Mathematical Modelling of Waves in Multi-Scale Structured Media

Alexander Movchan is a Professor at the University of Liverpool, Natasha Movchan is a Professor at the University of Liverpool, Ian Jones is a Professor at Liverpool John Moores University and an Honorary Fellow at the University of Liverpool, and Daniel Colquitt is a Lecturer at the University of Liverpool. The authors have worked on wave propagation in multi-scale elastic media over many years and have developed novel modelling approaches, which have opened efficient ways to design and study the dynamic response of multi-scale structures known as elastic metamaterials introduced within the last decade.

Introduction and literature survey. Foundations, methods of analysis of waves and analytical approaches to modelling of multi-scale solids. Linear differential operators and distributions. Waves in 1D, 2D and 3D media. Fundamental solutions. Integral transforms and their applications in modelling of waves. Dispersion. Periodic systems, Floquet-Bloch waves. Lattice dynamics. Asymptotic analysis and singular perturbations. Waves in structured media with thin ligaments and disintegrating junctions. Multi-resonator systems. Singular perturbation analysis of fields in solids with disintegrating junctions. Floquet-Bloch waves in periodic multi-resonator systems. Standing waves. Asymptotic estimates for their frequencies. Tuneable multi-resonator systems. Transmission of waves by structured interfaces containing multi-scale resonators. Negative refraction and localisation of waves in structured solids with thin ligaments. Dynamic response of elastic lattices and discretised elastic membranes. Lattice Green’s functions in dynamics. Dynamic anisotropy and localisation near defects. Stop-band Green’s functions and strong exponential localisation. Localisation near cracks/inclusions in a lattice. Cloaking and channelling of elastic waves in structured solids. A cloak is not a shield. Cloaking as a channelling method for incident waves. Boundary conditions on the interior contour of a cloak. Cloaking in elastic plates. Singular perturbation analysis of an approximate cloak. Example of an "impossible cloaking" problem. Structured interfaces in dynamics of elastic solids. Structured interfaces as filters and polarisers. Vortex-type resonators and chiral polarisers of elastic waves. Coated inclusions in dynamics, scattering and neutrality. Lattice approximations of filters and polarisers. Electronic Appendix: Illustrative animations