Wave propagation in elastic lattices subjected to a local harmonic loading. I. A quasi-one-dimensional problem

The anti-plane dynamics of infinite (-∞ < x < ∞,-∞ < y < ∞) material-bond rectangular lattices subjected to a uniform monochromatic excitation of the x = 0 line nodes is studied. A quasi-onedimensional model is formulated: the original lattice is considered as an infinite waveguide in the x-direction with periodically joined bonds bounded in the y-direction. In such a structure, the wave pattern consists of waves propagated along x-axis and standing waves along y-axis. Steady and unsteady processes are investigated. Dispersion relations are analyzed and resonance points are detected. A combined analytical-numerical approach is used to describe (i) the quasi-steady propagation of waves when the source frequency is within the pass-band, (ii) development of resonance waves, and (iii) percolation of perturbations to the periphery when the excitation frequency is within the stop-band. Long-wave and short-wave components of solutions are compared with those for a simplified 1D mass-spring lattice (MSL) model.