Analytical solutions for Bloch waves in resonant phononic crystals

I will discuss the canonical problem of wave scattering by arrays of Neumann inclusions. Firstly, we consider the wavefield to be governed by the Helmholtz equation. We apply the method of matched asymptotic expansions to show how small scatterers can be modelled as singular perturbations to the free space. Analytical expressions then follow in terms of singular Green’s functions (and their derivatives), from which we can construct an eigenvalue problem to consider Floquet-Bloch waves, or we can consider scattering problems as an extension to Foldy’s method. The methods presented allow for efficient, rapid, and accurate computations.These methods will then be applied in an elastic setting, to consider waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales. We cross-verify against finite element numerical simulations to gain further confidence in our approach. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect. These topologically non-trivial states exist near flexural resonances of the constituent beams of the phononic crystal and hence can be tuned into a deep-subwavelength regime.