Anti-plane interfacial waves in a square lattice

Using the lattice dynamics approach, we discussed the propagation of interfacial waves localized near the interface in an infinite square lattice. The interface has been modeled as a single-particle layer of material particles with masses and elastic bonds different from those in the bulk. In this lattice structure there were anti-plane interface waves, i.e., waves that decayed exponentially with distance from the interface. Such waves could be useful for determining material properties in the vicinity of the interface. We obtained equations of motion and analyzed the corresponding dispersion relations for steady-state solutions. Here, the dispersion equation related the circular frequency to the wave number. In addition, we provided a comparison of the dispersion relations with those derived within the Gurtin-Murdoch surface elasticity. To do this, we have used the scaling law that links the continuum and discrete models. Unlike the continuum model, in the discrete model the wave number was limited by the first Brillouin zone, whereas in the continuum model it took a range from zero to infinity. The detailed parametric analysis was given for the discrete model. Finally, other models of interfaces in the case of a square lattice were discussed.