Asymptotics for Rayleigh--Bloch Waves along Lattice Line Defects

High-frequency homogenization is applied herein to develop asymptotics for waves propagating along line defects in lattices; the approaches developed are anticipated to be of wide application to many other systems that exhibit surface waves created or directed by microstructure. With the aim being to create a long-scale continuum representation of the line defect that nonetheless accurately incorporates the microscale information, this development uses the microstructural information embedded within, potentially high-frequency, standing wave solutions. A two-scaled approach is utilized for a simple line defect and demonstrated versus exact solutions for quasi-periodic systems and versus numerical solutions for line defects that are themselves perturbed or altered. In particular, Rayleigh--Bloch states propagating along the line defect, and localized defect states, are identified both asymptotically and numerically. Additionally, numerical simulations of large-scale lattice systems illustrate the physics underlying the propagation of waves through the lattice at different frequencies.