1525:(due to impedance differences) through the materials that form the composite. Phononic crystals effectively reduce low-frequency noise, since their locally resonant systems act as spatial frequency filters. However, they have narrow band gaps, impose additional weight on the primary system, and work only at the adjusted frequency range. For widening band gaps, the unit cells must be large in size or contain dense materials. As a solution to the disadvantages mentioned above of phononic crystals, proposes a novel three-dimensional lightweight re-entrant meta-structure composed of a cross-shaped beam scatterer embedded in a host plate with holes based on the square lattice metamaterial. By combining the re-entry networks mechanism and the FloquetâBloch theory, on the basis of cross-shaped beam theory and perforation mechanism, it was demonstrated that such a lightweight phononic structure can filter elastic waves across a broad frequency range (not just a specific narrow region) while simultaneously reducing structure weight to a significant degree.
1829:(instead of two) creates the low-frequency resonances to achieve double negativity. With monopolar resonance, the spheres expand, which produces a phase shift between the waves passing through rubber and water. This creates a negative response. The dipolar resonance creates a negative response such that the frequency of the center of mass of the spheres is out of phase with the wave vector of the sound wave (acoustic signal). If these negative responses are large enough to compensate the background fluid, one can have both negative effective bulk modulus and negative effective density.
805:
1987:. This causes sound waves to vary their speed from ring to ring. The sound waves propagate around the outer ring, guided by the channels in the circuits, which bend the waves to wrap them around the outer layers. This device has been described as an array of cavities which actually slow the speed of the propagating sound waves. An experimental cylinder was submerged in a tank, and made to disappear from sonar detection. Other objects of various shapes and densities were also hidden from sonar.
1323:
902:
2011:(CNOT) gate, a key component in quantum computing, have been demonstrated. By employing a nonlinear acoustic metamaterial, consisting of three elastically coupled waveguides, the team created classical qubit analogues called logical phi-bits. This approach allows for scalable, systematic, and predictable CNOT gate operations using a simple physical manipulation. This innovation brings promise to the field of quantum-like computing using acoustic metamaterials.
1542:
1550:
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881:, and how to measure some other physical properties using sound. With acoustic metamaterials the direction of sound through the medium can be controlled by manipulating the acoustic refractive index. Therefore, the capabilities of traditional acoustic technologies are extended, for example, eventually being able to cloak certain objects from acoustic detection.
36:
969:
1952:
sheet of nonlinear acoustic materialâone whose sound speed varies with air pressure. An example of such a material is a collection of grains or beads, which becomes stiffer as it is squeezed. The second component is a filter that allows the doubled frequency to pass through but reflects the original.
1174:
at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. In comparison, infrasonic frequencies range from 20 Hz down to 0.001 Hz, audible frequencies are 20 Hz to 20 kHz and the ultrasonic range is above 20 kHz.
1931:
have a negative dynamic modulus for ultrasound waves. A point source of 60.5 kHz sound was focused to a spot roughly the width of half a wavelength, and there is potential of improving the spatial resolution even further. Result were in agreement with the transmission line model, which derived
1449:
In order to speed up the calculation of the frequency band structure, the
Reduced Bloch Mode Expansion (RBME) method can be used. The RBME applies "on top" of any of the primary expansion numerical methods mentioned above. For large unit cell models, the RBME method can reduce the time for computing
844:
Research employing acoustic metamaterials began in 2000 with the fabrication and demonstration of sonic crystals in a liquid. This was followed by transposing the behavior of the split-ring resonator to research in acoustic metamaterials. After this, double negative parameters (negative bulk modulus
1832:
Both the mass density and the reciprocal of the bulk modulus decrease in magnitude fast enough for the group velocity to become negative (double negativity). This gives rise to the desired results of negative refraction. The double negativity is a consequence of resonance and the resulting negative
1828:
This behavior is analogous to low-frequency resonances produced in SRRs (electromagnetic metamaterial). The wires and split rings create intrinsic electric dipolar and magnetic dipolar response. With this artificially constructed acoustic metamaterial of rubber spheres and water, only one structure
1326:
Copper split-ring resonators and wires mounted on interlocking sheets of fiberglass circuit board. A split-ring resonator consists of an inner square with a split on one side embedded in an outer square with a split on the other side. The split-ring resonators are on the front and right surfaces of
1982:
A laboratory metamaterial device that is applicable to ultrasound waves was demonstrated in 2011 for frequencies from 40 to 80 kHz. The metamaterial acoustic cloak was designed to hide objects submerged in water, bending and twists sound waves. The cloaking mechanism consists of 16 concentric
1824:
is speed of the acoustic signal. The effective bulk modulus and density near the static limit are positive as predicted. The monopolar resonance creates a negative bulk modulus above the normalized frequency at about 0.035 while the dipolar resonance creates a negative density above the normalized
1796:
As an example, acoustic double negativity is theoretically demonstrated with a composite of soft, silicone rubber spheres suspended in water. In soft rubber, sound travels much slower than through the water. The high velocity contrast of sound speeds between the rubber spheres and the water allows
928:
per unit volume and is expressed in grams per cubic centimeter (g/cm). In all three classic states of matterâgas, liquid, or solidâthe density varies with a change in temperature or pressure, with gases being the most susceptible to those changes. The spectrum of densities is wide-ranging: from 10
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Negative bulk modulus is achieved through monopolar resonances of the BWS series. Negative mass density is achieved with dipolar resonances of the gold sphere series. Rather than rubber spheres in liquid, this is a solid based material. This is also as yet a realization of simultaneously negative
1951:
was introduced in 2009, which converts sound to a different frequency and blocks backward flow of the original frequency. This device could provide more flexibility for designing ultrasonic sources like those used in medical imaging. The proposed structure combines two components: The first is a
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Even for composite materials, the effective bulk modulus and density should be normally bounded by the values of the constituents, i.e., the derivation of lower and upper bounds for the elastic moduli of the medium. The expectation for positive bulk modulus and positive density is intrinsic. For
840:
produced the basic elements of metamaterials in the late 1990s. His materials were combined, with negative index materials first realized in 2000, broadening the possible optical and material responses. Research in acoustic metamaterials has the same goal of broader material responses with sound
1354:
which operate in a certain frequency range. Elements which interact and resonate in their respective localized area are embedded throughout the material. In acoustic metamaterials, locally resonant elements would be the interaction of a single 1-cm rubber sphere with the surrounding liquid. The
812:
is large-scale example of a phononic crystal: it consists of a periodic array of cylinders in air (the 'metamaterial' or 'crystal structure') and its dimensions and pattern is designed such that sound waves at a frequency of 1670 Hz are strongly attenuated. It became the first evidence for the
1867:
Double C resonators (DCRs) are rings cut in half, which can be arranged in multiple cell configurations, similarly to the SRRS. Each cell consists of a large rigid disk and two thin ligaments, and acts as a tiny oscillator connected by springs. One spring anchors the oscillator, and the other
1310:
The amplitudes of the sound waves entering the surface were compared with the sound waves at the center of the structure. The oscillations of the coated spheres absorbed sonic energy, which created the frequency gap; the sound energy was absorbed exponentially as the thickness of the material
1359:
and band-gap frequencies can be controlled by choosing the size, types of materials, and the integration of microscopic structures which control the modulation of the frequencies. These materials are then able to shield acoustic signals and attenuate the effects of anti-plane shear waves. By
1002:
The simplest realization of an acoustic metamaterial would constitute the propagation of a pressure wave through a slab with a periodically modified refractive index in one dimension. In that case, the behavior of the wave through the slab or 'stack' can be predicted and analyzed using
1919:
presented the design and test results of an ultrasonic metamaterial lens for focusing 60 kHz (~2 cm wavelength) sound waves under water. The lens was made of sub-wavelength elements, potentially more compact than phononic lenses operating in the same frequency range.
1307:. Transmission was measured as a function of frequency from 250 to 1600 Hz for a four-layer sonic crystal. A two-centimeter slab absorbed sound that normally would require a much thicker material, at 400 Hz. A drop in amplitude was observed at 400 and 1100 Hz.
1296:, which exhibit spectral gaps two orders of magnitude smaller than the wavelength of sound. The spectral gaps prevent the transmission of waves at prescribed frequencies. The frequency can be tuned to desired parameters by varying the size and geometry.
1778:. This requires negativity in bulk modulus and density. Natural materials do not have a negative density or a negative bulk modulus, but, negative values are mathematically possible, and can be demonstrated when dispersing soft rubber in a liquid.
1887:. The DCR design produced a suitable band with a negative slope in a range of frequencies. This band was obtained by hybridizing the modes of a DCR with the modes of thin stiff bars. Calculations have shown that at these frequencies:
1577:
structures that exhibit effective negative permittivity and negative permeability for some frequency ranges. In contrast, it is difficult to build composite acoustic materials with built-in resonances such that the two effective
1520:
The position of the band gap in frequency space for a phononic crystal is controlled by the size and arrangement of the elements comprising the crystal. The width of the band gap is generally related to the difference in the
1974:
are needed. Making such a metamaterial for a sound means modifying the acoustic analogues to permittivity and permeability in light waves, which are the material's mass density and its elastic constant. Researchers from
777:, but also extremely small-scale phenomena like atoms. The latter is possible due to band gap engineering: acoustic metamaterials can be designed such that they exhibit band gaps for phonons, similar to the existence of
1418:
and mass). One of their main properties is the possibility of having a phononic band gap. A phononic crystal with phononic band gap prevents phonons of selected ranges of frequencies from being transmitted through the
1782:
example, dispersing spherical solid particles in a fluid result in the ratio governed by the specific gravity when interacting with the long acoustic wavelength (sound). Mathematically, it can be proven that β
1512:
through that medium. Likewise, when the advancing wave-front meets a low impedance medium it will slow down. This concept can be exploited with periodic arrangements of impedance-mismatched elements to affect
892:
Properties of acoustic metamaterials usually arise from structure rather than composition, with techniques such as the controlled fabrication of small inhomogeneities to enact effective macroscopic behavior.
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1970:
homes, advanced concert halls, or stealth warships. The idea of acoustic cloaking is simply to deviate the sounds waves around the object that has to be cloaked, but realizing has been difficult since
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of artificially created SRRs, paralleled an analysis of sonic crystals. The band gap properties of SRRs were related to sonic crystal band gap properties. Inherent in this inquiry is a description of
1883:
A phononic band gap occurs in association with the resonance of the split cylinder ring. There is a phononic band gap within a range of normalized frequencies. This is when the inclusion moves as a
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This lens could improve acoustic imaging techniques, since the spatial resolution of the conventional methods is restricted by the incident ultrasound wavelength. This is due to the quickly fading
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1744:
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For acoustic materials and acoustic metamaterials, both bulk modulus and density are component parameters, which define their refractive index. The acoustic refractive index is similar to the
1402:
This method can be used to tune band gaps inherent in the material, and to create new low-frequency band gaps. It is also applicable for designing low-frequency phononic crystal waveguides.
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and the bulk modulus β parameters, which are analogous to permittivity and permeability, respectively. The sonic (or phononic) metamaterials are sonic crystals. These crystals have a solid
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Comparison of 1D, 2D and 3D phononic crystal structures where the metamaterial exhibits a periodic variation of sound speed in 1, 2 and 3 dimensions (from left to right, respectively).
1927:
that oscillate at certain frequencies. Similar to a network of inductors and capacitors in an electromagnetic metamaterial, the arrangement of
Helmholtz cavities designed by Zhang
2100:
54:
984:), the phononic crystal comprises pressure waves (phonons) propagating through a material with a periodically modified acoustic refractive index, resulting in a modified
3785:
859:) were produced by this type of medium. Then a group of researchers presented the design and test results of an ultrasonic metamaterial lens for focusing 60 kHz.
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for the transmission of very low monopolar and dipolar frequencies. This is an analogue to analytical solution for the scattering of electromagnetic radiation, or
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In electromagnetic metamaterials negative permittivity can be found in natural materials. However, negative permeability has to be intentionally created in the
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Shelby, R. A.; Smith, D. R.; Nemat-Nasser, S. C.; Schultz, S. (2001). "Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial".
1880:
and shear modulus Îź. Although linear elasticity is considered, the problem is mainly defined by shear waves directed at angles to the plane of the cylinders.
1213:
are negligible. Because of the bonding between them, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration
1808:
Hence, there is a narrow range of normalized frequencies 0.035 < Ďa/(2Ďc) < 0.04 where the bulk modulus and negative density are both negative. Here
980:: instead of electromagnetic waves (photons) propagating through a material with a periodically modified optical refractive index (resulting in a modified
3243:
Ravanbod, Mohammad (2023). "Innovative
Lightweight Re-Entrant Cross-like Beam Phononic Crystal with Perforated Host for Broadband Vibration Attenuation".
1375:. Methods which can be applied to two-dimensional stopband and band gap control with either photonic or sonic structures have been developed. Similar to
139:. They can be engineered to either transmit, or trap and amplify sound waves at certain frequencies. In the latter case, the material is an acoustic
1299:
The fabricated material consisted of high-density solid lead balls as the core, one centimeter in size and coated with a 2.5-mm layer of rubber
797:. Important branches of physics and technology that rely heavily on acoustic metamaterials are negative refractive index material research, and
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array of bubble-contained-water spheres (BWSs) and another relatively shifted fcc array of rubber-coated-gold spheres (RGSs) in special epoxy.
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moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through metamaterial
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In 2007 a metamaterial was reported which simultaneously possesses a negative bulk modulus and negative mass density. This metamaterial is a
3487:(synopsis for "Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems" by Bin Liang, Bo Yuan, and Jian-chun Cheng)
1335:(SRR) became the object of acoustic metamaterial research. An analysis of the frequency band gap characteristics, derived from the inherent
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Ding, Yiqun; Liu, Zhengyou; Qiu, Chunyin; Shi, Jing (2007). "Metamaterial with
Simultaneously Negative Bulk Modulus and Mass Density".
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Zhengyou Liu, Liu; Xixiang Zhang; Yiwei Mao; Y. Y. Zhu; Zhiyu Yang; C. T. Chan; Ping Sheng (2000). "Locally
Resonant Sonic Materials".
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the effective mass density and compressibility. This metamaterial lens also displays variable focal length at different frequencies.
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72:
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curves. Movchan and
Guenneau analyzed and presented low-frequency band gaps and localized wave interactions of the coated spheres.
2007:
Researchers have demonstrated a quantum-like computing method using acoustic metamaterials. Recently operations similar to the
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1303:. These were arranged in an 8 Ă 8 Ă 8 cube crystal lattice structure. The balls were cemented into the cubic structure with an
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in atoms. However, unlike atoms and natural materials, the properties of metamaterials can be fine-tuned (for example through
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of capacitance, C, and inductance, L, and resonant frequency â1/(LC). The speed of sound in the matrix is expressed as c = â
423:
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An acoustic cloak is a hypothetical device that would make objects impervious to sound waves. This could be used to build
794:
1979:, China in a 2007 paper reported a metamaterial which simultaneously possessed a negative bulk modulus and mass density.
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Fang, Nicholas; Xi, Dongjuan; Xu, Jianyi; Ambati, Muralidhar; Srituravanich, Werayut; Sun, Cheng; Zhang, Xiang (2006).
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Physics of
Negative Refraction and Negative Index Materials: Optical and Electronic Aspects and Diversified Approaches
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and electromagnetic metamaterial fabrication, a sonic metamaterial is embedded with localized sources of mass density
1276:. They also have a variety of engineering applications, for example they are widely used as a mechanical component in
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1260:. Phononic crystals can be engineered to exhibit band gaps for phonons, similar to the existence of band gaps for
1703:. With constant density and bulk modulus as constituents of the medium, the refractive index is expressed as n =
1261:
1007:. This method is ubiquitous in optics, where it is used for the description of light waves propagating through a
957:
782:
334:
163:
2313:
Shelby, R. A.; Smith, D. R.; Schultz, S. (2001). "Experimental verification of a negative index of refraction".
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between atoms. Any wavelength shorter than this can be mapped onto a long wavelength, due to effects similar to
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of an artificially fabricated transmission medium, and such negative values are an anomalous response. Negative
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increased. The key result was the negative elastic constant created from resonant frequencies of the material.
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624:
239:
3944:"Implementation of Deutsch and DeutschâJozsa-like algorithms involving classical entanglement of elastic bits"
2504:
413:
406:
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Ding, Yiqun; et al. (2007). "Metamaterial with
Simultaneously Negative Bulk Modulus and Mass Density".
1045:, is similar to electromagnetic metamaterials. The double negative parameters are a result of low-frequency
884:
The first successful industrial applications of acoustic metamaterials were tested for aircraft insulation.
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687:
356:
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2175:
D.T., Emerson (December 1997). "The work of
Jagadis Chandra Bose: 100 years of millimeter-wave research".
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In addition to the parallel concepts of refractive index and crystal structure, electromagnetic waves and
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244:
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extrapolating these properties to larger scales it could be possible to create seismic wave filters (see
3392:
Zhang, Shu; Leilei Yin; Nicholas Fang (2009). "Focusing
Ultrasound with Acoustic Metamaterial Network".
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3871:"Demonstration of a two-bit controlled-NOT quantum-like gate using classical acoustic qubit-analogues"
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Zhang, Shu; Xia, Chunguang; Fang, Nicholas (2011). "Broadband Acoustic Cloak for Ultrasound Waves".
3190:
M.I. Hussein (2009). "Reduced Bloch mode expansion for periodic media band structure calculations".
2335:
769:
Acoustic metamaterials are used to model and research extremely large-scale acoustic phenomena like
2560:
Thomas, Jessica; Yin, Leilei; Fang, Nicholas (2009-05-15). "Metamaterial brings sound into focus".
2218:(1898-01-01). "On the Rotation of Plane of Polarisation of Electric Waves by a Twisted Structure".
2021:
2008:
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1162:(negative modulus), and accelerates to the left when being pushed to the right (negative density).
1143:
1067:, is an equation for refractive index as sound waves interact with acoustic metamaterials (below):
707:
555:
448:
154:
3615:
Zhang, Shu; Chunguang Xia; Nicholas Fang (2011). "Broadband Acoustic Cloak for Ultrasound Waves".
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of different thickness than the other parts. (The illustration on the right ignores the effect of
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are definitely positive for natural materials. The exception occurs at low resonant frequencies.
1707:/ β. In order to develop a propagating plane wave through the material, it is necessary for both
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in atoms. That has also made the phononic crystal an increasingly widely researched component in
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While electromagnetic waves can travel in vacuum, acoustic wave propagation requires a medium.
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is a measure of a substance's resistance to uniform compression. It is defined as the ratio of
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of the wave is given by the displacements of the atoms from their equilibrium positions. The
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Adler, Robert; Acoustic metamaterials., Negative refraction. Earthquake protection. (2008).
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bulk modulus and mass density in a solid based material, which is an important distinction.
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Guenneau, SĂŠbastien; Alexander Movchan; Gunnar PĂŠtursson; S. Anantha Ramakrishna (2007).
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rings in a cylindrical configuration, each ring having acoustic circuits and a different
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a flat slab of the metamaterial can image a source across the slab like a Veselago lens,
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propagating through the lattice. One such wave is shown in the figure to the right. The
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Acoustic metamaterials or phononic crystals can be understood as the acoustic analog of
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the image formed by the flat slab has considerable sub-wavelength image resolution, and
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the phase vector in the medium possesses real and imaginary parts with opposite signs,
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Projected applications of sonic crystals are seismic wave reflection and ultrasonics.
17:
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Gorishnyy, Taras, Martin Maldovan, Chaitanya Ullal, and Edwin Thomas. "Sound ideas."
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nor negative β are found in naturally occurring materials; they are derived from the
1033:, the effective mass density and bulk modulus may become negative. This results in a
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Air Force Inst. of Tech Wright-Patterson AFB OH School of Engineering and Management
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Li, Baowen; Wang, L; Casati, G (2004). "Thermal Diode: Rectification of Heat Flux".
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1272:). For that reason, they constitute a potential testbed for fundamental physics and
945:, which is a density divided by the density of a reference material, such as water.
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Applications of acoustic metamaterial research include seismic wave reflection and
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The electromagnetic spectrum extends from low frequencies used for modern radio to
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Runge, Keith; Hasan, M. Arif; Levine, Joshua A.; Deymier, Pierre A. (2022-08-18).
3770:"Phononic Metamaterials for Thermal Management: An Atomistic Computational Study."
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3367:
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1194:
121:). Sound wave control is accomplished through manipulating parameters such as the
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The correlation in band gap capabilities includes locally resonant elements and
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the square grid and the single vertical wires are on the back and left surfaces.
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Gorishnyy, Taras; Martin Maldovan; Chaitanya Ullal; Edwin Thomas (2005-12-01).
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2433:"Composite Medium with Simultaneously Negative Permeability and Permittivity"
933:, 1.00 g/cm for water, to 1.2Ă10 g/cm for air. Other relevant parameters are
3748:(Online). 2011 U.S. News & World Report. January 7, 2011. Archived from
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in solids, acoustic metamaterials may be designed to control heat transfer.
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or β means that at certain frequencies the medium expands when experiencing
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Acoustic metamaterials: negative refraction, imaging, lensing and cloaking.
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3716:. University of Illinois (Urbana-Champaign). April 21, 2009. Archived from
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a double corner of the metamaterial can act as an open resonator for sound.
1193:
In a rigid lattice structure, atoms exert force on each other, maintaining
1135:*Îľ) are possible for wave propagation as the negative or positive state of
968:
3319:
Basics of fluid mechanics and introduction to computational fluid dynamics
2431:
Smith, D. R.; Padilla, WJ; Vier, DC; Nemat-Nasser, SC; Schultz, S (2000).
2163:
1714:
When the negative parameters are achieved, the mathematical result of the
1186:
3525:
1434:). Several numerical methods are available for this problem, such as the
1388:
1376:
1356:
1300:
1237:
1151:
914:
778:
585:
490:
470:
456:
1465:
in the same way that an elastic wave would propagate along a lattice of
828:
Acoustic metamaterials have developed from the research and findings in
2621:
1837:
Metamaterial with simultaneously negative bulk modulus and mass density
1508:
meets a material with very high impedance it will tend to increase its
1482:
1293:
1210:
921:
339:
3092:
3025:
2201:
A facility of the NSF provides added material to the original paper -
2196:
1593:
and bulk modulus β are position dependent. Using the formulation of a
4067:
3803:"Experimental classical entanglement in a 16 acoustic qubit-analogue"
1257:
953:
480:
110:
102:
1504:
elements comprising the crystal and the surrounding medium. When an
1891:
a beam of sound negatively refracts across a slab of such a medium,
4112:
3629:
3406:
3204:
2574:
1948:
1548:
1540:
1532:
1474:
1430:
is applied on a single unit cell in the reciprocal lattice space (
1304:
967:
900:
874:
384:
114:
1897:
the medium is well impedance-matched with the surrounding medium,
1384:
1347:
for sonic crystals, as a macroscopically homogeneous substance.
1214:
1131:). Hence within the last equation, Veselago-type solutions (n =
925:
3942:
Deymier, Pierre A.; Runge, Keith; Hasan, M. Arif (2022-08-01).
3801:
Hasan, M. Arif; Runge, Keith; Deymier, Pierre A. (2021-12-20).
2622:"New Acoustic Insulation Metamaterial Technology for Aerospace"
2203:
The Work of Jagadis Chandra Bose: 100 Years of MM-Wave Research
1923:
The lens consists of a network of fluid-filled cavities called
2913:
Lee, Jae-Hwang; Singer, Jonathan P.; Thomas, Edwin L. (2012).
1462:
1426:
To obtain the frequency band structure of a phononic crystal,
917:
increase needed to cause a given relative decrease in volume.
520:
106:
29:
4155:"An acoustic metafluid: realizing a broadband acoustic cloak"
3678:"New metamaterial could render submarines invisible to sonar"
2626:
New Acoustic Insulation Metamaterial Technology for Aerospace
1185:
3742:"Newly Developed Cloak Hides Underwater Objects From Sonar"
2713:"Acoustic metamaterials for sound focusing and confinement"
1699:
being the propagation speed of acoustic signal through the
1113:
The inherent parameters of the medium are the mass density
1139:
and β determine the forward or backward wave propagation.
97:
is a material designed to control, direct, and manipulate
3451:
3449:
1812:
is the lattice constant if the spheres are arranged in a
3786:"Investigation of Thermal Management and Metamaterials."
3458:"Acoustic 'superlens' could mean finer ultrasound scans"
2976:
Lu, Ming-Hui; Feng, Liang; Chen, Yan-Feng (2009-12-01).
1573:
Electromagnetic (isotropic) metamaterials have built-in
3341:
3339:
1667:{\displaystyle {\vec {k}}={\frac {\ |n|\omega }{c}}.\,}
1123:. Chirality, or handedness, determines the polarity of
813:
existence of phononic band gaps in periodic structures.
50:
2825:. World Scientific Publishing Company. pp. 3â11.
2768:. New York: Springer-Verlag. p. 183 (Chapter 8).
1756:
1724:
1496:
A key factor for acoustic band gap engineering is the
1395:
due to the coated spheres which result in almost flat
1292:
paved the way to acoustic metamaterials through sonic
1014:
Further information on acoustic wave propagation:
3157:
3155:
3153:
3151:
3116:
3114:
3112:
3110:
2259:
2257:
1755:
1723:
1616:
1450:
the band structure by up to two orders of magnitude.
1076:
1020:
Transfer-matrix method (optics) § Acoustic waves
836:
in 1967, but not realized until some 33 years later.
166:
3387:
3385:
2814:
2812:
2789:
Lavis, David Anthony; George Macdonald Bell (1999).
2177:
IEEE Transactions on Microwave Theory and Techniques
1939:
which carry the sub-wavelength features of objects.
1410:
Phononic crystals are synthetic materials formed by
3035:
3033:
2706:
2704:
2702:
2555:
2553:
2551:
2549:
2267:
Metamaterials: physics and engineering explorations
2264:Nader, Engheta; Richard W. Ziolkowski (June 2006).
2101:
Metamaterials: Physics and Engineering Explorations
1582:are negative within the capability or range of the
1391:coating. The sonic crystals had built-in localized
45:
may be too technical for most readers to understand
3714:Information for Mechanical Science and Engineering
2792:Statistical Mechanics Of Lattice Systems. Volume 2
2656:. The research group of D.R. Smith. Archived from
2498:
2496:
2003:Quantum-like computing with acoustic metamaterials
1771:{\displaystyle \scriptstyle {\overrightarrow {k}}}
1770:
1738:
1666:
1414:of the acoustic properties of the material (i.e.,
1102:
905:Bulk modulus - illustration of uniform compression
201:
3710:"Acoustic cloaking could hide objects from sonar"
2688:. Grolier. Vol. Online. Scholastic Inc. 2009
2676:
2674:
2494:
2492:
2490:
2488:
2486:
2484:
2482:
2480:
2478:
2476:
1739:{\displaystyle \scriptstyle {\overleftarrow {s}}}
4041:"Ultrasonic metamaterials with negative modulus"
2915:"Micro-/Nanostructured Mechanical Metamaterials"
2757:
2755:
2753:
2751:
1318:Split-ring resonators for acoustic metamaterials
1232:wavelength, given by the equilibrium separation
1025:Negative refractive index acoustic metamaterials
924:(or just "density") of a material is defined as
2819:Brulin, Olof; Richard Kin Tchang Hsieh (1982).
2380:
2378:
2376:
2374:
2372:
2370:
3185:
3183:
2978:"Phononic crystals and acoustic metamaterials"
832:. A novel material was originally proposed by
4004:"Focus on Cloaking and Transformation Optics"
3316:Petrila, Titus; Damian Trif (December 2004).
1991:Phononic metamaterials for thermal management
1569:whose effect increases over large distances).
1493:are more complicated than this simple model.
1453:The basis of phononic crystals dates back to
1103:{\displaystyle n^{2}={\frac {\rho }{\beta }}}
937:which is mass over a (two-dimensional) area,
750:
8:
1868:connects to the mass. It is analogous to an
1367:Arrayed metamaterials can create filters or
3284:
3282:
2859:"Split-ring resonators and localized modes"
2795:. New York: Springer-Verlag. pp. 1â4.
202:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
27:Material designed to manipulate sound waves
2300:U Penn Dept. Of Elec. And Sys. Engineering
1146:. For acoustic materials neither negative
757:
743:
590:
380:
223:
145:
4178:
4111:
4029:
3967:
3910:
3842:
3628:
3524:
3405:
3203:
2993:
2739:
2650:"What are Electromagnetic Metamaterials?"
2573:
2456:
2334:
1757:
1754:
1725:
1722:
1711:and β to be either positive or negative.
1663:
1646:
1638:
1632:
1618:
1617:
1615:
1090:
1081:
1075:
995:are both mathematically described by the
179:
165:
73:Learn how and when to remove this message
57:, without removing the technical details.
3292:Tap water as a hydraulic pressure medium
2762:Krowne, Clifford M.; Yong Zhang (2007).
1321:
941:- mass over a one-dimensional line, and
803:
4002:Leonhardt, Ulf; Smith, David R (2008).
3016:Eichenfield, M., Chan, J., Camacho, R.
2505:"Double-negative acoustic metamaterial"
2164:https://physicsworld.com/a/sound-ideas/
2151:
1197:. Most of these atomic forces, such as
1166:Electromagnetic field vs acoustic field
614:
569:
519:
479:
383:
252:
226:
153:
2852:
2850:
2848:
2846:
2844:
2842:
2426:
2424:
3864:
3862:
3042:Sonic crystals make the sound barrier
1799:electromagnetic plane wave scattering
1529:Double-negative acoustic metamaterial
1049:. In combination with a well-defined
55:make it understandable to non-experts
7:
2857:Movchan, A. B.; S. Guenneau (2004).
1746:is in the opposite direction of the
4215:Negative refractive index materials
3026:https://doi.org/10.1038/nature08524
3192:Proceedings of the Royal Society A
25:
3676:Nelson, Bryn (January 19, 2011).
3322:. Springer-Verlag New York, LLC.
2270:. Wiley & Sons. pp. xv.
808:The artwork "Ărgano" by sculptor
4153:Pendry, J B; Li, Jensen (2008).
3484:"One-way Mirror for Sound Waves"
2220:Proceedings of the Royal Society
1387:core and a softer, more elastic
1252:control technologies related to
1117:, bulk modulus β, and chirality
34:
2654:Novel Electromagnetic Materials
2503:Li, Jensen; C. T. Chan (2004).
1995:As phonons are responsible for
1911:Acoustic metamaterial superlens
4180:10.1088/1367-2630/10/11/115032
4130:10.1103/PhysRevLett.106.024301
4031:10.1088/1367-2630/10/11/115019
3969:10.1016/j.wavemoti.2022.102977
3647:10.1103/PhysRevLett.106.024301
3424:10.1103/PhysRevLett.102.194301
3289:Trostmann, Erik (2000-11-17).
2592:10.1103/PhysRevLett.102.194301
1647:
1639:
1623:
1457:who imagined that sound waves
1205:, are of electric nature. The
1144:artificial transmission medium
952:, but it concerns pressure or
1:
3775:vol. 49, no. 1 February 2011.
3594:10.1103/PhysRevLett.99.093904
3543:10.1103/PhysRevLett.93.184301
3368:10.1103/PhysRevLett.99.093904
2995:10.1016/S1369-7021(09)70315-3
2822:Mechanics of micropolar media
2407:10.1126/science.289.5485.1734
2291:Engheta, Nader (2004-04-29).
1477:constant is identical to the
1469:connected by springs with an
1371:of either electromagnetic or
1288:In 2000, the research of Liu
897:Bulk modulus and mass density
4187:Richard V. Craster, et al.:
3049:. 2000-09-07. Archived from
2047:Negative index metamaterials
1845:structure consisting of one
1485:. With phononic crystals of
865:is typically concerned with
3493:. American Physical Society
2458:10.1103/PhysRevLett.84.4184
1801:, by spherical particles -
1489:with differing modulus the
1009:distributed Bragg reflector
793:and experiments that probe
4256:
4191:Springer, Dordrecht 2013,
3895:10.1038/s41598-022-18314-5
3827:10.1038/s41598-021-03789-5
3773:Chinese Journal of Physics
3482:Monroe, Don (2009-08-25).
3265:10.1007/s00339-022-06339-6
3020:Optomechanical crystals .
2886:10.1103/PhysRevB.70.125116
2741:10.1088/1367-2630/9/11/399
2532:10.1103/PhysRevE.70.055602
1959:
1855:
1436:planewave expansion method
1182:Mechanics of lattice waves
1013:
821:
3295:. CRC Press. p. 36.
1825:frequency at about 0.04.
1820:is angular frequency and
1244:Research and applications
1053:during wave propagation;
1035:negative refractive index
2108:Metamaterials scientists
1972:mechanical metamaterials
1444:finite difference method
1264:and to the existence of
824:History of metamaterials
261:ClausiusâDuhem (entropy)
211:Fick's laws of diffusion
4100:Physical Review Letters
3574:Physical Review Letters
3513:Physical Review Letters
3162:G.P Srivastava (1990).
3073:Applied Physics Letters
2437:Physical Review Letters
2345:10.1126/science.1058847
2162:18, no. 12 (2005): 24.
1833:refraction properties.
799:(quantum) optomechanics
419:NavierâStokes equations
357:Material failure theory
4159:New Journal of Physics
4009:New Journal of Physics
3222:10.1098/rspa.2008.0471
3164:The Physics of Phonons
2939:10.1002/adma.201201644
2720:New Journal of Physics
2686:Encyclopedia Americana
2302:. Lecture. p. 99.
2240:10.1098/rspl.1898.0019
2096:Metamaterials Handbook
2052:Photonic metamaterials
1772:
1740:
1668:
1570:
1546:
1538:
1328:
1190:
1104:
1041:, which can result in
973:
950:concept used in optics
906:
863:Acoustical engineering
814:
203:
18:Acoustic metamaterials
3491:Physical Review Focus
2216:Bose, Jagadis Chunder
2115:Richard W. Ziolkowski
2082:Transformation optics
2077:Tunable metamaterials
2062:Seismic metamaterials
2042:Metamaterial antennas
2037:Metamaterial absorber
2027:Metamaterial cloaking
1962:Metamaterial cloaking
1856:Further information:
1773:
1741:
1669:
1552:
1544:
1536:
1440:finite element method
1362:Seismic metamaterials
1341:mechanical properties
1333:split-ring resonators
1325:
1189:
1105:
971:
958:electromagnetic waves
904:
852:and negative density
807:
414:Bernoulli's principle
407:Archimedes' principle
204:
87:acoustic metamaterial
3752:on February 17, 2011
3131:Institute of Physics
3047:Institute of Physics
2067:Split-ring resonator
1925:Helmholtz resonators
1753:
1721:
1614:
1597:the wave vector is:
1506:advancing wave-front
1274:quantum technologies
1152:resonant frequencies
1074:
791:quantum technologies
506:Cohesion (chemistry)
328:Infinitesimal strain
164:
4171:2008NJPh...10k5032P
4122:2011PhRvL.106b4301Z
4060:2006NatMa...5..452F
4022:2008NJPh...10k5019L
3960:2022WaMot.11302977D
3887:2022NatSR..1214066R
3819:2021NatSR..1124248H
3746:U.S. News - Science
3691:on January 22, 2011
3639:2011PhRvL.106b4301Z
3586:2007PhRvL..99i3904D
3535:2004PhRvL..93r4301L
3416:2009PhRvL.102s4301Z
3360:2007PhRvL..99i3904D
3257:2023ApPhA.129..102R
3214:2009RSPSA.465.2825H
3198:(2109): 2825â2848.
3085:2001ApPhL..78..489S
3024:462, 78â82 (2009).
2931:2012AdM....24.4782L
2878:2004PhRvB..70l5116M
2732:2007NJPh....9..399G
2584:2009PhRvL.102s4301Z
2524:2004PhRvE..70e5602L
2449:2000PhRvL..84.4184S
2399:2000Sci...289.1734L
2393:(5485): 1734â1736.
2327:2001Sci...292...77S
2232:1898RSPS...63..146C
2189:1997ITMTT..45.2267E
2022:Acoustic dispersion
1985:index of refraction
1863:Double C resonators
1814:face-centered cubic
1584:transmission medium
1345:continuum mechanics
1337:limiting properties
1262:electrons in solids
1209:, and the force of
783:electrons in solids
424:Poiseuille equation
155:Continuum mechanics
149:Part of a series on
3875:Scientific Reports
3807:Scientific Reports
3723:on August 27, 2009
3462:New Scientist Tech
2919:Advanced Materials
2726:(399): 1367â2630.
1997:thermal conduction
1915:In 2009 Shu Zhang
1768:
1767:
1736:
1735:
1701:homogeneous medium
1664:
1580:response functions
1571:
1547:
1545:Out-of-phase waves
1539:
1412:periodic variation
1329:
1191:
1100:
1039:Flat slab focusing
974:
907:
879:sound reproduction
871:medical ultrasound
815:
630:Magnetorheological
625:Electrorheological
362:Fracture mechanics
199:
4220:Acoustic cloaking
4210:Phononic crystals
4197:978-94-007-4812-5
4093:on June 23, 2010.
3784:Roman, Calvin T.
3329:978-0-387-23837-1
3302:978-0-8247-0505-3
3245:Applied Physics A
3173:978-0-85274-153-5
3093:10.1063/1.1343489
2925:(36): 4782â4810.
2832:978-9971-950-02-6
2802:978-3-540-64436-1
2775:978-3-540-72131-4
2277:978-0-471-76102-0
2197:10.1109/22.643830
1956:Acoustic cloaking
1937:evanescent fields
1765:
1733:
1689:angular frequency
1658:
1637:
1626:
1589:The mass density
1500:mismatch between
1473:constant E. This
1406:Phononic crystals
1266:electron orbitals
1258:precision sensing
1098:
1005:transfer matrices
978:photonic crystals
964:Theoretical model
909:The bulk modulus
795:quantum mechanics
787:electron orbitals
767:
766:
642:
641:
576:
575:
345:Contact mechanics
268:
267:
197:
83:
82:
75:
16:(Redirected from
4247:
4184:
4182:
4149:
4115:
4094:
4092:
4086:. Archived from
4068:10.1038/nmat1644
4048:Nature Materials
4045:
4035:
4033:
3990:
3989:
3971:
3939:
3933:
3932:
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3792:
3782:
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3767:
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3758:
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3738:
3732:
3731:
3729:
3728:
3722:
3706:
3700:
3699:
3697:
3696:
3690:
3684:. Archived from
3673:
3667:
3666:
3632:
3612:
3606:
3605:
3569:
3563:
3562:
3528:
3526:cond-mat/0407093
3508:
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3501:
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3453:
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3307:
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3286:
3277:
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3240:
3234:
3233:
3207:
3187:
3178:
3177:
3159:
3146:
3145:
3143:
3142:
3133:. Archived from
3127:Physicsworld.com
3118:
3105:
3104:
3068:
3062:
3061:
3059:
3058:
3037:
3028:
3014:
3008:
3007:
2997:
2973:
2967:
2966:
2910:
2904:
2903:
2901:
2900:
2894:
2888:. Archived from
2863:
2854:
2837:
2836:
2816:
2807:
2806:
2786:
2780:
2779:
2759:
2746:
2745:
2743:
2717:
2708:
2697:
2696:
2694:
2693:
2678:
2669:
2668:
2666:
2665:
2660:on July 20, 2009
2642:
2636:
2635:
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2618:
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2544:
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2500:
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2288:
2282:
2281:
2261:
2252:
2251:
2212:
2206:
2200:
2172:
2166:
2156:
2140:Vladimir Shalaev
2057:Photonic crystal
1977:Wuhan University
1876:/Îź with density
1777:
1775:
1774:
1769:
1766:
1758:
1745:
1743:
1742:
1737:
1734:
1726:
1673:
1671:
1670:
1665:
1659:
1654:
1650:
1642:
1635:
1633:
1628:
1627:
1619:
1517:in the crystal.
1343:and problems of
1270:microfabrication
1230:minimum possible
1125:wave propagation
1109:
1107:
1106:
1101:
1099:
1091:
1086:
1085:
1043:super resolution
943:relative density
888:Basic principles
759:
752:
745:
591:
556:Gay-Lussac's law
546:Combined gas law
496:Capillary action
381:
224:
208:
206:
205:
200:
198:
196:
188:
180:
146:
119:crystal lattices
95:phononic crystal
78:
71:
67:
64:
58:
38:
37:
30:
21:
4255:
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4225:
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4206:
4152:
4097:
4090:
4043:
4038:
4001:
3998:
3996:Further reading
3993:
3941:
3940:
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3868:
3867:
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3800:
3799:
3795:
3783:
3779:
3768:
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3735:
3726:
3724:
3720:
3708:
3707:
3703:
3694:
3692:
3688:
3675:
3674:
3670:
3617:Phys. Rev. Lett
3614:
3613:
3609:
3571:
3570:
3566:
3510:
3509:
3505:
3496:
3494:
3486:
3481:
3480:
3476:
3467:
3465:
3455:
3454:
3447:
3394:Phys. Rev. Lett
3391:
3390:
3383:
3348:Phys. Rev. Lett
3345:
3344:
3337:
3330:
3315:
3314:
3310:
3303:
3288:
3287:
3280:
3242:
3241:
3237:
3189:
3188:
3181:
3174:
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3149:
3140:
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3119:
3108:
3070:
3069:
3065:
3056:
3054:
3039:
3038:
3031:
3015:
3011:
2982:Materials Today
2975:
2974:
2970:
2912:
2911:
2907:
2898:
2896:
2892:
2861:
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2855:
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2818:
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2810:
2803:
2788:
2787:
2783:
2776:
2761:
2760:
2749:
2715:
2710:
2709:
2700:
2691:
2689:
2680:
2679:
2672:
2663:
2661:
2646:Smith, David R.
2644:
2643:
2639:
2630:
2628:
2620:
2619:
2615:
2559:
2558:
2547:
2507:
2502:
2501:
2474:
2430:
2429:
2422:
2384:
2383:
2368:
2336:10.1.1.119.1617
2321:(5514): 77â79.
2312:
2311:
2307:
2295:
2293:"Metamaterials"
2290:
2289:
2285:
2278:
2263:
2262:
2255:
2214:
2213:
2209:
2174:
2173:
2169:
2157:
2153:
2149:
2144:
2105:
2086:
2017:
2005:
1993:
1964:
1958:
1945:
1913:
1865:
1860:
1858:Poisson's ratio
1839:
1816:(fcc) lattice;
1792:
1785:
1751:
1750:
1719:
1718:
1716:Poynting vector
1691:represented by
1634:
1612:
1611:
1563:inhomogeneities
1531:
1428:Bloch's theorem
1408:
1320:
1286:
1246:
1184:
1172:gamma radiation
1168:
1077:
1072:
1071:
1031:frequency bands
1027:
1022:
966:
899:
890:
858:
851:
834:Victor Veselago
826:
820:
810:Eusebio Sempere
763:
734:
733:
732:
652:
644:
643:
597:Viscoelasticity
588:
578:
577:
565:
515:
511:Surface tension
475:
378:
376:Fluid mechanics
368:
367:
366:
280:
278:Solid mechanics
270:
269:
221:
213:
189:
181:
162:
161:
79:
68:
62:
59:
51:help improve it
48:
39:
35:
28:
23:
22:
15:
12:
11:
5:
4253:
4251:
4243:
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4237:
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4217:
4212:
4205:
4204:External links
4202:
4201:
4200:
4185:
4165:(11): 115032.
4150:
4095:
4036:
4016:(11): 115019.
3997:
3994:
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3858:
3793:
3777:
3762:
3733:
3701:
3682:Defense Update
3668:
3607:
3564:
3519:(18): 184301.
3503:
3474:
3445:
3400:(19): 194301.
3381:
3335:
3328:
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2872:(12): 125116.
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2781:
2774:
2747:
2698:
2670:
2648:(2006-06-10).
2637:
2613:
2568:(19): 194301.
2545:
2472:
2443:(18): 4184â7.
2420:
2366:
2305:
2283:
2276:
2253:
2226:(1): 146â152.
2207:
2167:
2150:
2148:
2145:
2143:
2142:
2137:
2132:
2130:David R. Smith
2127:
2122:
2120:Pierre Deymier
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2018:
2016:
2013:
2009:Controlled-NOT
2004:
2001:
1992:
1989:
1960:Main article:
1957:
1954:
1944:
1943:Acoustic diode
1941:
1912:
1909:
1908:
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1653:
1649:
1645:
1641:
1631:
1625:
1622:
1537:In-phase waves
1530:
1527:
1523:speed of sound
1515:acoustic waves
1510:phase velocity
1432:Brillouin zone
1407:
1404:
1355:values of the
1352:elastic moduli
1319:
1316:
1285:
1284:Sonic crystals
1282:
1278:optomechanical
1245:
1242:
1207:magnetic force
1183:
1180:
1167:
1164:
1111:
1110:
1097:
1094:
1089:
1084:
1080:
1026:
1023:
993:acoustic waves
986:speed of sound
982:speed of light
965:
962:
939:linear density
898:
895:
889:
886:
856:
849:
822:Main article:
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501:Chromatography
498:
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454:
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40:
33:
26:
24:
14:
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10:
9:
6:
4:
3:
2:
4252:
4241:
4240:Metamaterials
4238:
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4221:
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4151:
4147:
4143:
4139:
4135:
4131:
4127:
4123:
4119:
4114:
4109:
4106:(2): 024301.
4105:
4101:
4096:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
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4019:
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3836:
3832:
3828:
3824:
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3816:
3812:
3808:
3804:
3797:
3794:
3791:, March 2010.
3790:
3787:
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3763:
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3669:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3636:
3631:
3626:
3623:(2): 024301.
3622:
3618:
3611:
3608:
3603:
3599:
3595:
3591:
3587:
3583:
3580:(9): 093904.
3579:
3575:
3568:
3565:
3560:
3556:
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3548:
3544:
3540:
3536:
3532:
3527:
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3399:
3395:
3388:
3386:
3382:
3377:
3373:
3369:
3365:
3361:
3357:
3354:(9): 093904.
3353:
3349:
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3340:
3336:
3331:
3325:
3321:
3320:
3312:
3309:
3304:
3298:
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3215:
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3206:
3201:
3197:
3193:
3186:
3184:
3180:
3175:
3169:
3166:. CRC Press.
3165:
3158:
3156:
3154:
3152:
3148:
3137:on 2012-04-03
3136:
3132:
3128:
3124:
3123:"Sound ideas"
3117:
3115:
3113:
3111:
3107:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3074:
3067:
3064:
3053:on 2010-03-15
3052:
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3034:
3030:
3027:
3023:
3019:
3013:
3010:
3005:
3001:
2996:
2991:
2988:(12): 34â42.
2987:
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2979:
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2969:
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2906:
2895:on 2016-02-22
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2556:
2554:
2552:
2550:
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2541:
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2533:
2529:
2525:
2521:
2518:(5): 055602.
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2160:Physics World
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2135:Nader Engheta
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1963:
1955:
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1516:
1511:
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1476:
1472:
1471:elastic force
1468:
1464:
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1456:
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1437:
1433:
1429:
1424:
1422:
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1413:
1405:
1403:
1400:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1373:elastic waves
1370:
1365:
1363:
1358:
1353:
1348:
1346:
1342:
1338:
1334:
1324:
1317:
1315:
1312:
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1302:
1297:
1295:
1291:
1283:
1281:
1279:
1275:
1271:
1267:
1263:
1259:
1256:, as well as
1255:
1251:
1243:
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1235:
1231:
1226:
1225:Îť is marked.
1224:
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1062:
1058:
1057:
1052:
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1044:
1040:
1036:
1032:
1024:
1021:
1017:
1016:Acoustic wave
1012:
1010:
1006:
1000:
998:
997:wave equation
994:
989:
987:
983:
979:
970:
963:
961:
959:
956:, instead of
955:
951:
946:
944:
940:
936:
932:
931:neutron stars
927:
923:
918:
916:
912:
903:
896:
894:
887:
885:
882:
880:
876:
872:
868:
867:noise control
864:
860:
855:
848:
842:
839:
835:
831:
830:metamaterials
825:
817:
811:
806:
802:
800:
796:
792:
788:
784:
780:
776:
772:
771:seismic waves
760:
755:
753:
748:
746:
741:
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738:
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729:
726:
724:
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587:
582:
581:
572:
568:
562:
559:
557:
554:
552:
549:
547:
544:
542:
541:Charles's law
539:
537:
534:
532:
529:
528:
526:
525:
522:
518:
512:
509:
507:
504:
502:
499:
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494:
492:
489:
488:
486:
485:
482:
478:
472:
469:
465:
462:
458:
455:
450:
449:non-Newtonian
447:
443:
439:
438:
437:
434:
432:
429:
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422:
420:
417:
415:
412:
408:
405:
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377:
372:
371:
363:
360:
358:
355:
351:
348:
347:
346:
343:
341:
338:
336:
335:Compatibility
333:
329:
326:
324:
323:Finite strain
321:
320:
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316:
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311:
309:
306:
304:
301:
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227:Conservations
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100:
96:
92:
91:sonic crystal
88:
77:
74:
66:
56:
52:
46:
43:This article
41:
32:
31:
19:
4188:
4162:
4158:
4103:
4099:
4088:the original
4054:(6): 452â6.
4051:
4047:
4013:
4007:
3951:
3947:
3937:
3881:(1): 14066.
3878:
3874:
3813:(1): 24248.
3810:
3806:
3796:
3788:
3780:
3772:
3765:
3754:. Retrieved
3750:the original
3745:
3736:
3725:. Retrieved
3718:the original
3713:
3704:
3693:. Retrieved
3686:the original
3681:
3671:
3620:
3616:
3610:
3577:
3573:
3567:
3516:
3512:
3506:
3495:. Retrieved
3490:
3477:
3466:. Retrieved
3461:
3397:
3393:
3351:
3347:
3318:
3311:
3291:
3248:
3244:
3238:
3195:
3191:
3163:
3139:. Retrieved
3135:the original
3126:
3076:
3072:
3066:
3055:. Retrieved
3051:the original
3041:
3021:
3017:
3012:
2985:
2981:
2971:
2922:
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2125:John Pendry
1968:sound proof
1843:zinc blende
1748:wave vector
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1254:earthquakes
1228:There is a
1203:ionic bonds
1195:equilibrium
1160:compression
1129:wave vector
1029:In certain
954:shear waves
838:John Pendry
775:earthquakes
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536:Boyle's law
308:Hooke's law
286:Deformation
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688:Gay-Lussac
651:Scientists
551:Fick's law
531:Atmosphere
350:frictional
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291:Elasticity
4235:Acoustics
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2682:"Density"
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