Phononic Crystals: Artificial Crystals for Sonic, Acoustic, and Elastic Waves

✍ Scribed by Vincent Laude

• Publisher: De Gruyter
• Year: 2015
• Tongue: English
• Leaves: 420
• Series: De Gruyter Studies in Mathematical Physics; 26
• Category: Library

✦ Synopsis

Phononic crystals are artificial periodic structures that can alter efficiently the flow of sound, acoustic waves, or elastic waves. They were introduced about twenty years ago and have gained increasing interest since then, both because of their amazing physical properties and because of their potential applications. The topic of phononic crystals stands as the cross-road of physics (condensed matter physics, wave propagation in inhomogeneous and periodic media) and engineering (acoustics, ultrasonics, mechanical engineering, electrical engineering). Phononic crystals cover a wide range of scales, from meter-size periodic structures for sound in air to nanometer-size structures for information processing or thermal phonon control in integrated circuits. Phononic crystals have a definite relation with the topic of photonic crystals in optics. The marriage of phononic and photonic crystals also provides a promising structural basis for enhanced sound and light interaction.

As the topic is getting popular, it is nowadays presented and discussed at various international conferences. After the first ten years during which the topic has remained mainly theoretical with a few proof-of-concept demonstrations in the literature, the evolution has been towards applications, instrumentation, and novel designs. The physical explanations for various effects are now well understood and efficient numerical methods and analysis tools have been developed.

The book contains a comprehensive set of finite element model (FEM) scripts for solving basic phononic crystal problems. The scripts are short, easy to read, and efficient, allowing the reader to generate for him(her)self band structures for 2D and 3D phononic crystals, to compute Bloch waves, waveguide and cavity modes, and more.

✦ Table of Contents

Preface
Contents
1 Introduction
Part I: Acoustic waves in sonic crystals
2 Scalar waves in periodic media
2.1 Scalar waves in homogeneous media
2.1.1 One-dimensional wave propagation
2.1.2 Three-dimensional wave propagation
2.2 Bloch’s theorem
2.3 Physical origin of band gaps
2.3.1 1D periodic media
2.3.2 Two- and three-dimensional cases
2.3.3 Local resonance
2.4 Lattices, Brillouin zones and the band structure
2.4.1 Bravais lattice
2.4.2 Primitive cell
2.4.3 Reciprocal lattice
3 Acoustic waves
3.1 Dynamical equations of acoustic waves
3.1.1 1D acoustic equations
3.1.2 3D acoustic equations
3.1.3 Poynting’s theorem for acoustic waves
3.1.4 Constants of fluids, loss
3.2 Reflection and refraction
3.3 Finite element modeling of scattering acoustic problems
3.3.1 Mesh, finite element space, weak form, problem solving
3.3.2 Weak form of the acoustic wave equation
3.3.3 Radiation boundary condition
3.3.4 Representation of an internal source of waves
3.3.5 Perfectly matched layer for monochromatic waves
3.3.6 Scattering of an incident plane wave
4 Sonic crystals
4.1 Modeling of sonic crystals
4.1.1 Dynamical equations
4.1.2 Plane wave expansion (PWE) method
4.1.3 Multiple scattering theory (MST and LMS)
4.1.4 Finite-difference time-domain (FDTD)
4.1.5 Finite element modeling (FEM)
4.1.6 Other methods
4.2 2D sonic crystal
4.2.1 Rigid cylinders in air
4.2.2 Steel cylinders in water
4.2.3 Deaf bands and unit cell symmetry
4.2.4 Sonic crystal design
4.3 3D sonic crystals
4.3.1 Air bubbles in water
4.3.2 Tungsten carbide beads in water
4.A Derivation of PWE equations
4.B Some properties of eigenvalue problems
Part II: Elastic waves in phononic crystals
5 Elastic waves
5.1 Elastodynamic equations
5.2 Bulk waves in elastic solids
5.3 Piezoelectric media
5.4 Bulk waves in piezoelectric media
5.5 Reflection and refraction
5.6 Plate waves
5.7 Surface waves
5.A Tensors
5.B Modeling bulk, plate, and surface waves with FEM
5.B.1 Variational formulation for elastic waves
5.B.2 Finite element implementation
6 Phononic crystals for bulk elastic waves
6.1 Modeling of phononic crystals
6.1.1 Dynamical equations
6.1.2 Plane wave expansion (PWE) method
6.1.3 Finite-difference time-domain method (FDTD)
6.1.4 Finite element modeling (FEM)
6.2 2D phononic crystals
6.2.1 Stiff-in-soft composition
6.2.2 Soft-in-stiff composition
6.2.3 Solid-void composition
6.2.4 Crystals containing anisotropic elastic solids
6.2.5 Crystals containing piezoelectric solids
6.3 3D phononic crystals
7 Phononic crystals for surface and plate waves
7.1 Bloch waves of phononic crystal slabs
7.1.1 Analysis with FEM
7.1.2 Analysis with PWE
7.2 Experiments with phononic crystal slabs
7.2.1 Holey phononic crystal slab
7.2.2 Solid-solid phononic crystal slab
7.3 Surface Bloch waves
7.3.1 Expansion in partial waves
7.3.2 Surface boundary conditions
7.3.3 Semi-infinite surface phononic crystals
7.3.4 Finite-depth surface phononic crystals
7.4 Surface phononic crystals
Part III: Wave phenomena in phononic crystals
8 Coupling of acoustic and elastic waves in phononic crystals
8.1 Coupling of acoustic and elastic waves
8.2 Sonic crystal of solid inclusions in a fluid
8.2.1 Solid rods in water
8.2.2 Nylon rods in water
8.3 Fluid-filled inclusions in 2D phononic crystals
8.3.1 Air holes in 2D phononic crystals
8.3.2 Liquid-filled inclusions as sensors
8.4 Corrugated surfaces and plates
9 Evanescent Bloch waves
9.1 Evanescent waves and Bloch’s theorem
9.2 Evanescent Bloch waves of sonic crystals
9.2.1 Analysis via the plane wave expansion
9.2.2 Finite element modeling
9.2.3 Complex band structure
9.3 Evanescent Bloch waves of phononic crystals
9.3.1 Analysis via the plane wave expansion
9.3.2 Analysis via the finite element method
9.3.3 Complex band structure
9.3.4 Viscoelastic losses
9.4 Supercells and defect modes
10 Locally-resonant crystals
10.1 Local resonance and Fano resonance
10.2 1D arrays of resonators grafted on waveguides
10.3 Locally-resonant sonic crystals
10.4 Locally-resonant phononic crystals
10.5 Phononic crystal slab with pillars
10.6 Surface phononic crystal of pillars
11 Mirrors, waveguides, and cavities
11.1 Phononic crystal functions
11.2 Mirrors
11.3 Defect cavities
11.4 Defect waveguides
11.4.1 Waveguides in 2D sonic crystals
11.4.2 Waveguides in phononic crystal slabs
11.4.3 Waveguides in surface phononic crystals
11.4.4 Coupled-resonator acoustic waveguides (CRAW)
11.4.5 The phononic crystal fiber
12 Spatial and temporal dispersion
12.1 Dispersion relations
12.2 Refractive sonic crystal lenses
12.3 Negative refraction in sonic crystals
12.4 Collimation
12.5 Gradient-index phononic crystals
12.6 Negative refraction in phononic crystals
12.7 Reflection and refraction at a crystal boundary
12.8 Sonic crystal as a diffraction grating
12.9 Temporal dispersion and tunneling
13 Conclusion
Bibliography
Index