Rayleigh–Bloch, topological edge and interface waves for structured elastic plates

Abstract

Galvanised by the emergent fields of metamaterials and topological wave physics, there is currently much interest in controlling wave propagation along structured arrays, and interfacial waves between geometrically different crystal arrangements. We model array and interface waves for structured thin elastic plates, so-called platonic crystals, that share many analogies with their electromagnetic and acoustic counterparts, photonic and phononic crystals, and much of what we present carries across to those systems. These crystals support several forms of edge or array-guided modes, that decay perpendicular to their direction of propagation. To rapidly, and accurately, characterise these modes and their decay we develop a spectral Galerkin method, using a Fourier-Hermite basis, to provide highly accurate dispersion diagrams and mode-shapes, that are confirmed with full scattering simulations. We illustrate this approach using Rayleigh Bloch modes, and generalise high frequency homogenisation, along a line array, to extract the envelope wavelength along the array. Rayleigh-Bloch modes along graded arrays of rings of point masses are investigated and novel forms of the rainbow trapping effect and wave hybridisation are demonstrated. Finally, the method is used to investigate the dispersion curves and mode-shapes of interfacial waves created by geometrical differences in adjoining media. arXiv:1812.07531v1 [physics.class-ph]

Topological Rainbow Trapping for Elastic Energy Harvesting in Graded Su-Schrieffer-Heeger Systems

“…To elucidate the design paradigm and conditions for existence of an edge mode, we firstly consider the simplified elastic model of a point-mass-loaded thin Kirchhoff-Love elastic plate. The expected existence of edge states in a one-dimensional SSH chain is confirmed through calculation of the Zak phase via an efficient numerical scheme [49], corroborated via Fourier spectral analysis of scattering simulations. The differences between the localized 1D edge states and that of conventional band-gap defect states are highlighted.…”

“…Further to this, the Green's function of the governing biharmonic wave equation is nonsingular and remains bounded, and as such numerical complications during the implementation of scattering simulations are side stepped, enabling efficient scattering calculations to be obtained by extending a method attributed to Foldy [56]. Recent advances for analyzing one-dimensional, infinite, periodic structures, in such systems, [49] have generated efficient methods for calculating their dispersion curves, enabling fast design and analysis. These features of the KL system, and the numerical ease of its solution, motivate its use as a powerful toolbox for quickly characterizing topological systems [18,28,[57][58][59][60].…”

“…Considering an infinite periodic line array of point masses, capable of supporting propagating Rayleigh-Bloch modes, which exponentially decay perpendicularly to the array, allows the governing equation to be formulated as a generalized eigenvalue problem; we do so by partitioning the array and plate into periodic infinite strips and by formulating the wavefield, w(x) as a combination of a Fourier series and a decaying basis, as in Ref. [49]. Employing Floquet-Bloch conditions, and invoking orthogonality, then characterizes the dispersion relation for an arbitrary periodic strip of width a.…”

“…The dispersion curves, from the spectral method [49], for the normalized frequency, (κ) are shown in white, with the dotted white line showing the free-space flexural "sound line," which, for KL plates is not dispersionless. Plotted along the wave numbers from κ = −X ≡ −π/a to κ = ≡ 0 are the curves for S1, with to κ = X ≡ π/a showing those for S1 .…”

Ultrathin entirely flat Umklapp lenses

“…The inspiration of many such designs stem from the rainbow trapping effect, originating in electromagnetism [13], whereby the speed and phase of localised array guided modes is manipulated by graded geometrical changes of the array components. Elementary resonant, often sub-wavelength, devices have been proposed [14,15], and built [16], for elastic media enabling trapping, and mode conversion transferring energy from surface to body waves, effects for array guided surface states for applications to energy harvesting [17] from vibration. Recently, a reversed conversion phenomena which emulates negative refraction by a line array [18] has been developed using a counter-intuitive effect hybridising both trapping and conversion; this relies upon band crossings and phase-matching all within the first Brillouin zone and uses adiabatic grading of an array all in the Flat lensing by source placed at −8λ, producing image at opposite focal spot.…”