Band Gap Formation in Acoustically Resonant Phononic Crystals

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Band gap formation in
acoustically resonant
phononic crystals
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• A Doctoral Thesis. Submitted in partial fulfillment of the requirements
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LOUGHBOROUGH UNIVERSITY
Band Gap Formation in Acoustically
Resonant Phononic Crystals
by
Daniel Peter Elford
A doctoral thesis submitted in partial fulfillment for the
degree of Doctor of Philosophy
in the
Department of Physics
Faculty of Science
Loughborough University
November 2010
⃝c by Daniel Peter Elford 2010
Acknowledgements
I wish to thank all the people who contributed directly or indirectly to the realisation
of this thesis. First of all I wish to extend my deepest gratitude to my PhD supervi-
sors Dr. Gerry Swallowe and Prof. Feodor Kusmartsev, the work presented here owes
much to their enthusiasm and careful guidance throughout my PhD. Dr Swallowe’s vast
experience in experimental physics and Prof. Kusmartsev’s distinguished theoretical
background was an ideal partnership behind this research and I have benefited greatly
fromworkingwiththem, notonlyonanacademiclevelbutonapersonalleveltoo. Iam
indebted to Prof. Kusmartsev for my knowledge gained through numerous discussions
along with seminars and international conferences I attended. He is a true scientist and
working with him has been a great honor. I wish to equally thank Dr. Swallowe for
his full support, patient guidance and advice during my research. During the numer-
ous meetings in the past years I have benefited greatly from his abundant knowledge,
professionalism and insight into physics.
I am extremely grateful to the other member of our research group, Luke Chalmers,
whose friendship and support have got me through my undergraduate and PhD studies.
Our lengthly discussions and his assistance with experimental measurements have been
invaluable in completing this research. Special thanks are due to the Loughborough
Physics departmental technicians, Phil Sutton and Bryan Dennis, who painstakingly
machined and prepared the experimental phononic crystal systems. I also wish to thank
Prof. Victor Krylov and Jochen Eisenblaetter from the Department of Aeronautical and
Automotive Engineering at Loughborough University who provided access to their ane-
choicchamberandofferedtheirguidanceinperformingtheexperimentalmeasurements.
My acknowledgments are extended to Dr. Robert Perrin, from Institute of Fundamental
Sciences, Massey University, Palmerston North, New Zealand, whose fruitful discussions
regarding the vibrational modes and the numerous hours spent with me analyzing and
characterizingmodeshapesfromfiniteelementpredictionsandESPIwereinvaluablefor
thecompletionofthisthesis. IwishtothankProf. ThomasMoore,fromtheDepartment
of Physics, Rollins College, Winter Park, Florida, USA, for his expertise and support
with using his Electronic Speckle Pattern Interferometer.
I wish to thank the rest of postgraduates and staff in the Department of Physics at
Loughborough for fruitful and not always scientific discussions, their friendship and
their time. Special thanks are given to my parents, family and friends who have proudly
supported me during my time at Loughborough, which was spent far from their side and
need.
i
LOUGHBOROUGH UNIVERSITY
Abstract
Department of Physics
Faculty of Science
Doctor of Philosophy
by Daniel Peter Elford
The work presented in this thesis is concerned with the propagation of acoustic waves
through phononic crystal systems and their ability to attenuate sound in the low fre-
quencyregime. Theplanewaveexpansionmethodandfiniteelementmethodareutilised
toinvestigatethepropertiesofconventionalphononiccrystalsystems. Theacousticband
structure and transmission measurements of such systems are computed and verified ex-
perimentally. Goodagreementbetweenbandgaplocationsfortheinvestigativemethods
detailed is found. The well known link between the frequency range a phononic crystal
can attenuate sound over and its lattice parameter is confirmed. This leads to a reduc-
tion in its usefulness as a viable noise barrier technology, due to the necessary increase
in overall crystal size. To overcome this restriction the concept of an acoustically reso-
nant phononic crystal system is proposed, which utilises acoustic resonances, similar to
Helmholtz resonance, to form additional band gaps that are decoupled from the lattice
periodicity of the phononic crystal system. An acoustically resonant phononic crystal
system is constructed and experimental transmission measurements carried out to ver-
ify the existence of separate attenuation mechanisms. Experimental attenuation levels
achieved by Bragg formation and resonance reach 25dB. The two separate attenuation
mechanisms present in the acoustically resonant phononic crystal, increase the efficiency
of its performance in the low frequency regime, whilst maintaining a reduced crystal size
for viable noise barrier technology. Methods to optimise acoustically resonant phononic
crystal systems and to increase their performance in the lower frequency regime are dis-
cussed, namely by introducing the Matryoshka acoustically resonant phononic crystal
system, where each scattering unit is composed of multiple concentric C-shape inclu-
sions.
Contents
Acknowledgements i
Abstract ii
List of Figures vi
List of Tables xii
Motivation xiii
1 Introduction 1
1.1 Elastic Waves in Homogeneous Materials. . . . . . . . . . . . . . . . . . . 1
1.2 Acoustic Waves in Homogeneous Fluid Materials . . . . . . . . . . . . . . 4
1.2.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Wave Equation and Helmholtz Equation . . . . . . . . . . . . . . . 6
1.2.5 Solutions of the Wave Equation . . . . . . . . . . . . . . . . . . . . 7
1.2.6 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.7 Complex number Notation . . . . . . . . . . . . . . . . . . . . . . 10
2 Phononic Crystals 11
2.1 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Band Gap Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Electronic Band Structure of Solids. . . . . . . . . . . . . . . . . . 14
2.2.2 Electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Dispersion Relation Construction . . . . . . . . . . . . . . . . . . . 17
2.2.4 Phononic Band Gap Formation . . . . . . . . . . . . . . . . . . . . 19
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Plane Wave Expansion Method 23
3.1 Plane Wave Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Plane Wave Expansion Results . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Conventional Phononic Crystal . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Packing Fraction Investigation . . . . . . . . . . . . . . . . . . . . 29
3.2.3 Lattice Parameter Investigation . . . . . . . . . . . . . . . . . . . . 32
3.2.4 Low Frequency Phononic Crystal . . . . . . . . . . . . . . . . . . . 33
iii
Contents iv
4 Finite Element Methods 35
4.1 Acoustic Wave Propagation in Comsol Multiphysics . . . . . . . . . . . . 37
4.2 FEM Computed Pressure Map Accuracy . . . . . . . . . . . . . . . . . . . 39
4.3 Acoustic Band Structure Construction . . . . . . . . . . . . . . . . . . . . 42
4.4 Finite Element Band Structure Calculations . . . . . . . . . . . . . . . . . 45
4.4.1 Packing Fraction Investigation . . . . . . . . . . . . . . . . . . . . 48
5 Experimental Methods 51
5.1 Review of Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.1 Conventional 2D Steel Cylinders in Air . . . . . . . . . . . . . . . 54
5.2.2 Low Frequency Phononic Crystal . . . . . . . . . . . . . . . . . . . 55
5.2.3 Acoustically Resonant Phononic Crystal . . . . . . . . . . . . . . . 56
5.3 Acoustic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.1 Transmission and Phase Measurements . . . . . . . . . . . . . . . 57
5.4 Testing the Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.1 Source Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Conventional Phononic Crystal Results 62
6.1 Transmission Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.1.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Low Frequency Phononic Crystal . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1 Band Structure Calculations . . . . . . . . . . . . . . . . . . . . . 71
6.2.2 Transmission Measurements . . . . . . . . . . . . . . . . . . . . . . 72
6.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Experimental Band Structure Construction . . . . . . . . . . . . . . . . . 79
6.3.1 Phase Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Locally Resonant Sonic Crystals 82
7.1 Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Scattering from a Helmholtz Resonator. . . . . . . . . . . . . . . . . . . . 86
7.3 Resonance Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.4 Acoustically Resonant Phononic Crystal . . . . . . . . . . . . . . . . . . . 88
8 Acoustically Resonant Phononic Crystal Results 90
8.1 Band Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.2 Transmission Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Investigation on Number of Layers . . . . . . . . . . . . . . . . . . . . . . 101
8.4 Orientation of Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.5 Enlargement of Band Gap Width . . . . . . . . . . . . . . . . . . . . . . . 104
8.5.1 Band Structure Calculations . . . . . . . . . . . . . . . . . . . . . 104
8.5.2 Transmission Calculations . . . . . . . . . . . . . . . . . . . . . . . 105
8.6 Matryoshka ‘Russian Doll’ Configuration . . . . . . . . . . . . . . . . . . . 108
8.6.1 Band Structure Calculations . . . . . . . . . . . . . . . . . . . . . 108
8.6.2 Transmission Measurements . . . . . . . . . . . . . . . . . . . . . . 109
8.7 Low Frequency Matryoshka System . . . . . . . . . . . . . . . . . . . . . . 111
Contents v
9 Conclusion 115
10 Further Work — Modal Analysis 118
10.1 Vibration of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.1.1 Donnell Shell theory . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.1.2 Inextensibility Condition . . . . . . . . . . . . . . . . . . . . . . . 122
10.2 The Symmetry Group of the Cylinder . . . . . . . . . . . . . . . . . . . . 124
10.2.1 Classification of Mode Types . . . . . . . . . . . . . . . . . . . . . 125
10.2.2 Radiation of Sound from a Cylinder . . . . . . . . . . . . . . . . . 127
10.2.3 Clamped-Free Modes Classes . . . . . . . . . . . . . . . . . . . . . 127
10.2.4 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.3 Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.3.1 Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.3.2 Finite Element Accuracy . . . . . . . . . . . . . . . . . . . . . . . 131
10.4 Slender Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.4.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.4.1.1 Transverse Modes . . . . . . . . . . . . . . . . . . . . . . 134
10.4.1.2 Torsional Modes . . . . . . . . . . . . . . . . . . . . . . . 136
10.4.1.3 Longitudinal Modes . . . . . . . . . . . . . . . . . . . . . 138
10.4.1.4 Breather Modes . . . . . . . . . . . . . . . . . . . . . . . 139
10.5 Broad Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.6 Electronic Speckle Pattern Interferometry . . . . . . . . . . . . . . . . . . 145
10.6.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.7 Interferograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.9 Laser Doppler Vibrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.9.1 Laser Doppler Vibrometer Measurements . . . . . . . . . . . . . . 152
10.10LDV Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.11Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A Calculation of Refractive Index 155
A.1 Reproduction of Published Results . . . . . . . . . . . . . . . . . . . . . . 155
A.2 Calculation of Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . 159
A.2.1 Acoustic Fresnel Biprism . . . . . . . . . . . . . . . . . . . . . . . 159
A.2.2 Acoustic Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B MATLAB Codes 162
B.1 PWE.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
B.2 SingleFT.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.3 Phase.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Bibliography 170
Publications 177
List of Figures
1.1 The propagation of transverse and longitudinal elastic plane waves. . . . . 2
1.2 A volume element of a fluid material with definition of normal direction.
n˜(𝑥) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Schematic of a simple harmonic wave in polar form. . . . . . . . . . . . . 10
2.1 A two dimensional triangular Bravais lattice is depicted with vector no-
tation for a primitive lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Plane lattice types for two dimensional Bravais lattices. . . . . . . . . . . 12
2.3 Schematic of the Brillouin zone. The first Brillouin zone is shaded with
the dots indicating reciprocal lattice points and the solid lines indicating
Bragg planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Bragg diffraction of an electron. . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Electron Energy vs. wavevector 𝑘 for free electrons in reciprocal space. . . 16
2.6 Variation of potential energy of a conduction electron in the field of the
ion cores of a linear lattice (top). Distribution of probability density in
the lattice, where the wavefunction 𝜓 piles up charge on the cores of the
+
positive ions and 𝜓 piles up charge in the region between the ions. It is
−
the differences in the potential energy of these standing waves that is key
to understanding the origin of an energy band gap. After Kittel [19] . . . 17
2.7 Dispersion relation for one dimensional linear homogeneous medium. . . . 18
2.8 Dispersion relation for a two dimensional periodic medium. . . . . . . . . 18
2.9 Path difference of Bragg scattered waves. According to the derivation,
the phase shift causes constructive or destructive interferences. . . . . . . 21
3.1 Plane wave expansion computed dispersion relation for a Homogeneous
Solid Material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Planewaveexpansioncomputeddispersionrelationforaphononiccrystal
consisting of steel scatterers embedded in air. . . . . . . . . . . . . . . . . 29
3.3 PWE Investigation of packing fraction on band gap width.. . . . . . . . . 30
3.4 PWEinvestigationintotheeffectoflatticeparameteronbandgaplocation. 32
3.5 PWE computed band structure for a low frequency phononic crystal sys-
temcomposedofcardboardcompositetubesfilledwithsandembeddedin
air (𝑎 = 91.9 mm, 𝑟 = 26.5 mm). The red shading indicates the presence
of a band gap.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Discretization of a domain using triangular shape finite elements. . . . . . 36
4.2 Comsol described geometry for a rectangular domain modelled as air and
an incoming plane wave source. . . . . . . . . . . . . . . . . . . . . . . . . 38
vi
List of Figures vii
4.3 Comsol computed pressure map for a rectangular domain modelled as air
and an incoming plane wave source. . . . . . . . . . . . . . . . . . . . . . 38
4.4 Comsol investigation of the maximum element size the subdomain is dis-
cretized into, against frequency convergence. . . . . . . . . . . . . . . . . . 40
4.5 Finite element method computed pressure map for a propagating wave
from left to right of the subdomain for different maximum element sizes. . 41
4.6 Singleunitcellofaninfinitephononiccrystalsystemwithperiodicbound-
ary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.7 Comsol and PWE computed dispersion relation for a homogeneous solid
material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.8 Comsol computed dispersion relation describing the first two Eigenbands
for a homogeneous solid material. . . . . . . . . . . . . . . . . . . . . . . . 44
4.9 Singleunitcellofaninfinitephononiccrystalsystemwithperiodicbound-
ary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.10 Comsol computed band structure (solid line) for a phononic crystal con-
sisting of steel scatterers embedded in air. . . . . . . . . . . . . . . . . . . 46
4.11 Comsolcomputedbandstructureforaphononiccrystalconsistingofsteel
scatterers embedded in air. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.12 Packing fraction investigation on band gap width. . . . . . . . . . . . . . 48
4.13 The effective speedofsound withina phononiccrystal arraywith increas-
ing packing fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Examples of 2D phononic crystals studied experimentally. . . . . . . . . . 52
5.2 Parametersofmaterialsusedforthefabricationofphononiccrystalsstud-
ied experimentally so far. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Schematicfora10×10conventionalphononiccrystalsysteminaperiodic
square lattice comprising steel scatterers embedded in air (𝑎 = 22 mm, 𝑟
= 6.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 Schematicofa6×10lowfrequencyphononiccrystal,arrangedinasquare
lattice composed of cardboard composite, sand filled scatterers embedded
in air (𝑎 = 91.9 mm, 𝑟 = 26.5 mm). . . . . . . . . . . . . . . . . . . . . . 55
5.5 Schematic of a 10× 10 acoustically resonant phononic crystal arrangesd
in a square array composed of steel C-shaped scatterers embedded in air
(𝑎 = 22 mm, 𝑟 = 6.5 mm, 𝑟 = 5 mm, 𝑠 = 4 mm). . . . . . . . . . . . . . 56
𝑒 𝑖
5.7 Experimental data acquisition Setup . . . . . . . . . . . . . . . . . . . . . 58
5.8 Frequency spectra obtained for 200, 400, 600, 800 Hz signals. . . . . . . . 59
5.9 Typical frequency spectrum for a control recording of a rising tone source. 60
5.10 Typical phase dispersion for a control recording of a rising tone source. . . 61
5.11 Plane wave approximation schematic showing that a diverging wave ap-
proaches a plane wave at large propagation distances. . . . . . . . . . . . 61
6.1 Comsoldescribedgeometryforsolidsteelcylindersinairindicatingpoints
where the corresponding plots were taken. . . . . . . . . . . . . . . . . . . 62
6.2 Comsol computed frequency spectra for a conventional phononic crystal. . 63
6.3 FEM computed pressure (left) and sound pressure level (right) maps for
solid steel cylinders in air taken at three frequencies, before, during and
after Bragg formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64