New Acoustics Based on Metamaterials

321 Pages·2018·6.14 MB·English

by  Woon Siong Gan (auth.)

Engineering Materials
Woon Siong Gan
New Acoustics
Based on
Metamaterials
Engineering Materials
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Woon Siong Gan
New Acoustics Based
on Metamaterials
123
WoonSiongGan
Acoustical Technologies Singapore PteLtd
Singapore
Singapore
ISSN 1612-1317 ISSN 1868-1212 (electronic)
Engineering Materials
ISBN978-981-10-6375-6 ISBN978-981-10-6376-3 (eBook)
https://doi.org/10.1007/978-981-10-6376-3
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To My Parents
Foreword
Acoustics is a classic field of inquiry that has enjoyed a strong revival during the
pasttwodecades,propelledmainlybytheadventofphononiccrystalsandacoustic
metamaterials, two newly developed research areas that focused on man-made
structures with acoustic properties not commonly found in nature. Whereas pho-
nonic crystals denote periodic structures exhibiting frequency bandgaps in which
there can be no propagating acoustic/elastic waves, acoustic metamaterials acquire
their exotic characteristics as collective manifestations of local resonators. Both
phononic crystals and acoustic metamaterials are composite structures comprising
materials of different mass densities and hardness. In the case of acoustic meta-
materials,however,theresponseofthecompositetoexternalexcitationscandiffer
fromarigidsolidbyhavinginternalrelativemotionsbetweenthedifferentmaterial
components. The past fifteen years have witnessed the novel capabilities that can
arise from such locally resonant sonic materials, which are characterized not only
by their subwavelength physical size, but also by their effective mass density and
bulk modulus that can exhibit negative values. The unusual phenomena exhibited
by the phononic crystals and acoustic metamaterials, as well as their underlying
physics, are the subjects of the present volume—New Acoustics.
Theauthor,Dr.WoonSiongGan,wastrainedasaphysicist,withaPh.D.degree
in acoustics from Imperial College London. After doing postdoc at International
CentreforTheoreticalPhysics,Trieste,Italy,hereturnedtoSingaporeandtaughtat
NanyangUniversityfrom1970to1979.Hewasapracticingacousticconsultantfor
ten years, from 1979 to 1989, and after that he founded Acoustical Technologies
SingaporePteLtd.Thecontentsofthepresentvolumeverymuchreflectthisrather
unique background of the author—a combination of basic theory and practical
applications. The initial few chapters lay the theoretical basis of the “new acous-
tics”, a term coined to denote the recent developments enabled by acoustic meta-
materials,followedbychapterswitheachonedevotedtosomespecificapplication.
Underlying these applications are some unifying principles, such as the coordinate
transformation of the acoustic wave equation and its one-to-one equivalence to a
system where the material constants of the transformed system can be point-wise
determinedbythoseoftheoriginaluntransformedsystem,plustheJacobianmatrix
vii
viii Foreword
of the coordinate transformation. This mathematical equivalence, denoted trans-
formationacoustics,offerstremendousfreedomindesigningstructuresthatcan,for
example, “cloak” objects and achieving effects that were thought impossible pre-
viously.However,abasicrequirementfor thesuccessfulimplementationofsucha
structure,withmathematically transformedmaterialconstants,istheavailabilityof
material properties that can take all possible values. That is where metamaterials
come in—they offer the freedom of material design not available before, such as
negativerefractiveindexandnegative(dynamic)massdensity.Thelattermayseem
counter-intuitiveatfirst,buttheeffectcanbeeasilydemonstratedwithamechanical
system comprising local resonators, so that when the external forcing is out of
phase with the internal resonances, large relative motion of the components can
result, with the momentum of the internal resonators opposing the externally
exerted force. Similarly, in an array of Helmholtz resonators, the overall bulk
moduluscanappear negative. Suchnegativevaluesofthematerial propertieshave
to be interpreted in an effective medium sense, where the intend structure of the
system are “homogenized”, i.e. averaged over. As far as an external observed is
concerned,thismaynotbeaproblemsinceonlytheexternalresponseofthesystem
issensed.Andifonecanrealizeasystemwhereboththeeffectivemassdensityand
bulk modulus are simultaneously negative, then negative index becomes possible.
As shown by the Russian physicist Veselago in 1967, if a material possesses
negative index, then the phase velocity and group velocity would be in opposite
directions.Hispredictionwasrealizedexperimentallyaboutfourdecadeslater,with
the realization of structures that can exhibit the strange behaviours implied by the
negative index, one of which is that an obliquely incident plane wave will bend to
thesamesideofthe(planar)interfacialnormalastheincidentwave.Subsequently,
J. Pendry at Imperial College London predicted the possibility of using negative
index materials to break the classical resolution limit that is imposed by the finite
wavelength.Thisworkhasstimulatedagreatdealofexperimentalinterestsinboth
optics and acoustics, and various schemes, some of them not even involving the
negativeindexmaterials,weredevisedtoshowthatresolutionbeyondtheclassical
limit is indeed possible. In fact, it may be said that the greatest contribution of
metamaterialsliesintheirliberatingeffectonthinkingaboutwhatispossibleinthe
manipulation of electromagnetic and acoustic waves. Above are just few of the
manydevelopmentsthatcanbetracedtothiseffect,andthepresentvolumegivesa
selection of those topics judged by the author to reflect not only the novelty, as
advertised by the title of the book, but also their potential importance in applica-
tions.Withtheveryfastadvanceofthiswholearea,thisbookcangivethereaders
not only a timely vignette of the current landscape, but also serve as the basis for
further development.
August 2016 Ping Sheng
HKUST, Clear Water Bay
Hong Kong
Contents
1 Symmetry Properties of Acoustic Fields . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sound Propagation in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Derivation of Linear Wave Equation of Motion and
Its Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 SymmetriesinLinearAcousticWaveEquationsand
the New Stress Field Equation . . . . . . . . . . . . . . . . . 3
1.3 Use of Gauge Potential Theory to Solve Acoustic Wave
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Gauge Theory Formulation of Sound Propagation in Solids . . . 6
1.4.1 Translational Symmetry . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 Introduction of Covariant Derivative to the
Infinitesimal Amplitude Sound Wave Equation. . . . . . 7
1.4.3 Introduction of Covariant Derivative to the Large
Amplitude Sound Wave Equation . . . . . . . . . . . . . . . 8
1.4.4 Local Rotational Symmetry. . . . . . . . . . . . . . . . . . . . 8
1.5 Symmetry Is the Theoretical Framework of Acoustical
Metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Rotational Symmetry and Theory of Elasticity . . . . . . 9
1.6 Local Gauge Invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Covariant Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Discovery of Anisotropy as a Form of Local Symmetry . . . . . . 11
1.9 Role of Symmetry Properties of Acoustic Field in the Design
of a Phononic Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 Phonon as a Goldstone Mode . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Symmetry Property of Turbulence Field. . . . . . . . . . . . . . . . . . 13
1.12 Time Reversal Symmetry in Acoustics. . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
ix
x Contents
2 Negative Refraction and Acoustical Cloaking . . . . . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Limitation of Veselago’s Theory . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 GaugeInvariance ofHomogeneous Electromagnetic
Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Gauge Invariance of Acoustic Field Equations . . . . . . 20
2.2.4 Acoustical Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5 Gauge Invariance of Nonlinear Homogeneous
Acoustic Wave Equation. . . . . . . . . . . . . . . . . . . . . . 22
2.2.6 MyImportantDiscoveryofNegativeRefractionIsa
Special Case of Coordinate Transformations or a
Unified Theory for Negative Refraction and
Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Multiple Scattering Approach to Perfect Acoustic Lens. . . . . . . 24
2.4 Acoustical Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Derivation of Transformation Acoustics. . . . . . . . . . . 30
2.4.3 Application to a Specific Example. . . . . . . . . . . . . . . 34
2.5 Acoustic Metamaterial with Simultaneous Negative Mass
Density and Negative Bulk Modulus . . . . . . . . . . . . . . . . . . . . 35
2.6 Acoustical Cloaking based on Nonlinear Coordinate
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Acoustical Cloaking of Underwater Objects . . . . . . . . . . . . . . . 41
2.8 Extension of Double Negativity to Nonlinear Acoustics . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Basic Mechanisms of Sound Propagation in Solids for Negative
Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Methods to Treat Multiple Scattering in Conventional Solids . . 47
3.2 T-Matrix of Multiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Application of T-Matrix to Multiple Scattering in Acoustical
Metamaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Low-Frequency Resonances Giving Rise to Locally Negative
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Acoustic Scatterers with Locally Negative Parameters. . . . . . . . 50
3.6 Multiple Scattering of Acoustic Waves in the Low-Frequency
Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Multiple Scattering Effects: The D Factor. . . . . . . . . . . . . . . . . 53
3.8 Suitability of the T-Matrix Method to Multiple Scattering in
Acoustic Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.9 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Diffraction by Negative Inclusion. . . . . . . . . . . . . . . . . . . . . . . 58