Effects of corrugation shape on frequency band-gaps for longitudinal wave motion in a periodic elastic layer

Abstract

The paper concerns determining frequency band-gaps for longitudinal wave motion in a periodic waveguide. The waveguide may be considered either as an elastic layer with variable thickness or as a rod with variable cross section. As a result, widths and locations of all frequency band-gaps are determined by means of the method of varying amplitudes. For the general symmetric corrugation shape, the width of each odd band-gap is controlled only by one harmonic in the corrugation series with its number being equal to the number of the band-gap. Widths of even band-gaps, however, are influenced by all the harmonics involved in the corrugation series, so that the lower frequency band-gaps can emerge. These are band-gaps located below the frequency corresponding to the lowest harmonic in the corrugation series. For the general non-symmetric corrugation shape, the mth band-gap is controlled only by one, the mth, harmonic in the corrugation series. The revealed insights into the mechanism of band-gap formation can be used to predict locations and widths of all frequency band-gaps featured by any corrugation shape. These insights are general and can be valid also for other types of wave motion in periodic structures, e.g., transverse or torsional vibration.

Wave propagation in axially moving periodic strings

“…the expressions (21)- (23), agree well with the classical results [24,25]. Employing the Method of Varying Amplitudes [18,19,29], we search a solution to (18) in the form of an infinite series:…”

“…. As is shown in the paper [29], such an approximation is valid for predicting at least the first (lowest) band-gap of a periodic structure, which is of primary interest in the present study. Taking into account that the considered string moves axially we get:…”

“…In the following section, its solution by the Method of Varying Amplitudes will be briefly described. A comparison of the MVA with the classical methods is given in the papers [18,19,28,29]. It is noted, in particular, that for linear equations with periodic coefficients the method gives the same results as the approaches based on the classical Floquet theory.…”

Interactions between the first mode and the second Bragg gap in a cylindrical waveguide with undulated walls

“…Due to the larger shape factor of the rectangular wall, the passband is compressed and the enlarged stopband provide the smooth zero transmission instead of the fluctuations for the waveguides with sinusoidal and triangular walls. Physically, the rectangular wall can provide more intense interaction, which has been quantified by the shape factor (23). The estimations on the sinusoidal and triangular wall profile ducts are in accordance with the numerical results while that on the rectangular profile has a few deviations.…”

Longitudinal wave propagation in a one-dimensional quasi-periodic waveguide

“…It can also apply to a different property as is suggested by the example above. The bandgap emerges due to Bragg scattering [1][2][3][4], induced since the structure is not uniform and, in this sense, is similar to bandgaps featured by pure periodic structures with spatially varying parameters [1][2][3][4]17,18]. Assuming, for example, H 1 = 0.9, P 1 = 0.8, N j = 0, j ∈ Z and H k = 0, P k = 0, k = 1, k ∈ Z, for L = 0.1L, we get the low-frequency bandgap atω com which is nine times smaller than the frequency corresponding to the lowest bandgap of the original pure periodic structure,ω 1 .…”