Three-dimensional Periodic Media Research Articles

Clusters of Bloch waves in three-dimensional periodic media

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.

Dispersion of waves in two and three-dimensional periodic media

We consider the propagation of acoustic time-harmonic waves in homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius a of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two- and three-dimensional cases as and first several terms of the expansions are evaluated explicitly. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Dirichlet, Neumann, and transmission boundary conditions are considered. In the former case, we estimate the cutoff wavelength supported by the periodic medium in two and three dimensions. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of the asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.

Waves in Structured Mediums or Metamaterials: A Review

Wave propagation through a structured medium has attracted the attention of researchers for centuries due to its relevance to problems in condensed matter physics, chemistry, optics, phononics, composite, acoustics and mechanics. Wave containing certain band of frequencies can either propagate, known as transmission, or attenuated, known as attenuation band. This band structure for a continuum and its equivalent lumped spring mass model are not identical, although continuum medium is often modelled as a chain of discrete periodic structures because the continuous and discrete is depends on the scale. These band characteristics are dependent on the properties of the units, thus the effects of different parameters, such as damping, stiffness and mass ratios, nonlinearity, on the bandwidth are compared with each other in this review. To cloak, modulate, guide, filter out or attenuate unwanted frequencies from the propagating waves, metamaterials are widely investigated as a special form of the periodic structures from the past 2 decades. The main aim of this review is to compare the bandwidth for one-dimensional periodic structures. Waves through two and three-dimensional periodic medium are not considered in the review because the key band characteristics of periodic system can be perceived in one dimensional. The methods for computing the wave transmission are evaluated in the non-dimensional domain and the band characteristics of different one-dimensional periodic structures are critically assessed in this review. This review will help to the future researchers to choose a proper periodic medium for getting a specific band phenomenon.

Analysis of an Algorithm for Computing Electromagnetic Bloch Modes Using Nedelec Spaces

AbstractThe problem of approximating the band structure of electromagnetic Bloch modes in a three-dimensional periodic medium is studied. We analyze a mixed finite element approximation technique based on a variation of Nedelec edge elements. The usual conditions for convergence of the static problem are first verified. Subsequently, convergence of approximate eigenvalues to those of the continuous system is proved.