Mechanism for slow waves near cutoff frequencies in periodic waveguides

Guenneau, Sébastien; Adams, Samuel D.M.

doi:10.1103/physrevb.79.045129

Cited by 25 publications

(22 citation statements)

References 17 publications

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“…Notably along AC in the Brillouin zone, k 1 = k 2 and the correction term vanishes so λ 2 ∼ λ 2 0 and the dispersion curve is locally flat exactly as in figure 7; this flat curve suggests that the group velocity is zero for this range of wavenumbers and has relevance to physical applications such as slow light (27). Case (iii) which is based around standing waves periodic in η 2 and anti-periodic in η 1 is a simple rotation of the present case, so is absorbed into this section and not treated separately.…”

Section: The Matrix Problem Is Only Solvable If Ymentioning

confidence: 98%

High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

Craster

Kaplunov

Postnova

2010

The Quarterly Journal of Mechanics and Applied Mathematics

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SummaryA two-scales approach, for discrete lattice structures, is developed that uses microscale information to find asymptotic homogenized continuum equations valid on the macroscale. The development recognises the importance of standing waves across an elementary cell of the lattice, on the microscale, and perturbs around the, potentially high frequency, standing wave solutions. For examples of infinite perfect periodic and doubly-periodic lattices the resulting asymptotic equations accurately reproduce the behaviour of all branches of the Bloch spectrum near each of the edges of the Brillouin zone. Lattices in which properties vary slowly upon the macroscale are also considered and the asymptotic technique identifies localised modes that are then compared with numerical simulations. IntroductionDiscrete mass-spring lattice systems form classical models of crystal vibrations in solid state physics (1, 2) and were used by Newton, Kelvin, Born, and many others, to model and interpret wave phenomena and these models were instrumental in the development of wave mechanics; a review of the historical literature is contained in Brillouin's monograph (2). Lattice models remain a valuable and instructive way of modelling and understanding fundamental wave phenomena in crystal lattices and cellular structures (3). A common characteristic behaviour of such models is that they exhibit band-gaps, (4), namely bands of frequencies for which waves do not propagate through the atomic lattice. More recently such discrete models have been used for engineering structures (5) with a view to designing smart structures capable of filtering out various frequencies. These discrete systems, exhibiting band gap behaviour, are closely related to periodic continuous media, for instance photonic crystal fibres (6,7,8), for which band-gaps occur and that have numerous and varied industrial applications; in some limits there is a direct analogy between the discrete and continuum models (9). In the mechanics of cellular structures and lattices, considerable knowledge has been gained about the static and low-frequency behaviour in terms of homogenized models, but comparatively less is known of their high frequency behaviour.A common feature of both discrete and continuum models, that are defect-free and infinite in extent, is that the periodicity of the structure leads to dispersion relations between

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Section: The Matrix Problem Is Only Solvable If Ymentioning

confidence: 98%

High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

Craster

Kaplunov

Postnova

2010

The Quarterly Journal of Mechanics and Applied Mathematics

Self Cite

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“…Asymptotic homogenization was also limited in application to describing only the behavior of the fundamental Bloch mode at low frequencies [72,73]. Recently, however, it has been extended to higher frequencies and higher Bloch modes [74][75][76][77] (see also [78]). Additionally, there have been other efforts to bridge the scales for the study of dispersive systems based upon variational formulations [79], micromechanical techniques [80], Fourier transform of the elastodynamic equations [81], and strain projection methods [82].…”

Section: Averaging Techniquesmentioning

confidence: 99%

Elastic metamaterials and dynamic homogenization: a review

Srivastava

2015

International Journal of Smart and Nano Materials

122

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In this paper, we review the recent advances which have taken place in the understanding and applications of acoustic/elastic metamaterials. Metamaterials are artificially created composite materials which exhibit unusual properties that are not found in nature. We begin with presenting arguments from discrete systems which support the case for the existence of unusual material properties such as tensorial and/or negative density. The arguments are then extended to elastic continuums through coherent averaging principles. The resulting coupled and nonlocal homogenized relations, called the Willis relations, are presented as the natural description of inhomogeneous elastodynamics. They are specialized to Bloch waves propagating in periodic composites and we show that the Willis properties display the unusual behavior which is often required in metamaterial applications such as the Veselago lens. We finally present the recent advances in the area of transformation elastodynamics, charting its inspirations from transformation optics, clarifying its particular challenges, and identifying its connection with the constitutive relations of the Willis and the Cosserat types.

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“…Guided waves have a wide range of applications in engineering and physics and they have been subjected to investigation for many years in the context of electromagnetic [1,2], acoustic [3,4] and elastic [5,6] waveguides as well as in the context of shallow oceans [7][8][9]. For uniform and infinite waveguides, separation of variables is possible and exact solutions of the guided wave propagation (modes) exist.…”

Section: Introductionmentioning

confidence: 99%

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods

2014

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We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems.

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Mechanism for slow waves near cutoff frequencies in periodic waveguides

Cited by 25 publications

References 17 publications

High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

Elastic metamaterials and dynamic homogenization: a review

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods

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