Longitudinal wave propagation in one-dimensional waveguides with sinusoidally varying depth

Abstract

In this paper, longitudinal elastic wave propagation in one-dimensional waveguides with sinusoidally varying depth is investigated. Furthermore, different types of such waveguide designs are explored to understand their capability to attenuate the group speeds as well as the velocity amplitudes. The plane waveguide configurations with sinusoidally varying depth, of three types, namely Convex, Concave and Full, along with their combinations and variants, whose segmental depth along the length of the waveguide is modeled by a sine function having an amplitude parameter α and the half-period p, which leads to a governing differential equation with variable coefficients, are considered in this study. To study the longitudinal wave propagation in these inhomogenous waveguides, a novel superconvergent finite element formulation is developed, which gives an exact stiffness matrix. In addition to the wave propagation analysis, static and free vibration behaviors in these waveguides are also studied. The implemented superconvergent finite element formulation for these studies is validated with the commercial finite element software Abaqus. In the first part of the wave propagation analysis, abilities of the three plane waveguide configurations with sinusoidally varying depth to attenuate high amplitude and frequency waves are investigated for different values of α. Next, four different waveguides composed of Convex-Concave combinations are studied with an aim to get better attenuation. Following this, the waveguides' segmental parameters, α and p, are varied across the segments along the length of the waveguide, using a sine function and polynomial power law to see if such graded variations give better energy absorption properties compared to the plane waveguides whose parameters are unvarying. In the last part, a developed inhomogenous rod of a certain length is inserted in the middle of a uniform waveguide to study the possibility of changing the longitudinal wave propagation characteristics of the uniform waveguide. The results from the analyses show that the group speed and amplitude of the longitudinal wave in these configurations change significantly with space as well as with frequencies, especially for high values of α. The Concave and Full waveguides delay the propagation of longitudinal waves significantly, which translates into the reflected waves appearing later in the chosen large time window. The Convex waveguide and its variants reduce the wave amplitudes significantly. Building on these results, some waveguides with notable attenuation characteristics are proposed.