ThesisPDF Available

Wave Propagation Control Using Active Acoustic Metamaterials

Thesis

Wave Propagation Control Using Active Acoustic Metamaterials

Abstract and Figures

Acoustic metamaterials (AMM) are artificial materials with engineered sub-wavelength structures that possess acoustic material properties which are not readily available in nature. The material properties of AMMs can be manipulated by embedding active elements inside their structure (active AMMs). This manipulation of properties is done by an external voltage signal and is hardly available in any natural material. In this work, existing designs for passive and active AMMs are reviewed and summarized. Existing homogenization techniques for the material properties of passive metamaterials are investigated and extended to be applied for active AMMs. Three new designs for active plate-type AMM with tunable density are proposed and verified analytically, numerically and experimentally. The first design is a one dimensional (1D) AMM consisting of clamped piezoelectric disks in air. The effective density of the material is controlled by an external static electric voltage. An analytic model based on the acoustic two-port theory, the theory of piezoelectricity and the pre-stressed thin plate theory is developed to predict the behavior of the material. The results are verified using the finite element method. Excellent agreement is found between both models for the studied frequency and voltage ranges. The results show that the density is tunable within orders of magnitude relative to the uncontrolled case. This is done with a limited effect on the bulk modulus of the material. The results also suggest that simple controllers could be used to program the material density. The first design was modified and extended to construct a two-dimensional AMM with controllable anisotropic density. The modified design consists of composite lead-lead zirconate titanate (PZT) plates clamped to an aluminum structure with air as the background fluid. The effective anisotropic density of the material is controlled, independently for two orthogonal directions, by means of an external static electric voltage signal. An analytic model based on the acoustic two-port theory, the theory of piezoelectricity, the laminated pre-stressed plate theory is developed to predict the behavior of the material. The results are verified also using the finite element method. Excellent agreement is found between both models for the studied frequency and voltage ranges. The results show that, below 1600 Hz, the density is controllable within orders of magnitude relative to the uncontrolled case. A reconfigurable wave guide was constructed using the developed material and its performance was evaluated numerically and analytically. The waveguide can control the direction of the acoustic waves propagating through it. The results obtained from the previous models were used to construct and experimentally verify a third design with a fully real-time controllable effective density. The effective density of the AMM can be programmed and set to any value ranging from -100 kg/m3 to 100 kg/m3 passing by near zero density conditions. This is done through an interactive graphical user interface and is achievable for any frequency between 500 and 1500 Hz. The modified design consists of clamped composite piezoelectric diaphragms suspended in air. The dynamics of the diaphragms are controlled by connecting a closed feedback control loop between the piezoelectric layers of the diaphragm. The density of the material is adjustable through an outer adaptive feedback loop that is implemented by the real-time estimation of the density of the material using the 4-microphone technique. Applications for the new material include programmable active acoustic filters, nonsymmetric acoustic transmission and programmable acoustic superlens.
Content may be subject to copyright.
AIN SHAMS UNIVERSITY
FACULTY OF ENGINEERING
Mechatronics Engineering
Wave Propagation Control Using Active
Acoustic Metamaterials
A Thesis submitted in partial fulfillment of the requirements of the degree of
Master of Science in Mechanical Engineering
(Mechatronics Engineering)
by
Ahmed Abdelshakour Abdelfattah Elhousseiny Allam
Bachelor of Science in Mechanical Engineering
(Mechatronics Engineering)
Faculty of Engineering, Ain Shams University, 2012
Supervised By
Prof. Wael Nabil Akl
Assoc. Prof. Adel Moneeb Elsabbagh
Cairo - (2017)
AIN SHAMS UNIVERSITY
FACULTY OF ENGINEERING
Wave propagation control using active acoustic
metamaterials
By
Ahmed Abdelshakour Abdelfattah Elhousseiny Allam
B.Sc., Mechanical Engineering, Mechatronics Section
Ain Shams University, 2012
EXAMINERS COMMITTEE
Name
Signature
Prof. Amr Mohamed Baz
Mechanical Engineering, University of Maryland
………………….
Prof. Amr Mohamed Ezzat Safwat
Electronics & Electrical Communication
Engineering, Ain Shams University
………………….
Prof. Wael Nabil Akl
Design and Production Engineering, Ain Shams
University
………………….
Date: 5/4/2017
Statement
This thesis is submitted as a partial fulfillment of Master of Science
in Mechanical Engineering, Faculty of Engineering, Ain shams Uni-
versity. The author carried out the work included in this thesis, and
no part of it has been submitted for a degree or a qualification at
any other scientific entity.
Ahmed Abdelshakour Abdelfattah Elhousseiny Allam
Signature
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Date: 05 04 2017
Researcher Data
Name: Ahmed Abdelshakour Abdelfattah Elhousseiny Allam
Date of Birth: 02/08/1990
Place of Birth: Giza, Egypt
Last academic degree: BSc. in mechanical Engineering
Field of specialization: Mechatronics Engineering
University issued the degree: Ain Shams University
Date of issued degree: 2012
Current job: Demonstrator at Mechatronics Department, Faculty of Engineering,
Ain Shams University
Summary
Acoustic metamaterials (AMM) are artificial materials with engineered sub-wavelength
structures that possess acoustic material properties which are not readily available in
nature. The material properties of AMMs can be manipulated by embedding active
elements inside their structure (active AMMs). This manipulation of properties is done
by an external voltage signal and is hardly available in any natural material.
In this work, existing designs for passive and active AMMs are reviewed and summarized.
Existing homogenization techniques for the material properties of passive metamaterials
are investigated and extended to be applied for active AMMs. Three new designs for
active plate-type AMM with tunable density are proposed and verified analytically,
numerically and experimentally.
The first design is a one dimensional (1D) AMM consisting of clamped piezoelectric
disks in air. The effective density of the material is controlled by an external static
electric voltage. An analytic model based on the acoustic two-port theory, the theory
of piezoelectricity and the pre-stressed thin plate theory is developed to predict the
behavior of the material. The results are verified using the finite element method.
Excellent agreement is found between both models for the studied frequency and voltage
ranges. The results show that the density is tunable within orders of magnitude relative
to the uncontrolled case. This is done with a limited effect on the bulk modulus of the
material. The results also suggest that simple controllers could be used to program the
material density.
The first design was modified and extended to construct a two-dimensional AMM with
controllable anisotropic density. The modified design consists of composite lead-lead
zirconate titanate (PZT) plates clamped to an aluminum structure with air as the back-
ground fluid. The effective anisotropic density of the material is controlled, indepen-
dently for two orthogonal directions, by means of an external static electric voltage
signal. An analytic model based on the acoustic two-port theory, the theory of piezo-
electricity, the laminated pre-stressed plate theory is developed to predict the behavior
of the material. The results are verified also using the finite element method. Excellent
agreement is found between both models for the studied frequency and voltage ranges.
The results show that, below 1600 Hz, the density is controllable within orders of mag-
nitude relative to the uncontrolled case. A reconfigurable wave guide was constructed
using the developed material and its performance was evaluated numerically and an-
alytically. The waveguide can control the direction of the acoustic waves propagating
through it.
x
The results obtained from the previous models were used to construct and experimentally
verify a third design with a fully real-time controllable effective density. The effective
density of the AMM can be programmed and set to any value ranging from -100 kg/m3to
100 kg/m3passing by near zero density conditions. This is done through an interactive
graphical user interface and is achievable for any frequency between 500 and 1500 Hz.
The modified design consists of clamped composite piezoelectric diaphragms suspended
in air. The dynamics of the diaphragms are controlled by connecting a closed feedback
control loop between the piezoelectric layers of the diaphragm. The density of the
material is adjustable through an outer adaptive feedback loop that is implemented by
the real-time estimation of the density of the material using the 4-microphone technique.
Applications for the new material include programmable active acoustic filters, non-
symmetric acoustic transmission and programmable acoustic superlens.
Keywords: Acoustic metamaterials, Piezoelectric materials, Feedback control,
Adaptive control
Acknowledgment
I would like to thank my supervisors Prof. Dr. Wael Akl and Dr. Adel Elasabbagh
for their devoted support throughout all the stages of this work and for providing me
with the tools and the working environment required for finishing it. I would like to
specially thank Prof. Dr. Wael for introducing the topic of the thesis to me and sharing
with me all his profound knowledge of it. I would like to thank Dr. Adel Elsabbagh for
interfering in the right moments whenever he felt that I have lost my thrust or went off
track.
I would like to thank Prof. Dr. Tamer Elnady for his support with the experimental
details especially those related to signal analysis and duct acoustics.
I would like to thank all my colleagues in the ASU group for advanced research in
dynamic systems (ASU-GARDS) and also my colleagues in mechatronics department, I
would specify Eng. Weam Elsahhar and Eng. Mohamed Talaat Harb for their valuable
help with the theoretical and experimental details of my work. I would like also to
thank Eng. Mohamed Ibrahim for providing me with the necessary background on
electromagnetic metamaterials, Eng. Ahmed Barakat for our fruitful discussions about
control systems, Eng. Ahmed Abosrea for helping me with the manufacturing of the cell
and Dr. Maaz Farouqi for our long discussions about metamaterials . I would like to
thank my friends and colleagues Eng. Yehia Zakaria, Eng. Ahmed Elrakaybi, Eng. Ali
Zein and Eng. Ahmed Hesham, you have always been there for me whenever I needed
any kind of support.
I also cannot forget the role of Mrs. Fatma. I appreciate her valuable support whenever
it came to any administrative tasks.
I would not have reached this point without the keen support of my family. I would
like to thank my mother and father for being there for me and for their continuous
encouragement to finish my work in the best way possible. I would also like to thank
my brother Mohamed and my sister Samaa for their support and for helping me with
the preparation of the data and thesis material.
Contents
Contents xii
List of Figures xv
List of Tables xix
Abbreviations xxi
1 Literature Review 1
1.1 Introduction to Acoustic metamaterials (AMM) . . . . . . . . . . . . . . . 1
1.2 SignInterpretation............................... 3
1.2.1 Single Negative Metamaterials (SNG) . . . . . . . . . . . . . . . . 5
1.2.2 Double Negative Metamaterials (DNG) . . . . . . . . . . . . . . . 5
1.3 ClassicationofAMM............................. 6
1.3.1 ResonantAMM............................. 6
1.3.1.1 Mass-in-mass AMM . . . . . . . . . . . . . . . . . . . . . 8
1.3.1.2 Acoustic resonator based AMM . . . . . . . . . . . . . . 9
1.3.1.3 Membrane/Plate-type AMM . . . . . . . . . . . . . . . . 9
1.3.1.4 DNG resonant AMM . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Non-resonant AMM . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2.1 Phononic crystal based AMM . . . . . . . . . . . . . . . 13
1.3.2.2 Space coiling AMM . . . . . . . . . . . . . . . . . . . . . 14
1.4 ApplicationsofAMM ............................. 15
1.4.1 Spatial Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1.1 Acoustic Cloaking . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1.2 Other spatial Devices . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Subwavelength acoustic imaging . . . . . . . . . . . . . . . . . . . 17
1.4.3 Perfect Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.4 Extraordinary Transmission . . . . . . . . . . . . . . . . . . . . . . 17
1.5 ActiveAMM .................................. 17
1.6 ProblemStatement............................... 20
1.7 WorkObjective................................. 20
1.8 ScopeofWork ................................. 20
1.9 ThesisSummary ................................ 22
2 Theoretical Background 23
2.1 Analytic modelling of AMM . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 VolumeAveraging ........................... 24
xiii
Table of Contents xiv
2.1.2 The multiple scattering theory . . . . . . . . . . . . . . . . . . . . 25
2.1.3 Acoustic two-port theory . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.4 Retrieval of the effective material properties . . . . . . . . . . . . . 28
2.2 Piezoelectricity ................................. 29
3 Open loop 1D active AMM 33
3.1 Theoretical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Characterizing the open loop AMM cell . . . . . . . . . . . . . . . 35
3.2 Theniteelementmodel............................ 39
3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Characterization of a single cell . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Characterization of multiple cells . . . . . . . . . . . . . . . . . . . 46
3.4 Conclusion ................................... 47
4 Open loop 2D AMM 49
4.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Characterizing the 2D active MAM cell . . . . . . . . . . . . . . . 51
4.1.2 Characterizing the 1D building block of the 2D active AMM . . . 52
4.2 NumericalModel................................ 56
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Applications................................... 63
4.5 Conclusion ................................... 64
5 Closed loop 1D AMM 67
5.1 Material Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.1 Acoustic impedance of the piezoelectric diaphragm . . . . . . . . . 68
5.3 StabilityoftheCell............................... 76
5.4 Characterization of the AMM cell . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Controller transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.6 Adaptive control of the cell density . . . . . . . . . . . . . . . . . . . . . . 81
5.7 Conclusion ................................... 84
6 Conclusion and Future Work 87
6.1 Conclusion ................................... 87
6.2 FutureWork .................................. 89
Bibliography 91
List of Figures
1.1 Construction of the first proposed metamaterials. (a) Thin wire structure
exhibiting negative ǫ/positive µ, (b)split ring resonators exhibiting neg-
ative µ/ positive ǫand (c) double split ring resonator exhibiting double
negativity. Taken from (Ref.[7]). . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Construction of the (a) unit cell and (b) structure of the first proposed
AMM consisting of silicon rubber coated lead balls in an epoxy matrix.
Takenfrom(Ref.[5])............................... 3
1.3 A simple mass spring damper system. . . . . . . . . . . . . . . . . . . . . 6
1.4 Bode plot of the effective mass of the system shown in Figure 1.3 with
M= 1, k = 1, b = 0.01.............................. 7
1.5 Comparison between resonators with (a) dipolar resonance and (b) monopo-
lar resonance . The neutral position is shown with the dotted line while
the resonator is represented by a block. The arrows represents the di-
rection motion of the block in case of the monopolar resonator and the
direction of deformation of the block in case of the dipolar resonator. . . . 8
1.6 Construction of the (a) unit cell and (b) structure of the first proposed
AMM to include Helmholtz resonators in its design. Taken from (Ref.[15]). 10
1.7 Construction of the (a) unit cell and (b)resonant modes of the decorated
membrane AMM introduced by Ma et al. Taken from (Ref.[32]). . . . . . 11
1.8 Construction of the composite structure to form first DNG AMM with (a)
negative density structure, (b)negative modulus structure, (c) composite
structure. Taken from (Ref.[37]). . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Fok and Zhang design for a DNG metamaterial showing (a) the spring
rod resonator, (b) the helmholtz resonator and (c) the construction of the
unit cell. Taken from (Ref.[39]). . . . . . . . . . . . . . . . . . . . . . . . . 13
1.10 AMM with anisotropic density suggested by Torrent and Sanchez-Dehesa.
Takenfrom(Ref.[54]). ............................. 14
1.11 The first space coiling AMM as (a) designed by Liang and Li[57] and (b)
fabricated and tested by Xie et al.[59] .................... 15
1.12 The first 2D acoustic cloak suggested by Cummer and Schurig[62]. . . . . 16
1.13 The first proposed active metamaterial with (a) controllable density and
(b) and controllable bulk modulus. Taken from (Ref.[75] and Ref.[77]) . . 18
1.14 (a) Construction and (b) feedback circuit of the first realized active AMM.
Takenfrom(Ref.[78]) ............................. 19
1.15 (a) Construction and (b) feedback circuit of the AMM cell capable of
non-reciprocal transmission as designed by Popa and Cumer. Taken from
(Ref.[87]) ................................... 20
xv
List of Figures xvi
2.1 Different relations between the wavelength and the feature size of the
medium it is propagating in. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Homogenization of AMM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Representation of two port networks using (a) the transfer matrix and
(b) the scattering matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Schematic of a layered AMM structure. . . . . . . . . . . . . . . . . . . . 27
2.5 The crystal structure of a piezoelectric material (a) before the poling
process and (b) after the poling process . . . . . . . . . . . . . . . . . . . 30
2.6 Piezolectric bimorph for the amplification of the displacement of piezo-
electricmaterial. ................................ 31
3.1 Construction of the suggested active open loop AMM cell. . . . . . . . . . 34
3.2 Material model of the suggested 1D active plate-type AMM. . . . . . . . . 34
3.3 Four microphone setup for estimating the reflection and transmission co-
efficients of an acoustic sample. . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Finite element mesh including the impedance tube and the open loop
AMM cell. The positions of the virtual microphones are indicated and
the open loop AMM cell is highlighted. . . . . . . . . . . . . . . . . . . . 41
3.5 Amplitude and phase of the complex pressure reflection coefficient with no
voltage applied to the piezoelectric disk (Black lines with circular mark-
ers) and subjected to a static voltage of 75V (Red lines with triangular
markers) and 150V (Blue lines with diamond markers). . . . . . . . . . . 42
3.6 Amplitude and phase of the complex pressure transmission coefficient
with different voltages applied to the piezoelectric disk V= 0,75,100V. . 42
3.7 Transmission loss calculated analytically (Solid) and using the FEM (Dashed)
with different voltages applied to the piezoelectric disk V= 0,75,100V. . 44
3.8 Real and imaginary components of the (a,b)effective density , (c,d) effec-
tive Bulk’s modulus and (e,f) effective speed of sound . With no voltage
applied to the piezoelectric disk (Black lines with circular markers) and
subjected to a static voltage of 75V (Red lines with triangular markers)
and 150V (Blue lines with diamond markers). . . . . . . . . . . . . . . . . 45
3.9 The effect of the applied voltage on the real component of (a) the effec-
tive density and (b) Bulk’s modulus calculated analytically (Solid) and
using the FEM (Dashed) at a constant frequency of 600Hz (Black lines
with circular markers), 1000Hz (Red lines with triangular markers) and
1400Hz (Blue lines with diamond markers). . . . . . . . . . . . . . . . . . 46
3.10 The effect of the variation of the number of samples on the effective density
of the material calculated analytically with no applied voltage to all cells.
In (a) a constant branch number m= 0 is used for all frequencies, while
in (b) it is chosen correctly using Kramers-Kronig relationship. . . . . . . 47
3.11 The effective density calculated from a sample consisting of seven cells
(N= 7) analytically (Solid) and using the FEM (Dashed) with different
voltages applied to the piezoelectric disk V= 0,75,100V. ......... 48
4.1 A new concept for a 2D active membrane-type metamaterial. (a) A visu-
alization for the construction of the suggested 2D AMM. (b) Schematic
representation for the 2D building block of the material. (c) Acoustic 2-
Port representation for the building block. (d) Schematic representation
of the construction of the 1D building block (1D AMM cell). . . . . . . . 52
List of Figures xvii
4.2 A cross section in the finite element mesh of the (a) 1D building block
sample with quarter symmetry placed in a square impedance tube, and (b)
2D AMM sample with half symmetry placed in a rectangular impedance
tube........................................ 58
4.3 The effect of applying different voltages on the amplitude of (a) Rand
(b) T. The analytic two-port values are compared to those obtained from
the FEM using one 1D cell in the incident wave propagation direction
(x-direction), as well as, one 2D cell with the voltage being varied on the
plates normal to the y-direction. . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 The analytic and numerical results for the real components of (a) the
effective density, (b) effective bulk modulus and (c) the effective speed of
sound estimated from (1, 4 and 7) 1D cells placed in the wave propagation
direction (x-direction), as well as, one 2D cell while varying the voltage
applied to the plates normal to the y-direction. . . . . . . . . . . . . . . 60
4.5 The TL estimated from a sample consisting of (a) only one 1D cell and
(b) four 1D cells in the propagation direction. The TL is calculated
analytically (Lines) and using the FEM (Markers) with different voltages
applied to the piezoelectric annulus V=0,150,300 V. . . . . . . . . . . . . 61
4.6 The effect of the applied voltage on the real component of the effec-
tive density, which is calculated analytically (Lines) and using the FEM
(Markers) for one 1D cell at three different frequencies namely 600,720
and800Hz.................................... 62
4.7 The developed 2-port network model for the reconfigurable waveguide.
The “Air” blocks indicate quarter cell sections terminated by the rigid
walls of the guide. Incident pressure is applied to the node donated Pin . 63
4.8 The normalized pressure inside the suggested waveguide for an incident
acoustic wave of frequency 727 Hz. The pressure is estimated using the
FEM (a) and (c) and the analytic network model (b) and (d). The incident
excitation and the propagation direction are marked with white arrows.
The incident wave is controlled to (a),(b) split between the two ducts and
(c),(d) exit from the upper duct only . . . . . . . . . . . . . . . . . . . . . 65
5.1 Schematic for the construction of the suggested 1D AMM unit cell. . . . . 69
5.2 An electrical circuit model for piezoelectric layer kconnected to an arbi-
trary circuit represented by its Thevenin’s equivalent . . . . . . . . . . . . 74
5.3 A block diagram representing the dynamics of the closed loop cell with
adaptivecontrol................................. 76
5.4 (a) Schematic for the test setup connections and the construction of the
AMM cell (b) Photo of the actual test setup. . . . . . . . . . . . . . . . . 78
5.5 The dispersion effective density of the developed AMM characterized an-
alytically and experimentally. The results obtained without any control
applied to the cell are compared to those obtained (a) using controller 1
(Equation (5.47)) with Kc=-1000 and Kc=2500 and (b) using controller
2 (Equation (5.48)) with fc=700 Hz and Kc= 4 ×106and controller 3
(Equation (5.49)) with fc=1300 Hz and Kc=-120............... 79
5.6 Flowchart for the procedure of the adaption of the cell density based on
theincidentexcitation.............................. 83
List of Figures xviii
5.7 Dispersion plot of the (a) effective density and (b) Bulk modulus of the
closed loop AMM cell with the adaptive control algorithm. Three different
set points for the controller are compared ρdesired=-100 kg/m3,ρdesired=0
kg/m3and ρdesired =100 kg/m3. The measured open loop effective density
is also plotted as a reference. . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.8 Step response of the closed loop AMM cell with the adaptive control
algorithm. The density set-point is initially set for zero density and later
stepped to 100 kg/m3. This is done for different excitation frequencies. . 85
List of Tables
3.1 PropertiesofPZT-5A.............................. 43
4.1 Summary of the dimensions of the 2D open loop active MAM cell. . . . . 50
4.2 Properties of the materials used in the construction of the 2D open loop
activeMAMcell................................. 50
5.1 Properties of the materials used in the construction of the AMM cell. . . 68
xix
Abbreviations
AMM Acoustic Metamaterial.
SNG Single Negative metamaterial.
DNG Double Negative metamaterial.
TL Transmission Loss
dB Decibel
FCC Face Centered Cube.
1D One Dimensional.
2D Two Dimensional.
3D Three Dimensional.
FEM The Finite Element Method.
DNZ Density Near Zero.
PZT Lead Zirconate Titanate.
MAM Membrane-type Acoustic Metamaterial.
xx
Chapter 1
Literature Review
1.1 Introduction to Acoustic metamaterials (AMM)
Metamaterials are the current focus of a lot of theoretical and experimental work in the
fields of electromagnetic, acoustic and elastic wave propagation. They are defined as
materials which possess material properties not readily available in nature. To be more
specific, the material properties here are those that directly affect the propagation of
energy waves inside the material. For electromagnetic waves, they are the permittivity
(ǫ) and the permeability (µ), for acoustic waves they are the mass density ρand the bulk
modulus B, and for elastic waves they are mass density and the elastic modulus (E).
For traditional materials, these properties are always positive and are usually isotropic;
however, in metamaterials and specifically from the point of view of wave propagation
they can have any sign combination. The origin of the concept of metamaterials dates
back to 1968 when Veselago[1] imagined the consequences of an electromagnetic material
having simultaneous negative permittivity and permeability. He discussed the unusual
phenomena such as reverse Doppler effect and reverse Snell’s law. At the time of his
publication Veselago admitted that there weren’t any experimental observations that
suggest that such material could exist; however, he discussed several approaches for
achieving this. 30 years later, Pendry and his colleagues succeeded in manufacturing
materials with negative ǫ/positive µ[2] and positive ǫ/ negative µ[3]. Later on, Smith
et al. succeeded in manufacturing the first double negative metamaterial (DNG)[4].
Pendry and Smith did not discover a physical material as Veselago had predicted, they
rather engineered structures with feature length much smaller than the wavelength of the
1
Chapter 1. Literature Review 2
waves propagating through them. These structures could be thought of as effectively
homogeneous materials with extraordinary material properties (metamaterials). The
structures of the first proposed materials are shown in Figure 1.1. Roughly the same
year, Liu et al. succeeded in manufacturing the first acoustic metamaterial (AMM)[5].
They used a sub-wavelength structure consisting of hard lead balls coated with soft
rubber (Figure 1.2). They claimed that their material could achieve a negative elastic
modulus due to the vibration of the lead balls in the rubber. They also demonstrated
that their material could break the mass law which states that the sound insulation of
ordinary materials, sound transmission loss (TL), increases by 6 decibels (dB) for each
doubling of the mass of the material or frequency[6].
(a)
(b)
p
y
(c)
x
y
z
Figure 1.1: Construction of the first proposed metamaterials. (a) Thin wire structure
exhibiting negative ǫ/positive µ, (b)split ring resonators exhibiting negative µ/ positive
ǫand (c) double split ring resonator exhibiting double negativity. Taken from (Ref.[7]).
Since then different structure designs for AMM have been suggested, studied and exper-
imentally validated. These designs can be classified into resonant AMM which depends
on the presence of subwavelength local resonators embedded in the structure of the
material, phononic crystal AMM which depends on the multiple scattering effects of
Chapter 1. Literature Review 3
subwavelength scatterers and space coiling AMM which depends on the effect of con-
structing a subwavelength maze-like structure to delay the acoustic waves propagating
through it.
Figure 1.2: Construction of the (a) unit cell and (b) structure of the first proposed
AMM consisting of silicon rubber coated lead balls in an epoxy matrix. Taken from
(Ref.[5]).
1.2 Sign Interpretation
Considering an acoustic pressure wave traveling through a homogeneous loss-less sta-
tionary fluid, the behavior of the wave could be described by the linearized equations of
conservation of mass and conservation of momentum which are given by:
ρ¯u
∂t +p= 0 (1.1a)
1
B
p
∂t +.¯u = 0 (1.1b)
where pis the acoustic pressure, ¯u is the acoustic particle velocity vector, ρ, B are
the density and compressibility of the fluid, is the gradient operator and (.) is the
divergence operator. If we are to consider harmonic fields with time dependence, ej ωt
then we could write:
¯u(¯r, t) = ¯u(¯r)ejωt (1.2a)
pr, t) = p(¯r)ejωt (1.2b)
Chapter 1. Literature Review 4
Substituting in Equation 1.1 and rearranging we arrive to:
¯u(¯r) = j
ωρ pr) (1.3a)
pr) = jB
ω.¯u(¯r) (1.3b)
For simplicity without loss of generality we will consider the previous equations in one
dimension (z) hence:
u=j1
ωρ
dp
dz (1.4a)
p=jB
ω
du
dz (1.4b)
These equations show the mechanism behind acoustic wave propagation in fluids. For
traditional materials (ρ, B) are both positive. Hence Spatial pressure gradients in Equa-
tion 1.4a induce velocity fields whose spatial gradients in turn produces pressure fields in
Equation 1.4b. The mechanism repeats as long as the wave propagates in the medium.
If we take the divergence of Equation 1.3a and substitute into Equation 1.3b we arrive
to the famous Helmholtz wave equation:
2p+ω2
c2p= 0 (1.5)
where cis the speed of sound propagation in the fluid medium and is given by:
c2=B
ρ, c =±sB
ρ(1.6)
For a plane wave traveling in zdirection, the solution of Equation 1.5 is well known and
is given by:
p(z) = Aejkz +Bejkz (1.7)
which represents two waves traveling in the positive and negative directions of zwith a
wave number kgiven by:
k=ω
c(1.8)
For a traditional medium (ρ, B) are both positive and frequency independent hence c
is positive and constant and kis always positive. Metamaterials on the other hand
Chapter 1. Literature Review 5
can be synthesized to have effective density ρeff and bulk’s modulus Beff with dif-
ferent sign combinations at specific frequency bands. The possible sign combinations
for (ρ, B) is Traditional Materials (TM)(+,+), Single Negative Metamaterials (SNG)
(+,)or(,+) and Double Negative Metamaterials (DNG) (,). The behavior of
traditional materials is well known so we will examine the other sign combinations.
1.2.1 Single Negative Metamaterials (SNG)
The behavior of SNG is almost the same for whether (ρ, B) are (,+) or (+,). In
either case, the speed of sound in the medium, Equation 1.6, is imaginary and hence the
wave number (k). The presence of an imaginary wave number in Equation 1.7 produces
real exponentials and hence evanescent wave propagation. This leads to the presence of
band gaps in the SNG in which the acoustic waves cannot propagate. The mechanism
behind this phenomenon is clear in Equation 1.4. Pressure gradients induces particle
velocities whose direction depends on the magnitude and sign of (ρ, B ) if we assume
(ρ < 0, B > 0) then the induced particle velocities will induce pressure fields such that
they oppose the fields producing them and hence the wave propagation decays.
1.2.2 Double Negative Metamaterials (DNG)
In case of DNG the situation is different. Since (ρ, B) are both negative then the speed
of sound cis real but negative. The negative sign here is with respect to the direction
of energy propagation. For traditional non-dispersive materials, crepresents the phase
speed of acoustic waves as well as their group velocity (i.e. speed of propagation in the
medium). The group velocity for DNG is given by:
vg= ( dk
)1=d
(ω
c(ω)) (1.9)
Since cis function of ω, then vgis no longer equal to cand cis no longer representing
the speed of propagation of the acoustic wave in the medium. The fact that cis negative
doesn’t mean that the waves are propagating towards the source rather than the direction
of the phase propagation is opposite to that of energy propagation.
Chapter 1. Literature Review 6
1.3 Classification of AMM
1.3.1 Resonant AMM
Resonant AMMs were the first type of AMMs to be realized [5]. They are also the most
common type of AMM. Resonant AMM are constructed by creating a metamaterial
cell with one or more resonators. These resonators could be mechanical resonators in
the form of vibrating elastic objects or acoustic resonators like Helmholtz resonators or
quarter wavelength resonators. The material is formed by repeating this unit cell in one
or more dimensions. Its properties are studied as if it was a single homogeneous material.
The presence of such resonators could produce materials whose acoustic properties are
very different from the properties of its individual components. To demonstrate this,
we will study a material consisting of a mass (M)- spring (k)- damper (b) resonator.
Assuming an incident acoustic pressure wave on the one dimensional (1D) mechanical
resonator shown in Figure 1.3, the acoustic pressure waves will induce a harmonic force
f(t) acting on the system. The vibration of this resonator is of the form:
M¨x+b˙x+kx =f(t) (1.10)
Assuming that the dimensions of the resonator is so small that it is considered as a
b
k
M
f(t)
x
Figure 1.3: A simple mass spring damper system.
material with only an effective mass Meff . This effective mass can be calculated from
the relation m=f
a. This can be done by converting Equation (1.10) to the frequency
Chapter 1. Literature Review 7
domain using Laplace transform:
F(s)
A(s)=Ms2+bs +k
s2=Mef f (s) (1.11)
Equation (1.11) shows that the effective mass of the material formed from this resonator
is not constant, but depends on the frequency of the incident acoustic excitation. Fig-
ure 1.4 shows a Bode plot of a material formed from such resonator. Below the resonance
frequency of the resonator, the phase of Mef f is 180 deg which indicates that the effec-
tive mass is, in fact, negative. This is observed, even though all the system properties of
the resonator are positive and can be achieved with any ordinary material organized to
behave as the resonator in Figure 1.4. Resonant AMM can be further classified depend-
Figure 1.4: Bode plot of the effective mass of the system shown in Figure 1.3 with
M= 1, k = 1, b = 0.01.
ing on the resonator type used to create the material. Resonators can be classified to
monopolar and dipolar resonators depending on their interaction with the propagating
acoustic waves. The classification of dipolar and monopolar resonators depends on the
interaction of the resonator with the background fluid as shown in Figure 1.5. It is
observed that the presence of a dipolar resonator in the structure of the material affects
mainly its effective density, while monopolar resonators affect its bulk modulus. So usu-
ally, if it is desired to fabricate DNG metamaterial, both types of resonators are used in
the construction of the material at the same time.
Chapter 1. Literature Review 8
1D Dipolar resonance 1D Monopolar resonance
(a) (b)
Figure 1.5: Comparison between resonators with (a) dipolar resonance and (b)
monopolar resonance . The neutral position is shown with the dotted line while the
resonator is represented by a block. The arrows represents the direction motion of the
block in case of the monopolar resonator and the direction of deformation of the block
in case of the dipolar resonator.
1.3.1.1 Mass-in-mass AMM
The term mass-in-mass AMM normally refers to the type of AMMs formed by construct-
ing an array of cells consisting of a combination of light material with high elasticity
and dense materials with low elasticity. This was first demonstrated by Liu et al.[5],
when they fabricated the first AMM. They constructed their material from dense lead
balls coated with a silicon rubber shell, and used epoxy as a hard background material
to join the balls. The experimental estimation of the TL of such material was found
to be exceptionally large at frequencies near the resonance of the lead/rubber balls.
They attributed this behavior to the assumption that their material achieved a neg-
ative effective bulk modulus; however, they didn’t provide any theoretical details or
quantitative estimation of the effective acoustic properties of their material. It wasn’t
until later, that they showed analytically that a material with such construction could
only achieve negative effective density but not negative effective bulk modulus as they
originally speculated[8, 9]. Ding et al.[10] proposed a design for a DNG metamaterial.
The design consisted of an epoxy background matrix that contains a face centered cube
(FCC) array of spheres made of water and another FCC array made from gold rub-
ber coated spheres. The gold rubber coated spheres, being a dipolar resonator, would
Chapter 1. Literature Review 9
achieve negative effective density, while the water spheres, being a monopolar resonator,
would cause the bulk modulus to be negative at certain frequency regions. The concept
of mass-in-mass in AMM was later extended to elastic wave propagation[11–13]. This
includes experimentally evaluating physical mass-spring systems to further analyze the
mechanism of wave propagation in materials with negative and zero mass densities[14].
1.3.1.2 Acoustic resonator based AMM
An acoustic resonator is usually incorporated in the design of this type of material. Fang
et al.[15] succeeded in fabricating the first AMM with negative bulk modulus using arrays
of Helmholtz resonators placed in a 1D waveguide. As shown in Figure 1.6, Helmhotz
resonators are acoustic cavities with a small neck opening. The air in the neck acts as
an oscillating mass, while the air in the cavity acts as a spring. Helmholtz resonators
are usually connected parallel to a waveguide; thus, they normally act as monopolar
resonators which affects mainly the effective bulk modulus of the material. The AMM
developed by Fang could achieve a negative bulk modulus in the ultrasonic frequency
range around 30 kHz. Different configurations and sizes with similar structures have been
later studied. Cheng et al.[16] studied the effect of the number of Helmhotz resonators
on the bandgap caused by the material. Hu et al.[17] extended the theoretical analysis
to 2D and 3D arrays of Helmhotz resonators. Lee et al.[18] suggested and experimentally
verified the first AMM to achieve negative bulk modulus in the audible frequency range
below 500 Hz. The structure of their AMM consisted of a 1D waveguide with side slits
instead of Helmholtz resonators. Zhang et al.[19] used 2D arrays of Helmholtz resonators
to construct a flat ultrasonic acoustic lens. Fey et al.[20] used an array of 1D detuned
Helmholtz resonators to achieve a series of acoustic band gaps. Lemoult et al.[21] used a
2D array of Helmhotz resonators made from soda cans to focus audible acoustic waves in
air. They showed later that such material could be organized arbitrarily to control the
propagation of sound in air, including the construction of subwavelength waveguides[22].
1.3.1.3 Membrane/Plate-type AMM
Elastic membranes and plates have been used extensively in the manufacturing reso-
nant AMM with various configurations[23]. The dipolar resonance associated with the
vibration of membranes facilitated the fabrication of AMM with negative dynamic mass