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.

1 Introduction [6 p.]Description and purpose of the book. Introduction of some elementary concepts. History of the phononic crystalconcept.2 Waves in periodic media [40 p.]A presentation of waves in periodic media devoid of complications like polarization, anisotropy, tensors, loss, etc.Self-contained presentation for scalar waves.2.1 Bloch theoremScalar wave theory. Scalar Helmholtz equation. Bloch theorem.2.2 Physical origin of band gaps1D periodic media. Scattering and diffraction. Bragg band gaps. Local and Fano resonances.2.3 Brillouin zoneDefinition. Direct and reciprocal lattice.2.4 The band structureFourier transforms. Wave vectors. Band structure. Dispersion, group velocity. Equifrequency contours. Analogywith phonons in atomic lattices.2.5 Appendix: Brillouin zones for 2D and 3D latticesGeometrical description of the most common lattices.3 Acoustic waves [20 p.]A synthetic presentation of the subject, with reference to other basic books.3.1 Dynamic equationsParticle velocity and pressure. Acoustic equations.3.2 Constants of fluidsConstants for fluids. Determination of bulk velocities.3.3 Loss and viscosityRepresentation of propagation loss in fluids. Modifications of equations (complex material constants).3.4 Reflection and refractionBrief review of reflection and refraction at the interface of 2 media. Fresnel formulas.4 Sonic crystals [50 p.]Introduce sonic crystals (that can be described by pressure waves), with accent on finite element modeling andbasic properties.4.1 Modeling of sonic crystals4.1.1 Analysis via plane wave expansion (PWE)4.1.2 Multiple scattering theory (MST and LMS)4.1.3 Finite element modeling (FEM)4.2 2D sonic crystalSteel cylinders in air. Steel cylinders in water. Measurement techniques. Comparison with experiment. Deafbands.4.3 3D sonic crystalsSteel beads in water.4.4 Tutorial: sonic crystal analysis with FEMGeneration of band structures. Plotting Bloch waves. Worked examples with ff++.4.5 Appendix: Weak form modeling of sonic crystals. Lagrange Finite elements. Blochwaves and FEM.5 Elastic waves [40 p.]A synthetic presentation of the subject, with reference to other basic books, plus an original part on FEMmodeling.5.1 Dynamic equationsStrain and Stress. Elastic constants. Elastodynamic equations.5.2 Christoffel equation for bulk wavesAnisotropy of wave propagation in crystalline solids. Slowness curve. Wave surface. Polarization. Group velocity.Poynting theorem and energy conservation.5.3 Piezoelectric mediaDescription of the effect. Generalization of the concepts of the previous section.5.4 Plate wavesLamb and other plate waves. Dispersion diagram.5.5 Surface wavesRayleigh and other surface waves. Radiation and leakage. Slowness curves for SAW.5.6 Tutorial: modeling plate waves with FEM5.7 Appendix: tensors6 Phononic crystals for bulk waves [50 p.]6.1 Modeling of phononic crystals6.1.1 Analysis via plane wave expansion (PWE)6.1.2 Finite element modeling (FEM)6.2 2D phononic crystalHoley and solid-solid PC, for most common material combinations. Comparison with experiments.6.3 3D phononic crystalsSteel beads in epoxy. Comparison with experiments.6.4 Tutorial: phononic crystal analysis with FEMGeneration of band structures. Plotting Bloch waves. Worked examples with ff++.6.5 Appendix: weak form modeling of phononic crystals7 Phononic crystals for surface and plate waves [40 p.]7.1 ModelingPresentation based on PWE and/or FEM. Surface boundary conditions and determinants.7.2 Phononic platesSpecific properties and discussion of various forms. Preferred example: holey silicon plate.7.3 Surface phononic crystalsSpecific properties and discussion of various forms. Preferred examples: holey silicon and lithium niobate. Thesound cone and leakage.7.4 Measurement methodsElectrical transduction. Optical transduction. Optical measurement of surface displacements. Comparison withexperiments.7.5 Tutorial: phononic plates and surface waves with FEM8 Coupling of acoustic and elastic waves in phononic crystals [20 p.]8.1 Phononic crystal of solid inclusions in fluid8.2 Phononic plates in water and air8.3 Corrugated surfaces and platesScholte-Stoneley wave. Conversion of bulk to surface waves.9 Evanescent Bloch waves [30 p.]9.1 TheoryP.E and FEM9.2 Sonic crystalsComplex band structure. Symmetry and deaf bands.9.3 Phononic crystalsComplex band structure. Polarization evolution across phononic band gap.9.4 Super-cells and defect modeDiscussion of wave confinement in relation with evanescence.10 Locally-resonant crystals [30 p.]10.1 Local resonance, Fano resonance, metamaterialsDiscussion of the difference between Bragg band gaps and locally-resonant band gaps.10.2 1D corrugated waveguidesResonances introduced by resonators grafted along a waveguide.