Closed-form existence conditions for band-gap resonances in a 1D linear elastic diatomic lattice with general boundary conditions

The elastic band structure of an infinite periodic lattice may feature band gaps where waves experience destructive interferences and do not propagate. However, upon lattice truncation, resonances may appear within the band gaps depending on the nature of the finite boundaries. In this work, closed-form existence conditions are derived for band-gap resonances in discrete diatomic chains with general boundary conditions and arbitrary chain parity. Our approach is based on setting up the model in the form of a perturbed 2-Teoplitz matrix, from which a characteristic equation is obtained in closed form. The approach is general in that it could accommodate different compositions in the unit cell. A selected case of a free-free even chain with an arbitrary additional mass at one end is presented as an example. Introducing such an arbitrary mass underscores a transition among a set of distinct existence conditions depending on the type of the chain boundaries and parity. This analysis is in principle applicable to the characterization of band-gap resonances in other physical systems beyond elastic chains.