Wright Derek W 201006 PhD Thesis

TIME-VARYING PHONONIC CRYSTALS

by

Derek Warren Wright

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

Collaborative Program with

The Institute for Biomaterials and Biomedical Engineering

University of Toronto

Copyright © 2010 by Derek Warren Wright

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ABSTRACT

Time-Varying Phononic Crystals

Derek Warren Wright, PhD, 2010

Edward S. Rogers Sr. Department of Electrical and Computer Engineering,

Institute for Biomaterials and Biomedical Engineering,

University of Toronto

Toronto, ON, Canada

The primary objective of this thesis was to gain a deeper understanding of acoustic wave

propagation in phononic crystals, particularly those that include materials whose properties can

be varied periodically in time. This research was accomplished in three ways.

First, a 2D phononic crystal was designed, created, and characterized. Its properties closely

matched those determined through simulation. The crystal demonstrated band gaps, dispersion,

and negative refraction. It served as a means of elucidating the practicalities of phononic crystal

design and construction and as a physical verification of their more interesting properties.

Next, the transmission matrix method for analyzing 1D phononic crystals was extended to

include the effects of time-varying material parameters. The method was then used to provide a

closed-form solution for the case of periodically time-varying material parameters. Some

intriguing results from the use of the extended method include dramatically altered transmission

properties and parametric amplification. New insights can be gained from the governing

equations and have helped to identify the conditions that lead to parametric amplification in these

structures.

Finally, 2D multiple scattering theory was modified to analyze scatterers with time-varying

material parameters. It is shown to be highly compatible with existing multiple scattering

theories. It allows the total scattered field from a 2D time-varying phononic crystal to be

determined.

It was shown that time-varying material parameters significantly affect the phononic crystal

transmission spectrum, and this was used to switch an incident monochromatic wave. Parametric

amplification can occur under certain circumstances, and this effect was investigated using the

closed-form solutions provided by the new 1D method.

Abstract iii

The complexity of the extended methods grows logarithmically as opposed linearly with

existing methods, resulting in superior computational complexity for large numbers of scatterers.

Also, since both extended methods provide analytic solutions, they may give further insights into

the factors that govern the behaviour of time-varying phononic crystals. These extended methods

may now be used to design an active phononic crystal that could demonstrate new or enhanced

properties.

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ACKNOWLEDGMENTS

First and foremost, I wish to express my most sincere gratitude to Prof. Richard Cobbold, my

primary thesis supervisor. He is astonishingly generous with his time, experience, guidance, and

friendship. Throughout this program he has become a role model for me both personally and

professionally, and I look forward to our continued rapport.

I would also like to thank Dr. Wayne Johnston, my thesis co-supervisor. He has provided

the valuable service of maintaining a big-picture view of my research. He has also given me

many helpful tips and suggestions on improving my presentation skills. I have thoroughly

enjoyed our conversations on a variety of unrelated topics and his great sense of humour.

Prof. Mohammad Mojahedi and Prof. Adrian Nachman have been kind enough to serve

on my thesis committee. They have routinely provided valuable insight into numerous aspects of

my research, and their advice has greatly contributed to the quality of my PhD experience.

Over the past four years, the members of Prof. Cobbold’s ultrasound group have become

friends in addition to colleagues. My rides to and from Toronto with Muris Mujagić and our

tennis, ping pong, and workout sessions became the highlights of my workweek. Alfred Yu,

Reneé Warriner, and Alexia Giannoula helped me get settled into the group and were great

technical resources. They have acted as my sounding board on countless occasions and I am very

grateful for our continued friendships and camaraderie.

The team lead by Prof. Peter Burns at Sunnybrook has been extremely generous with their

time, equipment and expertise. In particular, Ross Williams and Ahthavan Sureshkumar spent

many hours helping me to characterize our phononic crystal. I learned a great deal from both of

them, and profited from their extensive practical experience.

Mark Wheeler, a fellow graduate student working with Prof. Mojahedi, was kind enough to

answer numerous questions from me that I’m sure were quite novice from his perspective. His

time and patience helped me get up to speed much faster than I would have been able to achieve

on my own. I am extremely grateful for his assistance and guidance.

Dr. Howard Ginsberg, who also co-supervises students in the ultrasound group, has been a

friend and strong advocate for my career. He introduced me to neurosurgery and allowed me to

see firsthand how ultrasound devices might improve a surgeon’s experience in the operating

room. I am also indebted to him for facilitating so many introductions to colleagues in the field

of ultrasound research.

Acknowledgments v

I wish to extend my thanks to my uncle and aunt, Neil and Judy Wright, with whom I lived

while in town. They provided me with a place to live and a sense of home and family that made

living in two cities possible and comfortable. I will always feel at home in 3B. I would also like

to thank my brother, David Wright, and his fiancée, Laura Tracey, and my brother and sister-

in-law, Frans and Natalie LeRoij. Their involvement in my Toronto life helped me balance the

stresses of the PhD program with fun and relaxation, and I shall forever be grateful for the

enjoyable times we had together. My parents, Warren and Annette Wright, and my wife’s

parents, Frans and Christy LeRoij, have always had my best interests in mind, and as such

have been supportive in countless ways throughout this experience.

I would like to acknowledge the scholarship support I received from an Ontario Graduate

Scholarship in Science and Technology, the Rogers Fellowship award, and the continual support

from Prof. Cobbold’s Natural Sciences and Engineering Research Council of Canada grant. This

support has been extremely generous and is much appreciated.

Finally, and most importantly, I wish to dedicate this thesis to my wife, Amy. Five years ago

she suggested that I might do well in a PhD program and that I should consider applying. This

was in spite of the personal inconveniences it would cause us for the duration of the program.

She has been entirely supportive and encouraging, and I truly would not have achieved this goal,

nor would my life be pointed in such a satisfying and fulfilling direction were it not for her love

and support. She is my lab partner for life – thank you. I would also like to thank our son,

William, for fitting so nicely in my arms as I typed this thesis.

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TABLE OF CONTENTS

Abstract .................................................................................................................................... ii

Acknowledgments.................................................................................................................... iv

List of Abbreviations and Symbols ......................................................................................... ix

List of Tables .......................................................................................................................... xv

List of Figures ........................................................................................................................ xvi

Chapter 1 Introduction ......................................................................................................... 1

1.1 Periodic materials ......................................................................................................... 2

1.1.1 Phononic crystals .................................................................................................. 3

1.1.2 Local resonance .................................................................................................... 5

1.2 Motivation.................................................................................................................... 6

1.2.1 Aims and scope ..................................................................................................... 7

1.3 Previous work .............................................................................................................. 8

1.3.1 Nonlinear photonics .............................................................................................. 9

1.3.2 Time-varying systems theory .............................................................................. 10

1.3.3 Active materials .................................................................................................. 11

1.4 Organization of thesis ................................................................................................. 13

Chapter 2 Static Phononic Crystals in One Dimension ..................................................... 14

2.1 Acoustic waves in one dimension ............................................................................... 14

2.2 Acoustic wave transmission and reflection in 1D ........................................................ 17

2.3 Periodic structures ...................................................................................................... 18

2.3.1 Band gaps ........................................................................................................... 18

2.3.2 Dispersion ........................................................................................................... 25

2.3.3 Corrugated tube waveguides................................................................................ 28

2.3.4 Anomalous Doppler effects ................................................................................. 29

2.3.5 Applications ........................................................................................................ 31

Chapter 3 Static Phononic Crystals in Two and Three Dimensions ................................. 32

3.1 Phononic crystal theory and effects ............................................................................ 32

3.1.1 Acoustic waves in two and three dimensions ....................................................... 32

3.1.2 Band gaps and waveguides .................................................................................. 34

3.1.3 Dispersion and refraction .................................................................................... 36

3.2 Phononic crystal design and experimental results ....................................................... 42

3.2.1 Crystal design ..................................................................................................... 42

3.2.2 Experimental methods ......................................................................................... 48

3.2.3 Results ................................................................................................................ 48

3.2.4 Summary ............................................................................................................. 53

Chapter 4 Simulation Methods .......................................................................................... 54

4.1 The finite-difference time-domain method .................................................................. 54

4.1.1 Derivation of a 2D algorithm ............................................................................... 56

4.1.2 Implementation ................................................................................................... 59

Table of Contents vii

4.1.3 Stability criteria ................................................................................................... 68

4.2 The transmission matrix method ................................................................................. 75

4.2.1 Example calculation ............................................................................................ 77

4.3 Multiple scattering theory ........................................................................................... 78

4.3.1 Scattering from one cylinder ............................................................................... 79

4.3.2 Scattering from two cylinders .............................................................................. 83

4.3.3 Scattering from N cylinders ................................................................................. 87

4.4 The plane wave expansion method ............................................................................. 88

4.4.1 Derivation of the plane wave expansion method in 3D ........................................ 89

4.4.2 Implementation details ........................................................................................ 94

4.5 Other simulation methods ........................................................................................... 95

4.6 Amenability to time-varying material parameters ....................................................... 96

Chapter 5 Dynamic Phononic Crystals in One Dimension ............................................... 99

5.1 The time-varying transmission matrix method ............................................................ 99

5.1.1 Transmission through one cell ........................................................................... 100

5.1.2 Time-varying phase velocity ............................................................................. 103

5.1.3 Transmission through multiple cells .................................................................. 105

5.1.4 Conversion back into a transmission spectrum................................................... 106

5.2 Solutions when the parameter variation is periodic ................................................... 106

5.2.1 Single-cell solution............................................................................................ 107

5.2.2 An n-cell solution .............................................................................................. 108

5.2.3 The discrete transmission spectrum ................................................................... 108

5.3 Closed-form solutions .............................................................................................. 110

5.3.1 A closed-form expression for one cell ............................................................... 111

5.3.2 A closed-form solution for two and n cells ........................................................ 115

5.4 Methodology and verification ................................................................................... 117

5.4.1 Computation method ......................................................................................... 117

5.4.2 The complexity of calculation ........................................................................... 118

5.4.3 Verification ....................................................................................................... 119

5.5 Results ..................................................................................................................... 121

5.5.1 Parametric amplification ................................................................................... 122

5.5.2 Signal switching ................................................................................................ 125

5.6 Summary and concluding discussion ........................................................................ 127

Chapter 6 Dynamic Phononic Crystals in Two Dimensions ........................................... 128

6.1 Time-varying multiple scattering theory ................................................................... 128

6.1.1 Scattering from one time-varying cylinder ......................................................... 128

6.1.2 Scattering from N periodically time-varying cylinders ....................................... 138

6.2 Verification .............................................................................................................. 139

6.2.1 One static cylinder ............................................................................................ 141

6.2.2 Several static cylinders ...................................................................................... 142

6.2.3 One periodically time-varying cylinder ............................................................. 142

6.2.4 Several periodically time-varying cylinders ....................................................... 145

6.2.5 The modified fundamental frequency field ........................................................ 148

6.2.6 Application to a time-varying phononic crystal ................................................. 148

6.3 Summary and concluding discussion ........................................................................ 150

Chapter 7 Summary and Conclusions ............................................................................. 152

7.1 Summary .................................................................................................................. 152

Table of Contents viii

7.1.1 Primary assumptions ......................................................................................... 152

7.2 Thesis contributions ................................................................................................. 153

7.2.1 Specific developments ....................................................................................... 153

7.2.2 Primary research contributions .......................................................................... 154

7.3 Suggestions for further work .................................................................................... 155

7.3.1 Analytical developments ................................................................................... 155

7.3.2 Implementation improvements .......................................................................... 157

7.3.3 Experimental work ............................................................................................ 158

7.3.4 Applications research ........................................................................................ 158

7.4 Concluding remarks ................................................................................................. 159

References ............................................................................................................................. 160

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LIST OF ABBREVIATIONS AND SYMBOLS

Abbreviations

1D One dimensional

2D Two dimensional

3D Three dimensional

AFC Active fiber composite

AO Acousto-optic

CPU Central processing unit

CW Continuous wave

DC Direct current

DFT Discrete Fourier transform

EAP Electroactive polymer

FDTD Finite-difference time-domain

FFT Fast Fourier transform

GPU Graphics processing unit

LHS Left-hand side

LTI Linear time-invariant

MFC Macro fiber composite

MIMO Multi-input multi-output

MST Multiple scattering theory

NR Negatively refracting

PML Perfectly matched layer

PWE Plane wave expansion

PZT Lead zirconate titanate

RHS Right-hand side

RLV Reciprocal lattice vector

SAW Surface acoustic wave

SNR Signal to noise ratio

TMM Transmission matrix method

TV-MST Time-varying multiple scattering theory

TV-TMM Time-varying transmission matrix method

List of Abbreviations and Symbols x

Roman-Based Symbols

A Matrix of coefficients in the PWE method

A PML coefficient matrix

A State transition matrix

A Vector of forward and reverse displacement wave amplitudes in the frequency domain

A Vector of Fourier coefficients of the inverse of a periodic time-domain function

A Cross-sectional area of a tube

A+, A-

Forward and reverse displacement wave amplitudes in the frequency domain

a Vector of forward and reverse displacement wave amplitudes

ai The ith vector of a phononic crystal lattice

a Cell spacing

a+, a- Forward and reverse displacement wave amplitudes

B PML coefficient matrix

B Function required in determining the scattering of a time-varying cylinder

b Vector of internal field coefficients

bi Coordinates of the centre of scatterer i

b+, b- Forward and reverse displacement wave amplitudes

bm Internal field coefficient of the mth order

Cn, Cm Matrices of Fourier coefficients

C Complexity of calculation

Cij Stiffness coefficient ij

c Elastic stiffness tensor

c Vector of scattered field coefficients

cg Vector group velocity

c0, cφ Phase velocity

cg Group velocity magnitude

cijmn Elements of the elastic stiffness tensor

cm Scattered field coefficient of the mth order

D Inverse of a periodic time-domain function

Di The ith Fourier coefficient of the inverse of a periodic time-domain function

d Vector of incident field coefficients

d Diameter

List of Abbreviations and Symbols xi

d Distance

d1, d2 PML damping functions

dm Incident field coefficient of the mth order

E1, E2 PML coefficient matrices

F Vector of Fourier coefficients of a periodic time-domain function

Fn Force at position n

f Frequency

fp Pumping frequency

fRx Received frequency

fTx Transmitted frequency

G Matrix required to find the Fourier coefficients of the inverse of a periodic time-domain function

G Reciprocal lattice vector

Hm Cylindrical Hankel function of the first kind and mth order

I Identity matrix

Iinc Incident field intensity

Ip Pump field intensity

i An integer

Jm Cylindrical Bessel function of the first kind and mth order

j 1−

j An integer

K Bulk modulus

k Wavevector

k An integer

k Spring constant

k Wave number

l An integer

L Length through a phononic crystal

M A point in the first Brillouin zone

M Inverse of a periodic time-domain function

M Maximum FDTD index in the x-direction

M Maximum order of the cylindrical wavefunctions considered

M, Mx, My Variables in the stability analysis of an FDTD simulation

Mi The ith Fourier coefficient of the inverse of a periodic time-domain function

List of Abbreviations and Symbols xii

m An integer

m Mass

N Maximum FDTD index in the y-direction

N Number of cylinders in a multiple scattering configuration

N, Nx, Ny Variables in the stability analysis of an FDTD simulation

Nt Number of data points per recorded waveform

Nx, Ny, Nz Number of points in the x, y, or z dimensions

n An integer

P Inverse of the time-varying density

P Momentum

P, Px, Py Variable in the stability analysis of an FDTD simulation

p An integer

q An integer

R Radius

R Reflection coefficient

r, r0, etc. Position vectors

r An integer

S Scattering matrix

S, S Separation matrices

S Cross-sectional area

Sij The ijth entry of a scattering matrix

Sij The ijth entry of a separation matrix

ijmnS The ijmn

th element of a time-varying separation matrix

Smn Stress in the mth dimension due to displacement in the nth dimension

Sn21 The n-cell transmission coefficient

s An integer

T Transition matrix

T Transmission matrix

TC Cell transmission matrix

TP Propagation transmission matrix

T Transmission coefficient

T0 Wave transit time through a material with a time-varying phase velocity

Tij The ijth entry of a transmission or matrix

Tmn The mnth element of a transition matrix

List of Abbreviations and Symbols xiii

ijmnT The ijmn

th element of a time-varying transition matrix

t Time

U Matrix of eigenvectors in the PWE method

u Acoustic particle vector displacement field

ul Longitudinal acoustic particle vector displacement field

ut Transverse acoustic particle vector displacement field

u, U Acoustic particle displacement

u, U Scalar velocity potential fields

um, Um Acoustic particle displacement in the mth dimension

Un Acoustic particle displacement at position n

Vm Matrix required in determining the scattering of a time-varying cylinder

V Volume

v PML state vector

v Velocity potential inside a penetrable cylindrical scatterer

vR Velocity of a reflector

Wm Matrix or vector required in determining the scattering of a time-varying cylinder

w1, w2 PML state vector

X A point in the first Brillouin zone

X Total FDTD simulation distance in the x-direction

x Position vector

xn Position in the nth dimension

Y Total FDTD simulation distance in the y-direction

Z Characteristic acoustic impedance

Z The set of all integers

Greek-Based Symbols

α Decay rate of the Fourier coefficients of the inverse of a periodic time-domain function

α01, α02 PML damping coefficients

β Variable in the stability analysis of an FDTD simulation

βm Coefficient in the T-matrix

Γ A point in the first Brillouin zone

∆t Resolution in the time-domain

∆x, ∆y, ∆z Resolution in the x, y, or z dimensions

List of Abbreviations and Symbols xiv

δ( ) Dirac delta function

δmn Kronecker delta

η Filling fraction

θ An angle

θin, θout Phase entering and exiting a region

κ Adiabatic compressibility

ΛΛΛΛ Matrix of eigenvalues

λ Wavelength

λ Eigenvalue

λ First Lamé coefficient

µ Second Lamé coefficient

µ Linear mass density

ρ Density

σσσσ Stress tensor

φ Scalar velocity potential

φ Phase angle

mψ , mψ Outgoing and regular cylindrical wavefunctions of order m

ψ , ψ Vectors of outgoing and regular cylindrical wavefunctions

ω An angular frequency

ωp Pumping angular frequency

Miscellaneous Symbols, Functions, and Abbreviations

Hamming distance

Floor operation

{ }ℑ Fourier transform

{ }1−ℑ Inverse Fourier transform

~ Overscore tilde denotes variables related to numerical dispersion

^ Overscore caret denotes a unit vector

* Convolution or complex conjugation

− Bar overscore denotes complex conjugation T Superscript T denotes the matrix transform

- xv -

LIST OF TABLES

Table 3.1. Transducer pair parameters ...................................................................................... 44

Table 3.2. Host and scatterer properties .................................................................................... 45

Table 3.3. Crystal lattice parameters ......................................................................................... 46

Table 3.4. Recorded acoustic field parameters .......................................................................... 51

Table 4.1. Calculable simulation regions and periodic boundaries ............................................ 62

Table 4.2. Continuous PML absorbing boundary equations ...................................................... 64

Table 4.3. Discrete PML absorbing boundary equations ........................................................... 65

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LIST OF FIGURES

Figure 1.1. The operating regime of periodic materials ............................................................... 2

Figure 1.2. One, two, and three dimensional phononic crystals ................................................... 3

Figure 1.3. A two dimensional phononic crystal ......................................................................... 4

Figure 1.4. The scattered field of two acoustically-hard cylinders ............................................... 4

Figure 1.5. An illustration of the role of resonance in metamaterials ........................................... 5

Figure 1.6. Two typical configurations of a nonlinear photonic crystal ..................................... 10

Figure 2.1. a) A fluid-filled rigid tube and b) its approximation as a linear chain of masses and springs ...................................................................................................................................... 15

Figure 2.2. Incident, transmitted and reflected waves at an interface ......................................... 17

Figure 2.3. The nomenclature for a two-component one dimensional phononic crystal ............. 18

Figure 2.4. Constructive and destructive interference of transmitted and reflected waves .......... 19

Figure 2.5. Simulated band gaps typical of a 1D phononic crystal............................................. 19

Figure 2.6. Band gap characteristics for various impedance ratios............................................. 20

Figure 2.7. The transmission spectra for various filling fractions (η) ........................................ 21

Figure 2.8. The transmission spectra for various phase velocity ratios ...................................... 22

Figure 2.9. The transmission spectrum for longitudinal waves in an aluminum-nickel phononic crystal ...................................................................................................................................... 22

Figure 2.10. Illustration of tunneling through a single barrier.................................................... 23

Figure 2.11. Illustration of modes allowed in an infinite double barrier ..................................... 24

Figure 2.12. The dispersion relation a) and group velocity b) for a simple periodic system ....... 26

Figure 2.13. The dispersion curves for an aluminum-nickel phononic crystal............................ 27

Figure 2.14. A corrugated tube waveguide ................................................................................ 28

Figure 2.15. A segment of a 1D periodic acoustic waveguide containing two sections .............. 29

Figure 2.16. Regular a) and anomalous b) Doppler effects ........................................................ 30

List of Figures xvii

Figure 3.1. Elastic wave propagation in two dimensions ........................................................... 33

Figure 3.2. Acoustic wave scattering off an infinitely dense cylinder in two dimensions ........... 34

Figure 3.3. The wavevectors at the interface between two isotropic media ................................ 37

Figure 3.4. The equifrequency curves used to calculate refraction at an interface ...................... 38

Figure 3.5. Two sets of equifrequency and corresponding dispersion curves ............................. 39

Figure 3.6. The dispersion curves for a 2D phononic crystal ..................................................... 40

Figure 3.7. Wave refraction under normal and negatively refracting conditions ........................ 41

Figure 3.8. An illustration of refocusing in an NR slab ............................................................. 41

Figure 3.9. The frequency response for three sets of transducer pairs ........................................ 43

Figure 3.10. Top view of the phononic crystal design ............................................................... 47

Figure 3.11. Top a) and side b) view of the fabricated phononic crystal .................................... 47

Figure 3.12. Experimental configuration ................................................................................... 48

Figure 3.13. Measured and simulated transmission spectrum of the phononic crystal ................ 49

Figure 3.14. Measured and simulated dispersion curves of the phononic crystal ....................... 50

Figure 3.15. The total and gate-only signals .............................................................................. 52

Figure 3.16. A diagram of the experimental configuration a), and the measured acoustic field .. 53

Figure 4.1. The offset grid implementation of the FDTD state variables ................................... 60

Figure 4.2. FDTD algorithm flow for periodic boundaries ........................................................ 62

Figure 4.3. Nomenclature used to describe displacement waves at the boundaries of a 1D scatterer .................................................................................................................................... 76

Figure 4.4. Calculated transmission spectrum for a 1D tube waveguide .................................... 78

Figure 4.5. The construction of the first irreducible Brillouin zone for a rectangular lattice ....... 94

Figure 5.1. A monotone is scattered and modulated at every time-varying interface................ 101

Figure 5.2. A monotone is modulated as it propagates through a time-varying slab of material 101

Figure 5.3. The time-varying cell transmission matrix ............................................................ 106

Figure 5.4. A flowchart illustrating the steps and equations necessary to simulate wave propagation through a time-varying tube waveguide ............................................................... 117

List of Figures xviii

Figure 5.5. A plot of the number of integrations required for n time-varying cells .................. 119

Figure 5.6. Comparison of transmission spectra using the new theory and a FDTD method .... 120

Figure 5.7. Fundamental frequency transmission coefficients for ten static and time-varying cells ............................................................................................................................................... 121

Figure 5.8. Another fundamental frequency transmission spectrum for ten static and time-varying cells ........................................................................................................................................ 122

Figure 5.9. Transmission spectra through ten time-varying 1D phononic crystal cells ............. 122

Figure 5.10. A surface composed of the fundamental frequency transmission spectra for various pumping frequencies .............................................................................................................. 123

Figure 5.11. The transmission spectra for two different pumping frequencies ......................... 126

Figure 5.12. The normalized short-time Fourier transform of a 2 kHz transmitted signal ........ 127

Figure 6.1. The time-varying scattered field coefficient vector................................................ 134

Figure 6.2. A discrete time-varying T-matrix .......................................................................... 135

Figure 6.3. The recorded FDTD source amplitude on the aperture plane ................................. 140

Figure 6.4. The simulated displacement fields for a single static cylinder ............................... 141

Figure 6.5. The simulated displacement fields for three static cylinders .................................. 142

Figure 6.6. The normalized displacement spectra of uz for a single cylinder ............................ 143

Figure 6.7. The simulated fundamental frequency displacement fields for one time-varying cylinder .................................................................................................................................. 144

Figure 6.8. The simulated displacement field harmonics for one time-varying cylinder .......... 145

Figure 6.9. The normalized displacement spectra of uz for three cylinders .............................. 146

Figure 6.10. The simulated fundamental frequency displacement fields for three time-varying cylinders ................................................................................................................................. 147

Figure 6.11. The simulated displacement field harmonics for three time-varying cylinders ..... 147

Figure 6.12. The change in scattered displacement field magnitude ........................................ 148

Figure 6.13. A a) static and b) dynamic phononic crystal ........................................................ 149

Figure 6.14. The displacement field harmonics present in the scattered field of the dynamic phononic crystal ..................................................................................................................... 149

- 1 -

Chapter 1

INTRODUCTION

1Periodic acoustic materials, called phononic crystals, are of interest because acoustic waves

behave differently in them than in homogeneous materials. These engineered materials have

been studied for over a century and the body of theory that describes them is extensive. The often

remarkable properties these crystals exhibit are set by the choice of constituent materials and the

size and spacing of those materials, collectively called the design parameters of the crystal. Until

now, the properties of phononic crystals were fixed and unchangeable because the design

parameters were fixed and unchangeable.

However, there many types of active materials that could be used in place of passive

materials in phononic crystals. Active materials have transductive properties and their material

parameters can be changed by applying an external stimulus. Such active materials in addition to

simpler mechanical means have been used in phononic crystals to change their design parameters

from one static state to another, thus enabling active crystals to have more than one mode of

operation. This thesis has taken the concept further and seeks to examine the situation when the

design parameters of a phononic crystal are continuously changing in time. The theory that

describes static phononic crystals is extended to handle time-varying material parameters, and

predictions based on the extended theories are presented.

This chapter introduces periodic materials in general and presents some of their capabilities.

Special attention is paid to phononic crystals as they are at the heart of this thesis. The possibility

of incorporating locally resonant scatterers in phononic crystals is also discussed because this

relates to the study of metamaterials. The motivation for using active materials in phononic

crystals is then examined, including the aims and scope of the thesis. A broad review of the

previous work accomplished in this field is offered, focusing on the work in nonlinear photonics,

time-varying systems theory, and active materials. Finally, an overview of the remainder of the

thesis is given.

1 This chapter contains material published in the 2009 paper by DW Wright and RSC Cobbold appearing in Ultrasound, 17(2) 68–73, entitled “The characteristics and applications of metamaterials”.

Chapter 1. Introduction 2

1.1 Periodic materials

Periodic materials are engineered composites that exhibit unique properties not usually found in

nature. Harry Potter-like invisibility cloaking [1], bending waves in the “wrong” direction [2],

and subwavelength imaging [3] are just a few of the more surprising capabilities. Periodic

materials can be designed to manipulate acoustic [4], electromagnetic [5], thermal [6], plasmonic

[7], or just about any other kind of bulk or surface waves of interest in new and potentially

important ways. They possess such unique abilities because they have characteristic feature sizes

that can be much less than, or on the same order as the wavelengths of interest. Not only does the

choice of constituent materials affect the macroscopic properties of a periodic material, but the

small-scale size and structure also play a significant role, which distinguishes them from regular

materials.

Periodic materials consisting of a host filled with scatterers are called photonic [8] or

phononic [9] crystals depending on whether they scatter electromagnetic or acoustic waves,

respectively. In these crystals, the feature sizes are similar to the wavelength, which leads to

strong interference effects from scattering off of the artificial crystal lattice. Effective material

parameters are not well defined in these structures. Instead, one considers the relationship

between spatial and temporal frequencies, called the dispersion relation. The focus of this thesis

is on phononic crystals.

Figure 1.1. The operating regime of periodic materials. Phononic crystals have feature sizes at or smaller than the wavelengths of interest. When the wavelength is large compared to the feature size of the phononic crystal, wave propagation can be characterized using effective material parameters, and the material appears to be homogeneous (called a metamaterial). As the wavelength approaches the feature size, scattering effects begin to dominate and the dispersion properties of the crystal come into play.


Periodic materials whose feature sizes are much smaller than the wavelengths of interest

consist of a host filled with discrete resonators and are often called metamaterials. The

wavelengths of interest are much larger than the individual resonators, so the wave does not

“see” them. Rather, it behaves as if travelling through a homogeneous medium with a particular

set of effective material properties, as shown in Figure 1.1, and those effective properties can be

adjusted to extreme values simply by tuning the resonators. These effective material properties

can be very large, very small, or even negative, which can lead to some interesting properties and

applications.

The distinction between resonant and scattering periodic materials is not abrupt, and quite

often structures may fall into both categories [10]. These structures are called locally resonant

phononic or photonic crystals. In these structures, each scatterer is also a resonator tuned for

certain frequencies. At shorter wavelengths their dispersion relation may resemble that of a

purely scattering phononic or photonic crystal, but at longer wavelengths may have additional

bands present due to the local resonances of the scatterers.

1.1.1 Phononic crystals

Phononic crystals are materials composed of regularly spaced acoustic scatterers. They can

be one, two, or three dimensional as illustrated in Figure 1.2. A two dimensional phononic

crystal created by our group is shown in Figure 1.3. The problem of wave propagation through

such a material has been considered since the mid-nineteenth century [11], and since then the

progress in considering ever more complicated scattering structures has been continuous. An

exceptionally clear and detailed account of the various approaches to determining wave

propagation through different scattering structures has been described in a recent book by Matrin

[12].

Figure 1.2. One, two, and three dimensional phononic crystals. In one dimension (1D), the crystal consists of infinite planes of alternating materials stacked together, as in a Bragg reflector. In two dimensions (2D), the crystal is made up of infinite rods regularly spaced and embedded in a host medium. In three dimensions (3D), the crystal is composed of discrete scatterers suspended in a crystal formation within a host medium.


Figure 1.3. A two dimensional phononic crystal consisting of eleven rows of 1/8” stainless steel cylinders packed in a square array and suspended in a water host. The filling fraction is 0.58, meaning that 58% of the volume within the crystal is stainless steel. This crystal exhibits negative refraction around 420 kHz because of the carefully designed shape of the dispersion curves.

The discrete and periodic nature of phononic crystals becomes increasingly important as the

wavelength approaches the same size scale as the constituent scatterers. Effective material

parameters are no longer easily defined, and interference effects come into play. The size and

spacing of the scatterers becomes increasingly relevant. As an example, consider the scattered

acoustic field caused by the two sound-hard cylinders shown in Figure 1.4a) for an incident

wavelength identical to the cylinder diameter. Strong interference effects can be seen as peaks

and nulls of energy radiating away from the cylinders. In Figure 1.4b), the spacing between the

cylinders has increased by just 10%, and now the scattered field is almost completely different,

demonstrating how important the feature size and spacing when considering scattering effects in

phononic crystals.

Figure 1.4. The scattered field of two acoustically-hard cylinders created by an incident plane wave. The incident field (not shown) has a wavelength identical to the cylinder diameter in both images. In a), interference effects can be seen as peaks and nulls in the scattered field. In b), the cylinders have been separated by an additional 10%, and the scattered field is dramatically different, indicating the importance of scatterer size and spacing.


Some of the interesting effects not normally found in nature that are demonstrated by

phononic crystals include transmission band gaps [13], negative wave refraction [14], ultrasound

focusing in two [9] and three [15] dimensions, tunneling [16], anomalous Doppler effects [17],

and subwavelength imaging capabilities [18].

1.1.2 Local resonance

Phononic crystals with homogeneous scatterers typically do not exhibit resonance effects at

longer wavelengths, and thus their properties arise exclusively from interference and dispersion

effects. However, if the scatterers consist of multiple materials designed to resonate at particular

wavelengths, then the phononic crystal may behave as a resonant metamaterial at longer

wavelengths and as a scattering crystal at shorter wavelengths ([10], [19]–[31]). Thus, these

locally resonant phononic or photonic crystals can be viewed as homogeneous resonant

structures for longer wavelengths and as discrete scattering structures for shorter wavelengths.

An “effective medium theory” is used to describe the characteristics of the metamaterial in terms

of effective material parameters, which is not possible for the shorter wavelengths that cause

scattering effects in phononic crystals.

Figure 1.5. An illustration of the role of resonance in metamaterials. Resonance can be used to achieve negative effective material parameters as shown above for a pendulum. At low frequencies, the mass moves together with the applied force. But as the frequency increases, the mass eventually slips out of phase and moves in opposition to the applied force. Thus, the pendulum exhibits a negative effective mass at that particular frequency.

In the homogeneous limit when the wavelength is much longer than the resonator size,

effective material parameters are well-defined and depend not only on the constituent materials

but also on the resonator properties. The effective parameters can be adjusted to extreme values

by tuning the subwavelength resonators. Properly tuned, these resonances can lead to negative

effective properties. A simple way to understand this concept is to envision a pendulum


consisting of a mass and a string, and being pushed and pulled about halfway up, as shown in

Figure 1.5. At low frequencies, energy is imparted to the pendulum and it moves in the direction

of applied force. However, as it is pushed and pulled faster and faster, the pendulum begins to

swing out of phase with the applied force. Eventually, the applied force is pushing forward while

the pendulum is moving backwards. In essence, the pendulum has a negative effective mass at

that frequency since it is moving in opposition to the direction of force.

Acoustic metamaterials have been postulated and demonstrated based on these principles.

Acoustically, a 3D resonant metamaterial can be created by embedding concentric spheres of

differing materials into a host medium. For example, rubber-coated stainless steel spheres can be

tuned so that at a particular frequency the inner sphere is moving in opposition to the outer

sphere, leading to negative effective values of density and compressibility. More directly, a

network of masses and springs could be created to achieve the same effect. Acoustic equivalents

of negative refraction [4], cloaking [32], and subwavelength imaging [33] have been postulated

and in some cases demonstrated experimentally.

1.2 Motivation

Phononic crystals are normally static in the sense that their properties are fixed in advance by

their design parameters, which can limit their functionality to very specific arrangements. So far,

useful phenomena in phononic crystals are limited by the fact that the effects are only

appreciable over rather narrow frequency ranges. For example, the phononic crystal described in

[15] was designed to focus ultrasound at 1.57 MHz, but when the frequency was changed to

1.60 MHz (a 1.9% change), the focusing effects were no longer apparent. As another example,

the band gap width was shown in [13] to be highly dependent on the crystal filling fraction.

These constraints appear to limit the usefulness and versatility of phononic crystals to

predesigned narrowband operating conditions.

We have proposed that it may be possible to dynamically alter the behaviour of phononic

crystals by changing the material parameters as a function of time [34]. Varying the material

parameters of the scatterers will modify the propagation of acoustic waves through a phononic

crystal. This may enable time-variant phononic crystals to have more robust or multiple

operating regimes as compared to their static counterparts.

One approach to determining the effects of time-varying material parameters is through the

use of the finite-difference time-domain (FDTD) method, which is a common and versatile

approach to acoustical simulation [35] and is discussed at length in Chapter 4. Within a

simulation region, the space domains are discretized and the material properties within this

region are represented on this discrete grid. The continuous acoustic wave equations are also


discretized and used to calculate the acoustical fields over incremental time steps. The

discretized equations are amenable to time-varying material parameters with little or no

modifications, which makes the FDTD method an attractive starting point for the analysis of

time-varying phononic crystals [36]. However, one major drawback of the FDTD method is that

it gives no analytical insight into why a particular result occurs – only what occurs. Moreover,

because the method is inherently an approximation of the governing equations, the accuracy

depends on the resolution in both the space and time domains. Thus, better results may take

much longer to generate than with an exact method.

To address these shortcomings, we have used the one dimensional transmission matrix

method (TMM) as an alternative starting point since it gives a closed-form solution to wave

propagation in periodic media [37]. In two dimensions we have used multiple scattering theory

(MST) since it is analogous to the TMM theory [12]. We extended these theories to handle time-

varying material parameters, which yielded solutions to acoustic wave propagation in time-

varying phononic crystals that are exact in principle.

1.2.1 Aims and scope

Aims

The ultimate goal of this thesis is to describe wave propagation in time-varying phononic

crystals. The practical outcomes of this research could be the development of new imaging

methods with improved resolution, materials whose transmittivity and reflectivity can be

adjusted in real-time, and materials with active damping properties. As of yet, these structures

have not been created; however, this thesis lays the groundwork for further explorations into this

topic.

To understand time-varying phononic crystals, we have extended the existing static phononic

crystal theory so as to handle time-varying material parameters and also to provide insight into

the factors that govern their behaviour. We then analyze the resulting governing equations to

gain insight into how to control these materials. The closed-form expressions2 are quite rich and

give insight into purposeful control of the wave transmission properties of time-varying

phononic crystals.

Scope

This thesis formulates the basic theory necessary to understand the properties of time-varying

phononic crystals. A static phononic crystal was created to verify that our static theories match

2 In this thesis we take closed-form to mean an equation consisting of a truncated set of terms. The exact solutions contain infinitely many terms and are therefore not closed-form.


the acquired experimental data. Then, once our dynamic theories were created, they were

analyzed to ensure that they collapsed to their static counterparts under static conditions. In this

way, a preliminary check was made upon our new theories to make sure that they are valid in the

static case.

The construction and testing of a time-varying phononic crystal is not within the scope of this

thesis since it was impossible to intelligibly design such a crystal without a theory of how to do

so. That theory is the contribution of this thesis, and it is left as a future project to construct and

characterize a dynamic crystal using the new theory. There is further discussion regarding how a

real-world time-varying phononic crystal might be constructed and operated. Possible

differences between the predictions of the theory in this thesis and future experimental results is

also discussed.

1.3 Previous work

Studies in the 1960s to 1970s have examined time-varying parameters in electromagnetic

systems [38]–[46], but none give adequate insight into the types of systems with which we are

concerned. Such efforts have included homogeneous time-varying media [38][39], time-varying

rough surfaces [40], infinite crystal wavevector approaches [41], and most notably work on

propagating time-variant parameters [42][43]. Time-varying scattering was considered for

moving scatterers in [44]–[46], but did not include the effects of time-varying material

parameters. A particularly interesting analysis of a time-varying Bragg reflector has been

presented by Wu et al [47], however, its derivation does not properly account for the amplitude

modulation of the time-varying impedance discontinuities. In photonics, Winn et al [48] showed

that a time-varying dielectric permittivity can modulate energy from one photon energy band to

another in an interband transition process. The dispersion properties of a static photonic crystal

were determined, and then the effect of perturbing this system by varying the dielectric constant

was examined to derive the underlying mathematics, which was then verified with a numerical

simulation.

In acoustics, there have been some recent studies in reconfigurable systems, which are quasi-

static in the sense that a particular parameter is changed and the properties of the phononic

crystal are re-measured [49]–[59]. In [49] and [50], the scatterers are square rods that are

physically rotated to adjust the phononic crystal properties. A dielectric elastomer is used as an

active material in a 1D phononic crystal in [51] and a 2D phononic crystal in [52]. In [53]–[59],

the theory and some experimental data for using various active materials as adjustable scatterers

in 1D and 2D phononic crystals is presented.


These studies are important because they characterise the individual static operating points of

a variable parameter system, but they do not capture the effects of varying those parameters in

time. They also demonstrate the value of using dynamic material parameters in phononic crystals

by illustrating such effects as increased transmission or attenuation, and altered band gaps and

dispersion. This thesis extends these results to cases where the material parameters are changing

as a function of time. It will be described how time-varying material parameters significantly

affect the phononic crystal transmission spectrum beyond what is possible with simple adjustable

scatterers, and how this was used to switch an incident monochromatic wave. Furthermore, the

closed-form solutions revealed the occurrence of and conditions necessary for parametric

amplification.

1.3.1 Nonlinear photonics

Some dielectric materials have a nonlinear response to the incident electric field at high field

intensities. Typically, there are two configurations used in nonlinear optical systems, as

illustrated in Figure 1.6 for a nonlinear photonic crystal. The first relies on the intensity of the

incident light to modulate the material properties of the nonlinear dielectric, shown in a). The

second relies on a separate beam whose sole purpose is to modulate the material properties of the

nonlinear dielectric, often called the pump as in b). Regardless of the configuration, if the

material is also periodic, then it behaves as a nonlinear photonic crystal. [60][61]

This second configuration bears the closest resemblance to the time-varying phononic

crystals, however, there are some notable differences. In the time-varying phononic crystals

described in this thesis, the pumping mechanism is controlled by a field that does not directly

interact with the acoustical field. For example, an electromagnetic field might be controlling the

material parameters, which would not directly interact with the acoustical field. Thus, the

intensity of the acoustical field is irrelevant within reasonable limits. In a pumped nonlinear

photonic crystal, the intensity of the incident wave also modulates the material parameters and

may come into play unless its magnitude is kept low enough to ignore its contribution. If the

intensity of the acoustical field is high enough to elicit a nonlinear response in the material, the

effect is fundamentally different than the modulation caused by pumping. Such effects are not

considered in this thesis.


Z1

c01

Z2(Iinc)

c02(Iinc)Incident (Iinc)

Z1

c01

Z2(Iinc)

c02(Iinc)

Z1

c01

Z2(Ip)

c02(Ip)Incident (Iinc)

Z1

c01

Z2(Ip)

c02(Ip)

Pump (Ip)

Ip >> Iinc

a)

b)

Figure 1.6. Two typical configurations of a nonlinear photonic crystal. In a), the intensity of the incident field (Iinc) stimulates the nonlinear response of material 2. In b), a separate pump field stimulates the nonlinear effect. The pump field is much more intense (Ip) than the incident field so that the intensity of the incident field does not significantly affect the nonlinear material parameters.

The other major difference is that the frequencies of interest in the time-varying phononic

crystals are orders of magnitude lower than in a nonlinear photonic crystal. In our acoustical

systems the material parameters may be varying at or even above the incident frequencies. The

dynamic materials can react to the pumping field on a time scale much shorter than one period of

the incident wave. In an optical nonlinear photonic crystal, the materials respond to the envelope

of the pumping signal as opposed to its instantaneous value, and so the change in material

parameters is considered slowly varying as compared to the incident frequencies. This

assumption is not made in our subsequent analyses.

1.3.2 Time-varying systems theory

The behaviour of linear systems with time-varying parameters has been previously explored

[62]. The Hill equation addresses second-order systems with periodically varying parameters,

and is classically stated as

( )( ) 02 =−+ ytqay ψ&& , (1.1)


where y is a state variable such as the acoustic particle displacement or the electric field, a is the

DC offset of the time-varying parameter, 2q is the magnitude of the time variation, and ( )tψ is a

periodic function with unit amplitude. The most widely used form of this equation is when

( ) tt 2cos=ψ which leads to the Mathieu equation:

( ) 02cos2 =−+ ytqay&& . (1.2)

These equations have been studied extensively and have resulted in numerous applications,

such as parametric amplifiers [63] and parametrons [64], antiquated digital storage devices that

have recently found a new niche [65]. These equations have also been applied to static photonic

crystals whose parameters are periodic in space instead of time. In this case, ( ) ( )tt 2sinsgn=ψ ,

which is a square wave where t represents space instead of time. This substitution into the Hill

equation is called the Meissner equation:

( )( ) 02sinsgn2 =−+ ytqay&& . (1.3)

Analysis of this spatially periodic system reveals the frequency pass and stop bands and

dispersion curves that one would expect from a photonic crystal [62].

As attractive as these equations and their accompanying body of theory are, they only deal

with periodicity in one dimension, either space or time, but not in both simultaneously. Thus, the

theory of periodically time-varying systems in its current form is unsuitable for analyzing time-

varying phononic crystals.

1.3.3 Active materials

Tacit in this discussion has been that there are active materials capable of being used to

construct time-varying phononic crystals. In fact, there are numerous materials using a variety of

transduction mechanisms that could be used as time-varying scatterers. Here we present a few of

the more notable candidates. The subsequent information is not intended as an in-depth or

exhaustive presentation of active materials, but rather is intended to show that a palette of active

materials exists for designing real time-varying phononic crystals. The following technologies

are some of the newest and most promising active materials that could be used in time-varying

phononic crystals

Macro and active fiber composites

Piezoelectric materials experience a strain in the presence of externally applied electric field

(the inverse piezoelectric effect). This effect and its dual, the piezoelectric effect, is why

piezoelectric materials are used in ultrasound transducers [66]. However, in spite of their ability

to deform in an electric field, their use in active materials is hampered by the “brittleness of


piezoceramic materials, poor conformability (particularly when applied to curvilinear surfaces),

isotropic strain actuation, and overall low strain energy density” [67].

To overcome these undesirable characteristics, composite materials have been created

consisting of sandwiches of piezoceramic fibres embedded in epoxy and covered with

interdigitated electrodes on either surface. If the fibers are square in cross-section then the

material is called a macro fiber composite (MFC) [67]. If the fibers are circular, then the material

is called an active fiber composite (AFC) [68]. These materials exhibit increased actuation strain

over monolithic transducer piezoelectric materials, such as PZT, and can conform to various

curved surfaces [69].

Electroactive polymers

There are certain polymers, called electroactive polymers (EAPs), whose shape is altered

under the influence of an electric field. Of the numerous varieties of EAPs, much of the focus

has been on dielectric elastomers [70]. There are numerous specific types of dielectric

elastomers, and many have desirable qualities, such as being lightweight, having large actuation

strain, and having the capability of generating a useful amount of force. These materials have

even been applied to a two dimensional phononic crystal as was previously mentioned [52], and

thus make a good candidate for a dynamic scattering material.

Magnetostrictive composites

The magnetostrictive effect occurs in certain materials when an external magnetic field

causes the magnetic domains to change their shape. Because of this effect, the material can

elongate in the presence of an applied magnetic field [71]. Recently, some of the problems

associated with traditional magnetostrictive materials, such as brittleness and losses due to eddy

currents, have been overcome by creating a magnetostrictive-epoxy composite [72]. These

composites appear to be much more durable, have better response, and do not sacrifice the strain

performance of purely magnetostrictive compounds.

Ferroelectric shape memory alloys

Shape memory alloys can be deformed and then reset to their original shape by heating them,

which causes them to undergo a material phase transition back to their original state. They have a

large actuation strain, but require time to be heated and cooled and thus react slowly. On the

other hand, magnetostrictive materials can react quickly but do not show nearly as much

actuation strain. A new type of material called a ferromagnetic shape memory alloy is a shape

memory alloy whose phase transition is caused by an applied magnetic field instead of by heat


[73]. Thus, these materials show the large actuation strain of a shape memory alloy and the fast

reaction time of a magnetostrictive material.

Carbon nanotubes

An aerogel made of multiwalled carbon nanotubes has recently been developed that is

extremely light, has very large actuation strain, can operate at very high frequencies, has a very

large operational temperature range, and a has very high work capacity [74]. This new material

seems to combine all the most desirable qualities of an active material and may become very

pervasive in systems requiring active materials.

1.4 Organization of thesis

This thesis is organized as follows. Chapter 2 introduces static phononic crystals and their

properties in one dimension. Chapter 3 extends these concepts to static phononic crystals in two

and three dimensions, and explores more complex issues such as wave dispersion and refraction.

The experimental results from a 2D phononic crystal that we manufactured are included in this

chapter. A variety of methods for analyzing phononic crystals and a discussion of their strengths

and weaknesses is presented in Chapter 4. The static theory is extended to handle time-varying

material parameters in one dimension in Chapter 5, and is extended to two dimensions in

Chapter 6. The bulk of new research contributions lie in these two chapters. A summary and

some conclusions of this thesis along with suggestions for future research are offered in

Chapter 7.

- 14 -

Chapter 2

STATIC PHONONIC CRYSTALS IN ONE

DIMENSION

This chapter explores acoustic wave propagation through phononic crystals in one dimension.

First, the 1D acoustic scalar wave equation is derived. Reflection and transmission through the

interfaces between differing acoustic materials are then discussed. Next, periodic acoustic

structures are introduced along with some of their typical transmission and dispersion properties.

A corrugated tube waveguide is presented as one type of phononic crystal and some of its useful

properties are explored. Finally, some applications of 1D phononic crystals are described1.

2.1 Acoustic waves in one dimension

In this section we derive the 1D acoustic scalar wave equation using an approximated model of a

fluid-filled rigid tube. Consider a rigid tube of cross-sectional area A filled with a lossless,

homogeneous fluid medium. The fluid has a density, ρ , and a longitudinal modulus, 11C . We

consider the case where the motion is constrained to one dimension, along the principle axis of

the tube. Thus, we can define the linear mass density as Aρµ = .

To approximate the motion of the fluid in the tube, we divide the tube into small sections of

length x∆ . As a first-order approximation, we represent each of these small fluid sections as a

mass with a value of xm ∆=∆ µ and a spring with a value of xACk ∆=∆ 11 . Figure 2.1a)

illustrates the fluid-filled rigid tube and Figure 2.1b) shows the corresponding mass-spring

approximation. The reader is reminded that this is a purely artificial construct that is introduced

for conceptual purposes alone. Two conditions must be met regarding the size of the fluid

segment: The first is it must be much larger than individual molecules and atoms so that the

underlying discrete nature of the material can be encapsulated in effective material parameters

such as density and longitudinal modulus. The second is that it must be much smaller than the

acoustic wavelengths under consideration to avoid the effects of dispersion, as will be discussed

later. Thus, the material can be viewed as a continuum of fluid segments with uniform effective

material parameters, which we have approximated as masses and springs.

1 Methods for analyzing wave propagation in phononic crystals are left to Chapter 4 due to the scope of this topic.

Chapter 2. Static Phononic Crystals in One Dimension 15

Figure 2.1. a) A fluid-filled rigid tube and b) its approximation as a linear chain of masses and springs. In the approximation, the masses are confined to longitudinal motion only.

In this configuration, we adopt a Lagrangian reference frame so that ( )txU , describes the

displacement from equilibrium of a mass at location x and at time t, as opposed to its absolute

position in space which is an Eulerian reference frame. That is, ( ) 0,Lagrangian =txU and

( ) xtxU =,Eulerian at equilibrium. The masses can be referenced using an index n, where n is an

integer and corresponds to the mass located at position xnx ∆= . Accordingly, ( )tU n refers to

the displacement ( )txnU ,∆ , which is the displacement from equilibrium of the centre of mass of

the nth fluid section.

The masses experience forces due to their displacement and that of their neighbours.

According to Hooke’s law, the force at a position n, nF , is

( ) ( )( )11

11

2 −+

+−

+−∆=

−∆+−∆=

nnn

nnnnn

UUUk

UUkUUkF, (2.1)

where the time dependence of U and F have been removed for clarity. Newton’s second law

states that

2

2

t

UmF n

n∂

∂∆= . (2.2)

If the density and stiffness are assumed to be locally constant, the two equations can be

coupled as

( )1111

2

2

2 −+ +−∆

=∂

∂∆ nnn

n UUUx

AC

t

UxAρ , (2.3)

which can be expressed more clearly as


( )

( )2112

02

2 2

x

UUUc

t

U nnnn

∆

+−=

∂

∂ −+ , (2.4)

where ρ110 Cc = is the longitudinal phase velocity. By taking xx ∂→∆ , the factor on the

right hand side of (2.4) approaches ( )

2

2 ,

x

txU

∂

∂, which can be seen by noting that the right hand

side factor is the central-difference approximation to the second derivative. This is a valid

approximation provided that x∂ is much smaller than the wavelengths of interest and much larger

than the individual constituent atoms and molecules of the fluid. In this approximation, the

equation becomes the one dimensional scalar wave equation in a homogeneous fluid medium:

( ) ( )

2

2202

2 ,,

x

txUc

t

txU

∂

∂=

∂

∂. (2.5)

For time-harmonic motion of a particular angular frequency 0ω , ( )txU , can be separated into

spatial and temporal components as

( ) ( ){ }tjexutxU 0Re, ω= , (2.6)

and thus a lower-case u indicates the spatially varying component of a monochromatic wave of

angular frequency 0ω . Upon inserting (2.6) into (2.5), the wave equation reduces to the one

dimensional Helmholtz equation,

( ) 0202

2

=

+

∂

∂xuk

x, (2.7)

where 000 ck ω= is the wavenumber. Solutions can be written as plane waves of the form

( ) ( ){ }txkjetxU 00Re, ω−±= . (2.8)

The choice of sign in (2.8) is arbitrary, but the convention chosen will be explicitly stated where

appropriate.

In deriving this simple solution to the wave equation, we have made a number of

assumptions, namely that the small-signal approximations are valid [66], and that the medium is

fluid, homogeneous, isotropic, and lossless. The issue of the validity of these assumptions will be

discussed in context of the problems and solutions presented in subsequent chapters. The 1D

scalar wave equation may also be derived beginning from a continuum description of a

homogeneous medium, and can be found in many texts, such as [75].


2.2 Acoustic wave transmission and reflection in 1D

Consider material i, to be homogeneous with density and bulk modulus iρ and iK , respectively.

Recall that ii

i CK κ111 == , where iC11 is the isotropic elastic stiffness coefficient (Voigt

notation [75]) and iκ is the adiabatic compressibility [66], all of material i. Further, it is lossless

and dispersionless, has a phase velocity of

iii Kc ρ= , (2.9)

and a characteristic acoustic impedance of

iii KZ ρ= , (2.10)

which, for brevity, will subsequently be referred to as the acoustic impedance.

Acoustic waves scatter when they impinge upon a boundary between differing materials, i

and j as illustrated in Figure 2.2. Two conditions must be met at the boundary: The pressures on

either side must be identical, and the normal component of the velocity must be equal. If either

of these conditions is not met, then there will be either a pressure or velocity source at the

boundary. In one dimension, these two conditions can be met and are captured by transmission

and reflection coefficients which, for a monochromatic wave, can be expressed in terms of the

characteristic impedances of the two media.

Figure 2.2. Incident, transmitted and reflected waves at an interface.

Considering displacement waves, the transmission coefficient from material i into material j

is

( )jiii ZZZT += 2 , (2.11)

and the reflection coefficient is

( ) ( )jijii ZZZZR +−= . (2.12)

For the displacement waves under consideration,

1=− ii RT . (2.13)


2.3 Periodic structures

As described in Figure 1.2, a phononic crystal in one dimension consists of infinite planes of

alternating materials stacked together. The transition between materials need not be abrupt, and

the stack may consist of many different materials provided that the structure is periodic.

Figure 2.3. The nomenclature for a two-component one dimensional phononic crystal consisting of a stack of infinite planes of alternating materials.

The nomenclature used for a two-component phononic crystal consisting of a stack of

alternating planes is provided in Figure 2.3. The minimum length at which the structure is

periodic is called the cell size, a. For a two-component system, a parameter called the filling

fraction, η, defines how much of host material is replaced with an inclusion. In the figure,

material 1 is the host and material 2 is the inclusion. This simple arrangement of alternating

materials leads to some notable transmission effects, such as band gaps, tunneling and

dispersion, all of which are briefly described in the next two subsections.

Research on phononic crystals has been primarily in 2D and 3D structures, however there has

been some analysis of purely 1D systems [76]–[88]. The transmission matrix method was

described in [76], and subsequent experiments demonstrated band gaps in 1D phononic crystals

[77]. A purely theoretical analysis of periodic one dimensional systems is given in [78]. A chain

of masses was explored experimentally in [79] and demonstrated the theoretically predicted band

gaps. Waveguides, structures that force waves to propagate in particular directions, were

analyzed in 1D in [80]. Bragg reflectors consisting of alternating planes of material were

investigated in [81] and [82]. Surface and Lamb waves have also been explored for structures

periodic in one dimension in [83]–[87]. Finally, bulk waves through a finite-thickness plate were

investigated in [88].

2.3.1 Band gaps

The total wave field at any given position within a phononic crystal is a summation of the

transmitted and reflected waves, which may interfere constructively or destructively as shown in

Figure 2.4. The nature of this interference is determined by the wavelength relative to the cell


spacing and filling fraction. The phenomenon of frequency band gaps in periodic structures is

known as Bragg reflection from the lattice [75] and results from this interference.

Figure 2.4. Constructive and destructive interference of transmitted and reflected waves in a phononic crystal. Note that the waves in question are longitudinal and not transverse. The arrows are curved to aid in visualizing the phase.

The net effect of all the constructive and destructive interference within a phononic crystal is

a transmission spectrum that contains band gaps, which occur when waves do not propagate

within a phononic crystal for specific ranges of frequencies. Waves of these particular

frequencies are attenuated as they propagate through a phononic crystal. Figure 2.5 shows that as

more cells are added to the phononic crystal, the frequencies within a band gap become

increasingly attenuated.

πka Figure 2.5. Simulated band gaps typical of a 1D phononic crystal. As more cells are added to a phononic crystal, the band gap attenuation increases.


The acoustic impedance determines the transmission and reflection at a material interface as

in (2.11) and (2.12), respectively. As the inclusion impedance increases relative to the host

material, the band gap width and attenuation increase as shown in Figure 2.6. Typically the band

gap width is measured at the -3 dB energy crossing points and is normalized by the gap centre

frequency. In the amplitude transmission graph of Figure 2.6, it corresponds to the -6 dB points.

Another way to consider this scenario of increasing impedance contrast is to envision the host

and inclusion consisting of identical materials, in which case there would be no reflection and a

wave would simply pass through unaffected by the interface. As the contrast is increased, the

reflection increases and interference becomes ever-more important. Eventually, the inclusion

impedance is so high that it can be considered infinite, and all of the wave energy is totally

reflected off of the first inclusion. A similar scenario holds if the host impedance is increased

with respect to the inclusion impedance. In fact, Figure 2.6 would be identical if the legend read

21 ZZ .

Transmission (dB)

πka Figure 2.6. Band gap characteristics for various impedance ratios. The simulated results for five cells shows that as the contrast between the impedances increases, the gap width and attenuation increases.

The filling fraction, η, and the phase velocity, c0, determine the phase change of a wave as it

traverses through each element. Thus, the thicknesses of the various sections that make up a cell

can be thought of in terms of wavelengths. For example, the thickness of an inclusion in terms of

number of wavelengths is 002 caft η= , where f0 is the frequency of interest. It is not surprising

from this assessment that adjusting the filling fraction and the phase velocity has very similar


(but inverse) effects on the transmission spectrum, as illustrated in Figure 2.7 and Figure 2.8

respectively. Decreasing the filling fraction reduces the relative physical thickness of the

inclusions, and increasing their phase velocity reduces the number of wavelengths that fit within

an inclusion for a given frequency.

πka Figure 2.7. The transmission spectra for various filling fractions (η). When the phononic crystal is symmetric ( 5.0=η ), the transmission spectrum is periodic. However, as the filling fraction is altered, new band gaps are introduced into the transmission spectrum, and the attenuation of the existing ones is changed.

Though the dimensionless frequency axis used so far is convenient for comparing various

phononic crystals, it is perhaps informative to examine the transmission spectrum of a crystal

made with real materials. Figure 2.9 shows the transmission spectrum for longitudinal waves in a

five-cell phononic crystal consisting of alternating plates of aluminum (host) and nickel

(inclusion). The cell size, a, is 20 mm, and the filling fraction is 35%. Thus, the aluminum plates

are 13 mm thick and the nickel plates are 7 mm thick. The material parameters are given in the

figure caption.


Transmission (dB)

πka Figure 2.8. The transmission spectra for various phase velocity ratios. Usually two different materials have differing phase velocities, and the degree to which they differ increasingly affects the transmission spectrum. The effect is similar to changing the filling fraction since in either case the phase shift between interfaces is altered.

Figure 2.9. The transmission spectrum for longitudinal waves in an aluminum-nickel phononic crystal. The crystal consists of five cells of an aluminum host with nickel inclusions. The aluminum is 13 mm thick and the nickel inclusions are 7 mm thick. The material parameters used for the aluminum are

-30 mkg2699 ⋅=ρ and -1

0 sm6320 ⋅=c , and for the nickel are -30 mkg8900 ⋅=ρ

and -10 sm5630 ⋅=c .


Tunneling

For a wave incident on a thin barrier, it is possible for some of the energy to pass through the

barrier. In this context, a barrier refers to a slab of material in which the wave vector is purely

imaginary (i.e., an evanescent wave); for the transmission to be easily measureable, the barrier

should be thin with respect to the wavelength. For example, tunnelling can occur when an

acoustic wave is incident normally on a phononic crystal at frequencies inside a band gap, and

when a wave is incident on material at angles greater than the critical angle at which total

internal reflection occurs. Since the wave is evanescent within the barrier, it will therefore reflect

off of the surface if the barrier is thick. However, if the barrier is thin, then some of the

evanescent energy may couple to the medium on the other side of the barrier and continue to

propagate. Thus, some of the energy may tunnel through the barrier.

Figure 2.10. Illustration of tunneling through a single barrier. A phonon has both particle and wave properties, and has a location determined by a spatial probability distribution. As such, when it is near a potential barrier it has a probability of existing on the other side of the barrier, and thus has a probability of tunneling. The thinner the barrier, the more likely it is that the phonon will tunnel. In the figure, PReflected and PTransmitted are the probabilities of phonon reflection and transmission, respectively.

It is possible also to view tunneling from the perspective of a single quantum mechanical

phonon, which is a quantized mode of vibration and has a spatial probability distribution

governed by the square of its wavefunction. Just as for an electron, it is not located at an exact

point, and hence may be represented pictorially as a fuzzy sphere with a higher density in the

centre as in Figure 2.10. In this case, as a phonon approaches a thin barrier, its wave function can

penetrate inside the barrier so that there is a finite probability that the phonon can be located on

the other side of the barrier. The thinner the barrier, the more likely that the phonon will be


transmitted instead of reflected. The fuzzy sphere is smaller in the transmitted case to represent

that the probability of transmission is lower than that of reflection for this barrier.

Another tunneling configuration is possible, using a double barrier instead of a single barrier.

In this case, standing waves solutions exist between the barriers, shown for the simple case of

barriers of infinite height in Figure 2.11. (Note that for real barriers, the height is finite, a

situation that is required for the waves to penetrate into the barrier, so that non-zero transmission

is possible.) In this setup, the spacing between the two barriers enables resonances to occur for

phonons whose wavelength “fits” inside, such that as illustrated in the figure. Because of the

strong resonant enhancement inside the cavity formed by the two barriers, the transmission

through the two barriers is 100%, in the absence of any dissipation, when this condition is

satisfied. For other wavelengths, the transmission is reduced not only because of the reduction

due to tunnelling through the two barriers but also because these phonons will not fit between the

barriers and are therefore rejected with increasing probability as their wavelength deviates from

those that are ideally matched with the barrier spacing.

Phononic crystals can act as phonon barriers at band gap frequencies, where acoustic waves

undergo destructive interference. Many researchers have used this property of to explore the

tunneling of phonons through phononic crystals at band gap frequencies [16][89]–[94]. Single

barrier [89]–[91], double barrier [92][93], and more complicated wave mixing schemes [94] have

been demonstrated experimentally.

d

n = 1

n = 2

n = 3

n = 4

Figure 2.11. Illustration of modes allowed in an infinite double barrier. Within a double barrier, sometimes called a quantum well, only specific wavelengths are allowed. In particular, a wave may exist within an infinite double barrier when an integer multiple of half of its wavelength is equal to the barrier width. Only these wavelengths can tunnel through such a barrier.


2.3.2 Dispersion

In free space, there is a linear relationship between spatial and temporal frequencies for plane

waves, meaning that a wave of a given frequency has a known wavelength that depends on the

medium in which the wave is travelling. This linear relationship can be stated as

0cf =λ , (2.14)

where f is the frequency, λ is the wavelength, and 0c is the phase velocity. Alternatively, this can

be stated as

0ck =ω , (2.15)

where ω is the angular frequency and k is the wavenumber.

In a periodic material, there is no longer a linear relationship between ω and k. This can be

demonstrated by considering the linear chain of masses and springs discussed earlier. By

assuming that the lattice cell spacing in (2.4) is a, so that ax =∆ and then substituting a plane

wave solution, ( ) ( )tknaj

neUkU ωω −= 0, , this gives

( )jkajkaee

a

c −+−=− 22

202ω , (2.16)

( )kaa

ccos1

22

202 −=ω . (2.17)

Since ( )2sin2cos1 2 θθ =− , this can be written as

( )2sin2 0 kaa

c=ω . (2.18)

Thus, we may define the phase velocity as the ratio of angular frequency to wavenumber:

( ) kkc ωφ = . (2.19)

This is called the dispersion relation of the system and it is plotted in Figure 2.12a).

Previously when deriving the wave equation, we had assumed that the wavelengths of interest

were much longer than the size of the discrete masses and springs used in our approximation to

the fluid medium. This assumption leads to the linear dispersion plotted in Figure 2.12. However,

as the wavelength becomes smaller and approaches the same size scale as the discrete masses

and springs, in this case, the individual molecules and atoms of the fluid, the dispersion begins to

deviate from the long wavelength assumption. The spatial structure of the system becomes

increasingly relevant. These same concepts apply to phononic crystals: At long wavelengths the

crystal appears to be a homogeneous medium with effective material parameters, but at shorter

wavelengths relative to the crystal spacing the spatial structure affects the dispersion.


Figure 2.12. The dispersion relation a) and group velocity b) for a simple periodic system. In the long wavelength limit, the dispersion matches that predicted by a wave equation in a continuous medium. However, as the wavelength approaches the size of the periodicity of the system the dispersion is affected by the spatial arrangement.

Another important measure of dispersion is the group velocity, which can be defined as the

rate of change of angular frequency with respect to wavenumber ( k∂∂ω ). For a sufficiently

narrowband pulse, the group velocity can also be considered as the propagation velocity of the

signal envelope. In a homogeneous material at long wavelengths, it is simply a constant equal to

the phase velocity, since 0kc=ω . Considering the linear chain of masses and springs,

( )2cos0 kack

cg

=∂

∂=

ω, (2.20)

which is shown in Figure 2.12b). At long wavelengths the group velocity is constant and equal to

the free-space longitudinal phase velocity. For shorter wavelengths the group velocity

approaches zero, corresponding to a standing wave.

The implication of periodicity

For an infinitely long phononic crystal, the field values within one cell are sufficient to

determine the field values at any position in the crystal. This is a very simplified way of stating

the Bloch theorem, sometimes called the Bloch-Floquet theorem [95]. As a consequence, only

wavenumbers from 0 to aπ are considered since shorter wavelengths (higher spatial

frequencies) are aliased into this range by the periodicity of the system. This can be envisioned

as the spatial equivalent of the Nyquist–Shannon sampling theorem.


πka

f 0(M

Hz)

Figure 2.13. The dispersion curves for an aluminum-nickel phononic crystal. The bands are shown for both positive and negative wavevectors. Frequency band gaps occur where the wavevector has a nonzero imaginary component, indicating that the wave will decay exponentially within the crystal. The material and crystal parameters are identical to those used in Figure 2.9.

To illustrate this concept, the dispersion curves for the same aluminum-nickel phononic

crystal as in Figure 2.9 is shown in Figure 2.13. Comparison between these two figures shows

that the band gaps in the five-cell finite crystal in Figure 2.9 occur at the same frequencies where

the imaginary component of the wavevector for the infinite crystal shown in Figure 2.13 is

nonzero. Thus, a wave with a frequency in one of the band gaps will be attenuated as it

propagates through the crystal, and the total attenuation will depend on the distance travelled and

the magnitude of the imaginary component at that frequency. This is why a real finite crystal will

still transmit some of the frequencies in the forbidden band gap and why adding more cells

further attenuates the transmission within the band gap.


2.3.3 Corrugated tube waveguides

Figure 2.14. A corrugated tube waveguide is a 3D structure that behaves as a 1D phononic crystal for wavelengths longer than the largest tube diameter.

A corrugated tube waveguide is a three dimensional structure that acts as a one dimensional

waveguide for long wavelengths and is illustrated in Figure 2.14. At a certain cut-off frequency,

transverse modes occur and the structure no longer behaves as a 1D waveguide [17]. A cross-

section of one cell of a corrugated tube acoustic waveguide is illustrated in Figure 2.15, where a

is the length of a segment, di is the diameter of section i, and η is the filling fraction. One cell

consists of a single corrugation of length a made up of both section 1 and section 2. The source-

free wave equation for this tube can be written as

( )

( )

∂

∂

∂

∂=

∂

∂

x

uxS

xxS

c

t

u20

2

2

, where ( )( )

( )

<≤−

−<≤=

axaS

axSxS

1

10

2

1

η

η, (2.21)

u is the acoustic particle displacement, c0 is the phase velocity in the propagation medium, and

42ii dS π= is the cross-sectional area of section i. Direct comparison of this wave equation with

the linear 1D wave equation as derived earlier leads to the definition of effective material

parameters, calculated as 20cSii =ρ and ii SK = , where ρi and Ki are the effective density and

bulk modulus respectively. This means that a 1D phononic crystal created with the same size and

filling fraction (a and η) and using materials with properties ρi and Ki will behave the same as a

corresponding corrugated tube waveguide with cross-sectional areas Si in a medium with phase

velocity c0. From these effective parameters, the acoustic impedance of each section can be

expressed as 0cSKZ iiii == ρ .


Figure 2.15. A segment of a 1D periodic acoustic waveguide containing two sections.

A corrugated tube waveguide is an attractive structure for analysis because it is exactly

analogous to a 1D phononic crystal in the static case, yet provides some simplifications in the

dynamic case which will be discussed in Chapter 5. In the static case, the phase velocity in a

corrugated tube waveguide filled with a homogeneous fluid is identical throughout the tube

( 0201 cc = ) whereas a phononic crystal may have 0201 cc ≠ . However, if alternating corrugated

tube sections are filled with differing fluids, then the phase velocities may also be different.

2.3.4 Anomalous Doppler effects

The well-known Doppler effect occurs when a source, receiver, or reflector are non-

stationary. In a homogeneous medium with a reflector that is moving collinear with a source and

receiver, the transmitted frequency will be up- or down-modulated so that

( )01 cvff RTxRx += , 0cvR < , (2.22)

where Txf and Rxf are the transmitted and received frequencies, respectively, and Rv is the

velocity of the reflector. This scenario is depicted in Figure 2.16a). This equation can be

understood by thinking of the reflector as moving through additional wavelengths per second

when moving towards the source, and moving through fewer wavelengths per second when

moving away from the source.


vR

vR

a)

c01 c02

b)

Figure 2.16. Regular a) and anomalous b) Doppler effects in homogeneous materials and phononic crystals. In a), the reflected wave from a moving reflector within a homogeneous material is up- or down-shifted in frequency with respect to the incident wave depending on the velocity of the reflector and the phase velocity of the medium. In the two-component phononic crystal shown in b), the reflected frequency shift depends on which medium the reflector is in at a particular time, and thus is a spectrum of frequencies instead of a single frequency in the homogeneous case. The source is shown as a disk on the right of the material, and only the incident wave is shown.

When the material is periodic, as in a phononic crystal, the phase velocity changes

periodically as the reflector moves from one medium to another. Thus, the reflected frequency

also changes periodically, and a reflected spectrum results. Figure 2.16b) illustrates this concept.

In [17], the authors used a slightly different setup that demonstrated this concept with a

corrugated tube waveguide standing on-end. They created a moving source by dropping an MP3

player with a speaker through the tube. Though the phase velocity is constant in the tube,

application of a moving source to the corrugated tube wave equation (2.21) results in a very

similar situation. Their results demonstrated anomalous Doppler shifts that accurately matched

their predictions.

In resonant metamaterials with a negative phase velocity, the expected Doppler shift is

reversed, and is called the inverse or reverse Doppler effect. It is evident by considering a

constant but negative phase velocity in (2.22), which differs from the anomalous Doppler effects

in phononic crystals shown in Figure 2.16b) where the phase velocity modulates between two


positive values. The reverse Doppler effect has been explored experimentally in both

electromagnetic [96]–[99] and acoustic [100] metamaterials.

2.3.5 Applications

The applications of 1D phononic crystals for bulk acoustic waves (BAW) are mostly limited

to filters [101]–[103], in particular, for isolation of radio frequency (RF) and microwave filters

and resonators in communication devices. Imaging and waveguiding applications are restricted

to 2D and 3D structures since wave refraction does not occur in one dimension. Bulk acoustic

wave filters are 1D phononic crystals designed to have a band gap at undesirable frequencies.

They can be fabricated directly underneath sensitive RF components on an integrated circuit (IC)

to help isolate the component from vibrations.

There has also been investigation into one dimensional surface acoustic wave (SAW)

phononic devices [104]–[107]. Surface acoustic wave transducers often take the form of

interdigitated piezoelectrics deposited on a substrate that resemble a phononic crystal, and their

transmission properties are explored in [105]. The transmission of Lamb waves (flexural and

extensional waves) through 1D periodic structures has also been investigated [106][107]. There

have also been interesting experiments that use acousto-optic (AO) interactions that allow a

SAW to modify the properties of periodic photonic devices (see for example [108]).

- 32 -

Chapter 3

STATIC PHONONIC CRYSTALS IN TWO AND

THREE DIMENSIONS

The basic concepts underlying phononic crystal effects in two and three dimensions are

described. Like their 1D counterparts, they also show band gaps and dispersion, but in addition,

they can also exhibit anomalous refraction, which offers the potential for unique focusing

capabilities. All of these effects and their underlying causes are described. Finally, we describe

the experimental design and characterization of a 2D crystal that exhibits band gaps and negative

refraction.

3.1 Phononic crystal theory and effects

The same principles of acoustic wave scattering leading to phononic crystal effects as presented

in one dimension also apply to two and three dimensional crystals. Constructive and destructive

interference leads to band gaps, and the periodicity of the system causes dispersion. However,

since we are no longer restricted to one dimension, a simple mathematical description of these

effects, such as that given for dispersion in Section 2.3.2, is usually not possible in higher

dimensions. These effects must be explored using a more complicated method, like the FDTD or

MST methods described in detail in Chapter 4.

3.1.1 Acoustic waves in two and three dimensions

Although the propagation of acoustic waves in two and three dimensions is similar to that

described in Section 2.1, there are some notable differences. One may still envision a simplified

model of differential volumes of material represented as masses and springs, as illustrated in

Figure 3.1 for two dimensions. Longitudinal waves are still supported in both solids and fluids.

However, there is now the possibility of a wave that propagates with displacements

perpendicular to the direction of propagation, called a shear or transverse wave. Shear waves will

propagate in solids but are not supported in inviscid fluids since shearing motions do not apply

forces to neighbouring differential volumes of material. Though shear waves can and do occur in

real 1D phononic crystals consisting of solids, they are usually not considered when the direction

of wave propagation is normal to the material interfaces, as was assumed in Chapter 2.

Chapter 3. Static Phononic Crystals in Two and Three Dimensions 33

Figure 3.1. Elastic wave propagation in two dimensions. The centre differential volume of material is displaced from its equilibrium position by an external force, indicated by a solid arrow. The motion of this material applies forces to neighbouring differential volumes of material in both longitudinal and transverse directions. The longitudinal direction is collinear with the direction of wave propagation and results in a wave with a pushing and pulling movement. The transverse direction is perpendicular to the direction of wave propagation and results in a wave with a shearing motion. Inviscid fluids do not support shear waves, but solids do.

The phrase “acoustic wave” can refer to any mechanical displacement wave in any

propagation medium. The phrase “elastic wave” is usually reserved for waves in solids that have

both longitudinal and transverse components. These waves cannot be described by a simple

scalar velocity potential, which can be seen by the following properties of vector calculus. The

acoustic particle displacement is given by the vector field ( )t,ru , and may be written as

( ) ( ) ( )ttt tl ,,, rururu += , (3.1)

where lu and tu are the longitudinal and transverse field components, respectively. The

longitudinal component of the field satisfies the condition is 0=×∇ lu and the transverse

component satisfies 0=⋅∇ tu [66]. Since the scalar velocity potential, φ , is defined as φ∇=u ,

then the transverse component must be zero since the curl of the gradient of any scalar field is

zero, ( ) 0=∇×∇ φ . Therefore, a scalar velocity potential is insufficient to describe a vector

displacement field containing transverse wave components [109].

The presence of transverse waves in solids complicates the analysis of phononic crystals

because of an effect called wave mode conversion. Consider a 2D or 3D phononic crystal

consisting of solid scatterers in a fluid. Some of the energy of the longitudinal waves originating

in the fluid external to the phononic crystal incident on the scatterers may be converted into

transverse waves within the scatterers. Similarly, some of the transverse wave energy exiting a

scatterer may be converted back into longitudinal waves. For these reasons, the models and


simulation methods used to analyze solid-fluid or solid-solid phononic crystals must account for

the vector nature of the acoustic waves.

Scattering

Acoustic wave scattering in two and three dimensions is much more complicated than in one

dimension as was illustrated in the scattered field for two sound-hard cylinders presented in

Figure 1.4. The total field consists of the incident and scattered fields as shown in Figure 3.2,

which was generated using multiple scattering theory. The scattered field profile for a single

scatterer is highly dependent on the wavelength of the incident field, and the overall transmission

properties of multiple scatterers are highly dependent on the spacing between the individual

scatterers.

Figure 3.2. Acoustic wave scattering off an infinitely dense cylinder in two dimensions. The total field consisting of the incident and scattered fields is shown in a). The complexity of the scattered field is evident in both the far-field b) and near-field c). The incident field d) wavelength is on the same order of magnitude as the scatterer diameter.

3.1.2 Band gaps and waveguides

The transmission spectrum for two and three dimensional phononic crystals will contain band

gaps under many circumstances. They occur at frequencies that result in destructive interference

between the incident and scattered fields. One additional consideration is that there is a


directional element to wave propagation in these crystals that was not present in 1D, and so a

frequency band gap may only be present in particular propagation directions. When the band gap

encompasses all possible directions of propagation, it is called a complete band gap.

A waveguide may be created in a phononic crystal at band gap frequencies that would

otherwise not propagate within the crystal. This is achieved by removing the scatterers from one

end of a crystal to the other along a path for which the wave will be guided. The wave will be

confined to propagate within the spaces of the missing scatterers and will not be able to enter the

phononic crystal. Another similar effect called localization occurs when scatterers are removed,

but there is no entrance or exit to the void. Should a wave be excited within the void, it will

remain stationary, or localized, and appears as a standing wave.

There has been and continues to be a vast amount of research pertaining to phononic crystals

in two [110]–[163] and three [15][16][89][164]–[174] dimensions. In two dimensions, analyses

have included a simple row of cylinders [110][111] to more comprehensive studies of 2D

phononic crystal properties, including the effects of scatter shape and size, the lattice shape,

material phase (solid or fluid), and material properties on the band gap width and centre

frequencies [112]–[129]. Other research has extended these general conclusions to include the

effects of viscosity [130] and magneto-electro-elastic materials that have coupling of the

magnetic, electric, and mechanical displacement fields [131]. Other groups have combined slabs

of differing phononic crystals to create very broadband reflectors [132][133]. An interesting

example is a sensor created by allowing holes in a phononic crystal lattice to fill with an

unknown liquid. The shift in its band gap properties helps to determine the properties of the

unknown liquid [134]. Bulk and bending wave modes for various phononic crystal plates and

slabs have also been extensively investigated [135]–[140]. The application of 2D phononic

crystal principles has been extended to Lamb waves [141]–[147] and surface acoustic waves

[148]–[152]. As well, there have been numerous simulations and demonstrations of waveguiding

and localization using 2D phononic crystals [153]–[162]. In particular, one group successfully

demonstrated the use of a 2D phononic crystal waveguide as a frequency demultiplexer [163].

Three dimensional phononic crystals have also been studied [15][16][89][164]–[173]. In

particular, phononic crystals were created and well characterized in [15] and [16] and

demonstrated many interesting effects, such as band gaps and negative refraction. Similar studies

of 3D phononic crystal properties as in the 2D case have been performed investigating how their

design parameters affect the band gap properties [89][164]–[171]. An investigation of the surface

acoustic waves present on a 3D phononic crystal has also been performed [172]. One particularly

unique application of the band gaps in phononic crystals was in the design of a crystal with a


band gap at thermal phonon frequencies [173]. Consequently, it demonstrated very low thermal

conductivity. Waveguiding through 3D phononic crystals was explored in [174].

3.1.3 Dispersion and refraction

Dispersion occurs when the phase velocity depends on the temporal frequency of a wave.

Acoustic wave dispersion in two and three dimensional phononic crystals arises from the spatial

periodicity of the system, just as in the one dimensional case described in Section 2.3.2.

However, there is now a directional component to both the group and phase velocities. As

previously discussed, wavevectors beyond a particular magnitude for a given direction are

aliased back into smaller wavevectors because of the periodicity of the system, which is called

“band folding”. Dispersion and refraction are intimately connected in two and three dimensional

phononic crystals, as will be illustrated.

Wave refraction occurs when the direction of propagation of a wave changes due to a change

in the phase velocity. It is important in phononic crystals because of their unusual dispersion

properties, which can lead to anomalous refraction. To consider wave refraction at the interface

between a normal homogeneous medium and a phononic crystal, it is first important to

understand wave refraction between two regular isotropic media. In a regular isotropic medium

with phase velocity 0c , a continuous plane wave of a given frequency 0ω will have a wavevector

rk ˆ00 k= , (3.2)

where 000 ck ω= is the magnitude of the wavevector and r is a unit vector in the direction of

propagation. When the wave is incident upon an interface with a second material having a phase

velocity 1c , there are a normal and a tangential component of the wavevector with respect to the

interface. Thus,

||||000 ˆˆ rrk kk += ⊥⊥ , (3.3)

where ⊥⊥r0k and ||||0 rk are the normal and tangential components, respectively. Also note that

2||0

2000 kkk +== ⊥k since the components are orthogonal.

To find the wavevector within the second material, 1k , three criteria must be met at this

interface: the tangential components of the wavevector must be equal, the magnitude of the

wavevector must correspond to the phase velocity within each medium, and the direction of

group velocity normal to the interface must be equal. From the first criterion, we have

||1||0 kk = . (3.4)

The second criterion is


101 ck ω= . (3.5)

And finally, the last criterion is satisfied because the group and phase velocity are parallel in

both media. Combining (3.4) and (3.5), and noting that 2||1

211 kkk += ⊥ , the normal component

of the wavevector is

2||02

1

20

1 kc

k −=⊥

ω. (3.6)

The results of this procedure for two materials is illustrated in Figure 3.3, where 5.110 =cc .

In the example, 0θ was chosen to be 30º. Note that the tangential components of the wavevectors

are equal, and that the magnitude of 1k has been scaled appropriately. The resulting angle, 1θ is

19.5º, which is in agreement with Snell’s law.

⊥r

||r

0k 1k

⊥0k

||0k

⊥1k

||1k

0θ 1θ

0c 5.101 cc =

Figure 3.3. The wavevectors at the interface between two isotropic media. The components of the wavevectors are normal and tangential to the interface and indicated as ⊥ik and ||ik , respectively. The tangential components of the

wavevectors are equal as is required for the wave to be continuous at the interface. The medium on the left has a phase velocity 1.5 times that of the medium on the right, which accounts for the increased wavevector on the right. There are more radians per metre on the right because of the slower phase velocity, and thus it has a longer wavevector. As a consequence, the angle of incidence changes from 0θ = 30º to 1θ = 19.5º, which is in accordance with

Snell’s law.

Another means of illustrating wave refraction at this interface is to use equifrequency curves,

as shown in Figure 3.4. These curves are generated by plotting k for a given 0ω in all possible

directions. For the homogeneous media under consideration, the curves are circles since the

magnitude of k is not a function of propagation direction. This may not be the case in phononic

crystals, as will be illustrated. The radius of a circle is the magnitude of k for a given 0ω in that

medium. The basis vectors with respect to the interface are also shown for clarity. To determine


the wave refraction from the first to second medium, the initial wavevector, 0k , is plotted. A line

is drawn marking the tangential component of this vector, and where it intersects the second

curve indicates the value of 1k . It is noteworthy that there are two values of 1k that will satisfy

the tangential matching condition, but one of them does not obey causality. Since the group

velocity is parallel to the wavevector in regular media, we must choose the wavevector on the

right since the one on the left would indicate that energy is somehow emanating from within the

second medium.

0k

1k

⊥r

||r

Figure 3.4. The equifrequency curves used to calculate refraction at an interface between two isotropic media. The curves are circles because the magnitude of the wavevectors are not functions of propagation direction. First, the two curves are drawn with the curve for second medium having a larger radius because there are more radians per metre for a given temporal frequency than in the first medium. The tangential components of the wavevectors are matched by drawing a line (the dashed line) marking the magnitude of the tangential component of 0k . Where

this line intersects the second curve denotes the location of 1k . There are two

possible choices, and the incorrect choice for 1k is shown with an ‘x’ though it as it does not obey causality. The correct choice is pointing to the right, away from the interface.

In a periodic material, the equifrequency curves are not necessarily circles, meaning that

wavevectors may have different magnitudes in different directions for a given 0ω . Also, the

group velocities may not be collinear with the wavevector. For an excellent visualization of these

properties, see Chapter 2 in [75]. The group velocity is

( )kc ωkg ∇= , (3.7)


or to state this in words, it is the gradient of the surface of ω as a function of k. For the

homogeneous case described, the gradient is in the same direction as k, since as the magnitude of

k increases, the radius of the equifrequency circle increases and k elongates in the same

direction. As a corollary, if the equifrequency curve is not a circle, the group velocity will not be

collinear with k.

M MX

Reduced Wave Vector

Transverse

Longitudinal

c)

M

X

a)

X

Mb)

Figure 3.5. Two sets of equifrequency and corresponding dispersion curves. Panels a) and b) show the first Brillouin zone for a fictitious 2D phononic crystal with a square lattice created for illustration purposes (see Section 4.4.1 for the construction of this zone). Each curve of a particular colour represents the set of allowed wavevectors for a given temporal frequency, whose colour scale is shown on the left in c). Panel a) shows the curves for the first longitudinal band and b) shows the curves for the first transverse band. The dispersion plot in c) is created by traversing from Γ to X to M and back to Γ , as shown by a red dashed arrow in a) and repeated on the bottom axis in c).

An example of the equifrequency and dispersion curves for a fictitious two-component

square lattice phononic crystal is shown in Figure 3.5. The equifrequency curves in a) and b)

correspond to the first longitudinal and transverse bands, respectively, shown in c). The colour of

each curve corresponds to the frequency colour bar in c). The dispersion plot is created by


following the path indicated by the red arrow in a) and redrawn on the bottom axis of c) for

clarity. Note that the curves are circles for low frequencies and become increasingly less circular

as the frequency increases. As a consequence, the group velocity becomes increasingly less

collinear with the wavevector.

In Figure 3.4, if the group velocity was antiparallel, or at least had an opposite sign in the

direction normal to the interface, then the alternative choice for 1k would have been the correct

choice, corresponding to negative refraction. This situation can occur when the equifrequency

curves contract towards the origin with increasing frequency. Just such a situation is shown in

Figure 3.6 which illustrates the calculated dispersion curves for a phononic crystal made up of Ni

rods in an Al host. The crystal is a square lattice with a filling fraction of 0.5. Outlined in red is a

band whose slope is decreasing as the magnitude of the wavevector increases leading to negative

refraction.

M MX

Reduced Wave Vector

0

10

20

30

40

4 kHz Band Gap

Figure 3.6. The dispersion curves for a 2D phononic crystal. The crystal is composed of a square lattice of Ni rods in an Al host spaced 10 cm apart and having a filling fraction of 50%. There is a complete 4 kHz band gap present between the first and second groups of bands. An area that displays negative refraction is outlined with a red dashed oval and is identifiable because the wavevector magnitude decreases as the frequency increases.

We have not used the term “negative refractive index” in the context of phononic crystals

because that particular value applies to homogeneous or effectively homogeneous media.

Instead, we simply refer to negative refraction. One must ensure that negative refraction is

actually occurring and that it is not just an artefact of the periodicity of the system. An excellent

discussion about the meaning of negative refraction in photonic crystals is given in [175], and the

discussion also applies to phononic crystals. Also one must be careful to consider the role of


diffraction in negative refraction experiments, since the diffraction pattern of a simple reflector

alone will contribute significantly to the appearance of focusing [176].

5.101 cc =0c 0c 1c = 5.10c

Figure 3.7. Wave refraction under normal and negatively refracting conditions. A normal interface is shown in a), and a negatively refracting (NR) interface is shown in b). The phase is indicated by the wavevector, k, and the energy flow is indicated by the dashed vector, S. The transverse component of the wavevector at the interface is also shown. The additional condition that the phase and energy are antiparallel in an NR material leads to the negative refraction observed at these interfaces.

The unusual situation of negative refraction is illustrated in Figure 3.7, which shows wave

refraction at a normal interface in a), and at a normal to negatively refracting interface in b).

Thus, unlike regular materials, a planar slab of negative refractive material can be used to

refocus a point source onto an image plane, as depicted in Figure 3.8. Whereas regular lenses

require a curved surface to accomplish focusing, negative refraction creates an internal and

external focus with a simple slab.

Figure 3.8. An illustration of refocusing in an NR slab. In a slab of normal material, shown in a), rays from a point source continue to diverge, but at a different rate within the material as determined by the ratio of phase velocities. In a slab of NR material, the rays undergo negative refraction upon entering and exiting the material, creating an internal and an external focus (after Pendry [2]).

The dispersion of waves propagating within phononic crystals has been extensively studied

in two [9][177]–[186] and three [15][16][89][177] dimensions. Negative refraction has been


theoretically and experimentally explored in 2D crystals [9][18][177]–[182]. Highly directional

wave mode coupling has accomplished by choosing a source frequency very close to a band edge

with only one allowed wavevector [183]. A unique time-reversal scheme for wave focusing with

a phononic crystal has been presented [184]. Interestingly, the effective aperture of the phononic

crystal is larger than the length of its surface, possibly because of the increased ray path lengths

due to multiple scattering. The dispersion properties of a phononic crystal with piezoelectric

scatterers that accounts for the E-field coupling have been investigated [185]. A phononic crystal

with spatially varying design parameters has even been shown to exhibit the same waveguiding

properties as a gradient index lens [186]. The dispersion and focusing properties of three

dimensional crystals have also been investigated [15][16][89][177].

3.2 Phononic crystal design and experimental results

We designed, fabricated, and characterized a two dimensional phononic crystal. The main

purpose of this exercise was to demonstrate some of the interesting phononic crystal effects,

including band gaps and negative refraction. It also served as a means of verifying that the

phononic crystal simulators described in Chapter 4 matched the experimental results. Also, many

of the considerations and difficulties encountered in the process of creating and characterizing a

phononic crystal have been illuminated through this exercise. The transmission and dispersion

properties of the crystal have been characterized in the same manner as described in [187].

3.2.1 Crystal design

There were a number of constraints that helped to determine the design parameters of the

phononic crystal. First, A 2D structure was chosen because a 1D crystal would not display wave

refraction, and a 3D structure was too difficult to fabricate in-house. We wanted to use the

submersion tank measuring apparatus in our lab, so water was chosen as the host material. This

also simplified the analysis because we did not need to measure transverse waves since fluids do

not support transverse waves [66]. A cylindrical scatterer shape was chosen because of its

simplicity. The remaining design parameters included:

• Desired frequencies of operation

• Scatterer materials

• Filling fraction

• Crystal arrangement

• Number of layers

• Other ancillary concerns


Frequency of operation

The desired range of frequencies for which the phononic crystal will exhibit interesting

effects determines, in part, the cell spacing and overall crystal size. Choosing a lower frequency

range means that the crystal will be easier to manufacture and that manufacturing tolerances will

be smaller compared to the wavelengths of interest. However, the size of the operating

environment limits the maximum size. In this case, the submersion tank is the operating

environment.

The desired frequency range was primarily set by the available transducers, which included

250 kHz, 500 kHz, 1 MHz, and several other piston transducers of higher frequencies. These

three sets of transducer pairs were characterized by placing them in the submersion tank in a

through-transmission configuration [66]. A Panametrics-NDT model 5800 pulser/receiver was

used to generate and receive an impulse. A Tektronix TDS 3012B oscilloscope was used to

capture the time-domain transmission waveforms which were then post-processed in MATLAB.

The received waveforms were averaged over many transmissions to reduce the effects of noise.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0

-10

-20

-30

-40

-50

-60

-70

Norm

alized Amplitude (dB)

Frequency (MHz)

500 kHz

1 MHz

250 kHz

Figure 3.9. The frequency response for three sets of transducer pairs. The transducers were placed in pairs a through-transmission configuration approximately 30 cm apart.

The results of the transducer characterization are shown in Figure 3.9. The 1 MHz and

500 kHz transducers displayed excellent characteristics, whereas the 250 kHz transducers were


very noisy. It was later determined that the 250 kHz transducer had two issues: They were

primarily designed to couple to transverse waves in solids, and one of the transducers had a crack

on its surface. For these reasons, the 250 kHz transducers were not longer used. The dashed line

in the figure represents the -6 dB point, and from this we can determine the transducer properties

summarized in Table 3.1. Therefore, the desired range of frequencies for the phononic crystal

would be within the 1 MHz and 500 kHz transducer response curves.

Table 3.1. Transducer pair parameters. Parameter 1 MHz Transducer 500 kHz Transducer

0 dB Frequency 1140 kHz 456 kHz

Mean Centre Frequency 1064 kHz 417 kHz

Lower -6 dB Frequency 816 kHz 228 kHz

Upper -6 dB Frequency 1312 kHz 606 kHz

Bandwidth 46.6% 90.6%

Aperture Diameter 12.7 mm 28.6 mm

Scatterer material

As described in Chapter 1, the scatterer size and placement in a phononic crystal dramatically

affects the crystal properties. The scatterer material choice is important to a lesser degree. The

most significant factor in choosing the scatterer material is that its acoustic impedance and phase

velocity be highly contrasting with the host material. Since we chose water as our host material,

any stiff solid would have drastically different properties. It was recommended that we use

stainless steel alloy 303 (abbreviated as “stainless 303”) because of its ability to resist corrosion

in water. Table 3.2 lists the material properties of the host and scatterer materials. The stiffnesses

given correspond to the Voigt notation for an isotropic material [75]. The phase velocity and

characteristic acoustic impedance were calculated from the density and bulk modulus.

Filling fraction

We decided to use a filling fraction of 58.0=η based on the 2D crystals presented and

characterized in [9][174][177]. This filling fraction was chosen to achieve the maximum band

gap width for a 2D square lattice.


Table 3.2. Host and scatterer properties. Parameter Host Scatterer

Name Water Stainless 303

Density ( ρ ) 998 kg·m-3 8030 kg·m-3

Stiffness

(C11) 2.2 GPa 231.6 GPa

(C12) 2.2 GPa 77.2 GPa

(C44) 0 GPa 77.2 GPa

Characteristic Acoustic Impedance (Z0)

1.48 MRayl 43.12 MRayl

Longitudinal Velocity (c0) 1485 m·s-1 5370 m·s-1

Shear Velocity (cs) N/A 3101 m·s-1

Crystal arrangement

We chose to use a simple square array as it is well-established that this structure exhibits

band gaps and negative refraction as discussed in Sections 3.1.2 and 3.1.3 and the references

therein. Both the ΓX and ΓM directions will demonstrate band gaps and negative refraction if the

crystal is properly designed, so we chose to fabricate the crystal in the ΓM direction.

As for lattice spacing, we decided to set this as a function of the scatterer radius and desired

filling fraction, so that

η

π 2R

a = , (3.8)

where a is the lattice spacing, R is the cylinder radius, and η is the filling fraction. We chose to

use 1/8th inch diameter rod (diameter tolerance = ±0.0015 inch) which was determined to

resonate in the range applicable to our available transducers, resulting in R = 1.575 mm. Thus,

the lattice spacing was a = 3.666 mm. The available manufacturing process had a tolerance of

0.0254 mm. These values are summarized in Table 3.3.

Number of layers

Ideally a phononic crystal would be of infinite extent in all directions, which is often an

assumption made in their theoretical analysis. However, the reality of fabricating a phononic

crystal is that a finite number of rows and columns must be chosen. Here we mean “rows” and

“columns” to be the number of planes of cylinders parallel and perpendicular to the direction of

transmission, respectively. The number of rows directly affects the degree of attenuation within


band gaps, as was illustrated in Figure 2.5 for one dimension. The number of columns affects the

degree to which edge effects and diffraction play a role in the received signals. To minimize

these effects, the number of columns in the crystal should be maximized. Also, the FDTD

simulations of the crystal assumed that it was infinitely repeating in the column direction, but

with a finite number of rows. Another consideration is the time and cost associated with

machining hundreds of cylinders out of long segments of rod. Weighing all these factors, it was

determined that 37 columns and 11 rows for a total of 204 cylinders would achieve a good

balance between these competing requirements.

Table 3.3. Crystal lattice parameters. Parameter Specified Value

Cell spacing (a) 3.666 mm

Scatterer Radius (R) 1.575 mm

Filling Fraction (η) 0.58

Lattice Type Square

Lattice Direction ΓM

Other ancillary concerns

In addition to the basic design parameters of the crystal, there are other factors that needed to

be considered. These included the cylinder mounting structure and the cylinder length. Since the

aperture of the 500 kHz transducer was the largest with a diameter of 28.6 mm, it was decided

that 50 mm long cylinders were sufficient to avoid excessive reflections from the mounting

structure.


Completed design

Figure 3.10. Top view of the phononic crystal design. The crosses located in the centres of the scatterer locations are drill guides.

A top view of the 2D phononic crystal design is illustrated in Figure 3.10 with the most

important dimensions labelled. The crosses present within the scatterer locations are drill guides

and are to aid the fabrication process. The finished fabricated crystal is shown in Figure 3.11.

Note that there is an additional region of acrylic jutting out in the top part of Figure 3.11a) that is

not present in Figure 3.10. This extra material was used as a location for a mounting hole into

which a threaded rod was placed to hold the crystal in position in the submersion tank.

Figure 3.11. Top a) and side b) view of the fabricated phononic crystal. There are five missing cylinders on either end of the crystal which occurred because of material loss during the fabrication of the cylinders. It was decided that their presence would not significantly affect the transmission or dispersion properties of the phononic crystal.


3.2.2 Experimental methods

Figure 3.12. Experimental configuration. In a), both the transmit (Tx) and receive (Rx) transducers are stationary. This was used for obtaining the transmission spectrum and dispersion curves. In b), the receive transducer is replaced with a hydrophone that is raster-scanned in the plane of the cylinders. This configuration was used to image the acoustic field.

Two experimental configurations were used to characterize the crystal. The first, depicted in

Figure 3.12a) was used to gain the transmission spectrum and dispersion properties of the crystal.

It used identical 500 kHz or 1 MHz transducers as both the transmit and receive element, and

they were located approximately 15 cm away from the crystal faces. The other configuration is

shown in Figure 3.12b). It is identical to the previous configuration except that the receive

transducer is replaced with a hydrophone that was raster-scanned in the plane of the cylinders. In

both cases the transducers and crystal were fixed spatially within the tank by supporting rods.

3.2.3 Results

The transmission spectrum

Using the configuration in Figure 3.12a), a reference pulse was sent into the tank and

recorded without the crystal. With the crystal in place, the transmitted pulse was recorded. The

spectrum of the received signal normalized by the spectrum of the reference signal reveals the

transmission spectrum of the crystal, and this is shown in Figure 3.13. The simulated spectrum is

the result of an FDTD simulation of a broadband pulse propagating through the crystal. Good

agreement between the primary features of the two spectra can be seen over the range from 200

to 800 kHz. Much beyond this, the wavelength approaches the size of manufacturing defects and

the crystal appears to be increasingly disordered which may account for the discrepancies. In

addition, a minor displacement of the band gap edges between the two spectra should be noted.


Transmission (dB)

Figure 3.13. Measured and simulated transmission spectrum of the phononic crystal. The measured spectrum is composed of the two spectra obtained from using both the 500 kHz and 1 MHz transducers and piecing the results together. The simulated spectrum was created from an FDTD simulation of a broadband pulse propagating through the crystal.

The dispersion properties

The dispersion properties of the phononic crystal can be determined by examining the phase

shift between the reference and crystal spectra. This uses the same data obtained for the

transmission spectrum experiment. Using the phase shift of the measured signal, the magnitude

of the wavevector as a function of angular frequency is

( ) ( )

0cLk

ωωφω +

∆= , (3.9)

where ( )ωφ∆ is the phase difference between the spectra at angular frequency ω and L is the

path length from one face of the crystal to the other. Values of k greater than aπ are folded

back into the first Brillouin zone as in [187].

The measured and simulated dispersion properties of the crystal are shown in Figure 3.14.

Since the crystal was created in the ΓM direction, the measured dispersion curve does not give

information about dispersion in the ΓX or XM directions. The FDTD simulation produces a

spectrum for every applied wavevector, and the peaks of that spectrum are indicated as red

diamonds in the figure. However, some peaks are of a greater magnitude than others and

represent the propagation modes that carry energy. These are outlined in blue for the modes

below 600 kHz. The dispersion curves calculated from the transmission spectrum using (3.9) are

shown in dashed green for the first two bands and band gap. It matches well with the predicted

dispersion. A portion of the transmission spectrum of Figure 3.13 is shown on its side on the

right of the dispersion curves to aid in identifying the band gaps. The first three band gaps are

indicated with red hash marks and are aligned to the FDTD results.


Frequency (kHz)

-40

-200

-60

Figure 3.14. Measured and simulated dispersion curves of the phononic crystal. The simulated dispersion curves were created from an FDTD simulation of the crystal for various applied wavevectors. The peaks of the frequency response for each wavevector are plotted as red diamonds. The strongest propagation modes from 0 to 600 kHz are outlined in blue. The dispersion calculated from the measured transmission spectrum of the crystal is shown in green for the first two bands and band gap. The transmission spectra are repeated on the right hand side of the figure to aid in visual comparison. The first three band gaps, aligned to the FDTD simulation, are indicated with red hash marks.

The slope of the wavevector can be seen to be decreasing with respect to frequency in the

second transmission band. It is in this frequency region that negative refraction should be

observed.

The acoustic field

Using the configuration in Figure 3.12b), the acoustic field in the plane of the cylinders was

measured using a membrane hydrophone. A Tektronix AWG520 pulser connected to a Ritec

RPR-4000 power amplifier was used to transmit an ultrasound pulse. The power amplifier was

used to increase the transmitted signal amplitude to achieve an acceptable SNR for the

hydrophone used. The hydrophone, which had a 40 µm diameter sensitive region, was a Sonora

Medical Systems 804 membrane hydrophone. It was connected to a Panametrics-NDT model


5670 preamp and then a National Instruments 5112 digitizer. A broadband pulse was sent with

the crystal in place and was recorded at regularly spaced grid points as detailed in Table 3.4. The

x and z directions are parallel and perpendicular to the face of the crystal, respectively.

Table 3.4. Recorded acoustic field parameters Parameter Value

Number of waveforms averaged per location 100

Points per waveform (Nt) 8000

Time step (∆t) 250 ns

Waveform duration (Tmax) 2 ms

Extent in the x direction 80 mm

Resolution in the x direction (∆x) 1 mm

Total number of points in the x direction (Nx) 81

Extent in the z direction 60 mm

Resolution in the z direction (∆z) 1 mm

Total number of points in the x direction (Nz) 61

Total number of spatial locations sampled 4941

Distance from transducer to crystal 80 mm

Distance from crystal to edge of scan region 10 mm

SNR 42.6 dB

The received signals were then converted to the frequency domain, and a colour-coded image

of the recorded spatial field for each frequency was generated. Unfortunately, the power

amplifier introduced a parasitic signal into the transmitted waveform due to the switching on and

off of a gate signal. The gate signal was required to enable and disable the power amplifier

output and could not simply be left on. It must be switched on for each transmitted pulse, then

switched off afterward to limit the duty cycle of the amplifier. It was this switching action that

created the parasitic signal shown in red in Figure 3.15. Ideally it should be zero, but the

transmitted pulse, shown in blue, clearly contains elements of the parasitic gate signal.


Norm


Figure 3.15. The total and gate-only signals. The total transmitted signal includes both the desired input pulse and the unwanted gate signal. A time-domain window was used to isolate the desired signal and to reject the parasitic gate signal.

One strategy to eliminate the gate signal was to simply subtract it from the total received

signal. Unfortunately, slight phase errors in the two sets of signals made this approach

unsuccessful. The strategy we opted to use was to filter the received signals in the time-domain

with a Kaiser-Bessel window centred about the signal peak. This successfully rejected the gate

signal.

Based on the dispersion properties of the crystal, we expected to find negative refraction

occurring slightly above 400 kHz, near the top of the second transmission band. To identify this,

we attempted to observe refocusing of the acoustic field external to the crystal. The experimental

configuration is shown in Figure 3.16a). The red dot between the transducer and the crystal

indicates the natural focus of the transducer, located 68 mm from the transducer surface. The red

dashed lines emanating from the focus illustrate how negative refraction leads to the generation

of a focal region in the scan area. Figure 3.16b) shows the acoustic field at 420 kHz and a focal

region can clearly be seen, demonstrating negative refraction. The focal region is misaligned in

the image because of offset errors in the experimental setup. This focusing effect was only

apparent from 409 kHz to 428 kHz, and Figure 3.16c) shows an example of this limited

bandwidth. The field at 360 kHz does not show the focusing present in the above panel. A more


definitive proof of negative refraction would be to create a wedge prism and observe the beam

angles exiting the wedge; however, this was not a possible configuration given the design of our

crystal.

60 m

m

80 m

m

Figure 3.16. A diagram of the experimental configuration a), and the measured acoustic field at 420 kHz b) and 360 kHz c). The experimental configuration is shown in a) (not to scale). The scan area is shifted to the right of the centreline because of experimental configuration offset errors. In b) and c), the image colour represents the logarithm of the normalized magnitude of the pressure with respect to the field pressure maximum when no crystal is present.

3.2.4 Summary

We successfully designed, fabricated, and characterized a two dimensional phononic crystal.

Our crystal demonstrated band gaps at the frequencies predicted by our simulations, thus helping

to confirm that our simulators were properly implemented. The crystal also demonstrated

dispersion that led to negative refraction. The broadband acoustic field was measured and

decomposed into its temporal frequency components. Negative refraction leading to refocusing

of acoustic waves at 420 kHz was demonstrated.

- 54 -

Chapter 4

SIMULATION METHODS

In this chapter, the best established models for simulating the behaviour of periodic acoustic

systems are presented: all assume that the media is lossless. In particular, we describe the finite-

difference time-domain (FDTD), transmission matrix method (TMM), multiple scattering theory

(MST), and plane-wave expansion (PWE) approaches. Each of these methods was implemented

in MATLAB and C as a means of exploring both theoretical systems and the experimental

phononic crystal described in Chapter 3. In particular, a great deal of work went into

implementing the FDTD simulator since it was the benchmark against which our new time-

varying methods were compared. The TMM and MST methods form the basis of our new time-

varying models, and the PWE method was primarily used to generate equifrequency curves for

our 2D phononic crystal. Finally, some alternative methods are briefly presented along with a

short discussion regarding the amenability of various methods to incorporating time-varying

material parameters.

In two and three dimensions, the vector nature of waves in solid-solid or solid-fluid phononic

crystals must be accounted for. The FDTD and PWE methods account for transverse waves, but

the MST method as presented does not. It can be modified to account for the vector nature of the

waves by including additional field values and boundary conditions, but the principles remain the

same. For the sake of clarity, we restrict the MST method to considering scalar velocity

potentials.

4.1 The finite-difference time-domain method

The FDTD method numerically simulates the partial differential equations that govern the

propagation of waves within a propagation medium. The FDTD method has many benefits: The

resulting difference equations are amenable to software implementation; the results mirror what

would be observed in a laboratory scenario; transmission and reflection coefficients are easily

determined from the results; modelling a variety of structures is possible; any realistic input

waveform can be used. Some of the drawbacks are as follows: There is no closed-form analytic

solution to elucidate the simulation results; there are stability criteria that mandate grid resolution

limits which can significantly increase the simulation run time. Thus, we have used the

Chapter 4. Simulation Methods 55

numerical results from the FDTD method to verify the results from our new time-varying

methods.

Periodic acoustic structures can be investigated with the FDTD method by simulating one

cell of the structure and implementing periodic boundary conditions. These boundary conditions

translate the outgoing waves at one boundary to incoming waves at the opposite boundary with

an appropriate phase shift. The net result appears to be an infinite crystal made up of identical

cells with an applied wavevector. The time domain displacement of a randomly chosen point on

the simulation grid is recorded for the duration of the simulation. Afterwards, a Fourier transform

of the signal will reveal which wave frequencies correspond with the imposed wavevector. The

dispersion relation of the phononic crystal can be compiled by repeating this process for many

different wavevectors.

Non-periodic structures are simulated by applying a zero-wavevector ( 0=k ) to the same

discrete equations as in the periodic case. Two types of boundary conditions are used: Reflecting

and absorbing. The reflecting boundary conditions force the boundary displacement to zero,

which entirely reflects incident waves. The absorbing conditions are an approximation to

simulating a region of infinite extent, meaning that outgoing waves of all spatial and temporal

frequencies will exit the simulation boundaries without any of the energy being reflected. This

idealized situation is not entirely possible, and numerous schemes exist for approximating this

condition. These three boundary conditions (periodic, reflecting, and absorbing) allow the

simulation of laboratory-type setups so that experimental data may be compared with FDTD

simulations.

The flexibility of the FDTD method is evident by its pervasiveness in both acoustics and

electromagnetics. It has also been applied to the field of seismology to study models of wave

propagation through the earth [188][189]. It has already been extensively used to study phononic

and photonic crystals (see, for example [190][191]). It is very amenable to code parallelization

[169], which is particularly relevant considering the trend towards parallel computation and

multi-core CPUs. The method has been modified to improve its accuracy by using higher-order

equations [192], adjusting the shape of the underlying discrete grid [193], improving

compatibility with absorbing boundary equations [194], and accounting for dispersive media

[195].

We created an FDTD simulator in the C language that uses MATLAB as a front-end for

providing data to and analyzing data from the simulator. This configuration allows the versatile

data analysis of MATLAB with the speed of execution of compiled C code. Further performance

gains are possible by rewriting the C-code to run on a graphics processor, but that project is


outside the scope of this thesis. We wrote the simulator because the commercially available

acoustic FDTD simulators were not capable of simulating time-varying material parameters.

4.1.1 Derivation of a 2D algorithm

Here we derive the discretized governing equations for the interior region of an FDTD

simulator. We then apply various boundary conditions to mimic different circumstances,

including periodic, reflecting, and absorbing boundaries. Periodicity is assumed in the derivation,

but the end result is a set of discrete-time equations that also work for non-periodic simulations.

The following conventions will be used interchangeably in the derivation, which are used to

remain consistent with [35]:

( ) ( )yxxx ,, 21 ==x , (4.1)

( ) ( )yx kkkk ,, 21 ==k , (4.2)

( ) ( )yx uuuu ,, 21 ==u , (4.3)

where x is the Cartesian coordinate vector, k is the wavevector, and u is the Lagrangian acoustic

particle displacement from equilibrium. Furthermore, we consider only lossless, isotropic media

with a scalar density and compressibility. We choose to solve for the vector displacement field

since this will permit the simulation of transverse waves in addition to longitudinal waves, as

opposed to a scalar field simulation which can only incorporate longitudinal waves [66].

There are two sets of equations required to solve for the displacement field, as was the case

in Section 2.1. The first is Newton’s second law of motion and the second is the stress-strain

relationship of the material. For a very clear and extensive derivation similar to the following,

the reader is referred to pp. 41 of [75]. We begin with the density version of Newton’s second

law,

σu ∇=&&ρ , (4.4)

where σσσσ is the stress tensor, ρ is the mass density, and u is the vector displacement. This can be

rewritten explicitly as

( )( ) ( )

∑∂

∂=

∂

∂

n n

mnm

x

t

t

tu ,,2

2 xxx

σρ . (4.5)

The periodicity of the system is then enforced, meaning that the interior simulation region is

assumed to be infinitely tiled in both the x and y directions, which is the practical implementation

Bloch-periodic condition for a given wavevector k [95]. The Bloch conditions on the

displacements and stresses are

( ) ( )tUetu m

j

m ,, xx xk ⋅= , (4.6)


and ( ) ( )tSet mn

j

mn ,, xx xk⋅=σ . (4.7)

Substituting back into the original equation gives

( ) ( ) ( )∑

∂

∂=

∂

∂ ⋅⋅

n n

mn

j

m

j

x

tSe

t

tUe ,,2

2 xxx

xkxk

ρ . (4.8)

Expanding the derivative and cancelling the common exponential,

( ) ( ) ( ) ( )∑

∂

∂+=

∂

∂ ⋅⋅⋅

n n

mnj

mn

j

nmj

x

tSetSejk

t

tUe

,,

,2

2 xx

xx xkxkxk ρ , (4.9)

( ) ( ) ( ) ( )∑

∂

∂+=

∂

∂

n n

mnmnn

m

x

tStSjk

t

tU ,,

,2

2 xx

xxρ . (4.10)

Expanding the summation over two dimensions and changing the notation to match [35] gives

the first two equations in continuous form:

∂

∂++

∂

∂+=

∂

∂

y

SSjk

x

SSjk

t

Utyx

tyxtyx

tyxtyx

yx:,

12:,122

:,11:,

1112

:,1

2,ρ , (4.11)

and

∂

∂++

∂

∂+=

∂

∂

y

SSjk

x

SSjk

t

Utyx

tyxtyx

tyxtyx

yx:,

22:,222

:,21:,

2112

:,2

2,ρ . (4.12)

To discretize (4.11), the following discrete approximations to the continuous derivatives are

used. They are chosen to centre the finite differences when combined with the second set of

equations for Sij. See [196] for more information on central finite differences. Also, x, y, and t are

taken to be linear indices in the discrete case, meaning that pdiscretecontinuous pp ∆= , where p is one

of those parameters and p∆ is the space or time resolution. This convention is used to simplify

comparison of the continuous and discrete equations. The discrete approximations are

2

1:,1

:,1

1:,1

2

:,1

2 2

t

tyxtyxtyxtyxUUU

t

U

∆

+−=

∂

∂ −+

, (4.13)

2

:,111

:,11:,

11

tyxtyxtyx SS

S−+

= , x

tyxtyxtyx SS

x

S

∆

−=

∂

∂ − :,111

:,11

:,11 , (4.14)

and 2

:1,12

:,12:,

12

tyxtyxtyx SS

S−+

= , y

tyxtyxtyx SS

y

S

∆

−=

∂

∂ − :1,12

:,12

:,12 . (4.15)

Applying these approximations to (4.11) gives

tyxtyxtyxtyx

t

tyxtyxtyxyx

SKSKSKSKUUU :1,

122:,

122:,1

111:,

1112

1:,1

:,1

1:,1, 2 −−+−−+

−+

+++=

∆

+−ρ , (4.16)


where

∆±=±

ix

ii

jkK

1

2. (4.17)

Rearranging to solve for U1,

( ) 1:,1

:,1

:1,122

:,122

:,1111

:,111,

21:,

1 2 −−−+−−++ −++++∆

= tyxtyxtyxtyxtyxtyx

yx

ttyxUUSKSKSKSKU

ρ. (4.18)

Equation (4.12) may also be discretized using the following discrete approximations and by

noting that 2112 SS = , which is a consequence of the symmetries described subsequently:

2

1:,2

:,2

1:,2

2

:,2

2 2

t

tyxtyxtyxtyxUUU

t

U

∆

+−=

∂

∂ −+

, (4.19)

2

:,12

:,112:,

12

tyxtyxtyx SS

S+

=+

, x

tyxtyxtyx SS

x

S

∆

−=

∂

∂ + :,12

:,112

:,12 , (4.20)

2

:,22

:1,22:,

22

tyxtyxtyx SS

S+

=+

, y

tyxtyxtyx SS

y

S

∆

−=

∂

∂ + :,22

:1,22

:,22 . (4.21)

Applying these approximations to (4.12) and rearranging gives

tyxtyxtyxtyx

t

tyxtyxtyxyx

SKSKSKSKUUU :,

222:1,

222:,

121:,1

1212

1:,2

:,2

1:,2, 2 −++−++

−+

+++=

∆

+−ρ , (4.22)

( ) 1:,2

:,2

:,222

:1,222

:,121

:,1121,

21:,

2 2 −−++−+++ −++++∆

= tyxtyxtyxtyxtyxtyx

yx

ttyxUUSKSKSKSKU

ρ. (4.23)

The second set of equations required to solve for the displacement field is the stress-strain

relationship, which is coupled by the position-dependent elastic stiffness tensor c so that

( ) ( ) ( )∑

∂

∂=

nm n

mijmnij

x

tuct

,

,,

xxxσ . (4.24)

Enforcing the system periodicity as before,

( ) ( ) ( )∑

∂

∂=

⋅⋅

nm n

m

j

ijmnij

j

x

tUectSe

,

,,

xxx

xkxk (4.25)

Expanding the derivative, cancelling the common exponential, and noting that j is an index as

well as 1− ,

( ) ( ) ( ) ( )∑

∂

∂+= ⋅⋅⋅

nm n

mj

m

j

nijmnij

j

x

tUetUejkctSe

,

,,,

xxxx xkxkxk , (4.26)

( ) ( ) ( )( )

∑

∂

∂+=

nm n

m

mnijmnijx

tUtUjkctS

,

,,,

xxxx . (4.27)


As described in [75], many of the stiffness tensor terms are redundant due to energy

conservation rules. Following the Voigt notation (summarized on pp. 41 of [75]), the relevant

terms are shown condensed in the middle matrix. Assuming isotropic properties (a concept

defined in [66]) which permit cubic symmetry rules, the coefficients reduce to those shown in the

matrix on the right (pp. 92 of [75]):

Symmetry Cubic1112

4444

4444

1211

NotationVoigt 22262612

26666616

26666616

12161611

2222212212221122

2221212112211121

2212211212121112

2211211112111111

00

00

00

00

=

=

CC

CC

CC

CC

CCCC

CCCC

CCCC

CCCC

cccc

cccc

cccc

cccc

. (4.28)

Expanding the summation in (4.27) over two dimensions and changing the notation to match

[35] gives the continuous form of the stress components:

∂

∂++

∂

∂+=

y

UUjkC

x

UUjkCS

tyxtyxx,y

tyxtyxx,ytyx

:,2:,

2212

:,1:,

1111:,

11 , (4.29)

∂

∂++

∂

∂+=

y

UUjkC

x

UUjkCS

tyxtyxx,y

tyxtyxx,ytyx

:,2:,

2211

:,1:,

1112:,

22 , (4.30)

and

∂

∂++

∂

∂+==

x

UUjkC

y

UUjkCSS

tyxtyxx,y

tyxtyxx,ytyxtyx

:,2:,

2144

:,1:,

1244:,

21:,

12 . (4.31)

As before, these equations are discretized using discrete approximations. The resulting

discrete equations, adjusted to achieve central differences, are

( ) ( )tyxtyxx,ytyxtyxx,ytyxUKUKCUKUKCS

:1,22

:,2212

:,11

:,11111

:,11

−−+−++ +++= , (4.32)

( ) ( )tyxtyxx,ytyxtyxx,ytyxUKUKCUKUKCS

:1,22

:,2211

:,11

:,11112

:,22

−−+−++ +++= , (4.33)

and ( )tyxtyxtyxtyxx,ytyxUKUKUKUKCS

:,12

:1,12

:,121

:,2144

:,12

−++−−+ +++= . (4.34)

Thus, five discrete equations ((4.18), (4.23), (4.32), (4.34), and (4.33)) and five unknown

state variables ( 1U , 2U , 11S , 22S , and 12S ) fully specify the interior region of the FDTD

simulation. For a periodic simulation with a given wavevector, k, iK in (4.17) is complex and

thus the interior region state variables are complex. For a non-periodic simulation, 0=k and the

state variables are real.

4.1.2 Implementation

Offset grid

The discrete equations are calculated using an offset grid, as shown in Figure 4.1. This allows

central differences to be used in the approximated derivatives, which results in improved


accuracy of the simulation [196][197]. The material parameters are also located off-grid as

shown in the right-hand portion of the figure, which must be accounted for when considering

very fine structures in relation to the grid spacing.

Figure 4.1. The offset grid implementation of the FDTD state variables. On the Cartesian grid shown to the left, the black dots indicate variables located directly on-grid. That is, they are located at xm∆ and yn∆ , where m and n are integers.

The white dots indicate variables located off-grid in either the x, y, or both axes. That is, m and/or n are integers 2

1± . This arrangement permits the use of central

differences used in the discrete approximations.

The discrete equations for the FDTD state variables may now be adjusted to explicitly

describe the offset grid used in Figure 4.1, which yields the following set of equations and are

the same as presented in the Appendix of [35]. Note that the equations in [35] for ±iK are

missing an i (or j using the notation in this thesis), an error that was confirmed with the authors.

( )

1:,1

:,1

:,122

:,122

:,111

:,111,

21:,

1

2

21

21

21

21

−

−−++−−+++

−+

+++∆

=

tyxtyx

tyxtyxtyxtyx

yx

ttyx

UU

SKSKSKSKUρ (4.35)

( )

1:,

2

:,

2

:,

222

:1,

222

:,

121

:,1

121,

21:,

2

21

21

21

21

21

21

21

21

21

21

21

21

2 −++++

+−++++−+++

++

+++

−+

+++∆

=

tyxtyx

tyxtyxtyxtyx

yx

ttyx

UU

SKSKSKSKUρ (4.36)

( ) ( )tyxtyx,yxtyxtyx,yxtyxUKUKCUKUKCS

:,

22

:,

2212:,

11:,1

1111

:,

1121

21

21

21

21

21

21 −+−++++−++++

+++= (4.37)

( ) ( )tyxtyx,yxtyxtyx,yxtyxUKUKCUKUKCS

:,

22

:,

2211:,

11:,1

1112

:,

2221

21

21

21

21

21

21 −+−++++−++++

+++= (4.38)

( )tyxtyxtyxtyxx,ytyxUKUKUKUKCS :,

12:1,

12

:,

21

:,

2144

:,

1221

21

21

21

21

21

−+++−−++++++++= (4.39)

The offset grid is of no practical consequence in the actual software implementation. It only

requires that the user is aware that the material parameter inputs and the field value outputs

correspond to points located on the offset grid.


Boundary conditions

In an FDTD simulation using the discrete equations above, the interior region may be

calculated entirely from provided initial conditions and material parameters. For the simulation

to be meaningful, two other pieces of information must be provided, which are the source and

boundary conditions. With respect to boundary conditions, there are three useful types that were

implemented in the FDTD simulator: Periodic, reflecting, and absorbing.

Periodic boundary conditions

Periodic boundary conditions take the field values from one end of the simulation region and

apply them to the other with a phase shift determined by the applied wavevector. The reason for

using periodic boundaries in a simulation is to make the interior simulated region appear to

repeat infinitely in the dimension of the periodic boundaries. For example, a phononic crystal can

be made to appear infinite in all directions by setting all four boundaries to be periodic. The

dispersion relation of such an infinite crystal can be discovered by applying a wavevector to the

simulation and measuring the frequency spectrum of a point in the interior. As an example of a

use of periodic conditions in only one dimension, the crystal that was constructed and described

in detail in Section 3.2 was simulated by assuming it had an infinite width and a finite depth,

when in reality it had 37 columns of width and 11 rows of depth. The simulated and measured

transmission spectrum matched quite well as was shown in Figure 3.13.

In each displacement and stress field, there is enough information to calculate two of the

boundaries, and the other two must be copied using periodic boundary conditions. Table 4.1

summarizes the regions that can be computed and the boundaries that must be copied. In the

table, X and Y represent the total x and y distances being simulated and M and N represent the

maximum x and y indices, respectively, so that xMX ∆= and yNY ∆= . The boundaries are

copied from the opposite side of the simulation region with a phase shift that depends on k and

the size of the simulation region. The unspecified corner values can be left undefined since they

do not influence any other cells.


Table 4.1. Calculable simulation regions and periodic boundaries.

Field Required Region Calculable

Indices

Boundaries

Boundary Conditions

U1

Mx

2,

Ny

2

yMXjky UeU x ,1

,11

−= NxYjkx UeU y ,

11,

1−

=

U2

1

1

−Mx ,

1

1

−Ny

yXjkyM UeU x ,12

,2 =

1,2

,2

xYjkNx UeU y=

S11/S22

1

1

−Mx ,

Ny

2

yXjkyM SeS x ,122/11

,22/11 =

NxYjkx SeS y ,22/11

1,22/11

−=

S12

Mx

2,

1

1

−Ny

yMXjky SeS x ,12

,112

−= 1,

12,

12xYjkNx SeS y=

The flow of the FDTD algorithm with periodic boundaries is depicted in Figure 4.2.

Figure 4.2. FDTD algorithm flow for periodic boundaries. The interior stress fields, S, are calculated, then periodic boundary conditions are applied. Next, the interior displacement fields are calculated and the periodic boundary conditions are applied. Finally, the time step is incremented and the process is repeated.


Reflecting boundary conditions

A boundary can be made reflective simply by setting all the field values to zero along the

boundary. This also works to create reflectors in the interior, if required. It is a convenient way to

simulate a highly contrasting material, such as an aluminum block in water.

Absorbing boundary conditions

The problem of making a wave propagate out of an FDTD simulation region without any

energy reflecting back in is not trivial and many schemes for achieving this have been proposed

[198]–[203]. It requires the presence of a perfectly matched layer (PML) that is non-reflective

for the spatial and temporal frequencies of interest and that can also dissipate the incident energy.

The problem is somewhat simplified when considering monochromatic sources that are normally

incident. However, outgoing waves that are increasingly parallel to the boundary become ever

more reflected back into the interior simulation region.

A new PML absorbing layer has been proposed that addresses many of the shortcoming of

previous models [204]. It has been shown to be perfectly matched and stable with proper

parameter choices [205], and has even been successfully applied to the nonlinear wave equation

[206]. For these reasons, this particular PML model was used in the FDTD simulator for

absorbing boundary conditions. It is worthwhile to explicitly restate the PML governing

equations in both continuous and discrete form since they are given in [204] as continuous and

dimensionless, whereas the FDTD simulator requires discrete and properly dimensioned

equations.

The 13 individual governing equations are given as three matrix equations, in which p∂ , for

example, denotes differentiation with respect to the variable p. The three matrix equations are

( )( ) ( )( )222111 wEvBwEvAv ydxd yxt +∂++∂=∂ , (4.40)

( )( ) 0101111 =++∂+∂ wvEw αxdx

T

t, (4.41)

and ( )( ) 0202222 =++∂+∂ wvEw αydy

T

t . (4.42)

In these equations, the state variables related to those in the interior simulation region are

contained in v and the PML state variables are introduced in w1 and w2. They are

∂

∂

=

3

2

1

2

1

σ

σ

σ

u

u

t

t

v ,

=

4,1

3,1

2,1

1,1

1

w

w

w

w

w , and

=

4,2

3,2

2,2

1,2

2

w

w

w

w

w . (4.43)

The matrices of the coefficients are


=

−

−

0000

0000

0000

0000

0000

44

12

11

1

1

C

C

C

ρ

ρ

A ,

=

−

−

0000

0000

0000

0000

0000

44

11

12

1

1

C

C

C

ρ

ρ

B , (4.44)

=

1000

0000

0100

0010

0001

1E , and

=

1000

0100

0000

0010

0001

2E . (4.45)

The damping functions, ( )xd1 and ( )yd2 , are positive and smooth and are zero at the boundary.

They have been set to a hyperbolic tangent function which increases in value from zero at the

boundary to maxd at the far edge of the PML. The parameters 01α and 02α help to prevent

growing oscillations as the simulation time progresses [204]. These governing equations are

expanded in Table 4.2 and are the basis for the discrete equations.

Table 4.2. Continuous PML absorbing boundary equations. Continuous PML equation

(4.46a) ( ) ( )

+

∂

∂++

∂

∂=

∂

∂ −4,22

33,11

1121

2

wydy

wxdxt

u σσρ

(4.46b) ( ) ( )

+

∂

∂++

∂

∂=

∂

∂ −3,22

24,11

312

22

wydy

wxdxt

u σσρ

(4.46c) ( ) ( )

+

∂∂

∂+

+

∂∂

∂=

∂

∂2,22

22

121,111

2

111 wyd

ty

uCwxd

tx

uC

t

σ

(4.46d) ( ) ( )

+

∂∂

∂+

+

∂∂

∂=

∂

∂2,22

22

111,111

2

122 wyd

ty

uCwxd

tx

uC

t

σ

(4.46e) ( ) ( )

+

∂∂

∂++

∂∂

∂=

∂

∂1,22

12

2,112

2

443 wyd

ty

uwxd

tx

uC

t

σ

(4.46f) ( )( ) 01,10111

21,1 =++

∂∂

∂+

∂

∂wxd

tx

u

t

wα

(4.46g) ( )( ) 02,10112

22,1 =++

∂∂

∂+

∂

∂wxd

tx

u

t

wα


Continuous PML equation

(4.46h) ( )( ) 03,101113,1 =++

∂

∂+

∂

∂wxd

xt

wα

σ

(4.46i) ( )( ) 04,101134,1 =++

∂

∂+

∂

∂wxd

xt

wα

σ

(4.46j) ( )( ) 01,20221

21,2

=++∂∂

∂+

∂

∂wyd

ty

u

t

wα

(4.46k) ( )( ) 02,20222

22,2

=++∂∂

∂+

∂

∂wyd

ty

u

t

wα

(4.46l) ( )( ) 03,202223,2

=++∂

∂+

∂

∂wyd

yt

wα

σ

(4.46m) ( )( ) 04,202234,2

=++∂

∂+

∂

∂wyd

yt

wα

σ

The continuous equations are discretized using central differences. The discrete equations are

summarized in Table 4.3, where the letter suffixes of the continuous and discrete equations

match.

Table 4.3. Discrete PML absorbing boundary equations. Discrete PML equation

(4.47a) ( ) ( )

1:,1

:,1

:,4,22

:1,3

:,3:,

3,11

:,11

:,1121:,

1

2 −

−−−+

−+

+

∆

−++

∆

−∆=

tyxtyx

tyx

y

tyxtyxtyx

x

tyxtyx

t

tyx

uu

wydwxduσσσσ

ρ

(4.47b) ( ) ( )

1:,2

:,2

:,3,22

:,2

:1,2:,

4,11

:,3

:,13121:,

2

2 −

++−+

−+

+

∆

−++

∆

−∆=

tyxtyx

tyx

y

tyxtyxtyx

x

tyxtyx

t

tyx

uu

wydwxduσσσσ

ρ

(4.47c)

( )

( )

tyx

tyx

t

y

tyxtyxtyxtyx

tyx

t

x

tyxtyxtyxtyx

tyx

wyduuuu

C

wxduuuu

C

:,1

:,2,22

1:1,2

1:,2

:1,2

:,2

12

:,1,11

1:,1

1:,11

:,1

:,11

11

1:,1 σσ +

∆+

∆

+−−+

∆+

∆

+−−

=−−−−

−−++

+


Discrete PML equation

(4.47d)

( )

( )

tyx

tyx

t

y

tyxtyxtyxtyx

tyx

t

x

tyxtyxtyxtyx

tyx

wyduuuu

C

wxduuuu

C

:,2

:,2,22

1:1,2

1:,2

:1,2

:,2

11

:,1,11

1:,1

1:,11

:,1

:,11

12

1:,2 σσ +

∆+

∆

+−−+

∆+

∆

+−−

=−−−−

−−++

+

(4.47e)

( )

( )

tyx

tyx

t

y

tyxtyxtyxtyx

tyx

t

x

tyxtyxtyxtyx

tyx

wyduuuu

wxduuuu

C :,3

:,1,22

1:,1

1:1,1

:,1

:1,1

:,2,11

1:,12

1:,2

:,12

:,2

441:,

3 σσ +

∆+∆

+−−+

∆+∆

+−−

=−−++

−−−−

+

(4.47f) ( )( )[ ]x

tyxtyxtyxtyxtyx

t

tyx uuuuwxdw

∆

+−−−+∆−=

−−+++

1:,1

1:,11

:,1

:,11:,

1,10111:,

1,1 1 α

(4.47g) ( )( )[ ]x

tyxtyxtyxtyxtyx

t

tyx uuuuwxdw

∆

+−−−+∆−=

−−−−+

1:,12

1:,2

:,12

:,2:,

2,10111:,

2,1 1 α

(4.47h) ( )( )[ ]x

tyxtyx

t

tyx

t

tyxwxdw

∆

−∆−+∆−=

−+

:,11

:,1:,

3,10111:,

3,1 1σσ

α

(4.47i) ( )( )[ ]x

tyxtyx

t

tyx

t

tyxwxdw

∆

−∆−+∆−=

++

:,3

:,13:,

4,10111:,

4,1 1σσ

α

(4.47j) ( )( )[ ]y

tyxtyxtyxtyxtyx

t

tyx uuuuwydw

∆

+−−−+∆−=

−−+++

1:,1

1:1,1

:,1

:1,1:,

1,20221:,

1,2 1 α

(4.47k) ( )( )[ ]y

tyxtyxtyxtyxtyx

t

tyx uuuuwydw

∆

+−−−+∆−=

−−−−+

1:1,2

1:,2

:1,2

:,2:,

2,20221:,

2,2 1 α

(4.47l) ( )( )[ ]y

tyxtyx

t

tyx

t

tyx wydw∆

−∆−+∆−=

++

:,2

:1,2:,

3,20221:,

3,2 1σσ

α

(4.47m) ( )( )[ ]y

tyxtyx

t

tyx

t

tyx wydw∆

−∆−+∆−=

−+

:1,3

:,3:,

4,20221:,

4,2 1σσ

α

A thickness is chosen for the PML and the discrete equations are implemented across this

region. The thicker the PML the more gradual the damping, which results in less reflection from

within the PML region. Noting that there are five update equations in the interior but 13 in the


PML, the trade-off is between improved absorption and faster simulation execution, and the

parameter that adjusts this trade-off is PML thickness.

Source conditions

In an FDTD simulation, a source may be used in conjunction with initial conditions to input

energy into the system. The source may take on any shape, but is ultimately composed of point

sources on the FDTD grid. There are two useful configurations: Hard and transparent. Also of

note is that an input waveform must not have a DC offset in 2D simulations [207].

Hard source configuration

In a hard source, the calculated displacement values for the source locations are overwritten

at each time step with the desired displacement values. This is extremely simple to implement in

software and can be used to simulate many real-world scenarios. The major drawback to this

configuration is that waves incident on the source, either from reflections or another source

location, will reflect off of it. This can be averted in 1D by shutting off the source after the input

pulse is done being transmitted. That is, after the finite-duration input pulse, the field values are

simply not overwritten. If the source is placed far away enough from any reflectors, then by the

time a wave is reflected back to the source location it has been shut off and is non-reflective.

This is not an option for 2D line sources, because when it is shut off it will become unstable and

begin to grow without bound. To avoid this effect, the source waveform must be zero-padded for

the remainder of the simulation after the source is shut off.

The hard source configuration is particularly useful in 1D because, when shut off it behaves

in a transparent manner. In 2D, a hard source configuration is a good model for an ultrasound

transducer, since reflections from the transducer face often occur in practice.

Transparent source configuration

Frequently, one wishes to simulate an incident pulse coming from the far-field to

approximate a plane wave while simultaneously minimizing the simulation size for improved

computational speed. To achieve this, a transparent source can be used [208][209]. A transparent

source must fulfill two criteria: It must set the field values to the desired input waveform for the

duration of the input, and it must allow incident waves to pass through unaffected. This is

accomplished by convolving the desired input waveform with the impulse response of the source

for a given simulation configuration. A simulation is first run where a discrete impulse is applied

to the source locations, and the response of the system to that impulse is recorded. Next, when

the actual simulation is run, the input waveform is convolved with the recorded impulse response


to achieve transparency while still outputting the correct waveform. Full implementation details

for acoustic FDTD simulations are given in [208].

4.1.3 Stability criteria

When using the FDTD method, a time step, t∆ , must be chosen. Choosing a small value

increases the numerical accuracy of the simulation, whereas choosing a large value decreases the

execution time of the simulation. However, if the time step is chosen too large, the field values

will grow exponentially with each step and the results will be non-physical. The numerical

stability limit of the FDTD simulation is the largest possible t∆ for a given set of simulation

parameters that will not result in non-physical exponential growth.

There are numerous methods for determining the stability of a discrete system. The most

common are the von Neumann method [210][211], examining the numerical dispersion relation

[197], and using linear time-invariant (LTI) system theory [212]–[214]. A von Neumann analysis

is carried out by assuming that there is only one Fourier spatial component present, usually

indicated by rk⋅~

je . This substitution facilitates the application of a discrete Fourier transform

(DFT) to the governing equations, eliminating the spatial variables and reducing the equations to

a set of recursion relations. Then, a z-transform can be applied to the discrete time domain and

the roots of z can be obtained. The system is stable when 1≤z for any [ ]iiik ∆∆−∈ ππ ,~

,

which are the possible wavevectors within a discrete system.

The numerical dispersion relation can also be used to determine the system stability. The

relationship between ω~ and k~

, the angular and spatial frequencies occurring in the FDTD

simulation, is called the numerical dispersion relation. This is an artefact of the FDTD method,

and is not representative of the physical dispersion whose variables are ω and k . The numerical

dispersion relation can be used to enforce the condition that all real wavevectors, k~

, must only

stimulate real frequencies, ω~ , and vice versa since a complex angular or spatial frequency would

indicate the exponential growth associated with an unstable FDTD scheme. [197]

Linear system theory can also be used to examine system stability. The FDTD scheme can be

represented as a multi-input multi-output (MIMO) system. The system can be set up as follows,

which is in accordance with traditional LTI system theory [212]:

[ ] [ ]kk Axx =+1 , [ ]TkMkkMk UUUU 1:1

1:11

:1

:11

−−= LLx

In this configuration, x is the state variable consisting of all displacement values from the

current ( k ) and past ( 1−k ) time steps, A is the state transition matrix, and the complex

eigenvalues of A must lie on or within the unit circle for the system to be at least marginally


stable. The state transition matrix will be of the form

−=

0λ

I

IAA . This makes finding the

eigenvalues particularly convenient and theorems, such as the Gershgorin Circle Theorem, can

be used, as was shown in [213]. Other approaches may be used under certain conditions, such as

the property that ( ) ∑ == i

N

itrace λ1A in conjunction with the fact that all the diagonal elements of

λA are equal.

To investigate the stability of this particular FDTD implementation, a von Neumann

approach was used since it is the most straight-forward. As a test, all three methods were used to

determine the stability limit for a 1D contrasting medium with 0=k , and all three came to the

same conclusion that is outlined below. The coupled first-order equations must be combined to

eliminate the strain fields and yield second-order displacement equations. They are shown

without offset grid notation for simplicity since it does not affect the stability analyses.

Practically, if the resulting limits are functions of space, then the appropriate procedure is to

determine the required t∆ at every location and to take the minimum of these values as the

simulation time step. We first determined the stability limits in 1D for all the relevant cases, and

then determined the limit for the most basic 2D acoustical case. These limits were verified by

simulation. To the best of our knowledge, determining the limit in the 2D case has not been

previously described in the literature.

A 1D homogenous medium with k = 0

The governing equation for U1 is formed by substituting the first-order strain equations,

(4.29) and (4.31), into the first-order strain-dependent equation for U1, (4.11), resulting in a

second-order equation for U1 alone. In the governing equations, we set 0=k , 0=∂

∂

x

ρ, and

0=∂

∂

x

Cij , and collapse the equation for U1 to 1D, which corresponds to a 1D homogeneous

medium with no applied wavevector. The continuous and discrete equations are

21

211

21

2

x

UC

t

U

∂

∂=

∂

∂

ρ, (4.48)

and [ ]txtxtx

x

ttxtxtxUUU

CUUU

:11

:1

:112

2111:

1:

11:

1 22 −+−+ +−∆

∆=+−

ρ. (4.49)

The stability of the discrete system can be determined in this simple case by assuming that

the solution is an eigenmode with wavevector k~

, then by taking a z-transform in time and a DFT

in space of the system. This assumption is validated by requiring that the system be stable for all


[ ]xxk ∆∆−∈ ππ ,~

, corresponding to all the possible wavevectors in a discrete system [212].

Taking the z-transform of (4.49),

[ ]:zx-x:z:zx

x

tx:zx:zx:zUUU

CzUzU

111

112

211

11

11 2U2 +−∆

∆=+− +−

ρ, (4.50)

where { }txx:zUZU

:11 ≡ and {}⋅Z is the z-transform, so that the argument indicates the domain.

Next, assuming that the solution is a discrete plane wave, a spatial DFT of the system gives

[ ]x:zkjx:zx:zkj

x

tx:zx:zx:zUeUUe

CUzUzU xx

1

~

11

~

2

211

11

11 22 ∆−∆− +−∆

∆=+−

ρ. (4.51)

Cancelling the displacement and simplifying,

[ ]xx kjkjeezz

∆−∆− +−=+−~~

1 22 β , where 2

211

x

tC

∆

∆=

ρβ . (4.52)

Pzz β=+− −12 , where ( )[ ]1~

cos2 −∆= xkP . (4.53)

The roots of this equation are stable when 1≤z , which occurs when 04 ≤≤− Pβ . Thus,

≤

≤−⇒≤≤−

0max

min404

P

PP

β

ββ . (4.54)

Since [ ]xxk ∆∆−∈ ππ ,~

, therefore ( ) [ ]1,1~

cos −∈∆ xk and [ ]0,4−∈P , which gives

≤

−≤−⇒≤≤−

−→0lim

4404

0P

P

Pβ

ββ . (4.55)

The first inequality means that 1≤β , and therefore

12

211 ≤

∆

∆

x

tC

ρ. (4.56)

Taking the positive root and noting that the phase velocity is ρ110 Cc = gives the expected

stability criteria [197],

011 cC xxt ∆=∆≤∆ ρ . (4.57)

The second inequality, which implies that 0≥β , is of some interest with respect to

metamaterials, which can have negative effective properties. Since 2t∆ and 2

x∆ are both positive,

therefore 11C and ρ must have the same sign. Interestingly, this conclusion is in agreement with

[66], which states that waves will not propagate within a medium if the signs of these parameters


are different. In conclusion, the FDTD stability requirements for a 1D homogeneous medium are

that 11Cxt ρ∆≤∆ and that 11C and ρ must have the same sign.

A 1D contrasting medium with k = 0

We begin as before by setting 0=k and by collapsing the second-order equation for U1 to

1D. However, we lift the restriction that 0=∂

∂

x

ρ and 0=

∂

∂

x

Cij so that the material parameters

may vary in space – that is, they are contrasting. The continuous and discrete equations are

x

U

x

C

x

UC

t

U

∂

∂

∂

∂+

∂

∂=

∂

∂ 1112

12

1121

2

ρ , (4.58)

and [ ] [ ]( )txtxxtxtxx

x

x

ttxtxtxUUCUUCUUU

:11

:1

111

:1

:11112

21:

1:

11:

1

12 −−+−+ −−−

∆

∆=+−

ρ. (4.59)

Applying a discrete-time z-transform, a spatial DFT, and simplifying as before gives

[ ] [ ]( )x:zkjx:zxx:zx:zkjx

x

t

x

x:zx:zx:zUeUCUUeCUzUzU xx

1

~

11

1111

~

112

2

11

11

12 ∆−−∆− −−−

∆

∆=+−

ρ, (4.60)

[ ] [ ]( )111

2~

111

~

112

21 −+−

∆

∆=+− ∆−−∆− xx kjxkjx

x

t

xeCeCzz

ρ, (4.61)

[ ] ( )[ ] [ ] ( )( )x

xx

x

xx

x

t

xkCCjkCCzz ∆−+−∆+

∆

∆=+− −−− ~

sin1~

cos1

2 11111

111112

21

ρ, (4.62)

Pzz β=+− −12 , (4.63)

where 2

211111

2 x

t

x

xxCC

∆

∆+=

−

ρβ and ( )[ ] ( )xxx

xx

x kCC

CCjkP ∆

+

−+−∆=

−

− ~sin21

~cos2

11111

11111 . We have introduced

a division and multiplication by two in β and P, respectively, to remain consistent with the

previous results for a homogeneous medium.

We can now take { } RealRe PP ββ = since we introduced the eigenmode as a complex

exponential instead of as a cosine. That is, the imaginary part of P was added to facilitate the

calculation and does not represent real energy. The roots of the equation are stable when 1≤z ,

which occurs when 04 Real ≤≤− Pβ . Since [ ]0,4Real −∈P , the previous limits apply, and therefore

12 2

211111 ≤

∆

∆+ −

x

t

x

xxCC

ρ, (4.64)

( )111112 −+∆≤∆ xxx

xtCCρ . (4.65)


This stability limit is in agreement with [213], which was derived in a different manner. The

second inequality, 0lim Real0Real

≤−→

PP

β gives the condition that 0≥β as in the homogeneous case.

A 1D contrasting medium with k ≠ 0

When using periodic boundary conditions in an FDTD simulation to investigate the

dispersion properties of an infinite phononic crystal, a desired wavevector is applied to the

interior region as well as to the boundaries. Here we wish to determine the stability limit in a 1D

contrasting medium when 0≠k . The continuous and discrete equations are

21

2

11111

111111

1112

121

2

2x

UC

x

U

x

CCjkU

x

CjkCk

t

U

∂

∂+

∂

∂

∂

∂++

∂

∂+−=

∂

∂ρ , (4.66)

and

+

∆−

∆−−

∆+

+

∆−

∆+−

∆∆=+−

−−

+

−+

tx

x

tx

xx

x

tx

x

tx

xx

x

x

ttxtxtx

Uk

Ujkk

C

Uk

Ujkk

C

UUU

:1

21

2:1

11

21

21

11

:1

21

2:1

11

21

2112

1:1

:1

1:1

4

1

4

1

4

1

4

1

2ρ

. (4.67)

Applying the z-transform and the DFT as before so that Pzz β=+− −12 gives

2

211111

2 x

t

x

xxCC

∆

∆+=

−

ρβ , (4.68)

and

( ) ( ) ( ) ( )

( ) ( ) ( )

∆

∆−+∆∆

+

−+

∆+−∆∆−∆

∆−

=

−

−

xx

xxxx

xx

x

xxx

x

kk

kkCC

CCj

kkkk

k

P~

sin4

1~

cos

41

~sin

~cos

41

22

111

1111

11111

21

1

21

. (4.69)

Noting that ( )

∆+−∈ 0,

2

4 21

Realxk

P , the stability limit is

( ) 1

11112

1

2

4

4−+

∆+∆≤∆

xx

x

x

xtCCk

ρ. (4.70)

A 2D homogeneous medium with k = 0

This scenario represents the simplest physically relevant 2D case. The second-order

displacement equations are first determined by substituting (4.29) through (4.30) into (4.11) and

(4.12). In these governing equations, we set 0=ik , 0=∂

∂

x

ρ, and 0=

∂

∂

x

Cij . The resulting

continuous and discrete equations are


( )yx

UCC

y

UC

x

UC

t

U

∂∂

∂++

∂

∂+

∂

∂=

∂

∂ 22

441221

2

4421

2

1121

2

ρ , (4.71)

( )yx

UCC

x

UC

y

UC

t

U

∂∂

∂++

∂

∂+

∂

∂=

∂

∂ 12

441222

2

4422

2

1122

2

ρ , (4.72)

( ) ( )[ ]( ) ( )[ ]

( )[ ]( )[ ]

∆−∆+∆−∆∆−

∆−∆+∆−∆∆+

−∆+−∆∆−

−∆+−∆∆

∆=

+

−

−−−−−−−−−

−+−−−−−

−−−−−−−

−−+−−

−

+

tyx

y

tyx

y

tyx

x

tyx

xy

tyx

y

tyx

y

tyx

x

tyx

xy

tyxtyx

y

tyxtyx

xx

tyxtyx

y

tyxtyx

xx

yx

t

tyx

tyx

tyx

UUUUC

UUUUC

UUCUUC

UUCUUC

U

U

U

:1,1

1:,1

1:1,12

1:1,2

144

1

:,1

1:1,1

1:,12

1:,2

144

1

:1,12

:,1212

1:,11

:,111

11

:1,2

:,212

1:,1

:,1111

11

,

2

1:,1

:,1

1:,1

2ρ

, (4.73)

and

( ) ( )[ ]( ) ( )[ ]

( )[ ]( )[ ]

∆−∆+∆−∆∆−

∆−∆+∆−∆∆+

−∆+−∆∆−

−∆+−∆∆

∆=

+

−

−+−−−−−

+−++−−+−−

+−−−−

+++−+−−

−

+

tyx

y

tyx

y

tyx

x

tyx

xx

tyx

y

tyx

y

tyx

x

tyx

xx

tyxtyx

x

tyxtyx

yy

tyxtyx

x

tyxtyx

yy

yx

t

tyx

tyx

tyx

UUUUC

UUUUC

UUCUUC

UUCUUC

U

U

U

:,1

1:1,1

1:,12

1:,2

144

1

:,11

1:1,11

1:,2

1:,12

144

1

:,1

:,1112

1:1,2

:,211

11

:1,1

:1,1112

1:,2

:1,211

11

,

2

1:,2

:,2

1:,2

2ρ

. (4.74)

The same procedure of applying a temporal z-transform followed by a spatial 2D DFT is still

applicable, except that each field has its own wavevector. That is, yxk ˆ~

ˆ~~

111 yx kk += is the

numerical wavevector of the U1 field, and yxk ˆ~

ˆ~~

222 yx kk += is the numerical wavevector of the U2

field. Beginning with the U1 displacement field,

( )

( ) ( )[ ]( ) ( )[ ]( ) ( )( )[ ]

( ) ( )( )[ ]

−∆+−∆∆−

−∆+−∆∆+

−∆+−∆∆−

−∆+−∆∆

∆=+−

∆−−∆−∆−−−

∆−∆−−−

∆−∆−−∆−−−

∆−−∆−−

−

y:zxkj

y

y:zxkjkj

xy

y:zxkj

y

y:zxkj

xy

y:zxkjkj

y

y:zxkj

xx

y:zxkj

y

y:zxkj

xx

ty:zx

UeUeeC

UeUeC

UeeCUeC

UeCUeC

Uzz

yyxxyy

yyxx

yyxxxx

yyxx

,1

~1,

2

~~1

441

,1

~1,

2

~1

441

,2

~~

121,

1

~

1111

,2

~

121,

1

~

1111

2,

11

122

12

221

21

11

11

11

11

2ρ

,(4.75)

( ) ( ) ( )

( )( )( ) y:zxkjkj

yx

t

y:zx

yy

y

xx

x

ty:zx

UeeCC

UkC

kC

Uzz

yyxx ,2

~~4412

2

,112

4412

112

,1

1

22 11

1~

cos1~

cos2

2

−−

∆∆

+∆+

−∆

∆+−∆

∆

∆=+−

∆−∆−

−

ρ

ρ, (4.76)

( )[ ] y:zx

y

y:zx

x UNUzMz ,2

,1

12 =++− − , (4.77)

where ( )( ) ( )( )

−∆

∆+−∆

∆

∆= 1

~cos1

~cos

212

4412

112

yy

y

xx

x

tx

kC

kC

Mρ

, (4.78)

and ( ) ( )( )

−−

∆∆

+∆=

∆−∆− 11 22

~~4412

2yyxx

kjkj

yx

t

yee

CCN

ρ. (4.79)


A similar analysis of the U2 displacement field yields

( )[ ] y:zx

x

y:zx

y UNUzMz,

1,

212 =++− − , (4.80)

where ( )( ) ( )( )

−∆

∆+−∆

∆

∆= 1

~cos1

~cos

222

4422

112

xx

x

yy

y

ty

kC

kC

Mρ

, (4.81)

and ( ) ( )( )

−−

∆∆

+∆=

∆−∆− 11 11

~~4412

2yyxx

kjkj

yx

tx

eeCC

Nρ

. (4.82)

This procedure has yielded two equations, (4.77) and (4.80), in two unknowns. For

convenience, one can choose kkk~~~

21 ≡= , since the stability limit is reached as both approach

the same bounds. In this case, NNN yx ≡= .

If the simulation parameters are chosen so that yx ∆=∆ , then MMM yx ≡= , and (4.77)

minus (4.80) gives

( )[ ][ ] [ ]y:zxy:zxy:zxy:zxUUNUUzMz

,2

,1

,2

,1

12 −−=−++− − (4.83)

NMzz −=+− −12 (4.84)

For convenience we can split M and N as follows:

MM

PM β2= , where 2

24411

x

tM

CC

∆

∆+=

ρβ , and ( ) 1

~cos −∆= xxM kP , (4.85)

2NN PN β= , where

2

24412

x

tN

CC

∆

∆+=

ρβ , and 1

~

−= ∆− xxkj

N eP . (4.86)

Comparison with (4.53) shows that the roots of (4.84) are stable when { } 0Re4 ≤−≤− NM ,

therefore

{ }

≤−

−≤−⇒≤−≤−

0minmax2

maxmin240Re4

2Real,Real,

2Real,Real,

NNMM

NNMM

PP

PPNM

ββ

ββ (4.87)

Since [ ]iiik ∆∆−∈ ππ ,~

, therefore ( ) [ ]1,1~

cos −∈∆ iik , [ ]0,2Real, −∈MP , and [ ]4,02Real, ∈NP . The

2

0lim NN

PP

N

β+→

term in the second inequality can be ignored since it is approaching zero in the

squared sense whereas the other term is linearly approaching zero. Therefore the limits can be

restated as

{ }

≤

−−≤−⇒≤−≤−

−→02lim

4440Re4

0MM

P

NM

PNM

M

β

ββ (4.88)

Simplification of the first inequality gives


1≤+ NM ββ (4.89)

12

24412

2

24411 ≤

∆

∆++

∆

∆+

x

t

x

t CCCC

ρρ (4.90)

12

2

2124411 ≤

∆

∆++

x

tCCC

ρ (4.91)

( )124411 2 CCCxt ++∆≤∆ ρ (4.92)

The second inequality, 02lim0

≤−→

MMP

PM

β gives the condition that 0≥M

β , which implies that

4411 CC + has the same sign as ρ .

4.2 The transmission matrix method

The transmission matrix method captures all the transmission and reflection information

regarding a single cell of a 1D phononic crystal into one 22× matrix, called the cell

transmission matrix. The transmission matrix for many cells is simply composed by multiplying

all the cell transmission matrices for the individual phononic crystal cells. The properties of the

transmission matrix can be used to investigate the dispersion relationship and the transmission

and reflection spectra of a phononic crystal. It is quite attractive for analysis because it provides

closed-form solutions to these problems, however, it is limited to one dimensional analyses.

The TMM approach has been applied in 1D electromagnetic systems [215], enabling such

effects as birefringence and phase matching to be thoroughly investigated [216]. Acoustically,

TMM theory has been described in [217], and has enabled experimental data to be verified [218].

Nonlinear effects and modelling were described in [219]. The method has also found use in more

exotic structures like ducts and nozzles [220] and in sinusoidally corrugated tubes [221].

The transmission matrix method for a phononic crystal captures the scattering and

propagation information in separate matrices, and then combines them to form the cell

transmission matrix. Beginning with scattering, Figure 4.3 shows the incident and scattered

displacement waves at the interfaces between planes of differing materials in a 1D phononic

crystal. In the case of a corrugated tube waveguide, the scatterer is an impedance discontinuity

which occurs at the boundaries between different cross-sectional areas, as was described in

Section 2.3.3. The reflection coefficient for an incident displacement wave traveling from

material 1 to material 2 is ( ) ( )21211 ZZZZR +−= , and from material 2 to material 1 is

( ) ( )211212 ZZZZRR +−=−= . For the displacement waves under consideration, 1=− ii RT , so

the transmission coefficients are given by ( )2111 2 ZZZT += , and ( )2122 2 ZZZT += .


a+

a–

b+

b–

Z1 Z2 Figure 4.3. Nomenclature used to describe displacement waves at the boundaries of a 1D scatterer.

A scattering matrix relates the incident and scattered waves as

=

−

+

+

−

b

a

SS

SS

b

a

2221

1211 . (4.93)

Scattering matrices are convenient because the terms of the matrix directly describe transmission

and reflection. At the interfaces between differing regions, the scattering matrices are given by

=→

21

21

RT

TR21S ,

=→

12

12

RT

TR12S . (4.94)

Alternatively, a transmission matrix relates the waves on one side of a scatterer to the waves

on the other:

=

−

+

−

+

a

a

TT

TT

b

b

2221

1211 , (4.95)

which can also written in matrix notation as Tab = . The effects of multiple scatterers can be

chained together simply by multiplying their corresponding transmission matrices. The scattering

and transmission matrix formats can be converted using the following relations, which are

derived directly from the matrix definitions:

−

−=

1

1

11

22

12 S

S

S

ST ,

−=

12

21

22

11T

T

T TS . (4.96)

Noting that 1−=S for both 21S → and 12S → , the transmission matrices for each interface can be

determined as

−=→ 1

11

1

2

2 R

R

T21T ,

−=→ 1

11

2

1

1 R

R

T12T , (4.97)

which can be rewritten entirely in terms of 1R as

−

−

−=→ 1

1

1

1

1

1

1 R

R

R21T ,

+=→ 1

1

1

1

1

1

1 R

R

R12T . (4.98)

Waves can be propagated spatially by a distance d through a medium by using the following

propagation transmission matrix:


( )

=

−

0

0

0

0,

cdj

cdj

e

ed

ω

ω

ωPT . (4.99)

This expression assumes that the propagation medium is lossless and that a wave function is

given by ( )kxtje −ω , where 0ck ω= . The cell transmission matrix, CT , encapsulates all the

scattering information within one cell, and is determined by sequentially combining the effects of

scattering and propagation as

( ) ( )( )aa ηωηω −= →→ 1,, P21P12C TTTTT . (4.100)

Thus, the transmission through one entire cell of phononic crystal for a particular ω ,

assuming 0a on the left inside section 1 and 1a one cell to the right, is given as 01 aTa C= . For n

cells connected together, the transmission matrix is given quite simply as n

CT . It should be noted

that CT is unimodular in the static case (i.e. 1=CT ) and therefore n

CT can be quickly found

using the method described in [222].

To extract the transmission coefficient, the transmission matrix must be converted back into a

scattering matrix using (4.96). The 21S value corresponds with the transmission coefficient from

one end of the cell to the other, and is given by

0

0

221

21

2221 1

11

caj

caj

CeR

ReTS

ωη

ω

−

−

−

−== . (4.101)

The transmission through n cells is 2221 1 n

CTS = , where 22n

CT is the (2,2) element of the n

CT

matrix.

4.2.1 Example calculation

The previous results can be applied to the corrugated tube described in both [17] and in

Section 2.3.3 to determine the transmission spectrum through the tube as the number of segments

is increased. The transmission spectrum near the first band gap is shown in Figure 4.4 for a tube

with 10 sm343 −⋅=c , mm85=a , 5.0=η , mm481 =d , and mm602 =d . As expected, the

band gap attenuation increases as more segments are attached. The examples shown in

Section 2.3 were also calculated using the TMM.


Transmission Coefficient

100

10-1

10-2

10-3

2.0 2.4 2.81.61.2

Frequency (kHz)

5

10

15

Segments

Figure 4.4. Calculated transmission spectrum for a 1D tube waveguide showing that as more segments are added to the tube, the band gap attenuation increases [34].

4.3 Multiple scattering theory

The transmission matrix method solves the problem of multiple scattering by separating the

acoustic wave scattering and propagation within a phononic crystal. Multiple scattering theory

uses a similar approach, except it is applicable in higher dimensions. Much of this section is

extended from the results summarized in [12], which is an excellent resource on the topic.

Accordingly, specific references to [12] will not be made since they form the basis of these

results.

An approach akin to multiple scattering theory was used by Lord Rayleigh in his first paper

on the subject in 1892 [223] and was later formalized in 1913 by Záviška [224]. It was further

developed by Faran [225] and Waterman [226]–[229]. Modern acoustic and electromagnetic

MST has progressed from hard-shell models to space-filling cell potentials [230] that allow

continuous material parameters. Acoustic MST [231] and its application to periodic structures

[232][233] have been extensively described in the literature. It has been extended from a scalar-

field theory to account for the vector nature of elastic waves [234], and has been applied to

elastic spheres [235] and plates [236]. Multiple scattering theory has been applied to numerous

unique acoustic configurations, including arrays of balloons [237], random [238] and period

media [233] [239]. Improving the efficiency of MST algorithms by truncating the set of basis

functions has also been described [240]. Subwavelength imaging has even been investigated

using MST as a means of describing this phenomenon [241][242].


4.3.1 Scattering from one cylinder

In two dimensions, consider a homogeneous fluid medium with density and bulk modulus

0ρ and 0K , respectively. Recall that 10110−== κCK , where 11C is the isotropic elastic stiffness

coefficient (Voigt notation) and 0κ is the adiabatic compressibility. Further, it is lossless and

dispersionless, has a longitudinal phase velocity 000 ρKc = , and a characteristic acoustic

impedance 000 KZ ρ= . Embedded within this homogeneous medium is a fluid cylinder of

radius R whose centre is located at the origin. For the time being, we are restricting our analysis

to fluids so that transverse waves are not considered and a scalar field may fully specify the

waves in question. The coordinates are either radial, ( )θ,r=r , or Cartesian, ( )yx,=x .

The field external to the cylinder can be divided into an incident field, ( )tU inc ,r , and a

scattered field, ( )tU sc ,r . Here, we choose U to be the scalar velocity potential. Thus, the total

external field is

( ) ( ) ( )tUtUtUU scinc ,,,00 rrr +== . (4.102)

The fields obey the wave equation,

UKt

U 202

2

0 ∇=∂

∂ρ . (4.103)

Considering only time-harmonic motion, where ( ){ }tjeuU 0Re ω−= r , 0ω is the angular

frequency, and ( )ruu = is a complex-valued spatial envelope, the problem is reduced to finding

solutions of the Helmholtz equation,

( ) 020

2 =+∇ uk , (4.104)

where 000 ck ω= is the wavenumber. In cylindrical coordinates, the Helmholtz equation has

separated solutions

( ) ( ) θψ jm

mm ekrH=r , and ( ) ( ) θψ jm

mm ekrJ=rˆ , Ζ∈m , (4.105)

called outgoing and regular cylindrical wavefunctions, respectively. The outgoing wavefunction,

( )rmψ , satisfies the Sommerfeld radiation condition at infinity but is singular at the origin,

whereas the regular wavefunction, ( )rmψ does not satisfy the Sommerfeld radiation condition

and is not singular at the origin.

The incident field for a plane wave at an angle α to the coordinate axes is

( ) ( )αθαα −+⋅ === cossincos 000 rjkyxjkj

inc eeeuxk . (4.106)


where αk ˆ00 k= and α is a unit vector in the direction of angle α . Note that we use ( )tje

ω−⋅xk0 as

the definition of a plane wave. Applying the Jacobi expansion,

( )∑=m

jmm

mjwewJje

ϕϕcos , (4.107)

we can rewrite the incident field as an expansion of regular wavefunctions, called a multipole

expansion:

( )∑=m

mminc du rψ , where αjmm

m ejd −= and [ ]MMm ,−∈ . (4.108)

The incident field coefficients, md , are known and in this case correspond to a plane wave.

The value of M should be chosen to be large enough to minimize the computational error while

small enough to compute the summation in a reasonable amount of time. We choose to represent

the scattered field as an expansion consisting of outgoing wavefunctions to satisfy the

Sommerfeld radiation condition:

( )∑=m

mmsc cu rψ . (4.109)

The scattered field coefficients, mc , are the required unknowns.

So far, we have satisfied the Helmholtz equation by composing the fields of its solutions, we

have satisfied the Sommerfeld radiation condition by choosing to use outgoing cylindrical

wavefunctions to compose the scattered field, and we have a known regular cylindrical

wavefunction decomposition of the incident plane wave. Now we wish to solve for the scattered

field coefficients as a function of the incident field coefficients, so that

Tdc = , or ∑=n

nmnm dTc . (4.110)

In (4.110), { }mc=c and { }md=d are column vectors containing the scattered and incident

field coefficients, and { }mnT=T is called the transition matrix, or simply just T-matrix. Note the

subtle nomenclature difference between the transition matrix used here and the transmission

matrix used in a one dimensional analysis. The T-matrix will be found by applying boundary

conditions at the interface between the scatterer and the surrounding medium.

A sound-hard cylinder

At a sound-hard interface, the Neumann boundary conditions apply so that

0=∂

∂

n

u on S, (4.111)

where S denotes on the boundary of the scatterer. For a cylinder of radius R , this becomes


0=∂

∂

=Rrr

u. (4.112)

Since ( ) ( )∑∑ +=+=m

mm

m

mmscinc cduuu rr ψψ , therefore

( ) ( )( ) 0ˆ =

+

∂

∂

=

∑Rrm

mmmm cdr

rr ψψ , (4.113)

( ) ( )( ) 000 =

+

∂

∂

=

∑Rrm

jm

mm

jm

mm erkHcerkJdr

θθ , (4.114)

( ) ( )( ) 000 =′+′∑m

jm

mmmm eRkHcRkJd θ . (4.115)

Because of the orthogonality of the exponential factors, the summation separates into 12 +M

independent equations:

( ) ( ) mRkHcRkJd mmmm ∀=′+′ 000 . (4.116)

Therefore, for sound-hard cylindrical scatterers the elements of the T-matrix are

( )( ) mn

m

mmn

RkH

RkJT δ

′

′−=

0

0 , (4.117)

where mnδ is the Kronecker delta. Note that ( ) ( ) ( ) ( )zFz

nzF

z

zFzF nn

nn −=

∂

∂=′

−1 , where nF is any

of the Bessel or Hankel functions (see 9.1.30 in [243]).

The Kronecker delta in (4.117) is noteworthy because it means that the various modes of the

cylindrical wavefunctions are completely decoupled. This is because we are considering

cylinders, whose fields naturally decompose into cylindrical wavefunctions, so the T-matrix is

diagonal. If the scatterer was non-cylindrical the T-matrix would not be diagonal and the spatial

modes would be coupled. The practical implication of this is that T-matrix may simply be

represented in software as a vector of coefficients.

A sound-soft cylinder

At a sound-soft interface, the Dirichlet boundary conditions apply so that

0=u on S (4.118)

which occurs at Rr = for a cylinder. Following a similar derivation as before leads to

( )( ) mn

m

mmn

kaH

kaJT δ

−= . (4.119)


A penetrable cylinder

In the previous two cases, the cylinder was impenetrable. For a penetrable cylinder, one must

solve the corresponding transmission problem. The cylinder’s density, bulk modulus, phase

velocity, impedance, and wavenumber are 1ρ , 1K , 1c , 1Z , and 1k respectively. The field within

the cylinder can be represented as a regular cylindrical wavefunction since it is not singular at the

origin, and does not need to satisfy the Sommerfeld radiation condition as the field only extends

to the boundary of the cylinder. The field, v , must also observe the Helmholtz equation within

the cylinder, ( ) 021

2 =+∇ vk , and thus can be written as

( )∑=m

mmbv rψ . (4.120)

Two transmission boundary conditions must be met at the interface between the cylinder and

the surrounding medium:

vu = on S, (4.121)

and n

v

n

u

∂

∂=

∂

∂γ on S. (4.122)

where γ is some coefficient depending on the definition of u and v. For our analysis, (4.122)

becomes

RrRr r

v

r

u

== ∂

∂=

∂

∂

10

11

ρρ. (4.123)

We have two equations, (4.121) and (4.123), with two unknowns ( mb and mc ). From

(4.121):

( ) ( ) ( )( ) 0ˆˆ =−+∑ =m

Rrmmmmmm bcd rrr ψψψ . (4.124)

From (4.123):

( ) ( )[ ] ( ) 0ˆ1

ˆ1

10

=

−+

∂

∂

=

∑Rr

mmmmmmm bcd

rrrr ψ

ρψψ

ρ, (4.125)

( ) ( )[ ] ( ) 011

11

000

=

−+

∂

∂

=

∑Rr

m

jmmm

jmmm

jmmm erkJberkHcerkJd

r

θθθ

ρρ, (4.126)

( ) ( )[ ] ( ) 011

111

00000

=

′−′+′∑

m

jmmm

jmmm

jmmm eRkJkbeRkHkceRkJkd

θθθ

ρρ. (4.127)

By orthogonality of the exponentials, each term must equal zero. Rearranging for mb :

( ) ( ) ( ) 0101

1000 =′−′+′ RkJb

k

kRkHcRkJd mmmmmm

ρ

ρ, (4.128)


( ) ( )[ ]( )RkJk

kRkHcRkJdb

m

mmmmm

110

0100

1′

′+′=ρ

ρ. (4.129)

Note that 0

1

00

11

100

001

10

01

Z

Z

c

c

c

c

k

k===

ρ

ρ

ωρ

ωρ

ρ

ρ. Substituting (4.129) into (4.124) and again noting

the orthogonality of the exponentials yields

( ) ( ) ( ) ( )[ ] 00000 =′+′−+ RkHcRkJdRkHcRkJd mmmmmmmmm β , (4.130)

where ( )( )RkJ

RkJ

Z

Z

m

mm

1

1

0

1

′=β . (4.131)

Finally, rearranging gives the T-matrix as

( ) ( )( ) ( ) mn

mmm

mmmmn

RkHRkH

RkJRkJT δ

β

β

′−

′−−=

00

00 . (4.132)

4.3.2 Scattering from two cylinders

Consider two sound-hard cylinders with the kth radius given as kR and centres located at kb .

The location 0b is the origin. The total field is

( ) ( ) ( )∑∑∑ ++=+=m

mmm

mmm

mmscinc ccduuu 22

11ˆ rrr ψψψ , (4.133)

where kk brr −= and the superscripts denote the scatterer number. To solve this system of

equations, it is necessary to express (4.133) about a single origin so that the exponentials use a

common angle and the terms of the summation are orthogonal. For the incident wave, this is

simply accomplished by expanding the incident field about point kb as

( )∑=m

kmkminc du rψ (4.134)

For the scattered fields, the translation of origin is made possible by Graf’s addition theorem,

which for our purposes can be written as

( ) ( ) ( )∑=n

knklmnlm S rbr ψψ ˆ , (4.135)

where lkkl bbb −= , or more compactly in matrix notation as

kkll ψSψ ˆ= , (4.136)

where ( ){ }lml rψ ψ= , ( ){ }knk rψ ψˆ = , and ( ){ }klmnkl S bS = .

The matrix, klS , is called a separation matrix and allows a cylindrical expansion about an

origin lb to be re-expanded about an origin kb . Note that (4.136) is only valid for klk br < ,


which is always valid in the case of non-overlapping cylinders since klk bR < . The separation

matrix elements are given quite simply as

( ) ( )klnmklmnS bb −=ψ . (4.137)

Putting it all together, (4.133) may be rewritten about both 1b and 2b as

( ) ( ) ( ) ( ) ( )∑ ∑

++=

m

n

n

mnmmmmm Sccdu 1122

11

11

1 ˆˆ rbrrr ψψψ , (4.138)

( ) ( ) ( ) ( ) ( )∑ ∑

++=

m

mmn

n

mnmmm cScdu 22

2211

22

2 ˆˆ rrbrr ψψψ . (4.139)

The inner summation may be rearranged so that all the wavefunctions are of the same order:

( ) ( ) ( ) ( ) ( )∑ ∑

++=

m n

nnmmmmmm cScdu2

12111

11

1 ˆˆ brrrr ψψψ , (4.140)

( ) ( ) ( ) ( ) ( )∑ ∑

++=

m

mm

n

nnmmmm ccSdu 221

21222

2 ˆˆ rbrrr ψψψ . (4.141)

The boundary equations (sound-hard or sound-soft) may now be applied to (4.140) and

(4.141). If the cylinders are penetrable, there will be an additional set of coefficients and an

additional set of boundary conditions, as subsequently described.

Two sound-hard cylinders

Applying the sound-hard boundary condition, (4.112), to (4.140) gives

( ) ( ) ( ) ( ) ( ) 0ˆˆ

1111

21211

11

1

11

1 =

++

∂

∂=

∂

∂

==

∑ ∑Rrm n

nnmmmmmm

Rr

cScdrr

ubrrr

rψψψ , (4.142)

( ) ( ) ( ) ( ) 011

111 21211

11

1

1

=

++

∂

∂∑ ∑

=m Rrn

nnm

jm

m

jm

mm

jm

mm cSekrJekrHcekrJdr

bθθθ , (4.143)

Since the exponentials are orthogonal, every term of the summation must be zero,

( ) ( ) ( ) ( ) 011

21211

11

1

1

=

++

∂

∂

=

∑Rrn

nnmmmmmm cSkrJkrHckrJdr

b , (4.144)

( ) ( ) ( ) ( ) 021211

11

1 =′+′+′ ∑n

nnmmmmmm cSkRJkRHckRJd b , (4.145)

which is the first set of equations. By the same procedure, the second set of equations is

( ) ( ) ( ) ( ) 0221

21222 =′+′+′ ∑ kRHccSkRJkRJd mm

n

nnmmmm b . (4.146)

The above two equations describe ( )122 +M equations with ( )122 +M unknowns.


Two sound-soft cylinders

Proceeding as in the sound-hard case, but using (4.118) as the boundary condition,

( ) ( ) ( ) ( ) ( ) 0ˆˆ11

11

21211

11

11 =

++=

=

= ∑ ∑Rrm n

nnmmmmmmRrcScdu brrrr ψψψ , (4.147)

( ) ( ) ( ) ( ) 011

111 21211

11

1 =

++∑ ∑

=m Rrn

nnm

jm

m

jm

mm

jm

mm cSekrJekrHcekrJd bθθθ , (4.148)

( ) ( ) ( ) ( ) 011

21211

11

1 =

++

=

∑Rrn

nnmmmmmm cSkrJkrHckrJd b , (4.149)

( ) ( ) ( ) ( ) 021211

11

1 =++ ∑n

nnmmmmmm cSkRJkRHckRJd b . (4.150)

By the same procedure, the second set of equations is

( ) ( ) ( ) ( ) 0221

21222 =++ ∑ kRHccSkRJkRJd mm

n

nnmmmm b . (4.151)

These also correspond to ( )122 +M equations with ( )122 +M unknowns.

Two penetrable cylinders

Modifying (4.140) and (4.141) to include an field internal to the cylinder gives

( ) ( ) ( ) ( ) ( ) ( )∑ ∑

−++=

m n

mmnnmmmmmm bcScdu 112

12111

11

1 ˆˆˆ rbrrrr ψψψψ , (4.152)

( ) ( ) ( ) ( ) ( ) ( )∑ ∑

−++=

m

mmmm

n

nnmmmm bccSdu 22

221

21222

2 ˆˆˆ rrbrrr ψψψψ . (4.153)

Applying the first boundary condition, (4.121), to (4.152),

( ) ( ) ( ) ( ) ( ) ( ) 0ˆˆˆ11

111

121211

11

11 =

−++=

=

= ∑ ∑Rrm

mm

n

nnmmmmmmRrbcScdu rbrrrr ψψψψ , (4.154)

( ) ( )

( ) ( ) ( ) 0

11

11

11

112

121

11

11

=

−+

+

∑ ∑=

m

Rr

jm

mm

n

nnm

jm

m

jm

mm

jm

mm

ekrJbcSekrJ

ekrHcekrJd

θθ

θθ

b, (4.155)

( ) ( ) ( ) ( ) ( ) 011

112

12111

11 =

−++

=

∑Rr

mm

n

nnmmmmmm krJbcSkrJkrHckrJd b , (4.156)

( ) ( ) ( ) ( ) ( ) 0112

12111

11 =−++ ∑ kRJbcSkRJkRHckRJd mm

n

nnmmmmmm b . (4.157)

The second set of equations is

( ) ( ) ( ) ( ) ( ) 022

221

21222 =−++ ∑ kRJbkRHccSkRJkRJd mmmm

n

nnmmmm b . (4.158)


Applying the second boundary condition, (4.123), to (4.152) gives

( ) ( ) ( ) ( ) ( ) ( ) 0ˆˆˆ

1111

112

12111

11

11

1 =

−++

∂

∂=

∂

∂

==

∑ ∑Rrm n

mmnnmmmmmm

Rr

bcScdrr

urbrrr

rψψψψ ,(4.159)

( ) ( )

( ) ( ) ( ) 0

11

11

11

112

121

11

11

1

=

−+

+

∂

∂∑ ∑

=

m

Rrn

jm

mmnnm

jm

m

jm

mm

jm

mm

ekrJbcSekrJ

ekrHcekrJd

rθθ

θθ

b, (4.160)

( ) ( ) ( ) ( ) ( ) 011

112

12111

11

1

=

−++

∂

∂

=

∑Rr

mm

n

nnmmmmmm krJbcSkrJkrHckrJdr

b , (4.161)

( ) ( ) ( ) ( ) ( ) 0112

12111

11 =′−′+′+′ ∑ kRJbcSkRJkRHckRJd mm

n

nnmmmmmm b . (4.162)

The final set of equations is

( ) ( ) ( ) ( ) ( ) 022

221

21222 =′−′+′+′ ∑ kRJbkRHccSkRJkRJd mmmm

n

nnmmmm b . (4.163)

Thus, (4.157), (4.158), (4.162), and (4.163) describe ( )124 +M equations with ( )124 +M

unknowns.

A two-cylinder solution using T-matrices

If the T-matrix for each cylinder k is known ( kT ), a cluster T-matrix or T-supermatrix, totT ,

can be defined that captures all of the scattering, so that

dTc tottot = . (4.164)

We begin by considering the field “incident” on S1 consisting of the incident field and the

scattered field from S2:

( ) ( )∑ ∑

+

m n

nnmmm cSd2

121

1ˆ brψ . (4.165)

Since 1T is known and characterizes the scattering of S1, and knowing that (4.165) is the field

incident on S1, therefore we can write

( )212111 cSdTc T+= , (4.166)

which uses the notation defined in (4.136), and where the transpose operation arises from the

change in summation first introduced in (4.140). Similarly,

( )121222 cSdTc T+= . (4.167)

Eliminating 2c from (4.166) and simplifying:

( )( )1212212111 cSdTSdTc TT ++= , (4.168)


121212122121111 cSTSTdTSTdTc TTT ++= , (4.169)

( ) ( )2212111212121 dTSdTcSTSTI TTT +=− , (4.170)

( )22121111 dTSdTBc T+= , where ( ) 1

2121211

−−= TT STSTIB . (4.171)

Similarly,

( )11212222 dTSdTBc T+= , where ( ) 1

1212122

−−= TT STSTIB . (4.172)

Next, the total scattered field can expressed as

202101ˆˆ cScSc += , (4.173)

where the matrices S are separation matrices valid for iji br > . The over-bars denote complex

conjugation. The definition is slightly different than in (4.137):

( ) ( )klnmklmnS bb −=ψˆ , and ( ){ }klmnkl S bS ˆˆ = . (4.174)

The incident field at the various cylinder centres can be expressed in terms of the incident

field centred about the origin as

dSd 0ˆ

kk = . (4.175)

Finally, substituting (4.175) into (4.171) and (4.172), then both into (4.173) gives the T-

supermatrix.

++

+= 1012120220220212101101

ˆˆˆˆˆˆ STSSTBSSTSSTBST TT

tot . (4.176)

4.3.3 Scattering from N cylinders

Consider N cylinders with centres located at kb and the kth radius given as kR , [ ]Nk ,1∈ .

Each cylinder has a scattered field, and the total scattered field is

( )∑∑=

=N

k mkm

kmsc cu

1rψ . (4.177)

Thus, for sound-hard or sound-soft cylinders there is one boundary condition per cylinder,

resulting in a system of ( )12 +MN boundary conditions with ( )12 +MN sets of unknowns

( 1mc … N

mc ). For transmission problems, there is also an acoustic field within each cylinder,

( )∑=m

kmkmk bv rψ , (4.178)

along with two boundary conditions per cylinder. This means there are ( )122 +MN boundary

conditions with ( )122 +MN sets of unknowns ( 1mc … N

mc and 1mb … N

mb ).


We may simply generalize the results from the two-cylinder cases to arrive at the N-cylinder

equations. Generalizing (4.145) to give the sound-hard equations,

( ) ( ) ( ) ( ) 01

1 =′+′+′ ∑∑≠=

N

lkk n

knlknmlmlm

lmm

lm cSkRJkRHckRJd b , [ ]Nl ,1∈ . (4.179)

Generalizing (4.150) to arrive at the sound-soft equations,

( ) ( ) ( ) ( ) 01

1 =++ ∑∑≠=

N

lkk n

knlknmlmlm

lmm

lm cSkRJkRHckRJd b , [ ]Nl ,1∈ (4.180)

Generalizing (4.157) and (4.162) to yield the penetrable cylinder equations,

( ) ( ) ( ) ( ) ( ) 01

=−++ ∑∑≠=

lmlm

N

lkk n

knlknmlmlm

lmlm

lm kRJbcSkRJkRHckRJd b , (4.181)

( ) ( ) ( ) ( ) ( ) 01

=′−′+′+′ ∑∑≠=

lmlm

N

lkk n

knlknmlmlm

lmlm

lm kRJbcSkRJkRHckRJd b . (4.182)

An N-cylinder solution using T-matrices

The previous results can be generalized to any number of scatterers whose T-matrices are

known. Generalizing (4.171) for N scatterers,

+= ∑≠=

N

lkk

kkTlkllll

1dTSdTBc , where

1

1

−

≠=

−= ∑N

lkk

Tklk

Tlkll STSTIB . (4.183)

This can be rewritten in the form

∑=

=N

kklkl

1dTc , where ( )[ ]k

Tlklklklllk TSITBT δδ −+= 1 . (4.184)

Using a generalized (4.175) and (4.173) gives the T-supermatrix as

∑ ∑= =

=N

l

N

kklkltot

1 100

ˆˆ STST . (4.185)

4.4 The plane wave expansion method

The plane wave expansion method is a well established method for determining the dispersion

properties of waves in solids [244]. It has been widely used for determining the electron band

structure of crystals, and has subsequently been applied to photonic and phononic crystals. For

phononic crystals, the displacement field and the material parameters are decomposed into plane

waves that are periodic with respect to the crystal lattice. The Bloch conditions are then applied

to the boundaries of a unit cell, which results in an eigenvalue problem that reveals the


dispersion relation. Transverse wave modes are present in the solution because the elastic wave

equation is used. The derivation presented below primarily combines the approaches of [245]

and [246], though it should be noted that [247][248] provide excellent background. We assume

that the crystal is composed of lossless dispersionless media.

The PWE method has been successfully applied to 2D phononic crystals [249][250],

including examining waveguiding and defect modes [251], phononic crystals with anisotropic

properties [252], surface waves on 2D crystals [253][254], the prediction of giant band gaps in

fluid-fluid phononic crystals [255], band gaps in fluid-solid phononic crystals [256], band gaps

in solid cylinders suspended in air with defects [257], plates with periodic inclusions [258], and

surface [259] and bulk [260][261] acoustic waves in piezoelectric phononic crystals. It has also

been applied to 3D structures, including periodic elastic composites [262][263], anisotropic

photonic crystals [264], fluid-fluid phononic crystals [265], rigid sphere-air phononic crystals

[266], and describing the energy bands in iron [267]. It has also been modified to include

evanescent modes and used for finding the transmission and reflection at an interface [268], to

correct for computational problems at fluid-solid boundaries [269], and to improve the

convergence when dealing with discontinuities [270].

4.4.1 Derivation of the plane wave expansion method in 3D

As in lower dimensions and as described in Figure 1.2, a 3D phononic crystal is any material

with a unit cell that is periodic in three dimensions. The unit cell could have continuous

properties, such as a linear gradient from one material to the other material within the unit cell.

The PWE method in three dimensions is easily reduced to 2D by removing extraneous terms and

factors.

We begin with the elastic wave equation in three dimensions using Lamé coefficients instead

of the stiffness terms we have used previously. This is to remain consistent with the notation and

derivations in [245] and [246]. Subscripts i and l reference the dimension, ρ is the density, and λ

and µ are the Lamé coefficients of the transmission medium, all of which are functions of space.

The elastic wave equation is

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

i

l

l

i

ll

l

i

i

x

u

x

u

xx

u

xt

uµλ

ρ

12

2

. (4.186)

Since the parameters, ρ, λ and µ, are functions of r, where ( )zyx ,,=r is a Cartesian coordinate,

we cannot treat them as constants as we have in a homogeneous material.

Next, we define a set of reciprocal lattice vectors (RLVs). These will be used in the

exponential basis functions of the spatial Fourier transforms. They can be envisioned as the


spatial equivalent of the discrete frequencies of the Fourier transform of a discrete-time signal. In

the equations, ai is the ith vector of the unit cell, and mi is any integer:

332211 bbbG mmm ++= , Z∈im (4.187)

and ( )321

321 2

aaa

aab

×⋅

×= π ,

( )132

132 2

aaa

aab

×⋅

×= π , and

( )213

213 2

aaa

aab

×⋅

×= π . (4.188)

Note that the denominator is equal in every case because of the cyclic nature of the scalar triple-

product which corresponds to the volume of the unit cell.

The material parameters are expanded as a spatial Fourier series using the RLVs in the

exponential to define the set of basis functions.

( ) ∑ ⋅−− =G

rG

Gr je11 ρρ , ( ) ∑ ⋅=G

rG

Gr jeλλ , and ( ) ∑ ⋅=G

rG

Gr jeµµ , (4.189)

where, fG is the magnitude of the Gth Fourier component in the expansion of the function f.

According to Bloch’s theorem [95], the solutions to the wave equation will be modulated

sinusoids of the form

( ) ( ) rk

k rr⋅= 0

0

jeuu , (4.190)

where ( )

+= ∑

= 3,2,100

i

iinuu arr kk , Z∈in . (4.191)

That is, ( )rk0u is unit-cell periodic. Thus, we can expand the function on the same unit-cell basis

set as before,

( ) ( )∑∑ ⋅+

+

⋅⋅+ =

=

G

rGk

Gk

rk

G

rG

Gkr 0

0

0

0

jjjeueeuu . (4.192)

In this equation, k0 is restricted to being within the first Brillouin zone, a concept that is

subsequently discussed in greater detail. Values of k0 that are outside this region may be reached

instead by adding a RLV to a smaller k0 that is within the first Brillouin zone, which is a

consequence of aliasing due to the periodicity of the system.

To explain (4.192) further, Bloch’s Theorem says that ( )rk0u is periodic with respect to the

crystal lattice, and therefore we can use the same basis set of exponentials as before for the

material parameters in (4.189). An intuitive explanation of this result of Bloch’s Theorem is as

follows. Since the crystal under consideration is infinite, there is no way to distinguish one unit

cell from another except by relative location; therefore the wave frequency components must be

the same in every unit cell, and only the phase changes between cells. If the crystal was finite,

then each cell could be distinguished by its position, and Bloch’s Theorem would not hold.


Now that we have orthogonal decompositions of the material parameters and wave solutions

all using the same basis set, we may now substitute the Fourier expansions into the wave

equation. The goal is to get both sides of the equation to have the same exponent, ( )rGk ⋅+0je . In the

following notation, ( )i

Gk +0 means the component of vector Gk +0 resolved in the i direction,

( ) ( ) iixGkGk ˆ00 ⋅+=+ . The terms in brackets result from taking a directional derivative of the

complex exponential. For example,

( ) ( ) ( ) ( )[ ] ( ) ( ) rGkGkGkGkrGk Gk ⋅++++++⋅+ +=∂

∂=

∂

∂00000

0j

y

zyxjjeje

ye

y

zyx . (4.193)

We begin by substituting the Fourier expansion of u into the wave equation (4.186),

∑

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂=− −

l l

i

li

l

ll

l

i

ix

u

xx

u

xx

u

xu µµλρω 12 . (4.194)

Next, the expansion for u is substituted into the equation,

( )

( )

( )

( )

∑

∑

∑

∑

∑

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂

=−

′

⋅′+′+

′

⋅′+′+

′

⋅′+′+

−⋅+

+l

ji

ll

jl

il

jl

li

ji

euxx

euxx

euxx

eu

G

rGk

Gk

G

rGk

Gk

G

rGk

Gk

G

rGk

Gk

0

0

0

0

0

0

0

0

12

µ

µ

λ

ρω . (4.195)

The primed symbol simply means that the summation on the RHS is not tied to the summation

on the LHS. They both refer to the set of RLVs. Computing the innermost derivatives,

( )

( ) ( )

( ) ( )

( ) ( )

∑

∑

∑

∑

∑

′+

∂

∂+

′+

∂

∂+

′+

∂

∂

=−

′

⋅′+′+

′

⋅′+′+

′

⋅′+′+

−⋅+

+l

j

l

i

l

j

i

l

l

j

l

l

i

ji

ejux

ejux

ejux

eu

G

rGk

Gk

G

rGk

Gk

G

rGk

Gk

G

rGk

Gk

Gk

Gk

Gk

0

0

0

0

0

0

0

0

0

0

0

12

µ

µ

λ

ρω . (4.196)

Expanding the Lamé coefficients,

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

∑

∑ ∑

∑ ∑

∑ ∑

∑

′+

∂

∂+

′+

∂

∂+

′+

∂

∂

=−

′−′′ ′

⋅′+′+

⋅′−′′′−′′

′−′′ ′

⋅′+′+

⋅′−′′′−′′

′−′′ ′

⋅′+′+

⋅′−′′′−′′

−⋅+

+l

j

l

ij

l

j

i

lj

l

j

l

lj

i

ji

euex

euex

euex

jeu

GG G

rGk

Gk

rGG

GG

GG G

rGk

Gk

rGG

GG

GG G

rGk

Gk

rGG

GG

G

rGk

Gk

Gk

Gk

Gk

0

0

0

0

0

0

0

0

0

0

0

12

µ

µ

λ

ρω , (4.197)


where the Lamé coefficients have been expanded about G ′′′ , chosen so that GGG ′−′′=′′′ .

Combining the exponentials to cancel out G′ ,

( )

( ) ( )

( ) ( )

( ) ( )

∑

∑ ∑

∑ ∑

∑ ∑

∑

′+

∂

∂+

′+

∂

∂+

′+

∂

∂

=−

′−′′ ′

⋅′′+′+′−′′

′−′′ ′

⋅′′+′+′−′′

′−′′ ′

⋅′′+′+′−′′

−⋅++

l

j

l

i

l

j

i

l

l

j

l

l

i

ji

eux

eux

eux

jeu

GG G

rGkGkGG

GG G

rGkGkGG

GG G

rGkGkGG

G

rGkGk

Gk

Gk

Gk

0

0

0

0

0

0

0

0

0

0

0

12

µ

µ

λ

ρω . (4.198)

Computing the final derivative,

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

∑

∑ ∑

∑ ∑

∑ ∑

∑

′′+′++

′′+′++

′′+′+

=

′−′′ ′

⋅′′+′+′−′′

′−′′ ′

⋅′′+′+′−′′

′−′′ ′

⋅′′+′+′−′′

−⋅++

l

j

ll

i

j

li

l

j

il

l

ji

eu

eu

eu

eu

GG G

rGkGkGG

GG G

rGkGkGG

GG G

rGkGkGG

G

rGkGk

GkGk

GkGk

GkGk

0

0

0

0

0

0

0

0

00

00

00

12

µ

µ

λ

ρω . (4.199)

Finally, the density coefficient is expanded as ∑′′′

⋅′′′′′′=

G

rG

G

jeρρ , where we now choose

GGG ′′−=′′′ to link the RHS with the LHS, giving

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )∑ ∑ ∑ ∑∑

′′− ′−′′ ′

⋅′′+

′+′−′′

′+′−′′

′+′−′′

⋅′′−−′′−

⋅++

′′+′++

′′+′++

′′+′+

=l

j

ll

i

li

l

il

l

jji e

u

u

u

eeuGG GG G

rGk

0GkGG

0GkGG

0GkGG

rGGGG

G

rGkGk

0

GkGk

GkGk

GkGk

0

0

0

12

0

0

0

0

0

µ

µ

λ

ρω . (4.200)

Combining the exponentials to cancel out G ′′ ,

( )

( ) ( )

( ) ( )

( ) ( )

( )∑ ∑ ∑ ∑∑′′− ′−′′ ′

⋅+

′+′−′′

′+′−′′

′+′−′′

−′′−

⋅++

′′+′++

′′+′++

′′+′+

=l

j

i

ll

l

li

l

il

ji e

u

u

u

euGG GG G

rGk

Gk00GG

Gk00GG

Gk00GG

GGG

rGkGk

0

0

0

GkGk

GkGk

GkGk

00

0

12

µ

µ

λ

ρω , (4.201)

which can be rearranged as

( ) ( )

( ) ( )

( ) ( )

( ) 02

,

1 =

−

′′+′++

′′+′++

′′+′+

⋅+

′+

′′

′+′−′′

′+′−′′

′+′−′′

−′′−∑∑ rGk

G

Gk

G

Gk00GG

Gk00GG

Gk00GG

GG0

0

0

0

0

GkGk

GkGk

GkGk

ji

li

ll

l

li

l

il

eu

u

u

u

ω

µ

µ

λ

ρ . (4.202)

The above equation must be valid for all r, therefore every term in the brackets must be zero

for every choice of G, which is an eigenvalue problem of the form UΛAU = . In this

configuration, A is a matrix of coefficients, U is a matrix of eigenvectors, and Λ is a diagonal


matrix of eigenvalues. Beginning with the A matrix, for a particular G and G′ ( iG and jG′ ) the

elements of the matrix are composed of 33× matrices, so that

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

∑

∑

′′

′−′′

′−′′

′−′′

−′′−′

′′+′++

′′+′+′′+′+′′+′+

′′+′+′′+′+′′+′+

′′+′+′′+′+′′+′+

+

′+′′+′+′′+′+′′+

′+′′+′+′′+′+′′+

′+′′+′+′′+′+′′+

=G

00GG

000000

000000

000000

GG

000000

000000

000000

GG

GGG,G

IGkGk

GkGkGkGkGkGk

GkGkGkGkGkGk

GkGkGkGkGkGk

GkGkGkGkGkGk

GkGkGkGkGkGk

GkGkGkGkGkGk

A

3

1

lllj

zzjyzjxzj

zyjyyjxyj

zxjyxjxxj

zjzyjzxjz

zjyyjyxjy

zjxyjxxjx

j

j

j

iji

µ

µ

λ

ρ

, (4.203)

where 3I is the 33× identity matrix. Considering a truncated set of M RLVs, the total matrix, A

is composed of the sub-matrices and is

[ ]Mjiji L1,, ∈

′= GGAA . (4.204)

The matrix of the eigenvectors is

=

−

−

−

−

−

−

−

−

−

−

−

−

zM

Gk

yM

Gk

xM

Gk

z

Gk

y

Gk

x

Gk

zM

Gk

yM

Gk

xM

Gk

z

Gk

y

Gk

x

Gk

M

M

M

M

M

M

u

u

u

u

u

u

u

u

u

u

u

u

,3

,3

,3

,1

,1

,1

,3

,3

,3

,1

,1

,1

0

0

0

0

0

0

10

10

10

10

10

10

L

MLM

L

U , (4.205)

and the matrix of the eigenvalues is

=23

21

0

0

Mω

ω

OΛ , (4.206)

where MM 33C ×∈ΛU,A, . The M3 eigenvalues correspond to the different energy bands of the

phononic crystal. The eigenvalues represent the squared angular frequencies allowed for each

chosen 0k and therefore allow one to determine the dispersion properties of the phononic

crystal.

The first Brillouin zone

In 3D, the first Brillouin zone is bounded by taking the surface enclosing a single repeated

lattice element (called the Wigner-Seitz cell and defined by a1, a2, and a3), and representing that


surface in reciprocal space (defined by the RLVs). Next, select any point in the reciprocal lattice

as the origin and connect it to neighbouring points. Construct the perpendicular bisectors of these

lines. The region of intersection is the first Brillouin zone. [271]

The construction of the first irreducible Brillouin zone is shown in Figure 4.5 for a 2D

rectangular lattice, and in [246] for a face-centred cubic cell in 3D. The points indicated on the

figure represent points of interest. The allowed frequencies along a continuous path of k-vectors

that connects some or all of these points is plotted, which explains the symbols often found on

the bottom axis of dispersion diagrams. In 1D, the path is from Γ ( 0=xk ) to X ( akx π= ). In

2D, the path is from Γ ( ( )0,0=k ) through X ( ( )0,1aπ=k ) and M ( ( )21 , aa ππ=k ), and back

to Γ.

Figure 4.5. The construction of the first irreducible Brillouin zone for a rectangular lattice in 2D including the major lattice points of interest. The primitive rectangular lattice vectors are shown in a), and the primitive reverse lattice vectors are shown in b). In c), the chosen origin is connected to its nearest neighbours. The perpendicular bisectors of these lines is shown in d) as defining the boundary of the first irreducible Brillouin zone, expanded in e) and labelled with the major points of interest.

Energy bands where 0→ω as 0→k are called acoustic bands and the others are called

optical bands. When the peak and trough of the lower and higher energy bands defining a band

gap align with each other, the band gap is called “direct”. If they do not align, it is called

“indirect”.

4.4.2 Implementation details

During simulation there are two possible approaches: The first is to compute the Fourier

coefficients at each step, which simplifies indexing but results in numerous redundant

calculations. The second method is to pre-compute all the necessary coefficients and use some

indexing method to track which coefficients are needed at each step. Either more than M

coefficients will be necessary or values outside of a certain range will be truncated. If M is an

odd number cubed, then the required number of pre-computed coefficients will be ( )33 12 −M .

This value is arrived at by observing that two RLVs are subtracted when computing the basis


function of the Fourier coefficients. Since the set of M RLVs are contained in a cube in inverse

space, every combination of subtracting two of them will result in a cube of subtracted RLVs

with a side length double the original. Once these values are computed and stored, they only

need to be indexed when computing the elements of A, which is an implementation-specific task.

In general, the set of RLVs may form a rhombic prism instead of a cube, but the principle

remains the same. For 1D systems the required number of pre-computed Fourier coefficients will

be 12 −M .

The 3D equations may be reduced to 2D by deleting every third row and column of matrix A,

which essentially removes the z dimension from the wave equation. If pre-computing of the

coefficients is required and M is an odd number squared, then the required number of pre-

computed coefficients will be ( )212 −M .

4.5 Other simulation methods

There are other simulation methods that were not explored in the course of this thesis work, and

some of these have been successfully applied to the analysis of phononic crystals. They include

layer multiple scattering, pseudospectral techniques, and the finite-element method (FEM) in

addition to other hybrid or unique models and computational methods. They are described briefly

here for the sake of completeness.

The layer multiple scattering method [170]–[172][231][233][272]–[279] is a hybrid between

the concepts of multiple scattering theory and the transmission matrix method. Multiple

scattering theory is used to determine the T-matrix for a particular unit of scattering material,

such as a slab of periodic scatterers. Next, the T-matrices for multiple slabs are combined

through methods very similar to the transmission matrix method, allowing larger-scale structures

to be composed out of smaller-scale scattering structures. Since the matrix methods presented in

Section 4.3 allow the construction of such macroscopic scattering structures, one may argue that

we have already presented layer multiple scattering. However, the methods present in the

references are quite formalized and comprehensive, and give specific instruction for dealing with

periodic scatterers (esp. see Equation (17) and (18) of [233]).

Pseudospectral methods [280]–[284] use a spatial Fourier transform to estimate the spatial

derivatives in the wave equation, allowing the Fast Fourier Transform (FFT) algorithm to be

used. Spatial derivatives calculated as in the FDTD method are subject to numerical dispersion

and lead to phase error at high spatial frequencies. Thus, higher resolution grids are required to

mitigate the numerical dispersion at high frequencies, resulting in longer simulation run times.

Pseudospectral methods sidestep this problem by calculating the derivatives in the k-space

domain (the spatial-frequency domain), where phase error is not present. However, the drawback


is that non-periodic structures are difficult to simulate because of the inherent periodicity of the

FFT.

The finite-element method is similar to the FDTD method, except that integral forms of the

underlying PDEs are used instead of differential forms as in the FDTD method [282]. The

simulation region is partitioned into elements, and the field values at the boundaries of these

elements are used to fit a local basis function to each element. Boundary conditions are applied

to the edge of the finite-element simulation region, as in the FDTD method. Then, the required

numerical integration is performed across the elements to determine the field values in question.

The finite-element approach has been applied to such diverse problems as investigating surface

resonant states and superlensing in phononic crystals [33], electrically pumped photonic-crystal

terahertz lasers [283], and metamaterials [284]–[286]. With respect to phononic crystals, the

finite-element method has been used to study surface acoustic waves in bulk phononic crystals

[287] and phononic crystal plates [105]–[107][288][289].

Finally, it is worth mentioning that the simulation and modelling methods presented are

neither exhaustive or exclusive. See [290] for a review of several of the methods presented in

addition to some others not discussed here. Some methods combine features of several different

methods, such as in [291] where the authors used a finite difference of the governing differential

equations for the eigenvalue problem. In [292], an FDTD approach was used in k-space instead

of Cartesian space. A lumped-mass model was used in [21] and [293] to derive the band

structure of phononic crystals. A transfer-matrix approach using Green's functions was employed

in [294]. In [295], a phase-space model was used. A method called eigenmode matching theory,

which bears a strong resemblance to the PWE method, was presented and used in [296]. Finally,

wavelet-based methods were used in a plane-wave expansion in [297] as an alternative to

Fourier-based expansions. It should be clear from these examples that the pallet of simulation

and modelling methods for periodic acoustic systems is quite rich.

4.6 Amenability to time-varying material parameters

To investigate which of the simulation methods presented are amenable to analyzing time-

varying material parameters, one must first consider the fundamental implications of such

changes. Considering the wave equation with a time-varying density in 1D and beginning with

momentum,

t

umP

∂

∂= , (4.207)

t

u

t

m

t

um

t

PF

∂

∂

∂

∂+

∂

∂=

∂

∂=

2

2

. (4.208)


Converting this to the density formulation,

xt

u

tt

u

∂

∂=

∂

∂

∂

∂+

∂

∂ 112

2 σρρ , (4.209)

and substituting the constitutive equation,

x

uC

∂

∂= 1111σ , (4.210)

gives

2

2

112

2

x

uC

t

u

tt

u

∂

∂=

∂

∂

∂

∂+

∂

∂ ρρ , (4.211)

which is the modified 1D wave equation for a time-varying density.

This formulation would appear to support the notion that a time-varying density modifies the

wave equation, but further consideration leads to a different conclusion. One could imagine a

cube of material travelling at a given velocity with no external forces. If the mass of this cube

somehow increases, the new additional mass would need to share the kinetic energy of the initial

mass, and the velocity would decrease. However, this argument does not seem to hold for a time-

varying density. If two cubes of equal mass and density are travelling at a given velocity,

squeezing the two masses together into the space occupied by one of the initial cubes would

double the density. However, since each cube had the same initial kinetic energy, there is no

energy transfer when the two cubes are squeezed together. So it would seem under these

circumstances that a time-varying density and a time-varying mass are not equivalent concepts

either from a momentum perspective or energy conservation perspective.

To put this more explicitly, the density is changing but mass is conserved, so 0≠∂

∂

t

ρ but

0=∂

∂

t

m. This is possible because Vm=ρ and therefore the change in density originates from

the change in volume and not a change in mass,

2V

t

Vm

t

mV

t

∂

∂−

∂

∂=

∂

∂ρ, (4.212)

t

V

V

m

t ∂

∂−=

∂

∂2

ρ. (4.213)

This shows the expected result, that a negative change in volume creates a positive change in

density without affecting the mass of the system. Thus, the wave equation is not fundamentally

altered by a time-varying density.

The wave equation remains unchanged when considering a time-varying compressibility,

( )tC11 , as well. This is because there are no time-derivatives present in the stress-strain


relationship, and therefore ( ) ( )x

utCt

∂

∂= 1111σ . Similar arguments also apply to the stress-strain

relationship in higher-dimensions.

In light of these arguments, there is nothing fundamentally prohibitive about any of the

presented simulation methods in the presence of time-varying material parameters. The question

is the how complicated is the implementation and what benefit would such a model and

simulator provide. The FDTD method can be used as-is, simply by updating the material

parameters at time-step in the simulation. However, it still suffers from the same benefits and

drawbacks as before. Namely, it cannot provide closed-form solutions for the behaviour of a

time-varying system and therefore does not give any insight into why a particular behaviour is

occurring. The TMM and MST methods can be modified to handle time-varying material

parameters, and will give closed-form solutions to wave propagation in time-varying structures.

This is the bulk of new research presented in this thesis in Chapters 5 and 6. The PWE method

could be modified to handle time-varying material parameters, but this avenue has not been

investigated because the complexity of the eigenvalue problem obfuscates the analytical

relationships of the system.

- 99 -

Chapter 5

DYNAMIC PHONONIC CRYSTALS IN ONE

DIMENSION

1The phononic crystal effects previously described are typically narrowband and highly

dependent on the size and spacing of the scatterers within the crystal – properties that are fixed

and unchangeable. We have proposed that it may be possible to dynamically alter the behaviour

of phononic crystals by varying their material parameters in time, possibly by using active

materials. In this chapter, we extend the existing 1D static phononic crystal theory, the

transmission matrix method detailed in Section 4.2, so as to handle time-varying material

parameters and also to provide insight into the factors governing time-varying phononic crystal

behaviour.

Using the transmission matrix method as a starting point, the transmission equations for both

continuous (including non-periodic) and periodic time-varying material parameters are derived.

We then develop a closed-form solution to the acoustic wave transmission through a 1D time-

varying phononic crystal that shows excellent agreement with FDTD simulations and with

execution times several orders of magnitude faster. We have called our new 1D method the time-

varying transmission matrix method (TV-TMM). Results show that transmission properties can

be significantly altered using time-varying material parameters as the driving mechanism. Effects

such as parametric amplification and signal switching are demonstrated in simulation. Finally,

some conclusions are drawn.

5.1 The time-varying transmission matrix method

To develop the equations for a time-varying phononic crystal, it is necessary to first consider

only the transmission through a single time-varying cell. A phononic crystal can be made time-

varying by making any of its defining parameters vary in time. However, this analysis will first

consider only a time-varying impedance.

In the dynamic case, a time-varying tube diameter in a corrugated tube waveguide is a

surrogate for time varying material parameters in a phononic crystal. The tube waveguide was

1 This chapter contains material published in the 2009 paper by DW Wright and RSC Cobbold appearing in Smart Materials and Structures, 18 015008, entitled “Acoustic wave transmission in time-varying phononic crystals”.

Chapter 5. Dynamic Phononic Crystals in One Dimension 100

specifically chosen as a model because it has the property that 0c is constant even when the tube

diameter changes, so that only the effective acoustic impedance changes. This is equivalent to

the density and bulk modulus of a particular material, i, in a phononic crystal having a constant

ratio since ( ) ( ) ( )ttKtc iii ρ=0 , but the impedance is free to change since ( ) ( ) ( )tKttZ iii ρ= .

The result is that the propagation matrices are time-invariant ( P1T and P2T ), but the transmission

matrices, ( )t21T → and ( )t12T → , are time-varying. This restriction was initially chosen to make the

problem tractable. Subsequently, we then generalize our method to include time-varying

propagation matrices.

5.1.1 Transmission through one cell

Referring to Section 4.2, we extend the variables and matrices to be time-varying. Let

( ) ( ) ( )[ ]Ttatat −+=a , (5.1)

( ) ( ) ( )[ ]TAA ωωω −+=A , (5.2)

and ( ){ } ( )ωAa =ℑ t , (5.3)

where ℑ denotes the Fourier transform. Also let

( ){ } ( )ω2121 TT →→ =ℑ t , (5.4)

so that the argument indicates whether the function is in the time or angular frequency domain.

Time-varying scattering

The elements of ji→T are time-varying when iZ is a function of time. Assuming that iZ is

frequency independent, the time-varying scattering analogous to the single frequency case shown

in Figure 4.3 is given as

( ) ( ) ( )ttt aTb 21→= . (5.5)

Thus, a monochromatic wave incident on a time-varying scatterer will be modulated resulting in

a scattered spectrum, as depicted in Figure 5.1.


Figure 5.1. A monotone is scattered and modulated at every time-varying interface.

Time-varying propagation

When the transmitted portion of a monochromatic wave passes through material whose phase

velocity is time-varying, the wave will undergo phase modulation, as shown in Figure 5.2.

Figure 5.2. A monotone is modulated as it propagates through a time-varying slab of material.

Two modulating processes occur within a material that has a time-varying phase velocity.

The first is that the instantaneous wave number ( )tki at the interface within material i becomes a

function of the instantaneous phase velocity ( )tc0 and the input angular frequency ω ′ , such that

( ) ( )tctk ii 0ω′= . (5.6)

The second modulating process is that the propagation time for a wave to travel a distance id

through material i becomes a function of time, ( )tT0 . It can be visualized as a conveyor belt of

length id with an instantaneous speed ( )tc i0 carrying acoustic particle displacements that entered

the conveyor belt at time ( )tTt 0− and exit the conveyor belt at time t. In this analogy, the

material itself is not being transported, only the displacements from one end to the other. This

second process is fairly complex. A comparison of the various calculation methods available is

given in [298], and these include analytical, semi-analytical, and numerical approaches. For the

time being, we consider only constant phase velocities and devote a subsequent discussion to this

problem.


The time-varying propagation matrix is expressed in the angular frequency domain,

( )ωω ′,iPT , and can be understood as a map of how much energy of an input frequency ω ′ is

modulated to an output frequency ω . Thus,

( ) ( ) ( ) ωωωωω ′′′= ∫ dATB P , . (5.7)

The time-varying cell transmission matrix

In the following development, ( )t0a and ( )t1a are separated by a distance a , and b , c , and

d are simply intermediate locations as described subsequently. To develop the time-varying cell

transmission matrix, we proceed from ( )t0a on the left inside of section 1 (see Figure 2.15) to

the right, as in the previous static case. Assuming a constant phase velocity, propagation from the

section entrance to the first interface is given as

( ) ( ) ( )ωωω 01 ATB P= , (5.8)

where the distance propagated is ( )ad η−= 11 . Propagation from section 1 into section 2 obeys

the transmission and reflection defined by the 21T → transmission matrix, however, this matrix is

now a function of time. Since the interface is infinitesimally small, there is no propagation delay

and the waves on the right-hand side of the interface can be written as

( ) ( ) ( )ttt bTc 21→= . (5.9)

Propagating the waves from this interface to the next can be written as

( ) ( ) ( )ωωω CTD P2= , (5.10)

where the distance propagated is ad η=2 . Finally, propagation from section 2 back into

section 1 is

( ) ( ) ( )ttt dTa 12→=1 . (5.11)

Combining all of these equations yields the wave propagation through one cell, given by

( ) ( ) ( ) ( ) ( ) ( ){ }{ }{ }{ }ωωωω 011

21

1 ATTTTA P21P12

−→

−→ ℑℑℑℑ= tt , (5.12)

in which 1−ℑ denotes the inverse Fourier transform. Expanding the Fourier integrals, combining

the interior exponentials and reordering the integration gives

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ,4

14

14

1

10111212

1011112222

21101112221

1121

221111

∫ ∫

∫ ∫ ∫∫

∫ ∫ ∫ ∫

′′′′−−=

′′′=

′′′=

→→

′−−

→

−−

→

−−′

→→

ωωωωωωωωωπ

ωωωωωπ

ωωωωωπ

ω

ωωωω

ωωωω

dd

dddtetdtet

dtededtedett

tjtj

tjtjtjtj

ATTTT

ATTTT

ATTTTA

P21P12

P21P12

P21P12

, (5.13)


where ω ′ and ω refer to the input and output angular frequencies, respectively. This simplifies

to

( ) ( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ,,4

14

1

012

011112121

∫

∫ ∫

′′′=

′′′′−−= →→

ωωωωπ

ωωωωωωωωωπ

ω

d

dd

AT

ATTTTA

C

P21P12

, (5.14)

where ( ) ( ) ( ) ( ) ( )∫ ′′−−=′→→ 1111211 , ωωωωωωωωω dP21P12C TTTTT . (5.15)

5.1.2 Time-varying phase velocity

With a time-varying phase velocity, ( )tc0 , the propagation matrices will be of the form in

(5.7), which is the mapping from an input frequency ω′ to an output frequency ω .

Consequently, the above equations can be rewritten using additional variables of integrations as

follows. Equation (5.12) becomes

( ) ( ) ( ) ( ) ( ) ( ){ }{ }{ }{ }∫ ∫ ′′′′′′′= −→

−→ ωωωωωωωω ωω ddFtFFtF tt 011

1122

121 ,,

1122ATTTTA P21P12 , (5.16)

where the subscripts in the Fourier and inverse Fourier transforms indicate which variable is

involved in the transform. Expanding the Fourier integrals, combining the interior exponentials

and reordering the integration gives

( ) ( ) ( ) ( )

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( )( ) ( )( )

( )∫ ∫∫

∫

∫ ∫ ∫ ∫∫

∫

∫ ∫ ∫ ∫ ∫ ∫

′′

′′′−′′×

′′−=

′′′

′′×

′′=

′′′′′

′′=

→

→

−′′−

→

−−

→

−′′−

→→

ωωωωωωωω

ωωωωω

π

ωωωωωωω

ωω

π

ωωωωωωω

ωωπ

ω

ωω

ωω

ωωωω

ddd

d

dddddtet

dtet

dtededdteded

tt

tj

tj

tjtjtjtj

0

1111

2222

2

21

01111

2222

2

2211011

122221

,

,

4

1

,

,

4

1

,

,4

1

11

22

222111

ATT

TT

ATT

TT

AT

TTTA

P21

P12

P21

P12

P

21P12

L

L

, (5.17)

where the ellipsis on the first line indicates that the equation is continued below for space

reasons. Therefore,

( )

( ) ( )( )( ) ( )( )

( )

( ) ( )∫

∫ ∫∫

∫

′′′=

′′′′

′−′′×

′′−=

→

→

ωωωωπ

ωωωωωωωω

ωωωωω

πω

d

ddd

d

012

0

1111

2222

21

,4

1

,

,

41

AT

ATT

TTA

C

P21

P12

, (5.18)

where ( ) ( ) ( )( ) ( ) ( )( )∫ ∫∫ ′′′−′′′′−=′→→ ωωωωωωωωωωωωω ddd 111122221 ,,, P21P12C TTTTT . (5.19)


Calculation of TPi

As for the actual calculation of ( )ωω ′,iPT , one may consider the phase of a wave exiting a

slab of material i of thickness id with a time varying phase velocity, ( )tc i0 . If the phase of a

wave entering the region is ( ) φωωθ +′=′ ttin , , where φ accounts for any phase offset, then the

wave at that location is defined as inje

θ . The phase exiting the region, ( )ωθ ′,tout , will be

( ) ( )( ) φωωθ +−⋅′=′ tTttout 0, , (5.20)

where ( )tT0 is the propagation time for a wave to travel from one end to the other. It is itself a

function of time because the phase velocity is changing. Thus, it is the solution to the equation,

( )( )

i

t

tTt

i ddc =∫− 0

0 ττ . (5.21)

The transformation from input to output is therefore

( ) ( )tTjjee inout 0ωθθ ′−− = , (5.22)

and so the spectrum at the output is ( ){ }tTj

t e 0ω′−ℑ .

The propagation matrix is therefore the output spectrum multiplied by the frequency-

dependent phase shift encountered by each frequency component, and is

( ) ( ){ }

ℑ=′

−′−

ii

ii

cdj

cdj

tTj

tie

ee

0

0

0

0

0,

ω

ωωωωPT . (5.23)

To check that the phase exiting the region with a constant phase velocity is the expected

result, ( ) φωωθ +−′=′iiout dktt 0, , the integral of (5.21) is performed with ic0 constant, giving

i

i

c

dT

00 = . (5.24)

Substituting this into the equation for the phase at the output gives

φω

φωθ

+−′=

+

−⋅′=

ii

i

i

out

dkt

c

dt

0

0 , (5.25)

which is the expected result of a phase delay of ii

dk0 and is equivalent to the retarded time [66]

and leads to the propagation matrix,

( ) ( )

′−=′

−

ii

ii

cdj

cdj

ie

e

0

0

0

02,

ω

ω

ωωπδωωPT . (5.26)

If the phase velocity is varying sinusoidally, then we may write

( ) ( )[ ]tvCtc pii ωcos100 += , (5.27)


where i

C0 is the time-average phase velocity, v is the variation factor, and pω is the pumping

angular frequency. In this case, (5.21) is

( )[ ]( )

i

t

tTtpi ddvC =+∫

− 0

cos10 ττω , (5.28)

( )( )

i

t

tTt

p

p

i dv

C =

+

−= 0

sin0

τ

τωω

τ , (5.29)

( ) ( ) ( )[ ]( )[ ] 000 sinsin ttTttv

tT pp

p

=−−+ ωωω

, (5.30)

where ii

Cdt 00 = . Unfortunately, this equation is not easily solvable for ( )tT0 and so

approximations or simplifications are necessary. At the very least, ( )ωω ′,iPT can be determined

numerically using the equations provided in this section.

5.1.3 Transmission through multiple cells

We will now revert back to the assumption of a constant phase velocity. To determine the

acoustic wave propagation through multiple time-varying cells, we begin by considering the

transmission through two cells and then we extend this result to n cells. For an input spectrum of

( )ω′0A , the output spectrum from the second cell can be obtained from (5.14) as

( ) ( ) ( )

( ) ( ) ( ) ( ) ,,,2

,4

1

1011112

1111122

∫ ∫

∫′′′=

=

− ωωωωωωωπ

ωωωωπ

ω

dd

d

nATT

ATA

CC

C

(5.31)

where 2=n for this case. Reordering the integration leads to

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( )∫

∫ ∫′′′=

′′′=

−

−

ωωωωπ

ωωωωωωωπω

d

dd

n

n

022

0111112

2

,2

,,2

AT

ATTA

C

CC

, (5.32)

where ( ) ( ) ( )∫ ′=′ 111112 ,,, ωωωωωωω dCCC TTT . Extending this result to express the relationship

between the input and output spectra at the two ends of an n-cell time-varying phononic crystal

yields

( ) ( ) ( ) ( )∫ ′′′=−

ωωωωπω dn

n

n 02 ,2 ATA C , (5.33)

where ( )ωω ′,nCT is the time-varying transmission matrix for n cells and is given by

( ) ( ) ( )∫ −−−−′=′

lnlnllnlnn dωωωωωωω ,,, CCC TTT , (5.34)


in which ln−ω is an intermediate integration variable. Figure 5.3 illustrates propagation through n

time-varying cells. Beginning with 1CT which has the closed-form solution given in (5.15), nCT

can be constructed by repeated integration using (5.34). If each cell is identical, then nCT may be

constructed exponentially ( 1CT , 2CT , 4CT , 8CT , etc.). However, if the cells are not identical, or

are undergoing different material parameter variations, then nCT may be constructed by repeated

application of (5.34) using the 1CT corresponding to each unique cell.

Figure 5.3. The time-varying cell transmission matrix provides an output spectrum, ω, for a given input frequency ω′.

5.1.4 Conversion back into a transmission spectrum

To extract the transmission coefficient, it is necessary to convert the transmission matrix

back into a scattering matrix. The conversion formula given in (4.96) is valid for the time-

varying case if one considers the instantaneous time domain solution for a particular input

frequency over which to solve the scattering matrix. Setting 0ωω =′ gives

( )( )

( )( ) ( )

−=

0120

021

022

0 ,,

1,

,

1,

ωω

ω

ωω

tTt

tT

tTt

Cnn

Cn

Cn

n

CTS , (5.35)

where ( ) ( )[ ]00 ,, ωωω tnn

SS ℑ= and ( ) ( )[ ]01

0 ,, ωωωnn

t CC TT −ℑ= . For an input signal spectrum of

( )ω′+0A , the transmission output spectrum from the nth cell can be determined by integrating over

all input frequencies, i.e.,

( ) ( ) ( )∫ ′′′= ++ ωωωωω dASA nn 021 , . (5.36)

Thus, the function ( )ωω ′,21nS can be thought of as a surface where each point relates how much

of an input frequency ω ′ is converted to the output frequency ω when n time-varying cells are

connected together.

5.2 Solutions when the parameter variation is periodic

We are particularly interested in situations when the material parameters vary periodically in

time which represents a mode of operation that has not yet been adequately explored. Here, we

assume that the phase velocity is constant to help simplify the analysis. However, using the

results of Section 5.1.2, periodic solutions for time-varying phase velocities could also be

incorporated.


5.2.1 Single-cell solution

Again assuming a constant phase velocity, let us determine the expression for ( )ωω ′,1CT

given a known periodic parameter variation signal (i.e. the elements of 21T → and 12T → are

known periodic functions of time). Since 21T → and 12T → are periodic, they can be represented by

the Fourier series

( ) ∑ →→ =

n

tjn

n

petω21

21 CT , and ( ) ∑ →→ =

m

tjm

m

petω12

12 CT , (5.37)

where pω is the fundamental material parameter variation angular frequency (commonly called

the pumping frequency) and

=

nn

nn

ncc

cc

2221

1211C are the corresponding matrices of complex

Fourier coefficients. The Fourier transforms are then given as

( ) ( )∑ −= →→

n

pnnωωδω 21

21 CT and ( ) ( )∑ −= →→

m

pmmωωδω 12

12 CT . (5.38)

Recalling (5.15) and substituting the known Fourier series spectra,

( ) ( ) ( ) ( ) ( )∫ ∑∑ ′−′−−−=′ →→11111 , ωωωωωδωωωωδωω dnm

n

pn

m

pm P1

21

P2

12

C TCTCT . (5.39)

Integration over either of the delta functions in (5.39) results in

( ) ( ) ( ) ( )( )∑∑ +−′−′+′=′ →→

m

p

n

npmnmn ωωωδωωωωω P1

21

P2

12

C TCTCT ,1 . (5.40)

Thus, given an input frequency ω ′ and a pumping frequency pω , the relevant output frequencies

over ω can be determined by choosing all m and n to make the argument of the delta function

in (5.40) equal to zero, corresponding to the condition: ( ) pnm ωωω ++′= . That is, choosing m

and n determines which output frequency receives that particular incremental contribution from

the integral, which is now expressed as a summation. To facilitate a “per-frequency” formulation

we change the indices to read

nmp += , nq −= , (5.41)

so that qpm += , qn −= . This allows p to set the frequency of ω , and q to facilitate the

summation of all relevant pairs of m and n corresponding to that frequency. Using this

substitution and noting that ppωωω +′= enables the transmission matrix for a single cell to be

expressed as

( ) ( )

( ) ( ) ( ).

,,

12

11

p

p q

qpqp

p

p

pq

p

ωωωδωωω

ωωωωω

−′−′−′=

′+′=′

∑∑

∑→

−→+ P

21

P

12

CC

TCTC

TT

(5.42)


5.2.2 An n-cell solution

The above single cell solution can be extended to n-cells by first obtaining the transmission

matrix for two cells. The two matrix terms within the square brackets of (5.32) are replaced with

summations as given by (5.42), i.e., ( ) ( )∑ +=1

111111 ,,p

pp ωωωωω CC TT , and

( ) ( )∑ ′+′=′2

,, 2111p

pp ωωωωω CC TT . Noting that pp ωωω 11 += and pp ωωω 21 +′= , so that

( ) ppp ωωω 21 ++′= , the integration in (5.32) can be replaced by

( ) ( )( ) ( )∑∑ ′+′+′++′=′1 2

,,, 2122112p p

ppppppp ωωωωωωωωω CCC TTT . (5.43)

Following a similar development to that used in Section 5.1.3, for an n-cell waveguide, the

transmission matrix for an input frequency of ω ′ and a pumping frequency of pω is given by

( ) ( )( ) ( )∑∑ ′+′+′++′=′−

1 2

,,, 2221p p

plpplnnpppp ωωωωωωωωω CCC TTT . (5.44)

5.2.3 The discrete transmission spectrum

For a monochromatic input, the output spectrum will be a discrete set of frequencies

consisting of the input fundamental and sum and difference frequencies with respect to the

pumping frequency, which we choose to call harmonics for the sake of brevity. With the help of

(5.36) and recalling that ppωωω +′= , the discrete output spectrum can be written as

( ) ( ) ( )∑ ′′+′=+′ ++

p

pnpnApSpA ωωωωωω 021 , . (5.45)

Consequently the output spectrum contains a series of delta functions at the relevant frequencies.

Recalling (5.35), we seek a closed-form solution for ( )ωω ′,21nS directly from the known

complex Fourier series coefficients of ( )ωω ′,nCT . Computation of the ( )ω′,21 tS

n time-domain

function is readily achievable for simulation purposes but is not desirable for analysis. In

calculating ( )ω′,21 tSn

, the required division by the 22CnT Fourier series obfuscates the

relationship between the scattering and transmission Fourier coefficients. Finding the Fourier

coefficients of ( )ω ′,tnCT is straightforward, and the Fourier coefficients of ( )ω′,22 tTCn

are

simply given in ( )ωω ′,22CnT . However, finding the Fourier coefficients of ( ) ( )ωω ′′ ,, 22 tTt CnnCT

directly from the constituent Fourier coefficients, without computing the division in the time

domain, is rather more challenging. In this regard, the method described by Duffin [299] for

calculating the Fourier coefficients of the reciprocal of a complex periodic function is of key


importance. The first step in the analysis is to extract from (5.35) the (2,1) element of the

scattering matrix, yielding

( ) ( ) ( )ωωω ′′=′ ,,, 2221 tTttS Cnnn CT . (5.46)

Its Fourier transform is given by

( ) ( ) ( ){ }ωωωω ′′ℑ=′ ,,,21 tDtS nn CT , (5.47)

where ( ) ( )ωω ′=′ ,1, 22 tTtDCn

, whose Fourier coefficients need to be found. The Fourier

transform in (5.47) is straightforward since the argument is itself a Fourier series, which can be

expanded in terms of the various Fourier coefficients.

A matrix method for an inverse Fourier series

Theorem 4 in [299] enables the Fourier coefficients of ( )ω′,tD , denoted by the vector

{ }k

α=Α , to be found directly from the Fourier coefficients of ( )ω′,22 tTCn

, denoted by the vector

{ }jf=F . The theorem can be restated as

FGΑ 1−= , (5.48)

where { }kjG ,=G is a matrix in which kjkj gG −=, and ∑ −+=

r

rrjjffg . Thus, a closed-form

solution for the Fourier coefficients of ( )ωω ′,21nS in terms of the known Fourier coefficients of

( )ωω ′,nCT can be found for periodic parameter variation signals.

The result expressed by (5.48) requires that the inverse of a matrix G be determined, which

can become non-trivial as the number of harmonics under consideration, and consequently the

size of G , increases. However, for the purpose of developing a closed-form equation to analyze

the effects of parameter variation on acoustic wave transmission, one may choose to examine

only the influence of the fundamental and first harmonic. In this case, G is a 33× matrix so that

its inverse may be found directly. Furthermore, when considering the transmission of real signals

only, G becomes a positive definite Hermitian Toeplitz matrix which further simplifies the

closed-form expressions.

A direct approach for a simple offset sinusoid

The matrices of Fourier coefficients in (5.37) can be calculated directly from the known

Fourier coefficients of the time-varying impedances if they are simple offset sinusoids. The

Fourier series for a periodic function ( )tZ is

( ) ∑=n

tjn

n

peZtZω

. (5.49)


Consider a simple offset sinusoid defined by

( ) ( )

tjtj

p

pp eZZeZ

tZZtZ

ωω

φω

101

10 cos2

++=

++=

−

−

, (5.50)

where ∗− = 11 ZZ in general since the impedance is real and 1Z∠=φ . If ( ) ( )tZtD 1= , the

spectrum of the reciprocal is

( ) ∑=n

tjn

n

peDtDω

, (5.51)

and has infinitely many components given by

( ) φωωαpp jnnn

n eeDD−

−= 01 , (5.52)

so every coefficient is known if 0D and α can be found. We can calculate α from the first two

coefficients

peDD

ωα−= 01 , (5.53)

therefore

−=

0

1ln1

D

D

pωα . (5.54)

and the problem reduces to finding 0D and 1D .

Let 0Za = and 1Zb = . From 3.5.4 in [300], the 0D coefficient is

220

1

baD

−= . (5.55)

From 5.12.3 in [301],

−−−=

221 11

ba

a

bD . (5.56)

Therefore,

−−=

−−

−−−=

10

00

22

22

1ln

1

1ln1

ZD

ZD

ba

a

b

ba

p

p

ω

ωα

. (5.57)

Thus, the matrices of Fourier coefficients in (5.37) can be specified entirely in terms of the

impedance Fourier coefficients.

5.3 Closed-form solutions

We wish to find the transmission coefficient, nS21 , directly from the equations for C1T so that we

may gain insight into the factors governing acoustic wave transmission through time-varying


phononic crystals in 1D. We consider only the fundamental and first harmonic to simplify the

analysis, and show that this is a good approximation for small material parameter variations. We

consider the case where 1Z is constant but 2Z is varying periodically in time with an angular

frequency of pω . That is,

( ) tjtj pp eZZeZtZωω 1

202

122 ++=

−− . (5.58)

We have assumed a constant phase velocity.

5.3.1 A closed-form expression for one cell

We wish to find the fundamental frequency transmission spectrum for one cell of a time-

varying phononic crystal. To do this, we must solve for 1CT and thus the scattering and

propagation matrices of interest including the fundamental and first harmonic. Since

( ) ( )∑ ′+′=′p

pp ωωωωω ,, 11 CC TT , we must find solutions for ( )ωωω ′+′ ,1 ppCT at the values

[ ]1,0,1−=p .

The time-varying scattering matrices

From (4.98), and substituting in the definition of 1R in terms of iZ ,

+−

−+=

−

−

−=→

1212

1212

21

1

1 2

11

1

1

1ZZZZ

ZZZZ

ZR

R

R21T , (5.59)

and

+−

−+=

+=→

2121

2121

11

1

1 2

11

1

1

1ZZZZ

ZZZZ

ZR

R

R12T . (5.60)

Since tjtj pp eeωω 121212

12 CCCT →→−→−→ ++= 101 , (5.61)

therefore the 12C →m matrices of Fourier coefficients are known directly as

−

−=

−→

− 11

11

2 1

12

1Z

Z12C ,

+−

−+=→

021

021

021

021

10 2

1

ZZZZ

ZZZZ

Z

12C , and

−

−=→

11

11

2 1

12

1Z

Z12C , (5.62)

where ∗→→

− = 1212 CC 11 in general.

To find 21C →n , we may use coefficients for the inverse of an offset sinusoid given in (5.55)

and (5.56). For brevity we shall leave the coefficients as nD . Thus,

+−

−+=

+−

−+=→

11

11

1212

1212

2 11

11

2

1

2

1DZDZ

DZDZ

ZZSZ

ZZSZ

Z21T , (5.63)

where ( ) ( )tZtD 21= . Therefore, the 21C →m matrices of Fourier coefficients are


−

−= −→

− 11

11

211

1

ZD21C ,

+−

−+=→

1010

10100 11

11

2

1ZDZD

ZDZD21

C , and

−

−=→

11

11

211

1

ZD21C . (5.64)

The time-varying propagation matrices

Here we illustrate a useful property of the propagation matrices. It allows a first-harmonic

propagation matrix to be expressed in terms of the propagation matrices of the fundamental and

pumping frequencies. Since

( )( )

( )

=

=

=

=+′

′

′−−

−

′

′−

′

−′−

+′

+′−

0

0

0

0

0

0

0

0

00

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0,

cdj

cdj

cdj

cdj

cdj

cdj

cdj

cdj

cdjcdj

cdjcdj

cdj

cdj

p

e

e

e

e

e

e

e

e

ee

ee

e

ed

p

p

p

p

p

p

p

p

ω

ω

ω

ω

ω

ω

ω

ω

ωω

ωω

ωω

ωω

ωωPT

, (5.65)

therefore ( ) ( ) ( )

( ) ( )dd

ddd

p

pp

,,

,,,

ωω

ωωωω

′=

′=+′

PP

PPP

TT

TTT. (5.66)

The 1−=p matrix

Substituting 1−=p into (5.42) and discarding terms containing factors other than the

fundamental or first harmonic gives

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ωωωωω

ωωωωω

ωωωωωω

′−′+′′=

′−′+′′=

′−′=′−′

→−

→→→−

→−

→→→−

→−

→−∑

P1

21

P2P2

12

P1

21

P2

12

P1

21

P2

12

P1

21

P2

12

P1

21

P2

12

C

TCTTCTCTC

TCTCTCTC

TCTCT

1001

1001

11 ,

p

p

q

qpqp q

, (5.67)

( )

−

−

+−

−++

+−

−+

−

−

=′−′∗∗

∗−

∗

−

a

a

c

c

b

b

ZZZZ

ZZZZD

ZDZD

ZDZD

b

b

Z

Z

p 0

0

11

11

0

0

0

0

11

11

0

0

11

11

4

1,

021

021

021

021

1

1010

1010

1

12

1 ωωωCT , (5.68)

where ( ) ajcajeea

θηω −−′− == 01 , 0cajeb

ηω′−= , and 0caj pecηω−

= . Further simplification eventually

leads to


( )( )

−

−+

−−+

−

−+

=′−′∗

∗

∗

∗∗

∗

∗

a

a

ZDZ

Zj

ZDZD

bcb

bcb

p 0

0

11

11sin

11

11sin

11

11coscos

2

1,

111

12

021

120

1

θθ

θθ

ωωωCT . (5.69)

The 0=p matrix

Following a similar development as before, we substitute 0=p into (5.42) and discard terms

containing factors other than the fundamental or first harmonic, giving

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )ωωω

ωωωωω

ωωω

ωωωωω

ωωωωω

′−′+

′′+′′=

′−′+

′′+′+′=

′−′=′′

→−

→

→→→→−

→−

→

→→→→−

→−

→∑

P1

21

P2P2

12

P1

21

P2

12

P1

21

P2P2

12

P1

21

P2

12

P1

21

P2

12

P1

21

P2

12

P1

21

P2

12

C

TCTTC

TCTCTCTTC

TCTC

TCTCTCTC

TCTCT

11

0011

11

0011

1 ,

p

p

p

p

q

qpqq

, (5.70)

( )

−

−

−

−+

+−

−+

+−

−++

−

−

−

−

=′′∗

∗

∗−

∗

∗∗

−

a

a

c

c

b

bZD

ZDZD

ZDZD

b

b

ZZZZ

ZZZZ

Z

c

c

b

bZD

0

0

11

11

0

0

0

0

11

11

11

11

0

01

11

11

0

0

0

0

11

11

4

1,

121

1010

1010

021

021

021

021

1

121

1 ωωCT , (5.71)

eventually simplifying to

( )

( )

−

−+

−−+

+

−

−++

=′′∗

∗∗

∗

a

a

ZDZ

Zj

ZDZDZD

bb

bbcbc

0

0

11

11

11

11sin

11

11cos

11

11coscoscos

2

1,

101

02

020

121

121

1

θθ

θθθ

ωωCT . (5.72)

The 1=p matrix

Here, we substitute 1=p into (5.42) giving

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ωωωωω

ωωωωω

ωωωωωω

′′+′′=

′′+′+′=

′−′=′+′

→→→→

→→→→

→−

→+∑

P1

21

P2

12

P1

21

P2P2

12

P1

21

P2

12

P1

21

P2

12

P1

21

P2

12

C

TCTCTCTTC

TCTCTCTC

TCTCT

0110

0110

11 ,

p

p

q

qpqpq

, (5.73)


( )

+−

−+

−

−+

−

−

+−

−+

=′+′∗

∗

∗∗

a

a

ZDZD

ZDZD

b

b

Z

Z

c

c

b

b

ZZZZ

ZZZZD

p 0

0

11

11

0

0

11

11

11

11

0

0

0

0

4

1,

1010

1010

1

12

021

021

021

021

1

1 ωωωCT , (5.74)

( )( )

−

−+

−−+

−

−+

=′+′∗a

a

ZDZ

Zj

ZDZD

bcb

bcb

p 0

0

11

11sin

11

11sin

11

11coscos

2

1,

111

12

021

120

1

θθ

θθ

ωωωCT . (5.75)

The transmission coefficient for one time-varying cell

To find the fundamental frequency transmission coefficient, ( )ωω ′′,21S , we must substitute

the previously calculated 1CT matrices into (5.46), where

( )( )( )

( ) ( )ωωω

ωω ′×′=

′

′=′ ,,

,

,, 1

221

121 tMt

tT

ttS

C

C

CT

T, (5.76)

where ( ) ( )ωω ′=′ ,1, 221 tTtMC

, and

( )

++++

++++=

++=′

−−−−

−−−−

−−

221122

0122

1121

1121

0121

11

121112

0112

1111

1111

0111

11

11

01

111 ,

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

tjtj

TeTTeTeTTe

TeTTeTeTTe

eet

pppp

pppp

pp

ωωωω

ωωωω

ωωω CCCC TTTT

, (5.77)

where ( )ωωω ′+′= ,11 p

p pCC TT as was previously calculated for [ ]1,0,1−=p . Next, expanding

the determinant of the matrix and simplifying,

( ) ( )( )

( )( )211121

0121

1112

1112

0112

11

221122

0122

1111

1111

0111

111 ,

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

TeTTeTeTTe

TeTTeTeTTet

pppp

pppp

ωωωω

ωωωωω

++++−

++++=′

−−−−

−−−−

CT, (5.78)

( )

( )( )

( )( )( )

−+

−−++

−−−+++

−+−+

−

=′ −−−−

−−−−−

−−−−−

211112

1122

1111

11

2

211112

0121

0112

1122

1111

0122

0111

11

211112

1121

0112

0121

1112

1122

1111

1122

0111

0122

1111

11

210112

1122

0111

1121

1112

0122

1111

01

211112

1122

1111

11

2

1 ,

CCCC

tj

CCCCCCCC

tj

CCCCCCCCCCCC

CCCCCCCC

tj

CCCC

tj

TTTTe

TTTTTTTTe

TTTTTTTTTTTT

TTTTTTTTe

TTTTe

t

p

p

p

p

ω

ω

ω

ω

ωCT .(5.79)

Considering only the fundamental and first harmonic of ( )ω′,tM , they may be calculated

using the inverse Fourier series method in (5.48) but not the direct method in (5.55) and (5.56)

since ( )ω′,221 tTC

may be complex and is not a simple offset sinusoid. The fundamental and first

harmonic of ( )ω′,221 tTC

can be envisioned as an ellipse on the complex plane parameterized by


t . The method of (5.48), limited to a 33× matrix is fairly accurate provided that the ellipse is

sufficiently displaced from the origin t∀ , or, to put it another way, ( )ω ′,221 tTC is an offset

sinusoid whose value is sufficiently displaced zero. For more extreme material parameter

variations, either a fast Fourier transform (FFT) must be performed on an over-sampled version

of the time-varying material parameters to improve accuracy, or the method of (5.48) may be

used including the higher-order zero coefficients. In any case, for analytical purposes we simply

use

( ) 101, MeMMetMtjtj pp ωω

ω ++=′−

− (5.80)

to elucidate the role of ( )ω′,221 tTC

in the transmission coefficient. Substituting this value into

(5.76) and simplifying gives

( ) ( )

( )( )

( )( )

( )101

211112

1122

1111

11

2

211112

0121

0112

1122

1111

0122

0111

11

211112

1121

0112

0121

1112

11

221111

1122

0111

0122

1111

11

210112

1122

0111

1121

1112

0122

1111

01

211112

1122

1111

11

2

1 ,,

MeMMe

TTTTe

TTTTTTTTe

TTTTTT

TTTTTT

TTTTTTTTe

TTTTe

tMt

tjtj

CCCC

tj

CCCCCCCC

tj

CCCCCC

CCCCCC

CCCCCCCC

tj

CCCC

tj

pp

p

p

p

p

ωω

ω

ω

ω

ω

ωω

++×

−+

−−++

−−−

+++

−+−+

−

=′×′

−

−

−−

−−

−−−−−

−−−−−

CT, (5.81)

We only wish to know the DC component of ( )ωω ′′,21S , so we choose terms whose exponentials

cancel, leading to

( ) ( )( )( ) 112

1121

0121

1112

0111

1122

0122

1111

01

0210112

0122

0111

0112

1121

1121

1112

1111

1122

1122

1111

11

1120121

1121

0112

1111

0122

1122

0111

1121 ,

MTTTTTTTT

MTTTTTTTTTTTT

MTTTTTTTTS

CCCCCCCC

CCCCCCCCCCCC

CCCCCCCC

−−−−

−−−−

−

−−++

−+−−++

−−+=′′ ωω

, (5.82)

which is the closed-form equation for acoustic transmission through one cell of a time-varying

phononic crystal.

5.3.2 A closed-form solution for two and n cells

To find the closed-form expression for the fundamental frequency transmission spectrum for

two cells of a time-varying phononic crystal, we must determine which values of 1CT are

required in the expression for 2CT . Since the closed-form solutions for fundamental and first-

harmonic values of 1CT were given in (5.69), (5.72), and (5.75), the expansion of 2CT is left in

terms of 1CT . We again consider only the fundamental and first harmonic of 2CT . From (5.43),


( ) ( ) ( )

( )( ) ( )∑

∑

′−′′′++′′=

′−′−′+′=′+′

s

pp

s

pppp

ssr

ssrr

ωωωωωω

ωωωωωωωωωω

,,

,,,

11

112

CC

CCC

TT

TTT

, (5.83)

where 21 ppr += , 2ps −= , and psωωω −′=′′ are substitutions to facilitate a per-frequency

formulation.

Considering that

( )

++++

++++=

++=′

−−−−

−−−−

−−

221

2220222

1221

1221

0221

12

121

2120212

1211

1211

0211

12

12

02

122 ,

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

C

tj

CC

tj

tjtj

TeTTeTeTTe

TeTTeTeTTe

eet

pppp

pppp

pp

ωωωω

ωωωω

ωωω CCCC TTTT

, (5.84)

we therefore require 12

−CT , 0

2CT , and 12CT where ( )ωωω ′+′= ,22 p

p pCC TT . They are found by

inserting the desired arguments into the summation in (5.83) and discarding terms with factors

other than the fundamental and first harmonic. Using the abbreviated notation

( ) ( )ωωωω ,11 p

p p+= CC TT , the three required matrices are

( )( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )ωωωωω

ωωωω

ωω

ωωωωωω

ωωω

′−′+′′=

′′′+′′′=

′′′=

′−′′′−+′′=

′−′=

−−

−−

−−

−

∑

∑

11

01

01

11

11

01

01

11

11

1

11

212

,,1

,

CCCC

CCCC

CC

CC

CC

TTTT

TTTT

TT

TT

TT

p

s

ss

s

pp

p

ss

, (5.85)

( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )ωωωωωωωω

ωωωωωω

ωω

ωωωωωω

ωω

′−′+′′+′+′=

′′′+′′′+′′′=

′′′=

′−′′′+′′=

′′=

−−

−−

−∑

∑

11

11

01

01

11

11

11

11

01

01

11

11

11

11

20

2

,,

,

CCCCCC

CCCCCC

CC

CC

CC

TTTTTT

TTTTTT

TT

TT

TT

pp

s

ss

s

ppss

, (5.86)

and

( )( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )ωωωωω

ωωωω

ωω

ωωωωωω

ωωω

′′+′+′=

′′′+′′′=

′′′=

′−′′′++′′=

′+′=

∑

∑−+

−

01

11

11

01

01

11

11

01

11

1

11

212

,,1

,

CCCC

CCCC

CC

CC

CC

TTTT

TTTT

TT

TT

TT

p

s

ss

s

pp

p

ss

. (5.87)


The transmission spectrum can be found by using the same equation in (5.82) except using

the elements of the p

2CT matrices and with ( ) ( )ωω ′=′ ,1, 222 tTtMC

. In fact, ( )ωω ′′,21nS can be

found by using (5.85)–(5.87) to provide the expansion for ( )ωω ′,nCT in (5.44). The p

1CT matrices

are simply replaced by ln−CT and

lCT . Equation (5.82) may be used again to find the

transmission spectrum with the proper substitutions as before.

5.4 Methodology and verification

A detailed walkthrough of the many steps of the new TV-TMM formulation is given here. It is

then shown that the extended method shows a logarithmic growth in computational complexity

as the number of time-varying cells is increased, as opposed to a linear growth in comparable

FDTD simulations. Finally, results from the extended method are compared against similar

FDTD simulations and show excellent agreement.

5.4.1 Computation method

( )tZ1

( )tZ 2

( )t21T →

( )t12T →12C →

m

21C →n

0ωω =′ ( )ωω ′,nCT ( )ωω ′,1CT

( )0,ωtnCT ( )021 ,ωtSn ( )021 ,ωωnS

Figure 5.4. A flowchart illustrating the steps and equations necessary to simulate wave propagation through a time-varying tube waveguide. Dashed boxes indicate the variables and matrices and solid boxes indicate the equations and operations required. ‘Repeat’ means that (5.44) should be applied as many times as necessary to achieve n time-varying cells.

A procedure for numerically calculating the transmission spectrum is illustrated in Figure 5.4

to help guide the reader through the many steps involved. Beginning with the time-varying

impedances, ( )tZ1 and ( )tZ 2 are substituted into (4.98) to find the time-varying transmission

matrices, ( )t21T → and ( )t12T → . Next, the matrices of complex Fourier coefficients, 21C →n

and

12C →m

, are found simply by applying an FFT to the transmission matrices, or by using the method

for the inverse of an offset sinusoid given in (5.55) and (5.56). The single-cell time-varying

transmission matrix, ( )ωω ′,1CT , can then be found directly from (5.42). If there are multiple

time-varying cells, then (5.44) may be applied as many times as is necessary to generate


( )ωω ′,nCT , which accounts for transmission through n cells. Choosing a particular input

frequency, 0ωω =′ , and performing an inverse FFT yields ( )0,ωtnCT . Finally, converting back

into a time-domain scattering matrix using (5.35) and choosing the (2,1) entry gives the time-

domain transmission coefficient, ( )021 ,ωtSn

. A final FFT gives the transmission spectrum for a

particular input frequency, namely ( )021 ,ωωn

S .

It should be noted that for the purpose of simulation, (5.48) was not used since the much

simpler FFT and inverse FFT may be used in its place. However, for small values of material

parameter variation, it is accurate enough in its 33× form for finding ( )ωω ′,21nS directly from

( )ωω ′,nCT . Also note that the restriction of choosing a particular input frequency 0ωω =′ is

lifted when deriving the general solution since the Fourier transform and its inverse do not need

to be performed.

5.4.2 The complexity of calculation

Though it is possible to sequentially calculate 1CT , 2CT , 3CT , etc. up to nCT , it is far more

efficient to take an exponential approach: 1CT , 2CT , 4CT , etc. To do this, we begin calculating

the time-varying transmission matrices exponentially, including all

( ){ }ωω ′,lCT , where [ ]n

l 2log2,1∈ . (5.88)

In the above equation, L means the floor operation which rounds down to the nearest integer.

Another way to state this is to calculate ( )ωω ′,nCT by powers of two up to the maximum

placeholder in the binary representation of n. For example, if ( )2101010 ==n , then calculate

1CT , 2CT , 4CT , and 8CT .

The intermediate matrices are then combined so that they add up to the correct n. Because the

matrices have been calculated as powers of two, the required matrices that need to be combined

via integration correspond to the locations in the binary representation of n where the digits are

1s. Using the above example of n = 10, we would combine ( )ωω ′,2CT and ( )ωω ′,8CT to arrive at

the desired ( )ωω ′,10CT .

This final step only required one integration since there were two 1s in the binary

representation of n. In general, the number of integrations required will be the number of 1s in

the binary representation of n (called the Hamming weight of n, and denoted as n ) minus one.

Thus, the total number of integrations required to generate ( )ωω ′,nCT (not including the

generation of ( )ωω ′,1CT ), denoted by ( )nC , is given by


( ) 1log2 −+= nnnC . (5.89)

A plot of ( )nC versus n shown in Figure 5.5 clearly shows a logarithmic trend, as expected.

This is particularly noteworthy given that the size of a comparable time-varying FDTD

simulation would grow linearly as more cells are added. Thus, this extended method is superior

to the FDTD method for simulations involving large numbers of time-varying cells.

Figure 5.5. A plot of the number of integrations required for n time-varying cells. It can be seen that the plot follows a logarithmic trend as the number of time-varying cells is increased. A comparable FDTD simulation run-time would grow linearly with the number of time-varying cells.

5.4.3 Verification

As noted earlier, the FDTD method is able to handle time-varying material parameters in

periodic structures. As such, it was used as a benchmark in order to verify that the new TV-

TMM formulation is correct and to compare simulation times.

A single time-varying cell was analyzed using the FDTD method and the TV-TMM

formulation using the method illustrated in Figure 5.4. The corrugated tube was designed to the

same specifications as in the example calculations in Section 4.2.1 and in [17]. Additionally, the

cross-sectional area ( 2S ) of section 2 was varied sinusoidally by %5± at a frequency of 600 Hz,

and the input acoustic wave was a 1.0 kHz monotone. In order to prevent reflections from a hard

source in the FDTD simulation, a transparent source implementation was used [208].

Shown in Figure 5.6a) is that the resulting transmission spectra for one time-varying cell are

in excellent agreement. The solid arrows indicate that the dashed lines are delta functions. Since

the input monotone is a real function, it has frequencies at ±1.0 kHz. The peaks at 200 Hz

correspond to the second harmonic of the negative input frequency ( Hz6002Hz1000 ×+−

Hz200= ). Additional harmonics are present, but have magnitudes that are too small to appear

in the figure. The continuous nature of the FDTD plot is due to the truncation of a finite signal


with many frequency components. In general it is difficult to determine a commensurate

sampling period that would smoothly truncate all the harmonics present in the sampled signal.

Another contributing factor is that the simulation begins with all acoustic particle displacements

equal to zero, so time must be allowed for resonances to build up in order to approximate the

steady-state case captured by the TV-TMM formulation. In spite of these constraints, it is clear

that the transmission peaks present in the FDTD simulation have deltas that correspond exactly

to the TV-TMM simulation. Furthermore, the TV-TMM simulation execution times were four to

five orders of magnitude faster than the FDTD simulations used to generate Figure 5.6. Shorter

FDTD run times are possible by reducing the simulated time duration and by reducing the space

and time resolutions. However, these reductions result in a poorer frequency-domain resolution,

greater susceptibility to signal processing artefacts, and less accuracy due to increased numerical

dispersion.

Shown in Figure 5.6b) are the transmission spectra for ten identical time-varying tube cells

whose parameters are the same as for Figure 5.6a). Here again, the TV-TMM formulation is seen

to be in good agreement with the FDTD simulation. In the time-varying case the presence of

transmission coefficients at 021 ≠+ pp shows that a single input frequency produces a

multiplicity of frequencies at the output. Thus, a linear combination of input frequencies causes a

linear combination of output spectra, which is encapsulated in (5.36).

0 500 1000 1500 2000 2500 3000

Figure 5.6. Comparison of transmission spectra using the new theory and a FDTD method for a pumping frequency of 600 Hz and an input frequency of 1.0 kHz. a) Single time-varying cell. b) Ten time-varying cells. Arrows indicate the presence of delta functions. Because the FDTD simulation was performed over a finite time period, the spectrum is continuous. Had the simulation been performed over an infinitely long time period, the spectrum would have been delta functions at the peak locations giving excellent agreement for the two sets of results.


5.5 Results

The fundamental frequency transmission coefficient ( 021 =+ pp in (5.44)) for the ten tube

cells is shown for both the static and time-varying cases in Figure 5.7, generated by varying ω ′

and plotting the resulting output spectrum. The static plot in Figure 5.7 is the same as that shown

in Figure 4.4 for a ten-cell static tube. In the dynamic plot, the parameters are the same as in

Figure 5.6b). The transmission spectrum now includes energy that has been modulated to and

from the harmonics by the time-varying transmission and reflection coefficients. Notice that the

shapes of the band gap and band gap edges have been significantly altered, becoming less

rounded and more irregular than in the static case. This raises the possibility of changing the

band gap characteristics using time-varying material parameters as the controlling mechanism.

One could envision a controllable notch filter using such an arrangement.

Transmission coefficient

Figure 5.7. Fundamental frequency transmission coefficients for ten static and time-varying cells. In the dynamic case, the time-varying material parameters modulate the incident waves at every impedance boundary, causing energy to be continuously modulated to and from the fundamental frequency. The pumping frequency is 600 Hz and ( )tS2 is varying by %5± .

The same plot is reproduced in Figure 5.8, except with a pumping frequency of 400 Hz. The

change in pumping frequency has markedly changed the transmission spectrum from that shown

in Figure 5.7. Whereas the band gap attenuation was slightly reduced in the 600 Hz case, it is

increased in the 400 Hz case.


Transmission coefficient

Figure 5.8. Another fundamental frequency transmission spectrum for ten static and time-varying cells. This is the same as in Figure 5.7, but with a pumping frequency of 400 Hz. Note the dramatic difference made by the change in pumping frequency.

What is not shown in these two figures are the harmonics present due to the modulation

occurring at each interface. Figure 5.9 shows the fundamental and first harmonic transmission

spectra through ten identical time-varying cells. The conditions were identical to the dynamic

case in Figure 5.8, and the blue line corresponds to the fundamental frequency spectrum of both

figures.

|S2

1|

100

10-1

10-2

02.0

1.0

fin (kHz)

2.0

0

fout (k

Hz)

3.0

4.0

Figure 5.9. Transmission spectra through ten time-varying 1D phononic crystal cells, including the fundamental and first harmonic. The blue line corresponds to the dynamic spectrum in Figure 5.8 and to 021 =+ pp in (5.44). The green line is

121 =+ pp , and the red line is 121 −=+ pp . There are infinitely many such lines in addition to the three shown.

5.5.1 Parametric amplification

We noted that certain pumping frequencies resulted in exponential growth of the FDTD

simulation field values and that improving the time and space resolution and increasing the


simulation duration did not mitigate the problem. Improving the resolution seemed to narrow the

range of pumping frequencies that would cause exponential growth. It was therefore suspected

that the growth was a function of the system and not a function of the FDTD simulator. Given

the periodically time-varying nature of the system, an effect called parametric amplification was

suspected at the frequencies of exponential growth [62]. It is well known that any pumping

frequency of np 02ωω = , where n is a positive integer, will result in parametric amplification,

but as n is increased the amount of energy conferred to the system decreases.

As discussed in Section 1.3.2, existing time-varying systems theory only deals with

periodicity in either the space or time dimensions, but not in both simultaneously. Consequently,

the starting point of our analysis was by means of the FDTD simulator which required no

fundamental changes to handle time-varying material parameters. One problem with using the

FDTD simulator to investigate time-varying phononic crystals is its inability to produce a truly

monochromatic input in a finite-duration simulation; hence the relative smoothness of the plots

in Figure 5.6. This is also why it was difficult to isolate the particular frequencies suspected of

producing parametric amplification since the bandwidth of the pumping and incident frequencies

are functions of the simulation duration.

|S2

1|(dB)

f p(kHz)

Figure 5.10. A surface composed of the fundamental frequency transmission spectra for various pumping frequencies. A front view is shown in a) and the top view is shown in b). The slice corresponding to 0=pf is the static transmission

spectrum for the corrugated tube waveguide in [17]. The remainder of the surface corresponds to the transmission spectrum for ( )tS2 varying by %5± as the pumping frequency is increased. Note the periodicity of the behaviour in both the

0f and pf directions.


In contrast, our extended method provides solutions for exact frequencies which permits a

more accurate and insightful analysis of parametric amplification within time-varying phononic

crystals. Figure 5.10 is a plot of the fundamental frequency transmission spectrum as a function

of pumping frequency. The slice of the plot along the 0out =f plane corresponds to the static

transmission spectrum in Figure 5.7 and Figure 5.8, and the slices along Hz600out =f and

Hz400out =f correspond to the dynamic plots in Figure 5.7 and Figure 5.8, respectively. The

points that grow and attenuate sharply are points of infinite growth and attenuation, which can be

seen when further simulations are run to zoom into these areas of the plot. It was found

numerically that the areas of the plot corresponding to infinite growth occur when 2201C

T is zero,

which leads to 0M in (5.80) (the time average of ( )ω′,1 221 tTC ) and ( )ωω ′′,21nS being infinite.

Conditions necessary for parametric amplification

The definition of 0M is explored further to determine under what conditions it goes to

infinity. Since we are considering only the fundamental and first harmonic,

( )

∫

∫

++=

′=

−−

π

φφ

π

φπ

φωπ

2

0 221122

0122

11

2

0 2210

1

2

1

,

1

2

1

deTTeT

dtT

M

j

CC

j

C

C, (5.90)

where ( ) ( )( ) ( ) ( )( )φφφφω sincossincos, 110011221 jjaajaajjaatTIRIRIRC

+++++−+=′−− , (5.91)

and iIiR

i

CjaaT +=221 . We can rewrite this as

∫ +=

π

φπ

2

0

0

1

2

1d

jIRM , (5.92)

where ( ) ( )RIIRR

aaaaaR 01111 sincos +−++= −− φφ , (5.93)

and ( ) ( )IRRII

aaaaaI 01111 sincos +−−+= −− φφ . (5.94)

0M will go to infinity when the denominator of (5.92) goes to zero for some angle 0φ , which

occurs when both ( )0φR and ( )0φI are simultaneously zero. Expressing R and I in the more

convenient form

RRR

CBAR −+= φφ sincos , (5.95)

and III

CBAI −+= φφ sincos , (5.96)


where the coefficients can be determined simply by comparison with (5.93) and (5.94). To solve

for the angle 0φ at which both terms of the denominator are zero, we rewrite (5.95) and (5.96) in

matrix form as

=

2

1

0

0

22

11

sin

cos

C

C

BA

BA

φ

φ, (5.97)

whose solution is

=

−

−

−=

sin

cos

1221

2112

12210

0 1sin

cos

x

x

CACA

CBCB

BABAφ

φ. (5.98)

Since ( )( ) 21arccossin xx −= , and if

2cossin 1 xx −= , (5.99)

then there is a [ ]πφ 2,00 ∈ that satisfies (5.98). It is clear from numerical simulations using the

new time-varying method that this is true at points of infinite growth. Since these points of

infinite growth are occurring in both the FDTD simulations and in our new time-varying method

at the same pumping frequencies and do not occur in the static case, we conclude that time-

varying material parameters can cause parametric amplification under the right conditions.

5.5.2 Signal switching

One possible application of the modified transmission properties in time-varying phononic

crystals is switching a CW signal by changing the pumping frequency. To demonstrate this

concept, two pumping frequencies were chosen that had opposite effects on the attenuation in the

centre of the first band gap of the time-varying corrugated tube waveguide. Figure 5.11 shows

the spectra for Hz180=pf and for Hz400=pf . The transmission within the band gap is

changed by approximately 12 dB according to the new time-varying method.


~12dB

Transmission Coefficient

Figure 5.11. The transmission spectra for two different pumping frequencies chosen to demonstrate switching of a 2 kHz CW signal. There is approximately a 12 dB difference between the transmission coefficients at 2 kHz.

Next, an FDTD simulation was run with an incident 2 kHz CW waveform and a pumping

frequency that alternated between 180 and 400 Hz. Figure 5.12 shows a short-time Fourier

transform of the transmitted signal that demonstrated the expected switching, albeit with a lower

dynamic range. This is due in part to the fact that the finite nature of the pumping signals means

that they were not perfectly narrowband as in the new TV-TMM formulation. Thus, the pumping

signal spectrum was smooth and not discrete in the FDTD simulation. Similarly, the incident

pulse was also not a discrete frequency, but was Tukey windowed to reduce signal artefacts. All

these simulation non-idealities mean that energy is present at locations of the surface in

Figure 5.10 other than the two points described by ( ) ( )Hz180,kHz2,0 =pff and

( )Hz400,kHz2 resulting in a lower switching dynamic range than predicted in the ideal case

illustrated in Figure 5.11. Nevertheless, this demonstration of CW signal switching as predicted

by our extended method indicates that time-varying phononic crystals may be able to switch

from a transmissive to reflective state simply by changing the pumping frequency of an active

material.


2 kHz Transmission (dB)

Figure 5.12. The normalized short-time Fourier transform of a 2 kHz transmitted signal for an FDTD simulation of a time-varying phononic crystal switching between the two states illustrated in Figure 5.11. The red lines indicate when the pumping signal was switched. The dynamic range of approximately 4 dB is less than the predicted 12 dB due to non-idealities in the FDTD simulation. The incident waveform frequency is in the centre of the first band gap.

5.6 Summary and concluding discussion

We have presented a new formulation for predicting the transmission properties of time-varying

phononic crystals. This method gives closed-form solutions to the transmission through single

and multiple time-varying cells. In particular, a periodic parameter-variation signal results in a

discrete summation of harmonics. At every scattering interface, the time-varying material

parameters modulate the energy of the propagating waves to and from the fundamental

frequency, resulting in dramatically altered transmission properties. It was shown that the

extended method closely matches the predictions of FDTD simulations with execution times

many orders of magnitude faster. It was also demonstrated that time-varying material parameters

can affect the nature of acoustic wave transmission in phononic crystals. These effects remain to

be demonstrated experimentally, and the method and results presented in this chapter indicates

that such future research may be worthwhile and fruitful.

- 128 -

Chapter 6

DYNAMIC PHONONIC CRYSTALS IN TWO

DIMENSIONS

In this chapter we use multiple scattering theory to extend the time-varying results to cylindrical

scatterers in two dimensions. Our analysis is based on the scalar velocity potential, thereby

limiting the method to fluid-only hosts and scatterers. We compare this time-varying multiple

scattering theory (TV-MST) to FDTD simulations for both one and multiple cylindrical

scatterers. It is shown that the two methods produce similar results, but that our extended method

is superior in many ways. Most notably, the computational complexity increases logarithmically

as opposed to linearly with the FDTD method, making it faster when a large number of scatterers

are involved. Finally, some conclusions are drawn and possible improvements are discussed.

6.1 Time-varying multiple scattering theory

Our analysis uses the same notation and assumptions as the static case presented in Section 4.3.

First, we consider a single time-varying cylinder with continuously time-varying material

parameters. The solution is discretized and an extended T-matrix is obtained that includes the

effects of coupling between incident and scattered temporal frequencies. Finally, the scattered

field of multiple time-varying cylinders is investigated and it is shown that the matrix equations

for multiple scatterers essentially remain unchanged.

6.1.1 Scattering from one time-varying cylinder

It is assumed that the host material parameters, 0c and 0Z , are constant. Further, it is

assumed in the development of the time-varying T-matrix that the scatterer has a constant phase

velocity, 1c but a time-varying impedance, ( )tZ1 . Thus, ( ) ( ) 111 ctZt =ρ and ( ) ( ) 111 ctZtK = .

The reason for this restriction is that it isolates the effects of time-varying material parameters to

the boundary conditions at the host-scatterer interface. While this restriction is present in our

development of the time-varying T-matrix for an individual cylinder, it is not present when

combining multiple time-varying T-matrices to determine the total system scattering. Other

methods may be used to find the time-varying T-matrix for the more complicated case of a time-

varying phase velocity. Then, the resulting time-varying T-matrices may be combined using the

Chapter 6. Dynamic Phononic Crystals in Two Dimensions 129

methods subsequently described. Various methods for calculating the response of cylinders with

time-varying phase velocities are reviewed in [298].

The time-dependent incident and scattered fields can be written in either the time or

frequency domain. For example, the incident field is

( ) ( ) ( )∑⇔m

mminc dtu ωψω ,ˆ, rr , (6.1)

where u is the velocity potential and the double arrow indicates the Fourier dual. That is, the time

domain field can be represented as a sum of a spectrum of orthogonal cylindrical wavefunctions.

Thus, the time domain function can be rewritten as the inverse Fourier transform of the spectrum

as

( ) ( ) ( ) ωωψω ωπ dedtu

tj

mmminc ∫∑= ,ˆ, 2

1 rr , (6.2)

noting that the summation and integral may be exchanged due to the orthogonality of the terms.

The boundary conditions must be satisfied at all times, and thus can be restated in the

presence of time-varying material parameters within the cylinder as

( ) ( ) ( )[ ] 0,,, =−+ =Rrscinc tvtutu rrr , (6.3)

and ( ) ( )[ ]( )

( ) 0,1

,,1

10

=

−+

∂

∂

=Rr

scinc tvt

tutur

rrrρρ

, (6.4)

where v is the velocity potential within the cylinder. Substituting the spectra into the first

boundary condition, (6.3), and simplifying:

( ) ( ) ( ) ( )

( ) ( )0

,ˆ

,,ˆ

21

21

21

=

−

+

=

∫∑

∫∑∫∑

Rr

tj

mmm

tj

mmm

tj

mmm

deb

decded

ωωψω

ωωψωωωψω

ωπ

ωπ

ωπ

r

rr

, (6.5)

( ) ( )

( )

02

1

1

00=

−

+

∑

∫

∫∫

m tjmm

tjmm

tjmm

jm

deRc

Jb

deRc

HcdeRc

Jd

e

ωω

ω

ωω

ωωω

ω

πω

ωω

θ . (6.6)

Because the exponentials, θjme , are orthogonal, (6.6) becomes m independent equations,

( ) ( ) ( ) 0100

=

−

+

∫ ω

ωω

ωω

ωω ω deR

cJbR

cHcR

cJd tj

mmmmmm . (6.7)

Again because the exponentials, tje

ω , are orthogonal, the inner expression must be zero for all

ω so that the integral is zero for all t, leading to


( ) ( ) ( ) 0100

=

−

+

R

cJbR

cHcR

cJd mmmmmm

ωω

ωω

ωω , (6.8)

which is the first boundary condition in the time-varying case.

The simplification of the second boundary condition will proceed as before, beginning by

substituting the spectra into the second boundary condition as given by (6.4). The time domain

multiplication of the last term in the brackets in (6.4) can be rewritten as a frequency domain

convolution,

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )∫ ∫ ∑

∫ ∑

∑

′′′′−=

′′′′−ℑ=

∗ℑ=

=

−

−

ωωωψωωω

ωωψωωω

ωψωω

ρ

ω

π

π

π

dedbP

dbP

bP

tbtPtbt

tj

mmm

mmm

mmm

,ˆ

,ˆ

,ˆ

,,1

141

11

21

11

21

11

2 r

r

r

rr

. (6.9)

where ( ) ( )ttP 11 1 ρ= . If the density is an offset sinusoidal function of t, the method presented in

Section 5.2.3 may be used to find the Fourier series of its reciprocal. Proceeding with the

substitutions,

( ) ( ) ( ) ( )

( ) ( ) ( )0

,ˆ

,,ˆ1

141

21

21

0

2

=

′′′′−−

+

∂

∂

=

∫ ∫ ∑

∫∑∫∑

Rr

tj

mmm

tj

mmm

tj

mmm

dedbP

decded

r ωωωψωωω

ωωψωωωψωρ

ω

π

ωπ

ωπ

r

rr

, (6.10)

( ) ( )

( ) ( )

0

1

2

1

112

1

000=

′

′′′−−

+

∂

∂

=

∑

∫ ∫

∫∫

Rr

mtj

mm

tjmm

tjmm

jm

dedrc

JbP

derc

Hcderc

Jd

re

ωωω

ωωω

ωω

ωωω

ωρ

πω

π

ωω

θ . (6.11)

Because the exponentials, θjme , are orthogonal, (6.11) becomes m independent equations,

( ) ( )

( ) ( )

0

1

112

1

000=

′

′′′−−

+

∂

∂

=

∫ ∫

∫∫

Rr

tjmm

tjmm

tjmm

dedrc

JbP

derc

Hcderc

Jd

rωω

ωωωω

ωω

ωωω

ωρ

ωπ

ωω

. (6.12)

Combining the integrals,


( ) ( )

( ) ( )

0

11

112

1

0000=

′

′′′−−

+

∂

∂

=

∫

∫Rr

tj

mm

mmmm

de

drc

JbP

rc

Hcrc

Jd

rω

ωω

ωωω

ωω

ρ

ωω

ρ ω

π

. (6.13)

Taking the derivative, which is independent of the integration,

( ) ( )

( ) ( )

0

11

11

121

000000=

′

′′′′−

′−

′+

′

∫

∫

ω

ωω

ωωωω

ωω

ω

ρ

ωω

ω

ρω

π

de

dRc

JbPc

Rc

Hcc

Rc

Jdc

tj

mm

mmmm

. (6.14)

Again because the exponentials, tje

ω , are orthogonal, the inner expression must be zero for all

ω , giving

( ) ( )

( ) ( ) 0

1

11

121

0000

=′

′′′′−

′−

′+

′

∫ ωω

ωωωω

ωω

ωω

ω

ρ

πdR

cJbP

c

Rc

HcRc

Jdc

mm

mmmm

. (6.15)

Noting that 000 Zc =ρ , this can be rewritten in terms of the characteristic acoustic impedance as

( ) ( ) ( ) ( ) 02 1

11

0

00

=′

′′′′′−−

′+

′ ∫ ω

ωωωωω

π

ωωω

ωωω dR

cJbP

c

ZR

cHcR

cJd mmmmmm

, (6.16)

which is the second boundary condition simplified.

Noting that (6.8) can be rewritten as,

( ) ( ) ( )

+

=

−

Rc

HcRc

JdRc

Jb mmmmmm

00

1

1

ωω

ωω

ωω , (6.17)

we can substitute (6.17) into (6.16) to arrive at

( ) ( )

( ) ( ) ( ) 0,00

00

=′

′′′+

′′′′−

′+

′

∫ ωω

ωωω

ωωωω

ωωω

ωωω

dRc

HcRc

JdB

Rc

HcRc

Jd

mmmm

mmmm

, (6.18)

where ( ) ( )

′

′′′−=′ R

cJR

cJP

c

ZB mm

111

1

0

2,

ωωωω

πωω . (6.19)


This factor can be rewritten in a form similar to (4.132), except that ( )ωω ′,B is related to the

inverse of β because of the order in which we combined the boundary conditions. Rearranging

gives

( ) ( ) ( )

( ) ( ) ( )

′

′′′′−

′−=

′

′′′′−

′

∫

∫

ωω

ωωωωω

ωω

ωω

ωωωωω

ωω

dRc

JdBRc

Jd

dRc

HcBRc

Hc

mmmm

mmmm

00

00

,

,

, (6.20)

which may be rewritten as

( ) ( ) ( )

( ) ( ) ( )∫

∫

′′

′′′−

′′′′−−=

′′

′′′−

′′′′−

ωωω

ωωωω

ωωωδ

ωωω

ωωωω

ωωωδ

ddRc

JBRc

J

dcRc

HBRc

H

mmm

mmm

00

00

,

,

. (6.21)

This is the continuous equation relating the incident and scattered fields in the time-varying case.

Verification of the static case

To verify that (6.21) collapses into (4.132) under static conditions, we simply choose

( ) 11 ρρ =t , and therefore ( ) { } ( ) 111 21 ρωπδρω =ℑ=P . Furthermore, we consider only a single

incident frequency, 0ωω = , so that ( ) ( )02 ωωδπω −= mm dd and ( ) ( )02 ωωδπω −= mm cc .

Substituting these values into (6.21) gives

( ) ( ) ( )

( ) ( ) ( )∫

∫

′−′

′′′−

′′′′−−=

′−′

′′′−

′′′′−

ωωωδω

ωωωω

ωωωδ

ωωωδω

ωωωω

ωωωδ

ddRc

JBRc

J

dcRc

HBRc

H

mmm

mmm

000

000

,

,

, (6.22)

( )

( )mmm

mmm

dRc

JBRc

J

cRc

HBRc

H

−

′−=

−

′

0

0000

0

00

0

0000

0

00

,

,

ωωωω

ωω

ωωωω

ωω

, (6.23)

( ) ( ) ( )[ ]( ) ( ) ( )[ ] m

mm

mmm d

RkHBRkH

RkJBRkJc

0000

0000

,

,

ωω

ωω

−′

−′−= . (6.24)

Since ( )( )( )RkJ

RkJ

Z

ZB

m

m

1

1

1

000 ,

′=ωω , (6.25)

therefore, after some rearrangement,


( )( )( )

( )

( )( )( )

( )m

m

m

mm

m

m

mm

m d

RkHRkJZ

RkJZRkH

RkJRkJZ

RkJZRkJ

c

′

′−

′

′−

−=

010

110

010

110

, (6.26)

which is identical to the solution in (4.132).

A discrete solution

Since every factor in (6.21) is known except for ( )ωmc , a solution to the equation should be

possible. One means of numerically solving this equation is to parameterize ω and ω ′ so that

ωωω ∆+= ii 0 , Ζ∈i , (6.27)

where 0ω is a chosen fundamental reference frequency and ω∆ is the desired frequency

resolution. Equation (6.21) may be rewritten using this notation as

( ) ( ) ( )

( ) ( ) ( )

∆

−

′−=

∆

−

′

∑

∑

ωωω

ωωωωω

ω

ωωω

ωωωωω

ω

j

jm

j

mjjiim

i

mi

j

jm

j

mjjiim

i

mi

dRc

JBdRc

J

cRc

HBcRc

H

00

00

,

,

, (6.28)

where, in practice the summation would be truncated to cover some desired frequency range.

This equation can be rewritten in matrix notation as

mmmm dWcV −= , (6.29)

where { } ( ) ( ) ( )[ ]Tmmmimm cccc LL 101 ωωω−==c , (6.30)

{ } ( ) ( ) ( )[ ]Tmmmj

mm dddd LL 101 ωωω−==d , (6.31)

{ }ji

mm V=V , ( ) ωω

ωωωδω

ω ∆

−

′= R

cHBR

cHV

j

mjjiij

i

mi

ji

m

00

, , (6.32)

and { }ij

mm W=W , ( ) ωω

ωωωδω

ω ∆

−

′= R

cJBR

cJW

j

mjjiij

i

mi

ij

m

00

, . (6.33)

Thus, for a truncated range of frequencies with a finite frequency resolution, the matrices are

finite and the system is solvable:

mmmm dWVc1−−= . (6.34)

Using (6.34), we may define a new time-varying T-matrix for scattering from one time-

varying cylinder as


{ } [ ]{ }ij

mnmm

ij δWVTT1−−== , (6.35)

where the mnδ indicates that the spatial cylindrical wavefunction modes are decoupled because

we are dealing with a cylindrical scatterer. The matrix is indexed as { }ijmnT=T and corresponds

to the scattered field angular frequency index (i), the incident field angular frequency index (j),

the scattered field cylindrical wavefunction mode (m), and the incident field cylindrical

wavefunction mode (n).

By extending our definitions of the scattered and incident vectors and the T-matrix, we can

write the relationship between scattered and incident fields as

Tdc = , (6.36)

which is the same notation as in the static case. The extended vectors are shown in Figure 6.1

and the extended T-matrix is shown in Figure 6.2. The elements of c and d can be partitioned and

considered as block vectors. On the larger scale, the block components of the vector correspond

to the various temporal harmonics1, and on the smaller scale each block consists of cylindrical

wave modes for a particular harmonic.

{ } ( ){ }

===

−

M

M

1

0

1

c

c

c

ccc ii ω

( ){ }== −1ωmc

−

−

−−

M

M

11

10

11

c

c

c

Figure 6.1. The time-varying scattered field coefficient vector. The time-varying column vectors of scattered and incident field coefficients in (6.30) and (6.31), respectively, can be visualized as column block vectors. The scattered field coefficients are illustrated here, but the incident field coefficients are configured in an identical manner. Shown left, the entries on a large scale correspond to the various harmonics. Each of those entries is itself a column vector of cylindrical wave modes for a particular harmonic.

The T-matrix can be envisioned as a two dimensional block matrix. In the larger matrix, each

block corresponds to the relationship between scattered and incident harmonics. Each block

contains elements that relate the scattered and incident cylindrical wave modes.

1 The reader is reminded that we use the term “harmonics” for brevity in reference to sum and difference frequencies with respect to the pumping frequency.


{ } ( ){ }

===−

−

−−−−

ON

NO

1,10,11,1

1,00,01,0

1,10,11,1

,

TTT

TTT

TTT

TTT ji

ij ωω

( ){ }

==

−

−

−−−−

ON

NO

0,11,1

0,10,1

0,11,1

0,11,0

0,10,0

0,11,0

0,11,1

0,10,1

0,11,1

010,1 ,

TTT

TTT

TTT

Tmn ωωT

Figure 6.2. A discrete time-varying T-matrix. On the larger scale, shown above, each element of the T-matrix relates scattered field angular frequency i with incident field angular frequency j. Each element of this matrix is itself a matrix that relates scattered cylindrical wave mode m with incident mode n.

Using the block notation, the solution, Tdc = , is

=

−

−

−

−−−−−

M

M

ON

NO

M

M

1

0

1

1,10,11,1

1,00,01,0

1,10,11,1

1

0

1

d

d

d

TTT

TTT

TTT

c

c

c

. (6.37)

Since we are dealing with a cylinder and we are using a cylindrical decomposition of the fields,

T is diagonal in the mn dimensions. That is, T is a block matrix of diagonal matrices.

Under these new definitions, the incident field is

( ) ( )drψr ˆ=incu , (6.38)

where ( ) ( ){ } ( ){ }jj ω,ˆˆˆ rψrψrψ == and ( ) ( ){ }rrψ j

nj ψˆ = , are both row vectors. Similarly, the

scattered field is

( ) ( )crψr =scu , (6.39)

where ( ) ( ){ } ( ){ }ii ω,rψrψrψ == and ( ) ( ){ }rrψ i

mi ψ= , are also both row vectors.

The scattered field from a non-periodic time-varying scatterer

When the material parameters are varying in a non-periodic manner, one cannot simply

consider the question in terms of harmonics. The T-matrix will still be discrete in the mn

dimension but will be a continuous function of the incident and scattered field angular

frequencies, ω and ω ′ respectively. Thus,


( ) ( ) ( )∫ ′′′= ωωωωω ddTc , (6.40)

This integration can be performed numerically by considering the T-matrix in (6.35) and

discretizing ( )ω′d so that Nj, the number of incident harmonics, is sufficiently large enough to

span the incident spectrum at a suitable resolution.

The scattered field from a periodic time-varying scatterer

If the material parameter variation is periodic, the continuous relationship in (6.21) will

degenerate into the equivalent discrete relationship in (6.29) since a periodic spectrum consists of

discrete components. We begin by considering a monochromatic incident field of angular

frequency 0ωω = , and time-varying material parameters of a particular pumping angular

frequency, pω . The scattered field will therefore consist of harmonics located at pi iωωω += 0 ,

which is same notation used in (6.27) with pωω =∆ . The incident field is simply specified by

the coefficients, ( ) ( )00 ωωδω −= mm dd , and the scattered field coefficients can be written as the

angular frequency-domain Fourier series,

( ) ( )∑ −=i

iimm cc ωωδω , (6.41)

where the usual factor of π2 has been absorbed into the coefficients. Similarly, the inverse of

the time-varying density of the cylinder and its Fourier transform may be written as

( ) ( ) ∑==s

tjss pePttPω

ρ 111 1 , and ( ) ( )∑ −=s

ps

sPP ωωδπω 11 2 . (6.42)

Substituting (6.41) into (6.20) gives

( ) ( ) ( )

( ) ( ) ( )

′

′′−′′−

′−−=

′

′′−′′−

′−

∫

∫ ∑∑

ωω

ωωωδωωω

ωωωδ

ωω

ωωωδωωω

ωωωδ

dRc

JdBRc

Jd

dRc

HcBRc

Hc

mmmm

mi

rimm

ii

im

00

0

00

0

00

,

,

, (6.43)

( ) ( ) ( )

( ) ( ) ( ) 0

00

00

00

,

,

mmm

i

immimi

ddRc

JBRc

J

cdRc

HBRc

H

′

′′−′′−

′−−=

′

′′−′′−

′−

∫

∑ ∫

ωω

ωωωδωωω

ωωωδ

ωω

ωωωδωωω

ωωωδ

, (6.44)

( ) ( )

( ) ( ) 0

0

000

00

00

,

,

mmm

i

im

imiimi

dRc

JBRc

J

cRc

HBRc

H

−

′−−=

−

′−∑

ωωωω

ωωωωδ

ωωωω

ωωωωδ

. (6.45)


Substituting (6.42) into (6.19) gives

( ) ( )∑

′−′−=′

s

mp

sR

cLsP

c

ZB

11

1

0,ω

ωωωδωω , (6.46)

where ( ) ( ) ( )xJxJxL mmm′= . (6.47)

Substituting this into (6.45) and simplifying yields

( ) ( )

( ) ( ) 0

0

00

1

01

1

0

00

011

1

0

0

mmms

ss

m

i

im

imi

im

ssi

smi

dRc

JRc

LPc

ZR

cJ

cRc

HRc

LPc

ZR

cH

−−

′−−=

−−

′−

∑

∑ ∑ +

ωω

ωωωδ

ωωωωδ

ωω

ωωωδ

ωωωωδ

. (6.48)

Rearranging this equation on a per-frequency basis by solving for jωω = gives

( ) ( )

( ) ( ) 0

0

00

1

01

1

0

00

011

1

0

0

mmms

sjsj

mjj

i

im

imi

im

ssij

sj

mjij

dRc

JRc

LPc

ZR

cJ

cRc

HRc

LPc

ZR

cH

−−

′−−=

−−

′−

∑

∑ ∑ +

ωω

ωωωδ

ωωωωδ

ωω

ωωωδ

ωωωωδ

. (6.49)

Because of the delta functions, the first summation over s is only nonzero when jsi =+ and the

second when js = . Discretizing the delta functions with the understanding that each discrete

component is the coefficient of a corresponding delta function in the continuous spectrum,

0

0

00

1

01

1

0

0

000

011

1

0

0

mmmj

mj

i

im

imi

im

ijimiij

dRc

JRc

LPc

ZR

cJ

cRc

HRc

LPc

ZR

cH

−

′−=

−

′∑ −

ωω

ωωωδ

ωω

ωωωδ

. (6.50)

As in the discretized solution, (6.50) can be written in a matrix format as

0mmmm dWcV −= , (6.51)

where { } ( ) ( ) ( )[ ]Tmmmimm cccc LL 101 ωωω−==c , (6.52)

{ }ji

mm V=V ,

−

′= −

Rc

HRc

LPc

ZR

cHV i

mi

i

m

iji

miij

ji

m

011

1

0

0

ωω

ωωωδ , (6.53)

and { }jmm W=W ,

−

′= R

cJR

cLP

c

ZR

cJW mm

jmj

jm

0

00

1

01

1

0

0

000

ωω

ωωωδ . (6.54)

Note that mW is a column vector since there is only one incident frequency. Thus,

01mmmm dWVc −−= . (6.55)


We can reorder this result to conform with the notation in (6.36), in which case

[ ] mnj

i

mmij

mnT δδ01WV

−−= , (6.56)

where [ ]imm WV 1− refers to the ith element of the mm WV 1− vector.

If the incident field is not monochromatic, then the solution will consist of a summation of

relevant harmonics for a particular scattered frequency. The relevant incident frequencies are

( ) impm did =+ ωω0 . (6.57)

Considering all the relevant incident frequencies up- or down-modulated to the frequency of

interest, the T-matrix will be

[ ] mn

ij

mmij

mnT δWV 1−−= , (6.58)

where [ ]ijmm WV 1− is the ith element of the mm WV 1− vector calculated with jωω = . The extended

T-matrix relationship, (6.36), may then be multiplied with the incident coefficient vector d.

This concludes our discussion of scattering from a single time-varying cylinder. Next, the

multiple scattering from many such time-varying scatterers is considered, and is shown to be

very similar to the formulation in static case.

6.1.2 Scattering from N periodically time-varying cylinders

Consider N periodically time-varying cylinders with known discrete time-varying T-matrices,

{ }ijmnpp T ,=T , [ ]Np ,1∈ where m and n are the scattered and incident cylindrical waves modes,

and i and j are the scattered and incident angular frequency components, respectively. We want

to find an equivalent time-varying T-supermatrix, { }ij

mntottot T ,=T , so that

dTc tottot = , (6.59)

where we have used the notation introduced in Figure 6.1 and Figure 6.2. To use the matrix

solutions presented in Section 4.3.3, we require a new time-varying definition of the separation

matrix. Essentially, each harmonic is individually moved using the block matrix

( ){ }klij

kl bSS = , where ( ) ( ){ }klijmnkl

ij S bbS = . (6.60)

The separation matrix elements are

( ) ( ) ijiklnmklijmnS δωψ ,bb −= . (6.61)

Using these new definitions, we may simply use the same T-matrix equations for multiple

scatterers as in the static case, restated here for convenience:


∑=

=N

kklkl

1dTc , where ( )[ ]k

Tlklklklllk TSITBT δδ −+= 1 . (6.62)

∑ ∑= =

=N

l

N

kklkltot

1 100

ˆˆ STST . (6.63)

The complexity of calculating Ttot

If all the individual T-matrices are identical, then the T-supermatrix may be created with a

logarithmic computational complexity with respect to the number of scatterers. The argument as

to why this is the case is essentially identical to that presented for the 1D case in Section 5.4.1.

The T-supermatrix is calculated for two, then four, then eight, etc. scatterers until the desired

T-supermatrix has been created. The procedure and complexity are also identical as in

Section 5.4.1, except that repeated applications of (6.63) replace the integrations in Figure 5.5

and the accompanying discussion. If the individual T-matrices are not identical, then the

T-supermatrix must be composed in a linear manner.

The extended method can be easily parallelized (multithreaded) to run across multiple CPU

cores or on a GPU. The elements of the various matrices can be generated in parallel, and the

majority of the remaining operations are matrix multiplications and inversions. Since these

operations can be computed in parallel [302], therefore this extended method will experience

similar computational gains as the FDTD method using parallel computation.

6.2 Verification

The extended method for periodically time-varying scatterers was compared against

analogous FDTD simulations for both static and dynamic cases. The simulation results match

quite well as the subsequent figures show. The configuration used in all the simulations was

cylinders with identical proportions and properties as in the phononic crystal design that was

discussed in Chapter 3. The only difference is that the stiffnesses of the stainless 303 were

adjusted to disable transverse waves since our extended method does not account for these. To

do this, we set 1112 CC = and 044 =C . The simulation region was 12R by 9R ( z by x ), where R

is the cylinder radius. The source was located 5.5R (8.638 mm) in the - z direction from the

centre of the simulation region. In all the subsequent figures of the displacement or velocity field

amplitudes, field values are normalized so the colour scale proceeds from blue to red and

represents values from zero to one.

To produce results that can be visually compared, two separate FDTD simulations were

executed to obtain the incident and scattered fields. The first simulation contains only the source

and provides the incident displacement field. The second simulation contains the source and


scatterers and provides the total field, consisting of the incident plus scattered displacement

fields. The new TV-MST method calculates the incident and scattered fields as the velocity

potential, which is then converted into the velocity to compare it with the FDTD results. The

FDTD results are displacements and not velocities, but since the fields are decomposed by

frequency and normalized, the two sets of results can be directly compared. For brevity, we shall

simply refer to both fields as displacement fields.

The FDTD simulation used a Gaussian-apodized transparent source. The input pulse was

Gaussian-modulated with a centre frequency of 350 kHz and a 20 kHz bandwidth instead of a

monotone which alleviated the need for signal windowing. The simulation region consisted of

256 points in the z direction and 192 points in the x direction. The time step was 5 ns, and the

simulation duration was 118.52 µs. This was the duration of two full input waveforms.

One source of error in the FDTD simulations arose because the transparent source

implementation has some artefacts present at its edges. Figure 6.3 shows the magnitude of the

displacement components along the aperture plane. The magnitude of the source spatial response

in the z direction, shown in blue, closely resembles the intended Gaussian shape from

approximately -6 to 6 mm. However, there are unintended sharp jumps at these points. Also, the

magnitude of the source spatial response in the x direction, shown in red, should be zero but has

a notable amplitude in the simulation. A solution to these problems was unclear, but their impact

on the overall results is minimal. It is still apparent that the results of the extended method

closely match those of the FDTD method.

-8 -6 -4 -2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aperture Plane (mm)

Norm

alized Source Amplitude

|uz|

|ux|

Figure 6.3. The recorded FDTD source amplitude on the aperture plane. The x and z directions are parallel and perpendicular to the source aperture plane, respectively.


To simulate the Gaussian-apodized source in the extended method, the spatial transform of

the source apodization function was used to generate plane waves of various angles and

amplitudes. Then, each plane wave component was converted into cylindrical wavefunction

coefficients using the Jacobi expansion in (4.108). The coefficients for each mode were then

summed together to find the set of incident wave coefficients needed to reconstruct the source.

There were 50 wavevectors used in the source aperture decomposition and 51 cylindrical wave

modes included in the summations. These choices were deemed sufficient because increasing

them did not result in any appreciable improvement in the quality of the simulation results. Four

harmonics plus the fundamental were solved for (i.e. [ ]2,2−∈i in iω ).

6.2.1 One static cylinder

FDTD

TV-M

ST

Figure 6.4. The simulated displacement fields for a single static cylinder. Panels a) through c) show the incident, total, and scattered fields acquired from the FDTD simulations. Panels d) through f) show the results obtained from the extended method, labelled TV-MST. The slight scalloping visible in the FDTD simulations may arise from source and absorbing boundary condition non-idealities.

As a first check, simulations of one static cylinder were performed using both the FDTD

method and the new time-varying method. It was shown analytically in Section 6.1.1 that the

extended method collapses to the standard multiple scattering theory for one cylinder if the

material parameters are constant. However, a check against the well-established FDTD method


was still warranted. The simulation results showing the normalized magnitude of the

displacement amplitude zu at 350 kHz are shown in Figure 6.4 and display excellent agreement.

We therefore concluded that our extended method produces correct results for a single static

cylinder.

6.2.2 Several static cylinders

A simulation of three static cylinders placed in a non-symmetric configuration was designed

to verify that the T-supermatrix formulation is correct in the static case. Figure 6.5 shows that the

two different simulation methods provide the same normalized zu displacement fields for three

static cylinders. There are slight discrepancies between the scattered and total fields which can

be attributed to the FDTD simulation non-idealities, but the two sets of simulations show

excellent qualitative agreement otherwise.

FDTD

TV-M

ST

Figure 6.5. The simulated displacement fields for three static cylinders. Panels a) through c) show the incident, total, and scattered fields acquired from the FDTD simulations. Panels d) through f) show the results obtained from the extended method, labelled TV-MST. There is excellent qualitative agreement between the two sets of figures. The slight discrepancies are attributable to non-idealities in the FDTD simulation.

6.2.3 One periodically time-varying cylinder

In this test, the formulation for the time-varying T-matrix for a single cylinder was verified.

The characteristic impedance of the cylinder, 2Z , was varying sinusoidally by ±20%, called the


variation factor. It was varying at a pumping frequency of 100 kHz. The variation factor was

chosen so that the first two harmonics above and below the fundamental frequency were greater

than the noise floor of the FDTD simulation. By noise floor, we mean the error introduced

through signal truncation due to a finite-duration simulation, imperfect absorbing boundary

conditions, and the discretization of the simulation region. Figure 6.6 shows the normalized

incident and total field magnitude spectra of zu for a point along the x centreline and halfway

between the aperture and the cylinder centre. The incident field spectrum is shown in blue, with

the 350 kHz Gaussian envelope clearly visible. The total field, which is the sum of the incident

and scattered fields, is shown in red. The presence of the first and second harmonics are evident

in the total signal. The incident pulse bandwidth and pumping frequency were chosen to

minimize the overlap of the incident pulse spectrum with the harmonics.

Norm


Figure 6.6. The normalized displacement spectra of uz for a single cylinder. The point measured was along the x centreline and halfway between the aperture and the cylinder centre. The blue spectrum is the incident pulse alone, and the red spectrum is for a dynamic cylinder with 2Z varying by ±20% at 100 kHz. The first two harmonics are visible in addition to the fundamental. The curves are normalized by the incident spectrum at 350 kHz.


The normalized incident, scattered, and total zu displacement field magnitudes at 350 kHz

are shown in Figure 6.7 for both the FDTD and new TV-MST methods. The results match quite

well, but appear to differ very little from the static case presented in Section 6.2.1. However,

once one examines the displacement field magnitudes at the harmonic frequencies shown in

Figure 6.8, it is clear that the extended method accurately matches the FDTD method under time-

varying conditions. There is excellent agreement between the two methods, demonstrating the

validity of the new TV-MST formulation for a single cylinder. Panel d) in Figure 6.8 shows

some distortion as compared to panel h), but this can be attributed to the poor signal to noise

ratio at 550 kHz shown in Figure 6.6.

Incident Scattered Total

FDTD

TV-M

ST

a) b) c)

d) e) f)

Figure 6.7. The simulated fundamental frequency displacement fields for one time-varying cylinder. The cylinder and input pulse are identical to those in Section 6.2.1 except that the impedance of the cylinder is varying sinusoidally by ±20% at a frequency of 100 kHz.


FDTD

TV-M

ST

Figure 6.8. The simulated displacement field harmonics for one time-varying cylinder. The cylinder is the same as shown in Figure 6.7.

6.2.4 Several periodically time-varying cylinders

In this final verification test, the formulation for the time-varying T-supermatrix was verified

for three cylinders. The cylinders are identical to the one in Section 6.2.3, including their time-

varying properties. Figure 6.9 shows the normalized incident and total field magnitude spectra of

zu at the same point chosen for Figure 6.6. It indicates that the choice of a variation factor of

20% again places the first two harmonics sufficiently above the noise floor that they may be

extracted from the FDTD simulation results. The magnitude of the total field spectrum at

350 kHz is reduced due to destructive interference between the incident and scattered fields at

the chosen measurement location.


Norm


Figure 6.9. The normalized displacement spectra of uz for three cylinders. The point measured was the same as in Figure 6.6. The blue spectrum is the incident pulse alone, and the red spectrum is the total field for three dynamic cylinders. Again, the first two harmonics are visible in addition to the fundamental frequency. The curves are normalized by the incident spectrum at 350 kHz.

The fundamental frequency fields are shown in Figure 6.10 for both the FDTD and extended

methods. Again the results match very well for the fundamental frequency. As in the static case,

there are slight discrepancies between the scattered and total fields which can be attributed to the

FDTD simulation non-idealities. The displacement fields at harmonic frequencies are presented

in Figure 6.11, and confirm that the extended method accurately matches the FDTD method for

more than one time-varying cylinder. There are also some slight discrepancies between the two

methods, but they are due to the FDTD simulation non-idealities; in particular, the source

aperture. Otherwise, there is excellent qualitative agreement between the two sets of simulation

results.


FDTD

TV-M

ST

Figure 6.10. The simulated fundamental frequency displacement fields for three time-varying cylinders.

FDTD

TV-M

ST

Figure 6.11. The simulated displacement field harmonics for three time-varying cylinders. The cylinders are the same as shown in Figure 6.10.


6.2.5 The modified fundamental frequency field

Since the scattered field did not appear to change significantly between the static and

dynamic cases for both one and three cylinders, one may question the value of attempting to

include active materials into phononic crystals. The change in magnitude between the static and

dynamic cases is shown in Figure 6.12. The figures were generated by normalizing the

magnitude of each point in the dynamic simulation with the magnitude of the same point in the

static simulation and plotting the results on a dB magnitude scale. It can be seen that the change

is only on the order of ±0.1 dB. However, our experience with the 1D time-varying method as

described in Chapter 5 is that even these small changes for individual scatterers can lead to large

changes in the transmission behaviour of a phononic crystal. Thus, it should not be taken as a

deterrent to future research in the application of active materials to phononic crystals.

Change in magnitude (dB)

Figure 6.12. The change in scattered displacement field magnitude at the fundamental frequency. Panel a) shows the change for one cylinder, and panel b) shows the change for three cylinders.

6.2.6 Application to a time-varying phononic crystal

The extended method was applied to a phononic crystal composed of static and periodically

time-varying cylinders identical to those used in Section 6.2.1 and 6.2.3, respectively. The

cylinder spacing and arrangement was identical to the static phononic crystal that was

experimentally investigated in Section 3.2. In the simulation, there were 44 cylinders arranged as

8 rows and 11 columns. A 350 kHz plane wave was incident on the crystal from the left. The

cylinder impedance was varying by ±20% at 100 kHz. The total simulation space encompassed

5 cm in both dimensions, and 41 cylindrical wave modes were used in the multipole expansions.


Change in amplitude (dB)

Norm

alized amplitude

Figure 6.13. A a) static and b) dynamic phononic crystal. Panel a) shows the normalized amplitude of the velocity potential of the scattered field at 350 kHz for a static phononic crystal whose cylinders are identical to those in Section 6.2.1. Panel b) shows the same field for a dynamic phononic crystal whose cylinders are identical to those in Section 6.2.3. The differences between the fields are revealed in panel c), which shows the change in amplitude from the static field to the dynamic field.

Figure 6.14. The displacement field harmonics present in the scattered field of the dynamic phononic crystal. The fields show the normalized amplitude of the velocity potential.


The normalized amplitudes of the velocity potential of the scattered field at the fundamental

frequency for both the static and dynamic phononic crystals are shown in Figure 6.13a) and b),

respectively. The differences between the two fields are nearly imperceptible, but become

apparent when the dynamic scattered field is normalized by the static scattered field and is

plotted on a logarithmic colour scale, as in Figure 6.13c). In addition to the fundamental

frequency, there are also scattered fields at harmonic frequencies as shown in Figure 6.14.

These simulations demonstrate that the new time-varying multiple scattering theory may be

successfully applied to time-varying phononic crystals. Comparable FDTD simulations would

take significantly longer to execute than the extended method and would suffer from all of the

drawbacks previously discussed. Future studies may therefore use this extended method to

investigate the transmission and dispersion properties of 2D time-varying phononic crystals.

6.3 Summary and concluding discussion

We have extended multiple scattering theory to handle cylindrical scatterers with time-varying

material parameters. We have also showed that the results of this extended method match those

of similar FDTD simulations for a variety of scenarios. The matrix methods for determining the

effect of multiple scatterers remain unchanged, save for the definitions of the matrices and

vectors, which makes the extended method easy to include in existing multiple scattering

implementations. The extended method retains all the benefits of multiple scattering theory.

There are some improvements and additions to the extended method that would be of great

value, and consist of the inclusion of transverse waves, the ability to model time-varying phase

velocities, and the application to three dimensions. The inclusion of transverse waves is

facilitated by additional vector fields that represent the transverse wave components. There is an

additional boundary condition relating to the traction at the host-scatterer interface [12]. This

modification does not present any fundamental problems, and its incorporation into the extended

method should be feasible.

The analysis of cylindrical scatterers with time-varying phase velocities is complicated, but

various methods have been proposed [298]. The formulation of the time-varying T-matrix for a

single scatterer with a time-varying phase velocity would need to account for the fact that time-

varying effects would no longer be isolated to the interface between the host and scatterer.

However, if the phase velocity varies periodically, then the scattered field will consist of

harmonics just as in the present analysis. Therefore, the method of determining the multiply

scattered field would remain the same once the individual T-matrices are known. The extended

method would greatly benefit from a deeper understating of the role of time-varying phase

velocities in scattering from a cylinder.


Finally, the method as presented in two dimensions may be easily extended into three

dimensions. There is no conceptual difference between two and three dimensional multiple

scattering theory, and only the implementation details differ. This would allow the analysis of 3D

phononic crystals with time-varying scatterers. Also, the use of vector as opposed to scalar

velocity potentials would permit the inclusion of transverse waves in the analysis, allowing the

extended method to be applied to time-varying phononic crystals that use solid materials.

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Chapter 7

SUMMARY AND CONCLUSIONS

The summary reviews the most pertinent details of the thesis material. Major contributions of the

thesis are reiterated and the advantages of the new time-varying methods over alternatives are

emphasized. Finally, some suggestions on possible future research and applications are provided.

7.1 Summary

This thesis has explored phononic crystals that include active materials. In particular, we have

extended two of the existing analytical models of phononic crystals to account for scattering by

active materials. Other previous works have examined phononic crystals with active materials,

but have only considered the case of using the active material to change from one static state to

another. In this thesis we have specifically investigated the effects of periodically varying the

material parameters in time, and we have discovered many interesting resultant effects.

A thorough introduction to periodic materials and phononic crystals has been provided. This

was followed by the design, fabrication, and characterization of a static two dimensional

phononic crystal. Many interesting effects experimentally observed were predicted theoretically

by our simulators, including band gaps, dispersion, and focusing due to negative refraction. Next,

we provided a review of the various methods and analytical models that we used to investigate

phononic crystals theoretically. We then extended the 1D transmission matrix method to handle

time-varying material parameters and presented verification of this extended method against the

FDTD method. The results of our extended method showed some interesting effects, including

dramatically altered transmission properties and parametric amplification. In two dimensions, we

extended multiple scattering theory to include the effects of time-varying material parameters.

Finally, we verified that the 2D method produced results in accordance with FDTD simulations.

Both of these extended methods are superior to the FDTD method for computing the acoustic

wave propagation through large numbers of scatterers as the computational complexity grows

logarithmically instead of linearly.

7.1.1 Primary assumptions

Here we list the primary assumptions made in the developments of the methods presented.

The constituent materials of a phononic crystal are:

Chapter 7. Summary and Conclusions 153

• Lossless. Consequently, all fluids are inviscid and do not support transverse waves.

• Dispersionless. Dispersion arises from spatial periodicity alone. Material parameters

are not functions of the incident wave frequency.

• Homogeneous.

• Static, except in the extended methods.

• Linear. The nonlinear response of the material to high-amplitude incident waves is

not considered.

It is also assumed that the time-varying material parameters are identical at every location within

an active material. That is, the material parameters change instantaneously everywhere and there

is no propagation delay.

7.2 Thesis contributions

7.2.1 Specific developments

The three major accomplishments of this thesis were the design, fabrication, and

characterization of a 2D phononic crystal, the development of the time-varying transmission

matrix method, and the development of the time-varying multiple scattering theory. They are

briefly reviewed here.

Creation of a 2D phononic crystal

A 2D phononic crystal was designed based on the information available in research literature

and on the equipment available for characterization. The final design consisted of 11 rows and 37

columns of 1/8th inch diameter stainless steel cylinders packed in a square crystal lattice. They

were held in place by an acrylic support on the tops and bottoms of the cylinders. The crystal

was designed to operate under water in a submersion tank scanner.

Characterization of the crystal provided results that matched very well with FDTD

simulations. The crystal transmission spectrum contains band gaps at a number of different

frequencies. The dispersion properties of the crystal indicated that negative refraction may occur

in the second transmission band of the crystal. Subsequent scanning of the acoustic field external

to the crystal revealed the presence of a focal region indicative of negative refraction.

The time-varying transmission matrix method

The transmission matrix method allows the modelling of wave propagation through periodic

structures in one dimension. The benefit of this method is that it provides an analytical solution

that can give insight into the factors that govern the transmission of acoustic waves in these

structures. Also, the computational complexity grows logarithmically with respect to the number

of scatterers, as opposed to the FDTD method which grows linearly. We modified this method


by including the effects of time-varying material parameters on the scattering and propagation of

acoustic waves. We provided solutions for both periodically and non-periodically time-varying

material parameters. These solutions were verified against FDTD simulations and showed

excellent agreement.

Several interesting effects were noted using our extended method to simulate 1D time-

varying phononic crystals. They included dramatically altered transmission properties and the

presence of parametric amplification. Since our extended method provides analytical solutions to

acoustic wave propagation in 1D time-varying phononic crystals, these solutions were used to

create a closed-form equation for the transmission through one and two time-varying phononic

crystal cells. This equation was investigated and provided insight into the factors leading to the

observation of parametric amplification.

The time-varying multiple scattering theory

Multiple scattering theory is a versatile two and three dimensional method for characterizing

the acoustic wave transmission through many scatterers. The benefits of this method over the

FDTD method are similar to those of the transmission matrix method in 1D: It gives analytical

solutions to wave propagation in scattering structures, and its computational complexity grows

logarithmically. We modified the 2D method by including the effects of time-varying material

parameters. We describe a new T-matrix that includes the effects of frequency modulation that

occurs in time-varying phononic crystals. These solutions were also verified against FDTD

simulations and showed excellent agreement.

The extended method will allow fast characterization of time-varying phononic crystals

without the need to resort to lengthy FDTD simulations. Also, the method of combining T-

matrices to form the T-supermatrix remains unchanged provided that the new matrix definitions

are used. Thus, it is quite compatible with existing implementations of multiple scattering theory.

This extended method may be easily extended to three dimensional multiple scattering theory.

7.2.2 Primary research contributions

The main research contributions of this thesis consist of simulation tools and analytical

models for time-varying phononic crystals in one and two dimensions. The following list

summarizes the primary original research contributions of this thesis. They include:

a) The creation and characterization of a 2D phononic crystal demonstrating many interesting

phononic crystal effects.

b) The creations of an FDTD simulator capable of handling time-varying material parameters

that includes numerous source and boundary conditions.


c) The development of the analytical solution to the FDTD simulation stability limits for a 2D

homogeneous medium.

d) The modification of the transmission matrix method to include the effects of time-varying


e) The development of a closed-form equation for acoustic wave transmission through one and

two time-varying 1D phononic crystal cells.

f) The observation in simulation of dramatically altered transmission properties of time-varying

1D phononic crystals using our extended method.

g) The observation in simulation of parametric amplification in the simulation of time-varying

1D phononic crystals and the subsequent analysis of its origins in the closed-form equations.

h) The modification of 2D multiple scattering theory to include the effects of time-varying


7.3 Suggestions for further work

We propose some possible areas for future research based on the promising results presented in

this thesis. We have divided the suggestions into various areas depending on the nature of the

research.

7.3.1 Analytical developments

Application to resonant metamaterials

The analysis methods presented and used in this thesis are also applicable to resonant

metamaterials as was discussed in Section 1.1.2 with respect to locally resonant phononic

crystals. It would be very interesting to determine the effects of time-varying material parameters

on resonant metamaterials, especially in the homogeneous limit where effective material

parameters are defined. Perhaps some of the effects described in this thesis, like altered

transmission properties and parametric amplification, may be used to address some of the

shortcomings of metamaterials. For example, the lossiness and narrowband performance of

resonant metamaterials may be improved by using active materials and the analysis methods

presented in this thesis.

Effects on phononic crystal dispersion properties

As was extensively explained in Chapter 3, the dispersion properties of a phononic crystal

determine the nature of its acoustic wave refraction. Perhaps the slope of the dispersion surfaces

may be altered in the presence of time-varying material properties, which would enable the

refraction of the crystal to be dynamically adjusted. This may lead to flat acoustic lenses with a

variable focal length.


Better solutions for time-varying phase velocities

The analysis of time-varying phononic crystals presented in this thesis primarily neglected

the effects of time-varying phase velocities, but suggested ways that it might be incorporated. A

deeper understanding of the effects of a time-varying phase velocities would help to make the

extended methods more robust and would more accurately reflect real time-varying phononic

crystals to a greater degree.

Non-cylindrical scatterers

Our two dimensional analysis only considered cylinders because of the orthogonality of the

cylindrical wave modes. This simplified the development, but there is nothing that prevents the

analysis of a time-varying scatterer of any sensible shape in 2D. The method for determining the

T-supermatrix of multiple scatterers would remain unchanged.

Inclusion of transverse waves

The current implementation of time-varying multiple scattering theory used a scalar velocity

potential to define the acoustical fields. This limited the method to fluid-only systems. One may

envision active fluid scatterers, such as liquid crystals or ferrofluids, but it is most likely that

either the host or scatterer would be solid. Thus, it would be beneficial to extend the theory to

include transverse waves by including a vector velocity potential. A means of accomplishing this

is given in Section 1.5.5 of [12].

Inclusion of moving boundaries

It is conceivable that time-varying phononic crystals may have active materials that

physically deform when operating. In this case, the boundaries between host and scatterer may

have time-varying locations which would need to be accounted for in the formulation. Previous

work by Censor [44]–[46] may be of value in incorporating the effects of time-varying boundary

locations in phononic crystals.

Extension to three dimensions

As was mentioned in Chapter 6, two and three dimensional multiple scattering theories are

fundamentally similar, and differ predominantly in the implementation details. The 2D time-

varying multiple scattering theory should be extended into three dimensions to allow the analysis

of 3D phononic crystals that include active materials in their crystal lattice.

Development of inverse methods

As presented, the material parameters of a time-varying phononic crystal are varied

sinusoidally with a particular amplitude and frequency, then the transmission spectrum is


determined. An inverse method would allow the reverse procedure: a desired transmission

spectrum would dictate the optimum way of varying the material-parameters to achieve those

transmission properties. Such an inverse method may be possible by rearranging the closed-form

equations presented in Chapter 5.

Application to photonics

All of the new and existing methods presented in this thesis are applicable to photonic

systems. Indeed, the static methods are routinely used to analyze photonic crystals. Hence, the

time-varying methods may be re-derived for electromagnetic wave propagation and applied to

active photonic crystals, such as those that include nonlinear materials.

Further investigation of FDTD stability limits

An indirect line of research in this thesis was the investigation of FDTD stability limits under

various conditions. It would be helpful to know the stability limits for inhomogeneous 2D

simulations, 2D simulations with an applied wavevector, and 2D simulations that include time-

varying material parameters. This would allow the use of a Courant factor [197] in determining

the quality of a simulation as opposed to our trial and error method.

7.3.2 Implementation improvements

Identification of symmetries

Many of the coefficients present in both the new 1D and 2D time-varying methods are

related to one another. For example, many of the matrices in our derivation of the closed-form

equations in Chapter 5 are Hermitian. The identification of these symmetries may enable

significant gains in computational complexity by eliminating redundant computations. For

example, in the 1D analysis the transmission spectra of the ±ith harmonics are mirrored and

inverted along the plane outin ff = . Thus, it suffices to calculate only one or the other.

FDTD code improvements

There are plenty of opportunities for improvements in the FDTD simulator code. It currently

exists in one large C file that is compiled and executed through MATLAB. It would greatly

benefit from object-orienting and code division for clarity and to improve data import and

export. The core itself is fairly optimized, but the FDTD code is ideally suited to parallelization.

Multi-threading the core or modifying it to execute on a GPU could dramatically reduce the

simulation run time. Automated determination of the absorbing boundary condition parameters

would also be a welcome improvement.


7.3.3 Experimental work

Creation of a time-varying phononic crystal

First and foremost, a time-varying phononic crystal should be created. The extended methods

presented in this thesis could be used to design a time-varying phononic crystal. Then subsequent

experiments would verify that the extended methods are correct. Perhaps the simplest means of

creating a time-varying phononic crystal would be a corrugated tube waveguide with

mechanically moveable sidewalls. This would not require any active materials and would have a

static phase velocity, as previously described.

Fabrication processes

The phononic crystal that we created and characterized in this thesis had a cumbersome

fabrication process. Other simpler and more creative means of fabricating phononic crystals with

and without active materials should be investigated. The possibilities range from simply drilling

holes in a substrate to using 3D printers to create elaborate structures.

7.3.4 Applications research

Dynamic reflectors

The CW switching action demonstrated in simulation in Chapter 5 should be explored

further. It may be possible to switch a signal with a larger bandwidth and dynamic range by

choosing different operating conditions. If an active phononic crystal is created, it may be

possible to demonstrate that it can be dynamically switched from reflective to transmissive. This

could also lead to dynamically switched waveguides that can actively route acoustic signals by

switching their reflectivity.

Dynamic filters and active damping

If inverse methods are sufficiently developed, it may be possible to create dynamic filters

whose transmission properties can be adjusted and optimized in real-time. This could be of use in

systems that rely on phononic crystals as a means of mechanical isolation dampers. The damper

could be adjusted to optimally reject the changing spectrum of unwanted interference.

Active materials as sensors

The active materials present in time-varying phononic crystals could also be used as sensors,

possibly enabling closed-loop feedback control of these structures. For example, if an active

material is being controlled by an applied voltage, then the current draw of the material could be

monitored to determine the nature of the acoustic waves impinging on its surface. This could


enable smart materials that are able to determine their status and adjust their properties in real-

time.

7.4 Concluding remarks

Phononic crystals are very interesting materials because their crystal structure leads to

unconventional properties. In this thesis, we have described how using active materials in

phononic crystals may add new and interesting capabilities. The theoretical framework of this

thesis may be further extended and applied in a variety of new ways. In particular, we wish to

broaden the applicability of the extended methods to incorporate resonant metamaterials and

three dimensional crystals. This initial foray into understanding the use of active materials in

phononic crystals has laid the groundwork for many future investigations into a variety of related

topics.

- 160 -

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Wright Derek W 201006 PhD Thesis