Preface; Reference; Contents; Chapter 1: Introduction to Spring Systems; 1.1 Introduction; 1.2 One-Dimensional Monatomic Harmonic Crystal; 1.3 Phase and Group Velocity; 1.4 One-Dimensional Diatomic Harmonic Crystal; 1.5 One-Dimensional Monatomic Crystal with Spatially Varying Stiffness; 1.6 Greenś Function Approach; 1.7 Monatomic Crystal with a Mass Defect; 1.7.1 Monatomic Harmonic Crystal with a General Perturbation; 1.7.2 Locally Resonant Structure; 1.8 Interface Response Theory; 1.8.1 Fundamental Equations of the Interface Response Theory

1.8.2 Greenś Function of the Cleaved 1-D Monatomic Crystal1.8.3 Finite Monatomic Crystal; 1.8.4 1-D Monatomic Crystal with One Side Branch; 1.8.5 1-D Monatomic Crystal with Multiple Side Branches; 1.9 Conclusion; Appendix 1: Code Based on Greenś Function Approach; References; Chapter 2: Phase and Topology; 2.1 Introduction; 2.2 Overview; 2.3 Harmonic Oscillator Model Systems; 2.3.1 Geometric Phase and Dynamical Phase of the Damped Harmonic Oscillator; 2.3.2 Geometric Phase of the Driven Harmonic Oscillator; 2.3.3 Topological Interpretation of the Geometric Phase

2.4 Elastic Superlattice Model System2.4.1 Geometric Phase of a One-Dimensional Elastic Superlattice: Zak Phase; 2.4.2 Topological Interpretation of the Zak Phase; 2.5 Greenś Function Approach; 2.5.1 Greenś Functions and Berry Connection; 2.5.2 The One-Dimensional Harmonic Crystal; 2.5.3 The One-Dimensional Harmonic Crystal with Side Branches; 2.6 Topological Modes at Interfaces Between Media with Different Zak Phases; 2.7 Conclusion; Appendix 1: Eigen Values and Eigen Vectors in One-Dimensional Elastic Superlattice

Appendix 2: Discrete One-Dimensional Monatomic Crystal with Spatially Varying StiffnessAppendix 3: Introduction to Greenś Function Formalism; Greenś Function and the Dyson Equation; Greenś Function and Density of States; References; Chapter 3: Topology and Duality of Sound and Elastic Waves; 3.1 Introduction; 3.2 Overview; 3.3 Intrinsic Topological Phononic Structures; 3.3.1 Two Coupled Mass-Spring Harmonic Crystals in the Long-Wavelength Limit; 3.3.2 Single Harmonic Crystal Grounded to a Rigid Substrate in the Long-Wavelength Limit

3.3.3 Single Discrete Harmonic Crystal Grounded to a Rigid Substrate Beyond the Long-Wavelength Limit3.3.3.1 Dirac-Like Equation for the Discrete Crystal and Spinor Solutions; 3.3.3.2 Non-conventional Topology of Elastic Waves in the Discrete 1D Mass-Spring Single Crystal System; 3.3.3.3 Analysis of Fermion-Like Behavior of Single Harmonic Crystal Grounded to a Substrate in the Long-Wavelength Limit; 3.3.3.3.1 Lagrangian Formalism; 3.3.3.3.2 Energy and Anticommutation; 3.3.3.3.3 Number Operators; 3.3.3.3.4 Measurement of the Spinor State and Information Encoding and Processing