On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials

Abstract An analysis of the linear waves in infinitely-long square and triangular lattice strips of identical particles with nearest neighbor-interactions for all combinations of fixed and free boundary conditions, as well as the periodic boundary, is presented. The expressions for the dispersion relations and the associated normal modes in these waveguides are provided in the paper; some of which are expressed implicitly in terms of certain linear combinations of the Chebyshev polynomials. The e↵ect of next-nearest-neighbour interaction is also attended for the square lattice waveguides. It is found that the localized propagating waves, so called surface wave modes, occur in the triangular lattice strips, as well as square lattice strips with next-nearest-neighbour interactions, when either or both boundaries are free. In this paper, the even and odd modes are also discussed separately, wherever applicable. Graphical illustrations of the dispersion curves are included for all waveguides. The discrete waveguides analyzed in the paper have broad applications in physics and engineering, including their merit in classical problems in elasticity, acoustics, and electromagnetism, as well as recent technological issues involving various transport phenomena in quasi-one-dimensional nano-structures.