Discrete scattering by a pair of parallel defects

Area of Science:

• Solid mechanics
• Wave propagation
• Materials science

Background:

• Scattering phenomena are crucial in understanding wave dynamics in structured media.
• Lattice structures exhibit unique wave propagation characteristics.
• Crack and rigid tips introduce stress concentrations and alter wave behavior.

Purpose of the Study:

• To analyze the scattering of anti-plane shear waves by crack and rigid tips on a square lattice.
• To simplify the scattering problem using structural symmetry and reduce it to scalar Wiener-Hopf equations.
• To investigate wave incidence from both the bulk lattice and a lattice waveguide.

Main Methods:

• Reduction of coupled equations to two scalar Wiener-Hopf equations.
• Analysis of four geometrically reduced problems on a lattice half-plane.
• Construction of exact solutions for different incidence scenarios.
• Superposition of solutions to address the main scattering problem.

Main Results:

• Exact solutions were derived for wave scattering from crack and rigid tips on a square lattice.
• The study successfully reduced complex scattering problems to simpler Wiener-Hopf equations.
• Wave incidence from both bulk lattice and waveguide scenarios were analyzed.

Conclusions:

• The discrete wave paradigm from waveguide incidence is applicable to modern nanoscale electronic and thermal transport investigations.
• The methodology provides a framework for analyzing dynamic phenomena in structured media.
• The findings contribute to the understanding of wave localization and transport in engineered materials.