Design of interface modes in canonical phononic waveguides

An interface mode is a localised vibration field at the interface between two waveguides that may be excited at a frequency sitting in a band gap that is in common between the two structures. For electromagnetic waves, the condition for the mode to occur is associated with certain properties of either the surface impedances of the two waveguides or the value of the Zak phase of the adjacent pass bands. In this work, we propose a novel, rigorous and simple method to predict the presence of interface modes at the join between two dissimilar, one-dimensional, periodic, two-phase phononic waveguides. In particular, we show that when the two rods have a canonical configuration it is possible to determine the band gaps of the frequency spectrum where this condition is satisfied. The value of the impedance for all band gaps of the spectrum is analysed through an extended version of the method of the universal toroidal manifold, recently adopted by the Authors to describe the dynamic properties of canonical structures. In terms of prediction, the outcome of the proposed approach is identical to that derived by calculating the Zak phase of the bulk bands for both the waveguides composing the system. By considering two specific combinations of finite-sized canonical rods and studying the associated reflection coefficients, we also determine the frequency of the interface mode in closed form. Our approach provides significant new insight to the mechanics of structured waveguides in order to design and optimise systems able to support interface modes avoiding the challenging numerical calculations normally required to estimate topological invariants.