On the behaviour of infinite, periodic, mono-coupled waveguides using a transmission coefficient phase method

Abstract

Abstract Wave propagation in a one dimensional, mono-coupled waveguide composed of an infinite wave-bearing structure supported by periodic discontinuities is analysed in terms of the transmission and reflection coefficients of a single discontinuity, expressions for which are often available in the literature. The general solution is presented and is applicable for arbitrary transmission and reflection coefficients. For conservative discontinuities, conservation of energy imposes relationships on the complex values of these coefficients and yields a simplified expression for the propagation constant, which is a function of the phase of the transmission coefficient, and a second phase angle which arises from the conditions of continuity and equilibrium at the discontinuity. Expressions for the bounding frequencies between pass and stop bands, the bandwidth of a stop band, and the upper bound of attenuation within a stop band are also given in terms of these phases. For some common systems, the phase of the transmission coefficient determines the behaviour of the entire system and thus provides a straightforward analysis. The proposed approach for mono-coupled waveguides can also be applied to some multi-coupled waveguides such as beams under the assumption that nearfield waves are negligible. The approach is then applied to an infinite periodic string on arbitrary supports, an Euler-Bernoulli beam on periodic arbitrary supports, and a system of thin beams that are hinged at periodic intervals with arbitrary supports.