Resonant-frequency primitive waveforms and star waves in lattices

Ayzenberg-Stepanenko, M. V.; Slepyan, Leonid I.

doi:10.1016/j.jsv.2007.11.047

Cited by 67 publications

(92 citation statements)

References 30 publications

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“…In Figure 9(a) the lattice is excited at Ω = 1.98; this is very close to a flat band in the dispersion diagram, which is of independent interest (see [4]), and standing waves form entirely along the diagonals, centered at the forcing point, of the bulk lattice. We observe that energy travels along the nondefective masses outside and inside the box, while the inner square also allows for this to happen.…”

Section: Illustrative Examplesmentioning

confidence: 83%

“…These dispersion curves show much that is of independent interest: a flat portion along AC in Figure 2 is of particular interest [4], as is the linear lowfrequency behavior near the origin for which classical homogenization applies, and there are standing waves at the edges of the Brillouin zone, coalescences of dispersion curves, and regions with negative group velocities. However, these are tangential to our purpose, which is to investigate the propagating modes that occur along a line defect introduced into this lattice.…”

Section: Perfect Line Defectmentioning

confidence: 97%

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Asymptotics for Rayleigh--Bloch Waves along Lattice Line Defects

Joseph¹

2013

Multiscale Model. Simul.

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Abstract. High-frequency homogenization is applied herein to develop asymptotics for waves propagating along line defects in lattices; the approaches developed are anticipated to be of wide application to many other systems that exhibit surface waves created or directed by microstructure. With the aim being to create a long-scale continuum representation of the line defect that nonetheless accurately incorporates the microscale information, this development uses the microstructural information embedded within, potentially high-frequency, standing wave solutions. A two-scaled approach is utilized for a simple line defect and demonstrated versus exact solutions for quasi-periodic systems and versus numerical solutions for line defects that are themselves perturbed or altered. In particular, Rayleigh-Bloch states propagating along the line defect, and localized defect states, are identified both asymptotically and numerically. Additionally, numerical simulations of large-scale lattice systems illustrate the physics underlying the propagation of waves through the lattice at different frequencies.

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Section: Illustrative Examplesmentioning

confidence: 83%

Section: Perfect Line Defectmentioning

confidence: 97%

Asymptotics for Rayleigh--Bloch Waves along Lattice Line Defects

Joseph¹

2013

Multiscale Model. Simul.

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“…For a monatomic, harmonic scalar lattice (with no spinners), the dispersion equation is given in Ayzenberg-Stepanenko & Slepyan (2008) as…”

Section: Ii) the Low-frequency Rangementioning

confidence: 99%

Vortex-type elastic structured media and dynamic shielding

2012

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The paper presents an approach to modelling a novel elastic metamaterial structure that possesses non-trivial dispersion features. A system of spinners has been embedded into a two-dimensional periodic lattice system. The analysis of the motion of the spinners is used to derive an expression for a 'chiral term' in the equations describing the dynamics of the lattice. Dispersion of elastic waves is shown to possess innovative filtering and polarization properties induced by the vortex-type nature of the structured media. The related effective behaviour in a continuous medium is implemented to build a shielding cloak around an obstacle. Analytical work is accompanied by numerical illustrations.

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“…5(a), and although the corresponding path for the circular holes is no longer perfectly straight, the physical phenomena from the mass-spring system persist. The flat curve has been spotted, in a different context, for the mass-spring system 13,14 and in that system its presence can be derived analytically; as a result the diagonal cross has been seen numerically in simple systems. 13,15 Such flat modes, or nearly flat modes, are of considerable interest in the context of slow light and slow sound.…”

Section: Formulation Of Continuous Modelmentioning

confidence: 99%

Dangers of using the edges of the Brillouin zone

et al. 2012

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In solid-state physics, including photonics and wherever periodic lattice structures occur, it is essential to establish the fundamental features associated with wave propagation through the lattice: This is achieved using Bloch waves, the reciprocal lattice, and the reduction, using periodicity, to consider the irreducible Brillouin zone. A general approach, although widely accepted as not being perfectly legitimate, is to plot the dispersion relations around the edges of the Brillouin zone. We show definitively that this can be dangerous and that an important mode of practical significance is missed if this is done in too cavalier a fashion: This missing mode is illustrated for the design of endoscopes based on spring-mass (discrete) periodic structures and photonic crystals.

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Resonant-frequency primitive waveforms and star waves in lattices

Cited by 67 publications

References 30 publications

Asymptotics for Rayleigh--Bloch Waves along Lattice Line Defects

Asymptotics for Rayleigh--Bloch Waves along Lattice Line Defects

Vortex-type elastic structured media and dynamic shielding

Dangers of using the edges of the Brillouin zone

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