Controlling flexural waves in semi-infinite platonic crystals

TL;DR

This paper investigates how flexural waves interact with semi-infinite arrays of point scatterers on plates, revealing ways to achieve perfect transmission and wave localization through a novel analytical approach.

Contribution

It introduces a new method linking Wiener-Hopf kernels to Green's functions to analyze wave control in semi-infinite platonic crystals.

Findings

Demonstrates dynamic neutrality (perfect transmission) at specific frequencies.

Identifies conditions for wave localization near the structured interface.

Reveals anisotropic wave effects in semi-infinite arrays.

Abstract

We address the problem of scattering and transmission of a plane flexural wave through a semi-infinite array of point scatterers/resonators, which take a variety of physically interesting forms. The mathematical model accounts for several classes of point defects, including mass-spring resonators attached to the top surface of the flexural plate and their limiting case of concentrated point masses. We also analyse the special case of resonators attached to opposite faces of the plate. The problem is reduced to a functional equation of the Wiener-Hopf type, whose kernel varies with the type of scatterer considered. A novel approach, which stems from the direct connection between the kernel function of the semi-infinite system and the quasi-periodic Green's functions for corresponding infinite systems, is used to identify special frequency regimes. We thereby demonstrate dynamically anisotropic wave effects in semi-infinite platonic crystals, with particular attention paid to designing systems to exhibit dynamic neutrality (perfect transmission) and localisation close to the structured interface.