Dynamic Green’s functions for lattice systems with frontiers possessing contrasting periodicity and material properties

We consider anti-plane waves propagating in a lattice, formed from periodically placed masses interconnected by springs, that possesses a bi-periodic frontier, constituting a boundary or interface with contrasting inertial and elastic properties. In particular, we study the response of such a system to the application of a point force, corresponding to Green’s function. This problem is studied using the Z -transform which yields Green’s function in an exact form in terms of a contour integral. We also demonstrate how this approach provides explicit dispersion relations for waves along the lattice frontier. We study the influence of the lattice frontier’s composition on the pass band structure linked to the existence of these waves. A low-frequency homogenisation model is also developed that incorporates the effects due to the elastic and inertial properties of the frontier. All analytical results are accompanied by numerical illustrations.