Semi-analytical Formulation for Guided Wave Propagation

Abstract

Guided wave is a type of wave propagation in which the waves are guided in plates, rods, pipes or elongated structures such as rails and I-beams. In order to extract the guided wave velocities and wavestructures, for waveguides of arbitrary cross section, a theoretical framework is developed. Here, a semi-analytical finite element (SAFE) method is used for the calculation of the wave propagation characteristics in elastic waveguides immersed in vacuum. The method couples an approximate displacement field over the cross-section of the waveguide and assumes time harmonic representation of the propagating waves along the length of the guide. The Hamilton’s principle and the finite element discretization lead to a discrete weak form of the energy balance equation. The wave propagation problem reduces to a system of algebraic equations, from which the dispersive equation can be obtained. The solution, which depends on both time t and propagation coordinate z, i.e. e i(ξz − ωt) results in a two-parameter eigensystem. By specifying a real axial wavenumber ξ, the eigenproblem permits real frequencies of propagating modes to be determined. Giving instead real frequency ω, both real and complex axial wavenumbers can be extracted, where real values pertain to propagating modes and the complex ones to the evanescent modes. The method allows us to model a generic cross-section of solid waveguide and it is well suited for computing the phase velocity, the group velocity and the wavestructure or cross-sectional mode shape.