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'''Acoustic metamaterials''' are artificially fabricated materials designed to control, direct, and manipulate [[sound]] in the form of [[sonic]], [[infrasonic]], or [[ultrasonic]] waves, as these might occur in [[gas]]es, [[liquid]]s, and [[solid]]s. The hereditary line into acoustic metamaterials follows from theory and research in electromagnetic metamaterials.<ref name=acousticmeta1107/> Furthermore, with acoustic metamaterials, sonic waves can now be extended to the [[negative refraction]] domain.<ref name=acousticmeta1107/>

Control of the various forms of sound waves is mostly accomplished through the bulk modulus ''β'', mass density ''ρ'', and [[Metamaterial#Artificial materials and handedness|chirality]]. The bulk modulus and density are counterparts of the intrinsic parameters, [[permittivity]] and [[permeability]], in electromagnetic [[metamaterial]]s. Related to this is the mechanics of wave propagation in a [[lattice]] structure. Also materials have [[mass]], and instrinsic degrees of [[stiffness]]. Together, these form a [[resonant]] system, and the mechanical (sonic) resonance may be excited by appropriate sonic frequencies (for example [[pulse]]s at audio frequencies).<ref name=acousticmeta1107/>

==History of the acoustic metamaterials==
The research on acoustic metamaterials began in the year 2000 with the fabrication and demonstration of sonic crystals in a liquid.<ref name=soniccrystals/> This was followed by transposing the behavior of the split-ring resonator to research in acoustic metamaterials.<ref name=comp/> After this double negative parameters (negative bulk modulus ''β''<sub>eff</sub> and negative density ''ρ''<sub>eff</sub>) were produced by this type of medium.<ref name=acousticdbleneg/> Then a group of researchers presented the design and tested results of an ultrasonic metamaterial lens for focusing 60&nbsp;kHz.<ref name=underwaterlens09/>

The earlier studies of acoustics in technology, which is called [[acoustical engineering]], are typically concerned with how to reduce unwanted sounds, [[noise control]], how to make useful sounds for the medical diagnosis,[[sonar]], and [[sound reproduction]] and how to measure some other physical properties using sound. 

Using acoustic metamaterials, the directions of sound through the medium can be controlled ,[[refraction index]], so the traditional acoustic technologies extend to controlling the sound wave and even cloak certain matters from acoustic detection.

==Basic Principle==
Since the acoustic metamaterials are one of the branch of the [[metamaterials]], the basic principle of the acoustic metamaterials is similar to the pricinple of metamaterials. These metamaterials usually gain their properties from structure rather than composition, using the inclusion of small inhomogeneities to enact effective macroscopic behavior.<ref name=physicsengineering1>{{cite book|last = Nader|first = Engheta|coauthors = Richard W. Ziolkowski
|title = Metamaterials: physics and engineering explorations|publisher = Wiley & Sons
|date = June 2006|pages = xv, 3–30, 37, 215–233, 240, 241
|url = https://books.google.com/books?id=51e0UkEuBP4C|isbn = 9780471761020}}</ref><ref name=smithmetamaterials1>{{cite web
|last = Smith|first = David R.| authorlink = David R. Smith|title = What are Electromagnetic Metamaterials? 
|work = Novel Electromagnetic Materials|publisher = The research group of D.R. Smith 
|date = 2006-06-10|url = http://people.ee.duke.edu/~drsmith/about_metamaterials.html
|accessdate = 2009-08-19}}</ref> 
Similar to metamaterials research, investigating materials with [[Negative index metamaterials]], the negative index acoustic metamaterials became the primary research.   Negative refractive index of acoustic materials can be achieved to change the [[bulk modulus]] and [[mass density]]. 

===Bulk modulus and mass density===
{{Continuum mechanics}}
Below, the [[bulk modulus]] ''β'' of a substance reflects the substance's resistance to uniform compression. It is defined in relation to the [[pressure]] increase needed to cause a given relative decrease in volume. 

The term ''[[Density|mass density]]'' of a material, is interchangeable with ''density''. The latter is defined as mass per unit volume and is expressed in grams per cubic centimeter (g/cm<sup>3</sup>).<ref name=densitygroiler> {{cite encyclopedia|title =Density| encyclopedia = Encyclopedia Americana|volume = Grolier Online|publisher =Scholastic Inc| date =2009|url =http://ea.grolier.com/article?id=0124170-00| accessdate =2009-09-06}}</ref> In all three classic states of matter — gas, [[liquid]], or [[solid]] — the density varies with a change in temperature or pressure, and [[gas]]es are the most susceptible to those changes. The spectrum of densities is wide ranging: from 10<sup>15</sup> g/cm<sup>3</sup> for [[neutron star]]s, 1.00 g/cm<sup>3</sup> for water to 1.2×10<sup>−3</sup> g/cm<sup>3</sup> for air.<ref name=densitygroiler/> Also relevant here are [[area density]] which is mass over a (two-dimensional) area, [[linear density]] - mass over a one-dimensional line, and [[relative density]], which is a density divided by the density of a reference material, such as water.

For acoustic materials and acoustic metamaterials, both bulk modulus and density are component parameters, which define their [[refractive index]].

===Acoustic metamaterial analogues===
[[Image:Isostatic pressure deformation.png|thumb|right|Bulk modulus - illustration of uniform compression]]
Scientific research revealed that acoustic metamaterials have analogues to electromagnetic metamaterials when exhibiting the following characteristics: 

In certain [[frequency band]]s, the ''[[density|effective mass density]]'' and ''[[bulk modulus]]'' may become negative. This results in a ''negative refractive index''. ''Flat slab focusing'', which can result in ''[[super resolution]]'', is similar to electromagnetic metamaterials. The double negative parameters are a result of low-frequency [[resonance]]s.<ref name=acousticmeta1107/> In combination with a well-defined [[polarization]] during wave propagation; '''''k''''' ''= |n|ω'', is an equation for refractive index as sound waves interact with acoustic metamaterials (below):<ref name=acousticbook1>{{cite book|last =Krowne| first = Clifford M.| coauthors = Yong Zhang
|title = Physics of Negative Refraction and Negative Index Materials: Optical and Electronic Aspects and Diversified Approaches|publisher =Springer-Verlag| date =2007
|location =New York|page=183 (Chapter 8)
|url = https://books.google.com/books?id=AjdbPVfK7rkC&pg=PA183|isbn = 9783540721314}}</ref>

:::<math>n^2=\frac{\rho}{\beta} </math> 

The inherent parameters of the medium are the mass density ''ρ'', bulk modulus β, and chirality '''''k'''''. Chirality, or handedness, determines the polarity of [[wave propagation]] ([[wave vector]]). Hence within the last equation, Veselago-type solutions (n<sup>2</sup> = ''u''*ε) are possible for wave propagation as the negative or positive state of ''ρ'' and β determine the forward or backward wave propagation.<ref name=acousticbook1/>

In negative refractive, electromagnetic metamaterials, negative permittivity can be found in natural materials. However, negative permeability has to be intentionally created in the [[Transmission medium|artificial transmission medium]]. Obtaining a negative refractive index with acoustic materials is different.<ref name=acousticbook1/>
Neither negative ''ρ'' nor negative β are found in naturally occurring materials;<ref name=acousticbook1/>
they are derived from the [[resonant|resonant frequencies]] of an artificially fabricated transmission medium (metamaterial), and such negative values are an anomalous response. Negative ''ρ'' or β means that at certain frequencies the medium expands when experiencing [[compression]] (negative modulus), and accelerates to the left when being pushed to the right (negative density).<ref name=acousticbook1/>

===Electromagnetic field vs acoustic field===
The electromagnetic spectrum extends from below frequencies used for modern radio to [[Gamma ray|gamma radiation]] at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. That would be wavelengths from 10<sup>3</sup> to 10<sup>-15</sup> kilometers. The long wavelength limit is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length, although in principle the spectrum is infinite and continuous.

Infrasonic frequencies range from 20 Hz down to 0.001 Hz. Audible frequencies are 20 Hz to 20&nbsp;kHz. Ultrasonic range is above 20&nbsp;kHz. Sound requires a medium. Electromagnetics radiation (EM waves) can travel in a vacuum.

===Mechanics of lattice waves===
[[Image:Lattice wave.svg|200px|right]]
An imaginary demonstration: A hypothetical rigid [[Crystal structure|lattice]] structure (solid) is composed of 10<sup>23</sup> atoms. However, in a real [[solid]] these particles could just as easily be [[ions]]. In a rigid lattice structure, atoms exert pressure, or a force, on each other in order to maintain [[wikt:equilibrium|equilibrium]]. Atomic forces maintain rigid lattice structure. Most of them, such as the [[covalent bond|covalent]] or [[ionic bond]]s, are of electric nature. The [[magnetic force]], and the force of [[gravity]] are negligible.<ref name=latticemechanics2>{{cite book| last =Lavis| first =David Anthony 
|coauthors =George Macdonald Bell
|title = Statistical Mechanics Of Lattice Systems. Volume 2 
|publisher = Springer-Verlag|date = 1999| location = New York
|pages = 1–4|url=https://books.google.com/books?id=2ztrXFKT2c0C&pg=PA1|isbn =9783540644361}}</ref> 
Because of bonding between [[atom]]s, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration [[wave]]s propagating through the [[Crystal structure|lattice]]. One such wave is shown in the figure to the right. The [[amplitude]] of the wave is given by the displacements of the atoms from their equilibrium positions. The [[wavelength]] λ is marked.<ref name=latticemechanics1>{{cite book| last = Brulin| first =Olof| coauthors =Richard Kin Tchang Hsieh
|title = Mechanics of micropolar media| publisher = World Scientific Publishing Company
|date = 1982|pages= 3–11
|url = https://books.google.com/books?id=6S4rbmb1IqUC&pg=PA3|isbn =9971950022}}</ref>

There is a ''minimum possible'' wavelength, given by the equilibrium separation ''a'' between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than ''a'', due to effects similar to that in [[aliasing]].<ref name=latticemechanics1/>{{-}}

===Phononic crystal===
In December of 2005, phononic crystals were proposed by a Physics World online article (Physicworld.com).  [[Physics World]] is a membership magazine of the [[society |science society]], [[Institute of Physics]]. In this article, phononic crystals were characterized as a revolution in acoustics. Below is a general overview of phononic crystals: <ref name=Phononic-crystal-1/>

'''Phononic crystals''' are [[metamaterial|synthetic materials]] that are formed by [[split-ring resonator |periodic variation]] of the [[acoustics|acoustic properties]] of the material (i.e., [[elasticity]] and [[mass]]). One of the main properties of the phononic crystals is the possibility of having a phononic [[bandgap]]. A phononic crystal with phononic bandgap prevents [[phonons]] of selected ranges of [[frequencies]] from being transmitted through the [[transmission medium|material]].<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>

The basis of phononic crystals dates back to [[Isaac Newton]] who imagined that [[sound waves]] [[wave propagation| propagated]] through [[air]] in the same way that an [[elastic wave]] would propagate along a [[lattice]] of [[point mass]]es connected by springs with an 
[[Elasticity (physics)| elastic force]] constant E.  This [[force]] constant is identical to the [[Bulk modulus| modulus]] of the [[silicon|material]]. Of course with phononic crystals of [[Fluid mechanics|materials]] with differing modulus the [[Continuum mechanics|calculations]] are a little more complicated than this simple model.<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>

Based on Newton’s observation we can conclude that a key factor for [[Earthquake engineering  |acoustic band-gap engineering]] is [[impedance]] mismatch between [[periodic]] elements comprising the crystal and the surrounding medium.  When an [[longitudinal wave |advancing wave-front]] meets a material with very high impedance it will tend to increase its [[phase velocity]] through that medium.  Likewise, when the advancing wave-front meets a low impedance medium it will slow down.  We can exploit this concept with periodic (and handcrafted) arrangements of impedance mismatched elements to affect [[sound waves| acoustic waves]] in the crystal – essentially band-gap [[engineering]].<ref name=Phononic-crystal-1/><ref name=Physics-of-Phonons/>  

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 [[speed of sound]] (due to impedance differences) through the materials that comprise the composite.<ref name=Phononic-crystal-1>
{{Cite news| last = Gorishnyy| first =Taras| coauthors =Martin Maldovan, Chaitanya Ullal and Edwin Thomas| title =Sound ideas| newspaper =Physicsworld.com| language =English| publisher =[[Institute of Physics]]| date =2005-12-01| url =http://physicsworld.com/cws/article/print/23671
  | accessdate =2009-11-05 }}</ref><ref name=Physics-of-Phonons>
{{cite book |title=The Physics of Phonons |author=G.P Srivastava |isbn=0852741537 |year=1990 |publisher=CRC Press}}</ref>

==Acoustic metamaterials analysis and experiments==
The current research on acoustic metamaterials is based not only on prior experience with [[Metamaterial | electromagnetic metamaterials]]. The key [[physics]] in [[acoustics]] are ''sound'', ''[[ultrasound]]'' and ''[[infrasound]]'', which are [[elastic waves|  mechanical waves]] in [[gas]]es, [[liquid]]s, and [[solid]]s. One objective of the inquiry into the properties of acoustic metamaterials is applications in [[seismic wave]] reflection and in [[vibration]] control technologies related to [[earthquake]]s.<ref name=acousticmeta1107/><ref name=soniccrystals/><ref name=srr/>

===Sonic crystals===
====Introduction====
[[Image : Sonic Crystal.jpg|thumb|right|200px|A)Cross section of a coated lead sphere that forms the basic structure unit B)For an 8 × 8 × 8 sonic crystal]]
Complete sound attenuation for a certain frequency range can be achieved through the concept of a classical wave spectral gap, originally introduced in relation to the electromagnetic wave, denoted the photonic band gap. Subsequently extended to elastic waves, the idea states that a strong periodic modulation in density and/or sound velocity can create spectral gaps that forbid wave propagation. However, the spatial modulation must be of the same order as the wavelength in the gap. It is thus not practical for shielding acoustic sound, because the structure would have to be the size of outdoor sculptures in order to shield environmental noises.

====Method====
In the year 2000 the research of Liu ''et al.'' paved the way to acoustic metamaterials through sonic [[crystal]]s. The latter 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 of the metamaterial.<ref name=soniccrystals>{{cite journal
|title = Locally Resonant Sonic Materials|journal=Science|volume = 289
|issue = 5485|pages=1734–1736| date = 2000|doi = 10.1126/science.289.5485.1734
| last1 = Liu
| first1 = Zhengyou
| last2 = Zhang
| first2 = Xixiang
| last3 = Mao
| first3 = Yiwei
| last4 = Zhu
| first4 = Y. Y.
| last5 = Yang
| first5 = Zhiyu
| last6 = Chan
| first6 = C. T.
| last7 = Sheng
| first7 = Ping
| pmid = 10976063
}}</ref> 

The fabricated material consisted of a high-density solid lead ball as the core, one centimeter in size, which was coated with a 2.5-mm layer of rubber [[silicone]]. These were arranged in a crystal lattice structure of an 8 × 8 × 8 cube. The balls were cemented into the cubic structure with an [[epoxy]]. Transmission was measured as a function of frequency from 250 to 1600 Hz for effectively 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.<ref name=soniccrystals/><ref name=iopsoniccrystals>{{Cite book
|publisher = [[Institute of Physics]]
|title = Sonic crystals make the sound barrier
|date =2000-09-07
|url = http://physicsworld.com/cws/article/news/2787
|accessdate = 2009-08-25}}</ref>

The amplitudes of the sound waves entering the surface were compared with the sound waves at the center of the metamaterial structure. The oscillations of the coated spheres absorbed sonic energy, which created the frequency gap; the sound energy is absorbed exponentially as the thickness of the material is increased. The key result here is a negative elastic constant created from resonant frequencies of the material. Its projected applications, with a future expanded frequency range in elastic wave systems, are seismic wave reflection and ultrasonics.<ref name=soniccrystals/><ref name=iopsoniccrystals/>

====Conclusion====
The latter exhibits 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 of the metamaterial. Disordered composites made from such localized resonant structures behave as a material with effective negative elastic constants and a total wave reflector within certain tunable sonic frequency ranges. A 2-centimeter slab of this composite material is shown to break the conventional mass-density law of sound transmission by one or more orders of magnitude at 400 hertz.

===Split-ring resonators for acoustic metamaterials===
====Introduction====
[[Image:Left-handed metamaterial array configuration.jpg|thumb|250px|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 the square grid and the single vertical wires are on the back and left surfaces.<ref name=comp>
{{cite journal|url=http://people.ee.duke.edu/~drsmith/pubs_smith_group/Smith_PRL_84_4184_(2000).pdf|doi=10.1103/PhysRevLett.84.4184|title=Composite Medium with Simultaneously Negative Permeability and Permittivity|year=2000|last1=Smith|first1=D. R.|journal=Physical Review Letters|volume=84|issue=18 |pages=4184–4187|pmid=10990641 }}</ref><ref name=shelby>{{cite journal|doi=10.1063/1.1343489|title=Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial|year=2001|last1=Shelby|first1=R. A.|journal=Applied Physics Letters|volume=78|issue=4 |pages=489–491}}</ref>]]

During recent years the models involving split-ring resonator(SRRs) and their applications in the design of left handed materials (LHM) were the subjects of publications and intensive discussions in physics. The original concept was proposed by Pendry and his colleagues. In the electromagnetic theory, materials with optical resonances are interpreted in the sense of negative effective dielectric permittivity for certain frequency intervals (photonic bandgap frequencies), as described in the classical work. The effective medium, associated with an array of double C resonators, shows a negative effective permeability region (photonic bandgap region) in a neighborhood of the resonance frequency.<ref name=srr/>
In 2004 [[split-ring resonator]]s (SRR) became the object of acoustic metamaterial research.<ref name=srr>{{cite journal| title = Split-ring resonators and localized modes
|journal =Phys. Rev. B| volume = 70|issue = 12|page=125116| date = 2004|url = http://www.maths.liv.ac.uk/~guenneau/prb2004a.pdf
|doi =10.1103/PhysRevB.70.125116|accessdate =2009-08-27
| last1 = Movchan
| first1 = A. B.
| last2 = Guenneau
| first2 = S.
}}</ref>
Prior research with SRRs fabricated as negative index ''electromagnetic'' metamaterials was referenced as the progenitor of further research in acoustic metamaterials.<ref name=srr/> 
An analysis of the frequency band gap characteristics, derived from the inherent [[Asymptotic analysis|limiting properties]] of artificially created SRRs, paralleled an analysis of sonic crystals. The band gap properties of SSRs were related to sonic crystal band gap properties.<ref name=srr/> Inherent in this inquiry is a description of [[List of materials properties#Mechanical properties|mechanical properties]] and problems
of [[continuum mechanics]] for sonic crystals, as a macroscopically homogeneous substance.<ref name=srr/>

====Method====
The correlation in [[bandgap]] capabilities includes locally resonant elements and [[elastic moduli]] which operate in a certain [[frequency range]].<ref name=srr/> Locally resonant elements in electromagnetic metamaterials are [[Negative index metamaterials#Orthogonal coordinate systems|microscopic sources]] 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 values of the stop band 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 extrapolating these properties to larger scales it could be possible to create seismic wave filters (see [[Metamaterial#Seismic metamaterials|Seismic metamaterials]]).<ref name=srr/>

According to [[research]] prior to this analysis, arrayed metamaterials can create filters or [[polarizer]]s of either [[Electromagnetic radiation|electromagnetic]] or [[elastic wave]]s.<ref name=srr/>
Here a method is shown which can be applied to two-dimensional [[stop band]] and [[bandgap]] control with either photonic or sonic structures.<ref name=srr/>
Similar to [[photonic]] and electromagnetic metamaterial fabrication, a sonic metamaterial is embedded with localized sources of mass density ''ρ'' and the (elastic) [[bulk modulus]] β parameters, which are analogous to permittivity and permeability, respectively. The sonic (or phononic) metamaterials are [[sonic crystals]], as in the previous section. These crystals have a solid [[lead]] core and a softer, more elastic [[silicone]] coating.<ref name=soniccrystals/> The sonic crystals had built-in localized resonances due to the coated spheres which resulted in almost flat [[dispersion]] curves. Low-frequency bandgaps and localized wave interactions of the coated spheres were analyzed and presented in.<ref name=srr/>

====Conclusion====
They propose a more general multiscale configuration. This method can be used to tune bandgaps inherent in the material and, also, create new low-frequency bandgaps. It is also applicable for designing low-frequency phononic crystal waveguides ([[radio frequency]]). Doubly [[periodic]] square array of SRRs are used to illustrate the methodology.<ref name=srr/>

===Ultrasonic metamaterials with negative modulus===
====Introduction====
[[Image : Subwavelength_Helmholtz_resonators.jpg|thumb|right|200px|Subwavelength_Helmholtz_resonators]]
The focus so far has been limited to the emergence of bandgap and applications for effective sound attenuation. Recently, studies on acoustic bandgap materials have shown refraction-like beam bending and refocusing, which requires the size of the building blocks to be close to the wavelength. A simulation predicted that high index lenses can be made of subwavelength metamaterials. Yet to the best of our knowledge, experimental realization of ultrasonic metamaterials as well as acoustic superlensing with engineered surface states still remains unexplored. 
Here is a new class of ultrasonic metamaterials consisting of an array of subwavelength Helmholtz resonators with designed acoustic inductance and capacitance. These materials have an effective dynamic modulus with negative values near the resonance frequency. 

====Method====
Ultrasonic metamaterials have strong dispersive characteristics of elastic modulus with subwavelength resonant structural units, which leads to a bulk mode propagating in the cluster of the units with group velocity antiparallel to the phase advancement. The building block of this ultrasonic metamaterial, the Helmholtz resonator, consists of a cavity of known volume with rigid walls and a small hole in one side. A pressure variation in the channel causes the plug of fluid in the hole to oscillate in and out, producing adiabatic compression and rarefaction of the liquid enclosed in the cavity. Such a resonator is analogous to an inductor–capacitor circuit, with the enclosed cavity acting as the capacitor with capacitance C ∼ V/ρc2, and the neck acting as the inductor (L∼ρ(L_/S)), where V is the volume of the cavity, ρ is the density of the fluid, c is the sound speed in the fluid, L_ is the effective length of the neck, and S is the cross-sectional area of the neck. Because the Helmholtz resonator does not use typical standing waves to create a resonance, the dimension of each element can be made much smaller than the acoustic wavelength (at 33 kHz, l=4.4 cm in water)

To characterize the unique propagation properties of acoustic waves in this ultrasonic metamaterial made of Helmholtz resonators, we conducted a set of underwater ultrasonictransmission experiments on one-dimensional daisy-chained samples. An appreciable range of negative group delay time is found at the anomalous dispersion region of the one-dimensional ultrasonic metamaterial ; this negative group delay time means that a sound pulse propagating along the channel of ultrasonic metamaterials emerges downstream significantly earlier than if it had propagated the same distance in a water channel, so the peak of the pulse seems to leave the duct before entering it. The negative group velocity observed is a clear indicator of the negative modulus of the ultrasonic metamaterials 

====Conclusion====
In conventional acoustic bandgap materials, the acoustic waves are strongly diffracted when the wavelength is close to the size of the scatterers. In contrast, the ultrasonic metamaterials act as a homogenized medium to the acoustic excitation because their lattice constants are much smaller than the wavelength. Therefore, ultrasonic metamaterials offer a new opportunity in making compact and tunable acoustical devices, such as small footprint sonar and medical ultrasonic scanners. In addition, negative modulus can introduce collective surface oscillations. These resonant surface modes have very large wave vectors at the interface between two materials of opposite signs of modulus, similar to the surface plasmon on the metal–dielectric interface. The emergence of these surface states allows the coupling between the evanescent spectrum from a subwavelength object into the ultrasonic metamaterials with strong enhancement. This strong evanescent enhancement could compensate evanescent decay in the free space, leading to a potential acoustic superlens that produces images well below the diffraction limit, analogous to electromagnetic superlensing. Instead, the loss of the system, mostly attributed to the viscous dissipation of the fluid and the wall, would eventually limit the imaging resolution, in a similar fashion to their electromagnetic counterparts. The artificial ultrasonic metamaterials with designed negative modulus and dispersion open a new horizon in acoustic imaging below the diffraction limit.
<ref>{{cite journal|doi=10.1038/nmat1644 |title=Ultrasonic metamaterials with negative modulus|last1=Fang|first1=Nicholas|journal=Nature Materials|volume=5|pages=452–456|year = 2006 |issue=6 |pmid=16648856 }}</ref>

===Double-negative acoustic metamaterial===
[[Image:Sine waves same phase.svg|thumb|In-phase waves]]
[[Image:Sine waves different phase.svg|thumb|Out-of-phase waves]]
[[Image:Phase-shift illustration.png|right|thumb|Left: the [[real part]] of a [[plane wave]] moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through metamaterial [[Homogeneity (physics)|inhomogeneities]] of different thickness than the other parts. (The illustration on the right ignores the effect of [[diffraction]] whose effect increases over large distances).]]
====Introduction====
Veselago’s medium is an isotropic homogeneous medium having a negative refractive index. The electromagnetic (EM) metamaterials that are realized in practice are composites with built-in resonance structures that exhibit effective negative permittivity and negative permeability for some frequency ranges. These double negative media have generated tremendous interest due to unique phenomena, such as negative refraction and subwavelength imaging. We note that the phenomena of negative refraction is closely related but not identical to double negativity. Negative refraction can be achieved through a band-folding effect due to Bragg scattering. In an acoustic wave, we can also engineer the dispersion surface of solids to achieve the phonon focusing. With this comparison, it becomes natural and interesting to ask whether we can have an analog of Veselago’s medium in an acoustic wave, namely a composite with built-in resonances such that the effective response functions are double negative in the effective medium limit.
<ref name=acousticdbleneg>{{cite journal| title = Double-negative acoustic metamaterial| journal = Phys. Rev. E| date = 2004| volume =70|issue =5|doi = 10.1103/PhysRevE.70.055602| year = 2004| pages = 055602| last1 = Li| first1 = Jensen| last2 = Chan| first2 = C. T.| pmid = 15600684}}</ref>

====Method====
The electromagnetic (isotropic) metamaterials have built-in [[resonance]] structures that exhibit effective negative permittivity and negative permeability for some frequency ranges. In contrast, it is difficult to build composite [[acoustic|acoustic materials]] with built-in resonances such that the two effective [[response function]]s are negative within the capability or range of the [[transmission medium]].<ref name=acousticdbleneg>{{cite journal| title = Double-negative acoustic metamaterial| journal = Phys. Rev. E| date = 2004| volume =70|issue =5|doi = 10.1103/PhysRevE.70.055602| year = 2004| pages = 055602| last1 = Li| first1 = Jensen| last2 = Chan| first2 = C. T.| pmid = 15600684}}</ref>

The mass density ''ρ'' and bulk modulus β are position dependent. Using the formulation of a [[plane wave]] the wave vector is:<ref name=acousticdbleneg/>

::::::<math> \vec{k} = \frac{\ |n| \omega}{c}. \,</math>

The [[angular frequency]] is represented by ''ω'' and ''c'' is the propagation speed of acoustic signal through the [[homogeneous|homogeneous medium]]. With constant density and bulk modulus as constituents of the medium, the refractive index is expressed as n<sup>2</sup> = ''ρ'' / β. In order to develop a propagating (plane) wave through the material, it is necessary for both ''ρ'' and β to be either positive or negative.<ref name=acousticdbleneg/>

When the negative parameters are achieved, the mathematical result of the [[Poynting vector]] <math> \scriptstyle \overleftarrow{s} </math>. is the opposite direction of the [[wave vector]] <math> \scriptstyle \overrightarrow{k}</math>. This requires negativity in bulk modulus and density. Physically, it means that the medium displays an anomalous response at some frequencies such that it expands upon compression (negative bulk modulus) and moves to the left when being pushed to the right (negative density) at the same time.<ref name=acousticdbleneg/> 

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.<ref name=acousticdbleneg/><ref name=tapwater>{{cite book| last =Trostmann|first =Erik| title =Tap water as a hydraulic pressure medium| publisher =CRC Press|date =200-11|pages =36|url =https://books.google.com/books?id=EiaCkItHNRAC&pg=PA46|isbn =9780824705053}}</ref><ref name=fluidmechbasic>{{cite book| last=Petrila|first = Titus| coauthors = Damian Trif|title =Basics of fluid mechanics and introduction to computational fluid dynamics|publisher =Springer-Verlag New York, LLC| date =December 2004|url =https://books.google.com/books?id=pMH_fEFdilMC|isbn =9780387238371}}</ref>

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. Intrinsic is the expectation for positive bulk modulus and positive density. For example, dispersing spherical solid particles in a fluid results in the ratio governed by the specific gravity when interacting with the long acoustic wavelength (sound). Mathematically, it can be easily proven that β<sub>eff</sub> and ''ρ''<sub>eff</sub> are definitely positive for natural materials.<ref name=acousticdbleneg/><ref name=tapwater/> The exception occurs at low resonant frequencies.<ref name=acousticdbleneg/> 

As an example, acoustic double negativity is theoretically demonstrated with a composite of soft, silicone rubber spheres suspended in water.<ref name=acousticdbleneg/> 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 for the transmission of very low monopolar and dipolar frequencies. This is an analogue to analytical solution for the scattering of electromagnetic radiation, or [[Mie theory|electromagnetic plane wave scattering]], by spherical particles - [[dielectric]] spheres.<ref name=acousticdbleneg/> 

Hence, there is a narrow range of normalized frequency 0.035 < ωa/(2πc) < 0.04 where the bulk modulus and negative density are both negative. Here ''a'' is the lattice constant if the spheres are arranged in a [[Cubic crystal system|face-centered cubic]] (fcc) lattice; ω is frequency and ''c'' 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 frequency at about 0.04.<ref name=acousticdbleneg/> 

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 (instead of two) creates the low-frequency resonances to achieve double negativity.<ref name=acousticdbleneg/>. With monopolar resonance, the spheres expand, which produces a phase shift between the waves passing through rubber and water. This creates the 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.<ref name=acousticdbleneg/>

Both the mass density and the reciprocal of the bulk modulus are decreasing in magnitude fast enough so that the group velocity becomes 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 refraction properties.

====Conclusion====
The theoretical concept of a double-negative acoustic medium (Poynting vector in opposite direction with wave vector) which has a simultaneously negative effective bulk modulus and density has been demonstrated. It is the acoustic analog of Veselago’s medium having simultaneously negative values of ε and μ. While the negative ε and negative μ in EM metamaterials are typically derived from two types of resonances, the negative modulus and density originate from the monopolar and dipolar resonances of the same structure, one example being soft rubber in water. The double negativity is a consequence of resonance and the resulting negative refraction properties are not a consequence of a band-folding effect due to Bragg scattering. It is important to emphasize some unique properties of a double-negative medium, such as negative refractive index and subwavelength focusing, are natural consequences. <ref name=acousticdbleneg/>

===Metamaterial with simultaneously negative bulk modulus and mass density===
====Introduction====
[[Image : Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density.jpg|thumb|right|200px|Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density]]
With the realization of DNM for EM waves, an interesting question arises: whether or not a double-negative material is possible for elastic or acoustic waves. If both the mass density and the elastic constant become negative, negative refraction for elastic waves could be expected similarly, with the refractive index.
Recently, it has been shown that a class of three-component phononic crystal with local resonant structure exhibits negative effective mass density (EMD) at (dipolar) resonant frequency. A ultrasonic metamaterial consisting of an array of subwavelength Helmholtz resonators was demonstrated to have a negative effective bulk modulus (EBM) at (monopolar) resonant frequency. A phononic crystal consisting of rubber spheres in water was reported to be a double-negative acoustic medium . However, this structure is not a complete or pure DNM. Here is the complete band structure for it, where solid circles are the band structure.<ref name=simultaneousnegative>{{cite journal|title = Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density| journal = Phys. Rev. Lett.| date = 2007| volume =99|issue = 9|doi = 10.1103/PhysRevLett.99.093904|year = 2007| pages = 093904| last1 = Ding| first1 = Yiqun| last2 = Liu| first2 = Zhengyou| last3 = Qiu| first3 = Chunyin| last4 = Shi| first4 = Jing| pmid = 17931008}}</ref>

====Method====
A metamaterial was reported which simultaneously possesses a negative bulk modulus and mass density. This metamaterial is a [[Cubic crystal system|zinc blende]] structure consisting of one [[Cubic crystal system|fcc]] array of bubble-contained-water spheres (BWSs) and another relatively shifted fcc array of rubber-coated-gold spheres (RGSs) in special epoxy.<ref name=simultaneousnegative>{{cite journal|title = Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density| journal = Phys. Rev. Lett.| date = 2007| volume =99|issue = 9|doi = 10.1103/PhysRevLett.99.093904|year = 2007| pages = 093904| last1 = Ding| first1 = Yiqun| last2 = Liu| first2 = Zhengyou| last3 = Qiu| first3 = Chunyin| last4 = Shi| first4 = Jing| pmid = 17931008}}</ref>

====Conclusion====
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 bulk modulus and mass density in a solid based material, which is an important distinction.<ref name=simultaneousnegative/>
{{See|Poisson's ratio}}

===Acoustic cloaking in three dimension===
A scheme to achieve two dimensional acoustic cloaking is proposed by Cummer and Schurig
<ref>{{cite journal|doi=10.1103/PhysRevLett.100.063903|title=Optical design of reflectionless complex media by finite embedded coordinate transformations
|date=15 Feb. 2008|last1=Rahm.M|journal=PRL|volume=100|issue=6
|pages=063903–1–4|pmid=18352474
}}</ref> by mapping the acoustic equations to [[Maxwell's equations]] of one polarization in the 2D geometry. Huanyang Chen and C. T. Chan found that the acoustic equation can be mapped to the direct current conductivity equation in three dimensions, which then allows the design of three-dimensional acoustic cloaking using the coordinate transformation scheme. The perfect cloaking effect is confirmed by solving for the scattering problem using the spherical-[[Bessel function]] series expansion method.<ref>{{cite journal|doi=10.1063/1.2803315 |title=Acoustic cloakinging in three dimensions using acoustic metamaterials|year=2007|last1=Chen|first1=Huanyang|journal=Appl. Phys.Lett|volume=91|issue=18 |pages=183518}}</ref>

===Double C resonators===
Double C resonator (DCR) is a ring cut in halves. In 2007, proposals have been made for arrays of DCRs and similar negative acoustic metamaterial.<ref name=acousticmeta1107> {{cite journal|title =Acoustic metamaterials for sound focusing and confinement
|journal =New Journal of Physics| volume =9| issue =399| pages =1367–2630|date =2007|format=free download pdf|doi =10.1088/1367-2630/9/11/399
|last1 =Guenneau
|first1 =Sébastien
|last2 =Movchan
|first2 =Alexander
|last3 =Pétursson
|first3 =Gunnar
|last4 =Anantha Ramakrishna
|first4 =S.
}}</ref> Although linear elasticity is mentioned, the problem is defined around shear waves directed at angles to the plane of the cylinders. The DCR was constructed similar to the SRRs in a multiple cell configuration. The DCR has been improved with stiffer material sheets.<ref name=acousticmeta1107/> Each cell consists of a large rigid disk and two
thin ligaments.<ref name=acousticmeta1107/> The DCR cell is a tiny oscillator connected by springs. One spring of the oscillator connects to the ''mass'' and is anchored by the other spring. The [[LC circuit|LC resonator]] has specified capacitance and inductance.<ref name=acousticmeta1107/> The limitations are expressed with appropriate mathematical equations. In addition to the intended limitations is that the speed of sound in the matrix is expressed as c = √''ρ''/µ with a matrix of density ''ρ'' and shear modulus μ. The resonant frequency is then expressed as √1/(LC).<ref name=acousticmeta1107/> 

A phononic bandgap 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 [[rigid body]].

The DCR design produced a suitable band with 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:
*a beam of sound negatively refracts across a slab of such a medium, 
*the phase vector in the medium possesses real and imaginary parts with opposite signs,
*the medium is well impedance-matched with the surrounding medium,
*a flat slab of the metamaterial can image a source across the slab like a Veselago lens, 
*the image formed by the flat slab has considerable sub-wavelength image resolution, and 
*a double corner of the metamaterial can act as an open resonator for sound.

===Acoustic metamaterial superlens===
[[Image : AcousticSuperlens.jpg|thumb|right|200px|AcousticSuperlens]]

In May 2009 Shu Zhang ''et al.'' presented the design and test results of an ultrasonic metamaterial lens for focusing 60&nbsp;kHz (~2&nbsp;cm wavelength) sound waves under water.<ref name=underwaterlens09>{{cite journal|last = Thomas| first = Jessica| title = Metamaterial brings sound into focus
|journal =Physics|format=synopsis for Phys. Rev. Lett. 102, 194301|publisher =American Physical Society |date = 2009-05-15
|url =http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.102.194301 
|accessdate = 2009-08-29|doi = 10.1103/PhysRevLett.102.194301|volume = 102|pages = 194301| pmid = 19518957}}</ref> The lens is made of sub-wavelength elements and is therefore potentially more compact than phononic lenses that operate in the same frequency range.<ref name=underwaterlens09/>

High-resolution acoustic imaging techniques are the essential tools for nondestructive testing and medical screening. However, the spatial resolution of the conventional acoustic
imaging methods is restricted by the incident wavelength of ultrasound. This is due to the quickly fading [[Evanescent wave|evanescent fields]] which carry the sub-wavelength features of objects.<ref name=acousticsuperlens>{{cite journal
|title = Focusing Ultrasound with Acoustic Metamaterial Network| journal =Phys. Rev. Lett. 
|volume = 102| issue = 19|publisher = American Physical Society|date = 2009
|doi = 10.1103/PhysRevLett.102.194301|pages = 194301| arxiv = 0903.5101| last1 = Zhang| first1 = Shu| last2 = Yin| first2 = Leilei| last3 = Fang| first3 = Nicholas| pmid = 19518957}}</ref>

[[File:AcousticSuperlens Focusing.jpg|thumb|right|200px|Acoustic Focus]]
The lens consists of a network of fluid-filled cavities called [[Helmholtz resonance|Helmholtz resonators]] that oscillate at certain sonic frequencies. Similar to a network of inductors and capacitors in electromagnetic [[metamaterial]], the arrangement of Helmholtz cavities designed by Zhang ''et al.'' have a negative dynamic modulus for ultrasound waves. Zhang ''et al.'' did focus a point source of 60.5&nbsp;kHz sound to a spot size that is roughly the width of half a wavelength and their design may allow to push the spatial resolution even further.<ref name=underwaterlens09/> This result is in excellent agreement with the numerical simulation by transmission line model, which derived the effective mass density and compressibility. This metamaterial lens also displays variable focal length at different frequencies.<ref name=acousticsuperlens/>

===Acoustic diode===
An acoustic diode was introduced in August 2009. An electrical [[diode]] allows current to flow in only one direction in a wire; it is an essential electronic device which had no analogues for sound waves. However, the reported design partially fills this role by converting sound to a new frequency and blocking any backwards flow of the original frequency. In practice, it could give designers new flexibility in making ultrasonic sources like those used in medical imaging. The proposed structure combines two components: The first is a sheet of nonlinear acoustic material—one 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.<ref name=acousticdiode>{{cite journal| last = Monroe| first =Don|title = One-way Mirror for Sound Waves|journal =Physical Review Focus 
|format=synopsis for "Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems" by Bin Liang, Bo Yuan, and Jian-chun Cheng|publisher =American Physical Society|date =2009-08-25| url =http://focus.aps.org/story/v24/st8#author| accessdate =2009-08-28}}</ref><ref>{{cite journal|doi=10.1103/PhysRevLett.93.184301|title=Thermal Diode: Rectification of Heat Flux|year=2004|last1=Li|first1=Baowen|journal=Physical Review Letters|volume=93|issue=18 |pages=184301|pmid=15525165 }}</ref>

==Proposed Applications==
Current acoustic devices must be larger than the wavelength of the waves with which they interact, but ones with metamaterial lenses could be much smaller and more compact. Using metamaterial lenses, two possible application are medical ultrasound imaging and sonar systems.
<ref name=Scinecereview>{{cite journal| last = Strubbe | first =David  |title = (Meta)material World|journal = Berkeley Science Review| url =http://sciencereview.berkeley.edu/articles.php?issue=11&article=briefs_5}}</ref>

The acoustic imaging, while not quite as convenient or accurate as optical imaging, is improving. Presently, however, acoustic imaging can’t detect objects smaller than a few millimeters. The best tool for tumor detection is still optical imaging but exposure to electromagnetic radiation, like those found in x-rays, does pose health risks, especially when used on pregnant women. Acoustic images don’t rely on electromagnetic radiation. Acoustic images may prove useful when detecting tumors in the body that are often surrounded by hard tissues and don’t permit clear detection by traditional optical imaging methods.
<ref name=superlens>{{cite journal| last = Shalton| first =Jeff|title = Researchers create acoustic metamaterial 'superlens'|journal =Gizmag| url =http://www.gizmag.com/superlens-acoustic-imaging/12110/ }}</ref>

Also the application of acoustic imaging technology could lead to other possibilities. Buildings and bridges could be checked for hairline cracks or other deeply embedded defects that are undetectable by optical imaging. Furthermore, the technology might even extend to underwater stealth technology, what it will be termed an “acoustic cloak” that would act as camouflage for submarines.
<ref name=superlens>{{cite journal| last = Shalton| first =Jeff|title = Researchers create acoustic metamaterial 'superlens'|journal =Gizmag| url =http://www.gizmag.com/superlens-acoustic-imaging/12110/ }}</ref>

===Doppler effect reversed by metamaterial===
[[Image : Doppler effect reversed by metamaterial.jpg|thumb|right|200px|Doppler effect reversed by metamaterial]]

Physicists have generated a lot of excitement in recent years by dreaming up specially structured materials with novel applications like invisibility cloaks. What’s more, some of these [[metamaterials]] have been built in the laboratory and shown to work over a narrow range of electromagnetic wavelengths. Now, a group of researchers from Korea and China has created an acoustic metamaterial that causes the bizarre effect of a reverse Doppler effect. This is an important stepping stone to an acoustic cloak, according to the researchers.<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref> 

As every physicist is taught in school, the Doppler effect is what causes a pedestrian to hear a high pitch siren as a police car speeds towards them, and a lowering pitch as it races away. Surprisingly, a new material has defied physics textbooks by reversing this effect.<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref>

Chul Koo Kim of Yonsei University and his colleagues have achieved this feat by creating an elastic tube that transmits sound with a negative phase velocity. “We have successfully fabricated an acoustic metamaterial whose acoustic refractive index can be controlled; the theoretical models can now be implemented to realize acoustic cloaking as well as other applications.”<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref>

Sound is passed into the tube from a moving source via holes pierced periodically along the device. Inside the tube a fixed detector receives the sound before sending an electrical signal to a loudspeaker. According to Kim, the major engineering hurdle was to develop effective absorbers at each end of the tube. “This enabled us to so as to prevent reflections and ensure the quality of data,” he said.<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref> 

Kim and colleagues tested the apparatus using sound of 350 Hz with a source moving 5 m/s towards then away from the direction of wave propagation. Contrary to classic Doppler experiments they found that frequency was down-shifted as the source moved towards the receiver and up-shifted as it moved away from it. What the experimenters heard was a decreasing pitch as the source moved towards the detector and increasing one as it moved away from it.<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref> 

The next stage of this research is to translate their “1D design” into various types of 2D and 3D acoustic metamaterials. “These developments may find uses in medicine and industry. Also easy control of acoustic refractive index will spur new research directions in fiber acoustics.<ref name=Physicsworld>{{cite journal| last = Kim| first =Chul Koo  |title = Doppler effect reversed by metamaterial|website = Physicsworld.com| url =http://physicsworld.com/cws/article/news/37622#comments}}</ref>

==See also==
{{colbegin|3}}
*[[Metamaterial]]
*[[Negative index metamaterials]]
*[[Metamaterial antennas]]
*[[Perfect lens]]
*[[Photonic metamaterials]]
*[[Photonic crystal]]
*[[Split-ring resonator]]
*[[Terahertz metamaterials]]
*[[Tunable metamaterials]]

===Material properties===
*[[Constitutive equation]]
*[[Dielectric]]
*[[Equation of state]]
*[[Linear elasticity]]
*[[Stress (mechanics)]]
*[[Thermodynamic state]]
*[[Viscosity]]
{{colend}}

==External links==
*[http://www.physics.fudan.edu.cn/tps/people/jzi/research/contents/materials/Physics%20Worl%20DEC05%20thomas%20sonic%20crystals.pdf Ideas underpinning sound]
*[http://appliedmath.arizona.edu/modeling-computation-seminar/2009-09-17--pierre-deymier Acoustic Metamaterials and Devices: Negative, Positive, and Zero Refraction and Super-lensing in Phononic Crystals]

==References==
{{reflist|2}}

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