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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 ClassiﬁcationofAMM............................. 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 Theﬁniteelementmodel............................ 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)

p

z

(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)

p(¯r, 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 ∇p(¯r) (1.3a)

p(¯r) = 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) = Ae−jkz +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

dω )−1=d

dω (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