Abstract
Phononic crystals (PCs) have been widely reported to exhibit band gaps, which for non-dissipative systems are well defined from the dispersion relation as a frequency range in which no propagating (i.e., non-decaying) wave modes exist. However, the notion of a band gap is less clear in dissipative systems, as all wave modes exhibit attenuation. Various measures have been proposed to quantify the “evanescence” of frequency ranges and/or wave propagation directions, but these measures are not based on measurable physical quantities. Furthermore, in finite systems created by truncating a PC, wave propagation is strongly attenuated but not completely forbidden, and a quantitative measure that predicts wave transmission in a finite PC from the infinite dispersion relation is elusive. In this paper, we propose an “evanescence indicator” for PCs with 1D periodicity that relates the decay component of the Bloch wavevector to the transmitted wave amplitude through a finite PC. When plotted over a frequency range of interest, this indicator reveals frequency regions of strongly attenuated wave propagation, which are dubbed “fuzzy band gaps” due to the smooth (rather than abrupt) transition between evanescent and propagating wave characteristics. The indicator is capable of identifying polarized fuzzy band gaps, including fuzzy band gaps which exists with respect to “hybrid” polarizations which consist of multiple simultaneous polarizations. We validate the indicator using simulations and experiments of wave transmission through highly viscoelastic and finite phononic crystals.
Highlights
According to the Bloch theorem, as adapted to elastodynamics, waves propagating in periodic structures can be described by a basis of eigenfunctions having the form u( x, t) = ei(k· x−λt) û( x), (1)where x and t are the spatial and time coordinates, respectively; u is the displacement field; û is a function with the same periodicity as the periodic structure; k is the Bloch wavevector; and λ is the angular frequency
As damped Phononic crystals (PCs) may exhibit strong wave attenuation due to Bragg scattering, yet possess no true band gaps due to the effect of dissipation on the eigenmodes, we introduce the term “fuzzy band gaps” for frequency ranges with strong spatial attenuation of waves which may or may not be true band gaps
We developed a figure of merit called an “evanescence indicator” which gives a quantitative measure of wave decay in damped phononic crystals (PCs)
Summary
Where x and t are the spatial and time coordinates, respectively; u is the displacement field; û is a function with the same periodicity as the periodic structure; k is the Bloch wavevector; and λ is the angular frequency. I.e., waves which are excited by a harmonic displacement or force and decay in space but not in time, the “direct” or k(ω ) method is used to solve the dispersion relation. This method proceeds by prescribing a real-valued angular frequency ω and solving the resulting eigenvalue problem for the Bloch wavevector k. For complex-valued k, the wave amplitude will decrease with increasing k I · x and the wave is known as an “evanescent” wave
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