach 100 band– 40600 dB. of Hz 50 can–160 reach Hz can 40 dB reach. nearly 30 dB, and the average sound insulation in the frequency band of 100–600 Hz can reach 40 dB.

Figure 19. The diagram of the ideally designed acoustic black hole superstructure [63]. FigureFigure 1 19.9. TThehe diagram diagram of the ideallyideally designeddesigned acoustic acoustic black black hole hole superstructure superstructure [63 ].[63]. Only some common configurations are listed here, and there are many other acoustic metamaterialOnly some configurations, common conﬁgurations all of which are can listed produce here, and band there gap ares to many suppress other acousticelastic propagation. metamaterial Only some common configurations are listed here, and there are many other acoustic However,conﬁgurations, acoustic all ofmetamaterials which can produce with lower band gapsfrequency to suppress band elasticand wider propagation. frequency However, range to acoustic meet metamaterial configurations, all of which can produce band gaps to suppress elastic propagation. engineeringmetamaterials needs with are lower still needed frequency. band and wider frequency range to meet engineering needs are However, acoustic metamaterials with lower frequency band and wider frequency range to meet still needed. engineering5. The Extraordinary needs are P ropertiesstill needed of .A coustic Metamaterials and Phononic Crystals 5. The Extraordinary Properties of Acoustic Metamaterials and Phononic Crystals 5. TheIn E physics,xtraordinary dielectric Properties constant of andAcoustic permeability Metamaterials are two and basic Phononic parameters Crystals that describe the propertiesIn physics, of electromagnetic dielectric constant fields and in permeability a homogeneous are two medium. basic parameters In recent that years, describe scholars the properties have designedof electromagneticIn physics, many electromagnetic dielectric fields in constant a homogeneous metamaterials and permeability medium. with negative In are recent permittivity two years, basic scholars and parameters negative have designed thatpermeability. describe many the propertiesTheseelectromagnetic electromagnetic of electromagnetic metamaterials metamaterials with fields negative havein a permittivity homogeneous special properties and negative medium. such permeability. as In negative recent These years, refractive electromagnetic scholars effect, have designedperfectmetamaterials lens many effect, haveelectromagnetic anomalous special properties Cherenkov metamaterials such radiation, as negative with and negative refractive anomalous permittivity effect, Doppler perfect and effect lens negative effect,s. Based anomalouspermeability. on the ThesesimilarityCherenkov electromagnetic of radiation, electromagnetic and metamaterials anomalous waves Doppler and have acoustic effects. special wa Based ves, properties on acoustic the similarity such metamaterials ofas electromagnetic negative have refractive become waves andan effect, perfectemergingacoustic lens waves, research effect, acoustic hotspot.anomalous metamaterials Acoustic Cherenkov metamaterials have become radiation, an with emerging and negative anomalous research bulk hotspot. modulus Doppler Acoustic and effect negative metamaterialss. Based mass on the similaritydensity also of electromagnetic exhibit unusual waves characteristics and acoustic different wa ves, from acoustic natural metamaterials materials such have as negative become an emergingrefraction, research anomalous hotspot. Doppler Acoustic effect, metamaterials amplified evanescent with negative wave, andbulk perfectmodulus sound and absorptionnegative mass densityeffect [16,64,65] also exhibit. unusual characteristics different from natural materials such as negative refraction, anomalous Doppler effect, amplified evanescent wave, and perfect sound absorption effect5.1. Negative [16,64,65] Refraction. Effect of Acoustic Waves [66]

5.1. Negative Refraction Effect of Acoustic Waves [66]

Crystals 2020, 10, 305 13 of 26 with negative bulk modulus and negative mass density also exhibit unusual characteristics different from natural materials such as negative refraction, anomalous Doppler effect, amplified evanescent wave, and perfect sound absorption effect [16,64,65].

5.1. Negative Refraction Eﬀect of Acoustic Waves [66] Crystals 2020, 10, x FOR PEER REVIEW 13 of 26 Optical metamaterials have negative refractive indices related with the permittivity and permeability, whileOptical the acoustic metamaterials metamaterials’ have negative negative refraction refractive is determined indices by related the effective with themass permittivity density and bulk and modulus.permeability, Sound while waves the have acoustic the nature metamaterials' of waves, negative and reflection refraction and refraction is determined occur by at the interface effective betweenmass density two media.and bulk And modulus. negative Sound refraction waves occurs have whenthe nature acoustic of waves, waves and scatter reflection at the surfacesand refraction in the phononicoccur at the crystals, interface leading between to strong two media. deformation Sound [66 waves]. Sound follow waves Snell's follow law Snell’s when lawthey when are refracted, they are refracted,that is, they that satisfy is, they the satisfy following the following equations: equations:

nn11sinsinθ1 1= n 2 2sin θ 2 2 (1(1)) n n   wherewhere n11and andn 2 are2 are the the refractive refractive indices indices of theof the two two media, media,θ1 and1 andθ2 are 2 the are angles the angles of incidence of incidence and refraction,and refraction, respectively. respectively. The materialThe material properties properties on both on both sides sides of the of interfacethe interface are diareﬀ erent,different, and and the naturethe nature of refraction of refraction is also is dialsoﬀerent. different. When When both sides bothare sides regular are regular materials, materials, both sides both have sides a positive have a refractivepositive refractive index and index positive and refraction positive occurs;refraction when occurs; one sidewhen is aone normal side mediumis a normal and medium the other and side the is aother special side medium is a special with medium a negative with refractive a negative index, refractive negative index, refraction negative will occurrefraction (see Figurewill occur 20). (see Figure 20).

Figure 20.20. Negative refraction eeffectﬀect [[63]63]..

AA lotlot ofof researchresearch has has been been done done on on the the negative negative refraction refraction eff ecteffect of phononic of phononic crystals. crystals. Yang Yang et al. et [67 al.] studied[67] studied a phononic a phononic focusing focusing phenomenon phenomenon in the passbandin the passband over a complete over a complete band gap band in a 3Dgap phononic in a 3D crystalphononic and crystal found and that found the wave that propagationthe wave propagation depends largelydepends on largely the frequency on the frequency and the direction and the ofdirection incidence, of incidence, and due toand the due anisotropy to the anisotropy of the propagation, of the propagation, very large very negative large negative refraction refraction occurs. Inoccurs. 2019, In Mokhtari 2019, Mokhtari et al. [68] et proposed al. [68] theproposed metamaterial the metamaterial properties ofproperties a two-phase of a unit two cell-phase traditionally unit cell consideredtraditionally to considered be a non-local to resonancebe a non-local type, resonance and demonstrated type, and a hexagonaldemonstrated unit a cell hexagonal composed unit of twocell materials.composed Theyof two found materials. that within They thefound appropriate that within frequency the appropriate range, the frequency effective shearrange, modulus the effective and densityshear modulus are both and negative density and are can both produce negative negative and refraction.can produce That negative same year, refraction. Nemat-Nasser That same et al. year, [69] provedNemat- throughNasser et experiments al. [69] proved that the through simple two-dimensional experiments that phononic the simple crystals two- composeddimensional of aluminum phononic matrixcrystals and composed circular of flexible aluminum inclusions matrix with and square circular patterns flexible have inclusions negative with refractive square characteristicspatterns have innegative the shear refractive mode characteristics and longitudinal in the mode. shear Dong mode et and al. longitudinal [56] proposed mode. a space-curled Dong et al. metamaterial[56] proposed baseda space on-curled a broadband metamaterial double-negative based on aresonator broadband and double spatial-negative winding resonator mechanism, and andspatial verified winding the double-negativemechanism, and characteristicsverified the double and- negativenegative refractioncharacteristics of the and material negative through refraction a large of the number material of numericalthrough a large simulations. number of numerical simulations.

5.2.5.2. Anomalous Doppler Effffectect TheThe DopplerDoppler eeffectffect ofof naturalnatural materialsmaterials isis describeddescribed as:as: thethe wavewave receivingreceiving frequencyfrequency becomesbecomes higherhigher when the wave source moves toward the observer, and the wave receiving frequency becomes lowerlower when when the the wave wave source source moves moves away away from from the the observer. observer. In In a a metamaterial metamaterial with with a a negative negative refractiverefractive index, an an anomalous anomalous Doppler Doppler effect effect occurs. occurs. When When the the wave wave source source faces faces the theobserver, observer, the thefrequency frequency received received by the by theobserver observer decreases, decreases, and and vice vice versa. versa. In 1968, Veselago [6] first theoretically predicted that metamaterials with negative refractive indices could produce anomalous Doppler phenomena. In 2010, Lee et al. [70] used one-dimensional long-period tubes to periodically arrange thin films and open holes on the side to prepare one- dimensional double-negative acoustic composites. The abnormal Doppler phenomenon was tested with a moving sound source. In 2015, Zhai et al. [71] designed an acoustic metamaterial with seven flute-like binary molecular clusters. Their research found that this metamaterial has both local elastic modulus and local density of mass density, and can achieve negative refraction over a wide frequency range, and also exhibit broadband reverse Doppler effect. In 2017, Liu et al. [72] further studied this

Crystals 2020, 10, 305 14 of 26

In 1968, Veselago [6] first theoretically predicted that metamaterials with negative refractive indices could produce anomalous Doppler phenomena. In 2010, Lee et al. [70] used one-dimensional long-period tubes to periodically arrange thin films and open holes on the side to prepare one- dimensional double-negative acoustic composites. The abnormal Doppler phenomenon was tested with a moving sound source. In 2015, Zhai et al. [71] designed an acoustic metamaterial with seven flute-like binary molecular clusters. Their research found that this metamaterial has both local elastic modulus and local density of mass density, and can achieve negative refraction over a wide frequency range,Crystals and2020,also 10, x exhibitFOR PEER broadband REVIEW reverse Doppler effect. In 2017, Liu et al. [72] further studied14 this of 26 structure and designed a test device to measure the Doppler effect. The experimental results show thatstructure in the and negative designed refraction a test regiondevice of to the measure acoustic the metamaterial, Doppler effect. when The the experimental sound source results frequency show isthat 2000 in Hz:the negative when the refraction sound source region is of close, the acoustic the ratio metamaterial, of the frequency when detectedthe sound by source the detector frequency to theis 2000 sound Hz source: when isthe reduced sound source by 0.73 is Hz; close, when the theratio sound of the source frequency is far detected away, the by detectorthe detector detects to the it comparedsound source to the is increase reduced of by 0.68 0.73 Hz. Hz In; 2018, when Zhai the et soun al. [73d ] source reported is anomalousfar away, the Doppler detector effects detects found it incompared two commonly to the usedincrease wind of instruments 0.68 Hz. In based 2018, onZhai flute-type et al. [73] acoustic reported metamaterials. anomalous They Doppler found effects that bothfound wind in two instruments commonly can used observe wind anomalous instruments Doppler based eff onects flute in the-type frequency acoustic range metamaterials. of two diff erentThey speedsfound andthat ascendingboth wind scales.instruments can observe anomalous Doppler effects in the frequency range of two different speeds and ascending scales. 5.3. Enlargement of the Evanescent Wave 5.3. Enlargement of the Evanescent Wave When sound waves propagate in both media, evanescent waves and propagating waves that carryWhen sound sound wave informationwaves propagate are generated. in both media, Because evanescent of the characteristic waves and propagating that evanescent waves waves that disappearcarry sound rapidly wave withinformation increasing are distance generated. during Because propagation, of the characteristic the accuracy that of evanescentacoustic imaging waves cannotdisappear be achieved rapidly with at sub-wavelength increasing distance size. In during order propagation,to break through the theaccurac sub-wavelengthy of acoustic imaging imaging problem,cannot be it achieved is necessary at sub to- realizewavelength that the size. acoustic In order wave to break information through of the the sub evanescent-wavelength wave imaging is not lost.problem, In the it field is necessary of electromagnetics, to realize that based the acoustic on the left-handed wave information material of with the aevanescent refractive indexwave ofis not1 − proposedlost. In the by field Veselago of electromagnetic [6], the flat lenss, ebasedffect can on bethe achieved. left-handed Pendry material [74] proposedwith a refractive that metamaterials index of −1 canproposed be designed by Veselago by using [6] metamaterials, the flat lens so that effect the can evanescent be achieved. wave is Pendry amplified [74] by proposed the superlens that propagation.metamaterials When can be the designed evanescent by using wave metamaterials leaves the perfect so that lens, the it evanescent begins to reduce wave is rapidly amplified to the by originalthe superlens level, andpropagation theoretically . When a perfect the evanescent image can bewave formed leaves (see the Figure perfect 21 ).lens, it begins to reduce rapidly to the original level, and theoretically a perfect image can be formed (see Figure 21).

(a) (b)

FigureFigure 21.21. ((aa)) Wavenumber-frequencyWavenumber-frequency representation representation of o thef the signals signals measured measured under under the the grating grating for anfor excitationan excitation frequency frequency belonging belonging to the gap; to (b the) Amplitude gap; (b) versusAmplitude wavenumber versus at wavenumber a frequency f = at396 a frequency kHz [75]. f=396kHz [75]. According to the analogy of optical super-lens research, it can be found that the method to achieve acousticAccording sub-wavelength to the analogy imaging of isoptical to enlarge super the-lens evanescent research, itwave can so be that found the that evanescent the method wave to achieve acoustic sub-wavelength imaging is to enlarge the evanescent wave so that the evanescent wave propagating in the near field can propagate to the far field imaging. In 2007, an acoustic metamaterial with negative equivalent mass density was proposed by Ambati et al. [76] that can amplify evanescent waves. Based on the above theory, in 2011, Park et al. [77] designed a metamaterial plate that can amplify evanescent waves. Park et al. believe that expanding the evanescent wave must reduce its loss, and requires that the imaginary part of its equivalent mass density is much smaller than the real part. They improved the negative mass density material composed of a fluid that can be embedded with a floating double resonator, and used a structure to cover the double resonator with a thin film to avoid losses. After the double resonator is covered with a thin film, the thin film blocks the liquid, allowing the liquid to move with the entire dual resonator,

Crystals 2020, 10, 305 15 of 26 propagating in the near field can propagate to the far field imaging. In 2007, an acoustic metamaterial with negative equivalent mass density was proposed by Ambati et al. [76] that can amplify evanescent waves. Based on the above theory, in 2011, Park et al. [77] designed a metamaterial plate that can amplify evanescent waves. Park et al. believe that expanding the evanescent wave must reduce its loss, and requires that the imaginary part of its equivalent mass density is much smaller than the real part. They improved the negative mass density material composed of a fluid that can be embedded with a floatingCrystals 20 double20, 10, x FOR resonator, PEER REVIEW and used a structure to cover the double resonator with a thin film to15 avoid of 26 losses. After the double resonator is covered with a thin film, the thin film blocks the liquid, allowing thethereby liquid reducing to move losses. with the This entire structure dual resonator,also has the thereby advantage reducing that the losses. thin This film structure can support also the has du theal advantageresonator and that make the thin it in film a suitable can support position. the dualFurther resonator research, and taking make out it in the a suitable dual resonator position. from Further this research,structure, taking still can out obtain the dual the resonatormaterial with from negative this structure, mass density. still can At obtain this time, the material the structure with negative made of massaCrystals stretched density. 2020, film10 At, x FORcan this PEERstill time, amplifyREVIEW the structure the evanescent made of a wave. stretched There film are can 16 still thin amplify film structures the evanescent arranged wave.15 of in 26 Therethe x aredirection 16 thin and film eight structures arranged arranged in the in z the direction x direction to fo andrm eight a metamaterial arranged in thesystem z direction with negative to form thereby reducing losses. This structure also has the advantage that the thin film can support the dual amass metamaterial density (see system Figure with 22 ( negatived)). After mass calculation, density the (see imaginary Figure 22 d).part After of the calculation, mass density the is imaginary about 4% resonator and make it in a suitable position. Further research, taking out the dual resonator from this partof the of thereal mass part, density the energy is about loss 4% is ofvery the small, real part, and the the energy net amplitude loss is very gain small, of andthe theevanescent net amplitude wave structure, still can obtain the material with negative mass density. At this time, the structure made of gainemerging of the from evanescent the surface wave of emerging the metamaterial from the system surface reaches of the metamaterial 17 times. system reaches 17 times. a stretched film can still amplify the evanescent wave. There are 16 thin film structures arranged in In 2019, Zhang et al. [78] proposed an acoustic metamaterial composed of rubber cylinders arranged the x direction and eight arranged in the z direction to form a metamaterial system with negative in a cubic lattice manner in an aqueous medium (see Figure 23). Using the finite-difference time-domain mass density (see Figure 22 (d)). After calculation, the imaginary part of the mass density is about 4% method (FDTD), they found that this interface mode can be used to enhance the transmission of of the real part, the energy loss is very small, and the net amplitude gain of the evanescent wave evanescent waves and achieve subwavelength imaging. emerging from the surface of the metamaterial system reaches 17 times.

(a) (b) (c)

(d) (e)

Figure 22. (a–c) Evolution of a floating double resonator to a thin film structure; (d) Negative mass density system; (e) Amplified evanescent wave [77].

In 2019, Zhang et al. [78] proposed an acoustic metamaterial composed of rubber cylinders (d) (e) arranged in a cubic lattice manner in an aqueous medium (see Figure 23). Using the finite-difference timeFigure-domainFigure 22. 22 method.( a(a––cc)) Evolution Evolution (FDTD), of of a a they floating floating found double double that resonator resonator this interface to to a a thin thin mode film film structure; structure; can be ( usedd(d)) Negative Negative to enhance mass mass the transmissiondensitydensity system; system; of evanescent ( e(e)) Amplified Amplified waves evanescent evanescent and achieve wave wave subwavelength [ 77[77]]. . imaging.

In 2019, Zhang et al. [78] proposed an acoustic metamaterial composed of rubber cylinders arranged in a cubic lattice manner in an aqueous medium (see Figure 23). Using the finite-difference time-domain method (FDTD), they found that this interface mode can be used to enhance the transmission of evanescent waves and achieve subwavelength imaging.

FigureFigure 23.23. Supercell forfor calculationscalculations ofof interfaceinterface modesmodes [[78]78]..

5.4. Perfect Sound Absorption Effects The near-perfect sound absorption effect exhibited by acoustic metamaterials has attracted the

interest of a large numberFigure of 23 researchers,. Supercell for and calculations the use of of interface its sound modes absorption [78]. characteristics to manufacture sound-absorbing materials has also become a research focus in recent years. In 2010, Pai [79]5.4. Perfectreported Sound that Absorption humans Effect absorbeds electromagnetic waves more than electromagnetic metamaterials, and theoretically proposed a wide-band elastic wave absorber to make it possible to The near-perfect sound absorption effect exhibited by acoustic metamaterials has attracted the use acoustic metamaterials to absorb sound waves. interest of a large number of researchers, and the use of its sound absorption characteristics to manufacture sound-absorbing materials has also become a research focus in recent years. In 2010, Pai [79] reported that humans absorbed electromagnetic waves more than electromagnetic metamaterials, and theoretically proposed a wide-band elastic wave absorber to make it possible to use acoustic metamaterials to absorb sound waves.

Crystals 2020, 10, 305 16 of 26

5.4. Perfect Sound Absorption Eﬀects The near-perfect sound absorption effect exhibited by acoustic metamaterials has attracted the interest of a large number of researchers, and the use of its sound absorption characteristics to manufacture sound-absorbing materials has also become a research focus in recent years. In 2010, Pai [79] reported that humans absorbed electromagnetic waves more than electromagnetic metamaterials, and theoretically proposed a wide-band elastic wave absorber to make it possible to use acoustic metamaterials to absorb sound waves. For the difficult problem of low frequency sound attenuation, Mei et al. [80] designed a structure with a thin semi-circular metal plate attached to the film (see Figure 24). The structure is geometrically open, similar to a cavity system. This structure achieves wide-band sound absorption in the low frequency range of 100–1000 Hz, and the sound absorption rate reaches 86% at a frequency of 172 Hz. When the structure is made into a double-layer film structure, the sound absorption rate can reach 99% at the lowestCrystals resonance 2020, 10, x FOR frequency. PEER REVIEW 16 of 26

(a) (b)

(f)

FigureFigure 24. 2(4a.) Film(a) Film structure structure diagram diagram with a semi-circularwith a semi- metalcircular sheet; metal (b) Broadband sheet; (b) sound Broadband absorption sound diagram;absorption (c–e) diagram; Sound absorption (c–e) Sound spectra absorption at 172 Hz,spectra 340 Hz,at 172 813 Hz Hz, and340 Hzdisplacement, 813 Hz and relationship displacement of therelationship ﬁlm system; of (thef) Graph film system; of sound (f) absorption Graph of sound and displacement absorption and [80 ].displacement [80].

InFor 2018, the Zhaodifficult et al.problem [81] designed of low frequency an ultra-thin sound sub-wavelength attenuation, Mei absorber et al. [80] based designed on aa regularstructure couplingwith a thin array semi that-circular can achieve metal low-frequency plate attached absorption. to the film (see They Figure used a24 di).ﬀ Theerential structure evolution is geometrically algorithm toopen, optimize similar the geometry to a cavity of system. the coupled This microslit, structure and achieves improved wide the-band absorption sound of absorption the microslit in the in the low 300–800frequency Hz range frequency of 100 band−1000 under Hz, and diﬀ theerent sound conditions. absorption The rate ratio reaches of the 86% absorption at a frequency thickness of to172 the Hz. wavelengthWhen the atstructure the absorption is made peak into wasa double less than-layer 3.4%, film whichstructure, proved the thatsound the absorption absorption rate peak can was reach in the99% deep at the Asia. lowest The roleresonance of the wavelengthfrequency. region. In 2018, Zhao et al. [81] designed an ultra-thin sub-wavelength absorber based on a regular coupling array that can achieve low-frequency absorption. They used a differential evolution algorithm to optimize the geometry of the coupled microslit, and improved the absorption of the microslit in the 300–800 Hz frequency band under different conditions. The ratio of the absorption thickness to the wavelength at the absorption peak was less than 3.4%, which proved that the absorption peak was in the deep Asia. The role of the wavelength region. In 2019, Wu [82] proposed an acoustic metamaterial for low-frequency sound absorption consisting of three nested square-opened tubes and opposite opening directions (see Figure 25). The research found that the sound absorption at the sound absorption peak exceeds 90%, and the sound absorption frequency can be effectively adjusted by adjusting the geometric parameters of the sound absorber.

Crystals 2020, 10, 305 17 of 26

In 2019, Wu [82] proposed an acoustic metamaterial for low-frequency sound absorption consisting of three nested square-opened tubes and opposite opening directions (see Figure 25). The research found that the sound absorption at the sound absorption peak exceeds 90%, and the sound absorption frequency can be eﬀectively adjusted by adjusting the geometric parameters of the sound absorber. Crystals 2020, 10, x FOR PEER REVIEW 17 of 26

FigureFigure 25. 25.The The diagram diagram of of the the proposed proposed structure structure [82 [82].].

InIn 2020, 2020, Ji Ji et et al. al. [83 [83]] compared compared the the porous porous acoustic acoustic metamaterial metamaterial with with an an inverted inverted wedge wedge shape shape onon the the plane plane interface interface with with a a uniform uniform melamine melamine layer layer and and a a classic classic wedge-shaped wedge-shaped sound sound absorption absorption materialmaterial of of the the same same quality. quality. Better Better sound sound absorption absorption at a range at a of range frequencies of frequencies and angles and of incidence.angles of incidence.The above properties demonstrate the uniqueness of acoustic metamaterials. It’s these extraordinary propertiesThe ofabove acoustic properties metamaterials demonstrate as well that the lay auniquene solid foundationss of foracoustic its special metamaterials. engineering application. It’s these Theseextraordinary characteristics properties fill the of gap acoust in existingic metamaterials conventional as materialwell that and lay point a solid out foundation the direction for for its further special research.engineering Some application. applications These of acoustic characteristics metamaterials fill the and gap phononic in existing crystals conventional are described material below. and point out the direction for further research. Some applications of acoustic metamaterials and phononic 6.crystals Application are described of Acoustic below. Metamaterials and Phononic Crystals

6.1.6. Application Acoustic Stealth of Acoustic Cloaks Metamaterials and Phononic Crystals In the field of electromagnetics, Leonhardt [84] and Pendry [85] et al. proposed in 2006 the use of6.1. conversion Acoustic opticsStealth to Cloaks achieve stealth. Using the method of conversion optics, the electromagnetic parametersIn the offield metamaterials of electromagnetics, are reasonably Leonhardt designed [84] an sod thatPendry electromagnetic [85] et al. proposed waves doin 2006 not scatterthe use whenof conversion they pass optics through to theachieve hidden stealth. area, Using and guide the method the light of around conversion the object. optics, After the electromagnetic bypassing the metamaterial,parameters of the metamaterials electromagnetic are reasonably wave then designed travels along so that its originalelectromagnetic route. Due waves to thedo not existence scatter ofwhen the cloak, they pass electromagnetic through the waveshidden arearea, transmitted and guide orthe reflected light around on the the cloak, object. and After do notbypassing enter the the areametamaterial, covered by the the electromagnetic cloak. When an wave object then is placed travels inside along the its cloak,original the route. light willDue notto the spread existence inside of thethe cloak, cloak, and electromagnetic the object will waves not be are scattered, transmitted thereby or reflected achieving on the the cloak, effect and of stealth. do not Inenter summary, the area thecovered working by principlethe cloak. of When the cloak an object is to constructis placed ainside virtual the space, cloak, that the is, light a wave-free will not space.spread inside the cloak,Based and onthe the object principle will not of conversionbe scattered, optics, thereby Cummer achieving et al. the [86 effect] determined of stealth. the In existence summary, of the a two-dimensionalworking principle equation of the solutioncloak is ofto transformconstruct acousticsa virtual withspace, anisotropic that is, a masswave-free in cylindrical space. coordinates, and theoretically discussed the possibility of acoustic stealth (see Figure 26). The acoustic transformation formula they proposed for the two-dimensional stealth cloak is as follows:

!2 r r R R r R ρ = , ρ = − 1 , λ 1 = 2 − 1 (2) r r R φ r − R R r − 1 2 − 1 where R1 is the radius of the stealth area and R2 is the outer radius of the stealth shell. Based on the above two-dimensional theory, Chen et al. [87] used the expansion of the Bessel function series in spherical coordinates to solve the scattering problem. In 2008, Cummer et al. [88] veriﬁed the existence of a 3D acoustic wave stealth solution in ﬂuids and the invariance of acoustic (a) (b) wave coordinate transformation, which established a theoretical basis for the proposed 3D stealth cloak.Figure Cheng 26. et ( al.a) Schematic [89] used diagram homogeneous of a stealth and cloak isotropic of 2D materials electromagnetic to achieve metamaterials; two-dimensional (b)Schematic acoustic diagram of a stealth cloak of 3D electromagnetic metamaterials [86].

Based on the principle of conversion optics, Cummer et al. [86] determined the existence of a two-dimensional equation solution of transform acoustics with anisotropic mass in cylindrical coordinates, and theoretically discussed the possibility of acoustic stealth (see Figure 26). The acoustic transformation formula they proposed for the two-dimensional stealth cloak is as follows:

Crystals 2020, 10, x FOR PEER REVIEW 17 of 26

Figure 25. The diagram of the proposed structure [82].

In 2020, Ji et al. [83] compared the porous acoustic metamaterial with an inverted wedge shape on the plane interface with a uniform melamine layer and a classic wedge-shaped sound absorption material of the same quality. Better sound absorption at a range of frequencies and angles of incidence. The above properties demonstrate the uniqueness of acoustic metamaterials. It’s these extraordinary properties of acoustic metamaterials as well that lay a solid foundation for its special engineering application. These characteristics fill the gap in existing conventional material and point out the direction for further research. Some applications of acoustic metamaterials and phononic crystals are described below.

6. Application of Acoustic Metamaterials and Phononic Crystals

6.1. Acoustic Stealth Cloaks In the field of electromagnetics, Leonhardt [84] and Pendry [85] et al. proposed in 2006 the use of conversion optics to achieve stealth. Using the method of conversion optics, the electromagnetic parameters of metamaterials are reasonably designed so that electromagnetic waves do not scatter when they pass through the hidden area, and guide the light around the object. After bypassing the Crystalsmetamaterial,2020, 10, 305 the electromagnetic wave then travels along its original route. Due to the existence18 of o 26f the cloak, electromagnetic waves are transmitted or reflected on the cloak, and do not enter the area covered by the cloak. When an object is placed inside the cloak, the light will not spread inside the stealth of concentric alternating layered structure. Torrent et al. [90] used two-dimensional composite cloak, and the object will not be scattered, thereby achieving the effect of stealth. In summary, the cylindrical structure to achieve two-dimensional acoustic stealth. working principle of the cloak is to construct a virtual space, that is, a wave-free space.

Crystals 2020, 10, x FOR PEER REVIEW 18 of 26 Crystals 2020, 10, x FOR PEER REVIEW 18 of 26 2 r r R1 -1 R2  r R 1 r  ,  = ,  = 2 (2) rr R1 rr R1 -1 R2R2 R 1  r r R 1 r  ,  = ,  =  (2) r R1 r R2 R 1  r where R1 is the radius of the stealth area and R2 is the outer radius of the stealth shell.

whereBased R1 is onthe the radius above of thetwo stealth-dimensional area and theory, R2 is the Chen outer et al.radius [87] ofused the thestealth expansion shell. of the Bessel functionBased series on thein sphericalabove two coordinates-dimensional to solvetheory, the Chen scattering et al. [87]problem. used theIn 2008, expansion Cummer of theet al.Bessel [88] functionverified theseries existence in spherical of a 3D coordinates acoustic(a) wave to solve stealth the solutionscattering in problem.fluid(b) s and In the 2008, invariance Cummer of et acoustic al. [88] verifiedwave coordinate the existence transformation, of a 3D acoustic which wave established stealth solution a theoretical in fluid basiss and for the the invariance proposed of3D acoustic stealth FigureFigure 26. 26(.a ()a) Schematic Schematic diagram diagram of of aa stealthstealth cloakcloak of 2D electromagnetic metamaterials; metamaterials; ( (b))SchematicSchematic wavecloak. coordinate Cheng et al.transformation, [89] used homogeneous which established and isotropic a theoretical materials basis tofor achieve the proposed two-dimensional 3D stealth diagramdiagram of of a stealtha stealth cloak cloak of of 3D 3D electromagnetic electromagnetic metamaterials metamaterials [ 86[86]]. . cloak.acoustic Cheng stealth et of al. concentric [89] used al homogeneousternating layered and structure. isotropic Torrent materials et al. to [90] achieve used two-dimensional acousticcompositeTheBased abovestealth cylindrical on aretheof concentric theoreticallyprinciple structure ofal ternatingtoconversion achieved achieve layered two acoustic optics,-dimensional structure. Cummer stealth, inTorrentacoustic et contrast, al. [86] et stealth. al. determined to [90] achieve used acoustictwothe -existencedimensional stealth of ina experimentscompositetwo-Thedimensional above cylindrical is relatively are equationtheoretically structure hard. solution Stealth to achieved achieve cloaksof transform twoacoustic usually-dimensional stealth, acoustics require in acoustic the contrast, with production stealth. anisotropic to achieve of anisotropic mass acoustic in cylindricalstealth materials. in Ifexperimentcoordinates, oneThe wants aboves to is and achieverelatively are theoretically theoretically acoustic hard. discussedStealth stealth achieved cloaks in the reality, acoustic possibilityusually one stealth, require must of acoustic ﬁndin the contrast, another production stealth to way. (seeachieve of InFigure anisotropic 2017, acoustic 26 Chen). The materials.stealth et acoustic al. [91in ] proposedexperimentIftransformation one want a ring-shapeds tois relativelyachieve formula acoustic underwater hard.they proposed Stealth stealth acoustic cloaks in for reality, the capeusually two one structure-dimensional requiremust find made the another productionstealth of aluminum way.cloak of In is anisotropic blocks 2017,as follows: Chen (see materials. Figureet al. [91]27), achievingIfproposed one want ana sring averageto achieve-shaped reduction acoustic underwater ofstealth6.3 acoustic dBin reality, in cape the frequencyone structur muste find made range another of of aluminum 9–15 way. kHz. In blocks 2017, (seeChen Figure et al. [91]27), proposedachieving aan ring average-shaped reduction underwater of −−6 acoustic.3 dB in capethe frequency structure rangemade of aluminum9–15 kHz. blocks(see Figure 27), achieving an average reduction of −6.3 dB in the frequency range of 9–15 kHz.

(a) (b) (a) (b) FigureFigure 27. 27.( a()a Image) Image of ofthe the cloak cloak machined machined from from an aluminum an aluminum block; block; (b) Enlarged (b) Enlarged view of view one segment, of one highlightedFiguresegment, 27. highlighted in(a) (aImage)[91]. in of (a the) [91] cloak. machined from an aluminum block; (b) Enlarged view of one segment, highlighted in (a) [91]. TheThesame same year,year, BiBiet et al.al. [[92]92] designeddesigned anan underwaterunderwater carpet cape composed composed of of multilayer multilayer brass brass platesplatesThe by by introducing sameintroducing year, Bi aa scaleetscale al. factorfactor[92] designed andand sacriﬁcingsacrificing an underwater impedance carpet matching cape composed (see (see Figure Figure of 228 8multilayer)).. Experimental Experimental brass resultsplatesresults showedby show introducinged thatthat thethe a scale stealthstealth factor carpetcarpet and can sacrificing hide the impedance uneven information information matching (seeon on the theFigure reflective reﬂective 28). Experimental surface surface in in aa wideresultswide frequency frequency showed range. thatrange. the stealth carpet can hide the uneven information on the reflective surface in a wide frequency range.

Figure 28. Experimental prototype of the carpet cloak [92]. FigureFigure 28.28. Experimental prototype of the the carpet carpet cloak cloak [92] [92].. In 2019, He et al. [93] proposed an ultra-thin planar stealth structure based on a periodic sound gridIn In structure 2019, 2019, He He(seeet et al.Figureal. [[93]93] proposedproposed 29.). Through anan ultra-thinultra software-thin planar simulation stealth results, structure structure it wasbased based found on on a a periodic that periodic when sound sound the gridgridincident structure structure frequency (see(see Figure Figurewas 300 29 2). 9kHz Through.). Through, the acoustic software software target simulation simulation intensity results, decreased results, it was it(TSR) found was − found that5.425 when dB that. the when incident the frequencyincident frequency was 300 kHz, was 300 the acoustickHz, the targetacoustic intensity target intensity decreased decreased (TSR) 5.425 (TSR) dB. −5.425 dB. −

CrystalsCrystals 20202020, 10, 10, ,x 305 FOR PEER REVIEW 19 of19 26 of 26 Crystals 2020, 10, x FOR PEER REVIEW 19 of 26

Figure 29. Acoustic metamaterial for ultrasound cloaking [93]. FigureFigure 29 29.. AAcousticcoustic metamaterial metamaterial for ultrasound cloakingcloaking [ 93[93]]. . 6.2.6.2. Extraordinary Extraordinary A Acousticcoustic A Abnormalbnormal TransmissionTransmission E Effffectects s 6.2. Extraordinary Acoustic Abnormal Transmission Effects InIn 1998, 1998, Ebbesen Ebbesen et et al. al. [94 ][94] realized realized the phenomenon the phenomenon of optical of extraordinary optical extraordinary transmission transmission through a In 1998, Ebbesen et al. [94] realized the phenomenon of optical extraordinary transmission throughsubwavelength a subwavelength hole array. hole Inspired array. by Ebbesen,Inspired Luby etEbbesen, al. [95] found Lu et thatal. [95] the soundfound wave’s that the extraordinary sound wave's through a subwavelength hole array. Inspired by Ebbesen, Lu et al. [95] found that the sound wave's extraordinarytransmission behavior transmission through behavior a one-dimensional through grating a one- achievesdimensional a sound grating transmittance achieves close a to sound 1. extraordinary transmission behavior through a one-dimensional grating achieves a sound transmittanceHe et al. [96] sculpted close to periodical 1. He et al. air grooves[96] sculpted on both periodical sides of a copper air grooves plate, andon both also achieved sides of extraordinarya copper plate, transmittance close to 1. He et al. [96] sculpted periodical air grooves on both sides of a copper plate, andsound also transmission. achieved extraordinary sound transmission. and also achieved extraordinary sound transmission. ByBy changing the the structural structural parameters, parameters, the curled the space curled structure space can structure also achieve can the also characteristics achieve the ofBy metamaterials changing the such structural as zero mass parameters, density, so the the curl curleded space space structure structure can can also also achieve achieve the the characteristics of metamaterials such as zero mass density, so the curled space structure can also characteristicsextraordinary of acoustic metamaterials transmission such e ff asect. zero In 2015, mass Cheng density, et al. so [97 the] designed curled space a Mie resonance structure systemcan also achieve the extraordinary acoustic transmission effect. In 2015, Cheng et al. [97] designed a Mie achievewith a the curly extraordinary space structure, acoustic as shown transmission in Figure 30 a. effect. This Mie In 2015, ultra-slow Cheng fluid-like et al. [97] unit designed has eight parts a Mie resonance system with a curly space structure, as shown in Figure 30(a). This Mie ultra-slow fluid- resonancedistributed system uniformly, with a as curly shown space in Figurestructure, 30b, as and shown the simplified in Figure structure 30(a). This is shown Mie ultra in Figure-slow 30fluidc. - like unit has eight parts distributed uniformly, as shown in Figure 30(b), and the simplified structure likeIn unit Figure has 30 eightd, a longparts and distributed tortuous uniformly, acoustic wave as shown transmission in Figure channel 30(b), is and set, andthe simplified the acoustic structure wave is shown in Figure 30(c). In Figure 30(d), a long and tortuous acoustic wave transmission channel is is shownpropagation in Figure path is30 compared(c). In Figure after the30(d acoustic), a long metamaterial and tortuous unit acoustic is placed. wave When transmission not placed, the channel sound is set, and the acoustic wave propagation path is compared after the acoustic metamaterial unit is set,wave andbasically the acoustic has no wave eye channel propagation transmission. path isAfter compared being placed, after most the ofacoustic the sound metamaterial wave propagates unit is placed. When not placed, the sound wave basically has no eye channel transmission. After being placed.through When the channel. not placed, The resultsthe sound of the wave comparative basically experiments has no eye clearly channel show thetransmission. ultrasonic transmission After being placed, most of the sound wave propagates through the channel. The results of the comparative placed,capability most of of the the Mie sound resonance wave unit. propagates through the channel. The results of the comparative experiments clearly show the ultrasonic transmission capability of the Mie resonance unit. experiments clearly show the ultrasonic transmission capability of the Mie resonance unit.

(a) (b) (c) (a) (b) (c)

(d) (d) FigureFigure 30 30.. (a(–ac–)c ) Mie Mie resonant resonant unit unit with coiled coiled space space structure; structure; ( d ()d Meta-acoustic) Meta-acoustic transmission transmission of of Figure 30. (a–c) Mie resonant unit with coiled space structure; (d) Meta-acoustic transmission of metamaterialsmetamaterials [97] [97].. metamaterials [97].

Crystals 2020, 10, 305 20 of 26 Crystals 2020, 10, x FOR PEER REVIEW 20 of 26

In 2018,In Dong 2018, Donget al. et[98] al. [98proposed] proposed an an acoustic acoustic supersuper lens lens based based on the on idea the ofidea far-field of far acoustic-field acoustic super-resolution (see Figure 31). It was found that the far-field super-resolution of λ/6 can be achieved super-resolutionCrystals 2020(see, 10, x FORFigure PEER 31 REVIEW). It was found that the far-field super-resolution of λ/6 can20 ofbe 26 achieved by using a multiple signal classification algorithm combined with a supersonic lens. by using a multipleIn 2018, Dong signal et al.classification [98] proposed algorithm an acoustic combinedsuper lens based with on a supersonicthe idea of far-field lens. acoustic super-resolution(see Figure 31). It was found that the far-field super-resolution of λ/6 can be achieved by using a multiple signal classification algorithm combined with a supersonic lens.

FigureFigureFigure 31. A31.31. schematic A schematic diagram diagram ofof anan an acousticacoustic acoustic superlenssuperlens superlens [[98].98]. [98].

In 2020, Dong et al. [99] designed a class of multi-cavity 2D and 3D acoustic metamaterials (AMMs), In 2020,In Dong 2020, Donget al. et [99] al. [99]designed designed a a class class of multi-cavity multi-cavity 2D 2Dand and3D acoustic 3D acoustic metamaterials metamaterial s and(AMMs), converted and converted the 3D structure the 3D intostructure a super into lens a super (see Figure lens (see 32). Figure 32). (AMMs), and converted the 3D structure into a super lens (see Figure 32).

(a) (b) FigureFigure 32.32. ((aa)) FabricatedFabricated 3D3D(a) microstructuremicrostructure sample; sample; ( b(b)) Photograph Photograph(b) of of a 3Da 3D superlens superlens containing containing 8 8 9× × 9 ×3 3 microstructures microstructures [99 [99].]. Figure 32.× (a) Fabricated 3D microstructure sample; (b) Photograph of a 3D superlens containing 8 × 9 × 3 microstructuresNumericalNumerical andand [99] experimentalexperimental. verificationsverifications showshow thatthat thethe intensityintensity distributiondistribution measuredmeasured byby thethe 3D3D superlenssuperlens indicatesindicates thatthat thethe sub-wavelengthsub-wavelength resolutionresolution isis 0.460.46λλ00,, andand twotwo well-focusedwell-focused pointspoints cancan Numericalbebe obtainedobtained and regardlessregardless experimental ofof thethe phasephase verifications shift.shift. Secondly, show duedue that toto thethe the highlyhighly intensity symmetricalsymmetrical distribution 3D3D AMM,AMM, measured acoustic by the imagingimaging cancan bebe achievedachieved withwith normalnormal incidenceincidence onon eacheach faceface ofof thethe sample.sample. 3D superlens indicates that the sub-wavelength resolution is 0.46λ0, and two well-focused points can be obtained6.3.6.3. Underwater regardless Sound Sound of the Absorption Absorption phase shift. Secondly, due to the highly symmetrical 3D AMM, acoustic imaging can be achieved with normal incidence on each face of the sample. InIn 2019,2019, Zhong Zhong et et al. al. [100 [100]] put put forward forward the twothe theoreticaltwo theoretical requirements requiremen for thets ultra-thinfor the ultra-thin acoustic surfaceacoustic to surface fully absorb to fully low-frequency absorb low-frequency underwater underwater acoustic broadband acoustic underbroadband the action under of the semi-infinite action of 6.3. Underwaterair,semi-infinite and based Sound air, on A this,andbsorption based designed on this, a rubber designed periodically a rubber embedded periodically in theembedded cavity (see in the Figure cavity 33). (see The Figure layer is acoustically super-surface and achieves perfect sound absorption at a frequency of 500 Hz. In 2019,33). The Zhong layer is etacoustically al. [100] super-surface put forward and theachieves two perfect theoretical sound absorption requirements at a frequency for the of ultra -thin 500 Hz.The same year, Shi et al. [101] designed a multilayer local resonance scattering body based on acousticthe surface local resonance to fully mechanismabsorb low and-frequency the idea of underwater coupled resonance acoustic (see Figure broadband 34). They under used finitethe action of semi-infiniteelement air, analysis and based to study on thethis, band designed gap characteristics a rubber periodically and sound absorption embedded characteristics in the cavity ofthe (see Figure 33). Theproposed layer is multilayeredacoustically locally super resonant-surface acoustic and achieves metamaterials perfect (M-LRAMs). sound Numericalabsorption results at a showfrequency of 500 Hz. that M-LRAMs have more obvious advantages in underwater sound absorption than traditional locally resonant acoustic metamaterials (LRAMs).

(a) (b)

Figure 33. (a) A schematic view of an underwater absorptive metasurface with a finite-thickness steel plate followed by air; (b) A physical realization of the proposed metasurface for perfect absorption [100].

Crystals 2020, 10, x FOR PEER REVIEW 20 of 26

In 2018, Dong et al. [98] proposed an acoustic super lens based on the idea of far-field acoustic super-resolution(see Figure 31). It was found that the far-field super-resolution of λ/6 can be achieved by using a multiple signal classification algorithm combined with a supersonic lens.

Figure 31. A schematic diagram of an acoustic superlens [98].

In 2020, Dong et al. [99] designed a class of multi-cavity 2D and 3D acoustic metamaterials (AMMs), and converted the 3D structure into a super lens (see Figure 32).

(a) (b)

Figure 32. (a) Fabricated 3D microstructure sample; (b) Photograph of a 3D superlens containing 8 × 9 × 3 microstructures [99].

Numerical and experimental verifications show that the intensity distribution measured by the 3D superlens indicates that the sub-wavelength resolution is 0.46λ0, and two well-focused points can be obtained regardless of the phase shift. Secondly, due to the highly symmetrical 3D AMM, acoustic imaging can be achieved with normal incidence on each face of the sample.

6.3. Underwater Sound Absorption In 2019, Zhong et al. [100] put forward the two theoretical requirements for the ultra-thin acoustic surface to fully absorb low-frequency underwater acoustic broadband under the action of semi-infinite air, and based on this, designed a rubber periodically embedded in the cavity (see Figure 33Crystals). The2020 layer, 10, 305is acoustically super-surface and achieves perfect sound absorption at a frequency21 of 26 of 500 Hz.

Crystals 2020, 10, x FOR PEER REVIEW 21 of 26

The same year, Shi et al. [101] designed a multilayer local resonance scattering body based on the local resonance mechanism and the idea of coupled resonance (see Figure 34). They used finite element analysis to study the band gap characteristics and sound absorption characteristics of the (a) (b) proposed multilayered locally resonant acoustic metamaterials (M-LRAMs). Numerical results show that MFigureFigure-LRAMs 33. ((a ) have A schematic more viewobviousview of of an an advantages underwater underwater absorptive inabsorptive underwater metasurface metasurface sound with with absorption a ﬁnite-thickness a finite-thickness than steel traditional steel locallyplateplate resonant followed acoustic by by air; air; (b metamaterials)(b A) physicalA physical realization realization (LRAMs of the) of. proposed the proposed metasurface metasurface for perfect for absorption perfect absorption [100]. [100].

(a) (b) (c)

FigureFigure 34. The cross-section cross-section for afor unit a cellunit of cell the threeof the proposed three proposed devices: (a )devices Conventional: (a) C LRAMs;onventional (b) LRAMs LRAMs; (withb) L double-layeredRAMs with double cylindrical-layered scatterers; cylindrical (c) LRAMs scatter wither three-layereds; (c) LRAMs cylindrical with three scatterers-layered [101]. cylindrical scatterers [101]. The same year, Zhong et al. [102] proposed a metamaterial sheet consisting of a particle-filled polyurethaneThe same damping year, Zhong material et andal. [102] a square proposed lattice of a a metamaterial spiral resonator. sheet Based consisting on theoretical of a calculations particle-filled polyurethaneand experimental damping results, material they found and that a square the average lattice sound of a absorption spiral resonator. coefficient Based of the on proposed theoretical calculationsmetamaterial and board experimental in the frequency results, range they of found 0.8–6 kHz that is the 0.54 average at normal sound atmospheric absorption pressure. coefficient Further of the proporesearchsed found metamaterial that the average board sound in the absorption frequency coefficient range of of0.8 the−6 metamaterialkHz is 0.54 at board normal can reach atmospheric 0.51 pressure.under the Further conditions research that the found pressure that is lowerthe average than 0.5 sound MPa and absorption the frequency coefficient range isof 1.5–6 the metamaterial kHz. board6.4. Other can reach Applications 0.51 under the conditions that the pressure is lower than 0.5 MPa and the frequency range is 1.5–6 kHz. Over the past ten years, due to the strange properties of acoustic metamaterials, acoustic metamaterials 6.4.have Other been A appliedpplications to many other aspects. Christensen et al. [103] achieved the acoustic collimation of longer-wavelength sound waves through a two-dimensional array of subwavelength holes in a metalOver ﬁlm. the Analogous past ten to years, optical due rainbows, to the Cox strange et al. properties [104] implemented of acoustic acoustic metamaterials, rainbows using acoustic metamaterialsmany thin perforated have been plates applied separated to many by half other a wavelength. aspects. Christensen Liang et al. et [al.105 [103],106] achieved have theoretically the acoustic collimationproposed and of longer experimentally-wavelength designed sound a simplewaves 1Dthrough superlattice a two- anddimensional an acoustic array diode of composed subwavelength of holesnonlinear in a metal materials, film. which Analogous can change to optical the original rainbows, frequency Cox etf toal.2f [104]to achieve implemented the frequency acoustic doubling rainbows usingfunction. many With thin the perforated rapid development plates separated of acoustic by metamaterials, half a wavelength. its unique Liang properties et al. are [105,106] constantly have theoreticallybeing discovered proposed and applied and experimentally to many new ﬁelds, designed such a as simple architectural 1D superlattice acoustics andand soan on. acoustic diode composed of nonlinear materials, which can change the original frequency f to 2f to achieve the frequency doubling function. With the rapid development of acoustic metamaterials, its unique properties are constantly being discovered and applied to many new fields, such as architectural acoustics and so on.

7. Concluding Remarks The study of phononic crystals and acoustic metamaterials not only has important scientific research value, but also great practical value. The discovery of phononic crystals and acoustic metamaterials provides a new solution for vibration isolation and sound absorption. Therefore, the research work on phononic crystals and acoustic metamaterials has attracted widespread attention

Crystals 2020, 10, 305 22 of 26

(1) Deepen the effect of lattice parameters, structural configurations, and material properties of phononic crystals and acoustic metamaterials on the band gap characteristics; (2) Based on existing research on phononic crystal forbidden bands, designing more phononic crystals with different materials, shapes and decreasing the lattice sizes to meet actual needs; (3) Continue to advance the research on the local resonance mechanism for acoustic metamaterials and strive to achieve the control of elastic waves in low frequency bands and broader width for forbidden bands; (4) Increase the experimental research of phononic crystals and acoustic metamateirals, and expand theoretical research to practical applications.

The exploration of phononic crystals is only a few decades old, and there are still many valuable materials and uses to be studied. With the deepening of research on phononic crystals, many practical acoustic metamaterials will be extended to promote the development of sound insulation and noise reduction.

Author Contributions: Writing and revision: J.L.; revision and editing: H.G.; conceptualization, review and editing: T.W. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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