Gauge Invariance Approach to Acoustic Fields (eBook)

Foreword 6
Contents 9
About the Author 15
1 Introduction 16
Reference 17
2 History of Gauge Theory 18
2.1 The Roots of Gauge Invariance 18
2.2 The Vector Potential "0245A 19
2.3 Milestones of the Development of Gauge Invariance Principle in Physics 20
2.3.1 Maxwell’s Role in Developing the Gauge Theory 20
2.3.2 The Role of Fock [1] in the History of Gauge Invariance [1926b]—The Quantum Era 21
2.3.3 The Role of Hermann Weyl [2] in the History of Gauge Invariance [1929] 21
2.3.4 The Yang–Mills Theory 22
2.3.5 The Glashow-Feinberg-Salam’s Electroweak Theory 23
2.3.6 Gell-Mann’s Quantum Chromodynamics for Strong Interaction 23
2.3.7 W. S. Gan’s Introduction of Gauge Invariance to Acoustic Wave Equation of Motion 24
2.3.8 Gauge Invariance in Acoustic Wave Equation in Curvilinear Spacetime 27
2.3.9 Application of Gauge Invariance to Gravitational Wave 27
References 27
3 Coordinate Systems as the Framework of Equations 29
3.1 General Covariance 29
3.2 Forms of Coordinate Systems 29
3.2.1 Cartesian Coordinate System 30
3.2.2 Cylindrical Coordinate System 32
3.2.3 Curvilinear Coordinate System 33
3.3 Coordinate Systems as the Framework of Equations 34
References 34
4 Gauge Fields 35
4.1 Introduction 35
4.2 Formulation of Electric Field and Magnetic Field in Terms of Gauge Fields 35
4.3 Gauge Fields and Gauge Transformation 36
4.4 Field Strength and Gauge Fields 36
4.5 Role of Gauge Field in Local Gauge Invariance 37
4.6 Gauge Field and Phase of the Wave Function 37
4.7 The Verification of the Physical Effect of the Gauge Field "0245A by the Aharanov–Bohm Effect 38
4.8 Gauge Fields in Acoustics 39
References 40
5 Covariant Derivative in Gauge Theory 41
5.1 Covariance 41
5.2 Covariant Derivative and Tensor Analysis 41
5.3 Covariant Derivative and Local Gauge Invariance 42
5.4 Gauge Covariant Derivative for Sound Propagation in Continuous Fluids 43
5.5 Gauge Covariant Derivative for Sound Propagation in Solids 43
Reference 44
6 Lie Groups 45
6.1 Group Theory 45
6.2 History of Lie Groups and Introduction 46
6.3 Unitary Group of Degree n U(n) 47
6.4 Special Unitary Group of Degree n SU(n) 48
6.5 Orthogonal Group of Degree n O(n) 49
6.6 Special Orthogonal Group of Degree n SO(n) 51
6.7 Group Properties of SO(n), with n ? 3 53
References 55
7 Global Gauge Invariance 56
7.1 What Is Gauge Invariance 56
7.2 The U(1) Group Symmetry 57
7.3 Gauge Invariance Approach to Acoustic Fields 58
Reference 58
8 Local Gauge Invariance 59
8.1 Understanding of Gauge Fields 59
8.2 The Lagrangian and the Path Integral Approach to Local Gauge Invariance 59
8.3 The Introduction of Covariant Derivative into Local Gauge Invariance 60
8.4 Local Gauge Invariance Treatment in Acoustic Fields 61
Reference 62
9 Gauge Fixing 63
Reference 65
10 Noether’s Theorem 66
Reference 69
11 Spontaneous Symmetry Breaking and Phonon as the Goldstone Mode 70
11.1 Introduction to Spontaneous Symmetry Breaking 70
11.2 Continuous Symmetry—Phonon as a Goldstone Mode 71
11.3 Goldstone’s Theorem 72
References 73
12 Time Reversal Acoustics and Superresolution 74
12.1 Introduction 74
12.2 Theory of Time Reversal Acoustics 74
12.2.1 Time Reversal Acoustics and Superresolution 80
12.3 Application of Time Reversal Acoustics to Medical Ultrasound Imaging 81
12.4 Application of Time Reversal Acoustics to Ultrasonic Nondestructive Testing 82
12.4.1 Theory of Time Reversal Acoustics for Liquid–Solid Interface [18] 84
12.4.2 Experimental Implementation of the TRM for Nondestructive Testing Works 86
12.4.3 Incoherent Summation 89
12.4.4 Time Record of Signals Coming from a Speckle Noise Zone 89
12.4.5 The Iterative Technique 90
12.5 Application of Time Reversal Acoustics to Landmine or Buried Object Detection 93
12.5.1 Introduction 93
12.5.2 Theory 94
12.5.3 Experimental Procedure 95
12.5.4 Experimental Set-Up 97
12.5.5 Wiener Filter 97
12.5.6 Experimental Results 98
12.6 Application of Time Reversal Acoustics to Underwater Acoustics 100
12.7 Application of Time Reversal Acoustics to Nonlinear Acoustic Imaging 100
12.8 Theory of Nonlinear Time Reversal Acoustics Nonlinear Acoustic Imaging 101
12.9 The Experimental Set-Up to Analyse the Nonlinear Time Reversal Acoustic Waves 102
References 103
13 Negative Refraction, Acoustical Metamaterials and Acoustical Cloaking 105
13.1 Introduction 105
13.2 Limitation of Veselago’s Theory 106
13.2.1 Introduction 106
13.2.2 Gauge Invariance of Homogeneous Electromagnetic Wave Equation 107
13.2.3 Gauge Invariance of Acoustic Field Equations 108
13.2.4 Acoustical Cloaking 109
13.2.5 Gauge Invariance of Nonlinear Homogeneous Acoustic Wave Equation 110
13.2.6 My Important Discovery of Negative Refraction Is a Special Case of Coordinate Transformations or a Unified Theory for Negative Refraction and Cloaking 110
13.2.7 Conclusions 112
13.3 Multiple Scattering Approach to Perfect Acoustic Lens 112
13.4 Acoustical Cloaking 118
13.5 Acoustic Metamaterial with Simultaneous Negative Mass Density and Negative Bulk Modulus 124
13.6 Acoustical Cloaking Based on Nonlinear Coordinate Transformations 128
13.7 Acoustical Cloaking of Underwater Objects 132
13.8 Extension of Double Negativity to Nonlinear Acoustics 132
References 133
14 New Acoustics Based on Metamaterials 135
14.1 Introduction 135
14.2 New Acoustics and Acoustical Imaging 137
14.3 Background of Phononic Crystals 138
14.4 Theory of Phononic Crystals—The Multiple Scattering Theory (MST) 139
14.5 Negative Refraction Derived from Gauge Invariance (Coordinate Transformations)—An Alternative Theory of Negative Refraction 144
14.5.1 Gauge Invariance as a Unified Theory of Negative Refraction and Cloaking 144
14.5.2 Generalized Form of Snell’s Law for Curvilinear Coordinates 147
14.5.3 Design of a Perfect Lens Using Coordinate Transformations 147
14.5.4 A General Cloaking Lens 147
14.6 Reflection and Transmission of the Sound Wave at Interface of Two Media with Different Parities 148
14.7 Theory of Diffraction by Negative Inclusion 149
14.7.1 Formulation of Forward Problem of Diffraction Tomography 149
14.7.2 Modelling Diffraction Procedure in a Negative Medium 154
14.7.3 Results of Numerical Simulation 156
14.7.4 Points to Take Care of During Numerical Simulation 162
14.8 Extension to Theory of Diffraction by Inclusion of General Form of Mass Density and Bulk Modulus Manipulated by Predetermined Direction of Sound Propagation 164
14.9 A New Approach to Diffraction Theory—A Rigorous Theory Based on the Material Parameters 165
14.10 Negative Refraction Derived from Reflection Invariance (Right–Left Symmetry)—A New Approach to Negative Refraction 165
14.11 A Unified Theory for the Symmetry of Acoustic Fields in Isotropic Solids, Negative Refraction and Cloaking 167
14.12 Application of New Acoustics to Acoustic Waveguide 168
14.13 New Elasticity 169
14.14 Nonlinear Acoustics Based on the Metamaterial 170
14.14.1 Principles 170
14.14.2 Nonlinear Acoustic Metamaterials for Sound Attenuation Applications 171
14.15 Ultrasonic Attenuation in Acoustic Metamaterial 172
14.15.1 Mechanism of Energy Transfer and Wave Attenuation 173
14.15.2 Applications 173
14.16 Applications of Phononic Crystal Devices 175
14.17 Comparison of the Significance of Role Played by Gauge Theory and Multiple Scattering Theory in Metamaterial—A Sum-up of the Theories of Metamaterial 176
14.18 Conclusions 176
References 177