  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.
