Localisation for a line defect in an infinite square lattice

TL;DR

This paper analyzes localized defect modes in an infinite square lattice caused by a finite line defect, using Green's functions, eigenvalue analysis, and asymptotic methods to understand their properties and frequencies.

Contribution

It introduces new analytical methods for characterizing defect modes in lattice structures, including Green's function representations and asymptotic expansions near band edges.

Findings

Eigenfrequencies depend on defect size and properties.

Asymptotic expressions accurately predict eigenfrequencies near band edges.

Dispersion relations for infinite defects are explicitly derived.

Abstract

Localised defect modes generated by a finite line defect composed of several masses, embedded an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenisation are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.