Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Localization for a line defect in an infinite square lattice

D. J. Colquitt

D. J. Colquitt

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

[email protected]

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M. J. Nieves

M. J. Nieves

School of Engineering, John Moores University, James Parsons Building, Byrom Street, Liverpool L3 3AF, UK

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I. S. Jones

I. S. Jones

School of Engineering, John Moores University, James Parsons Building, Byrom Street, Liverpool L3 3AF, UK

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A. B. Movchan

A. B. Movchan

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

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N. V. Movchan

N. V. Movchan

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

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Published:https://doi.org/10.1098/rspa.2012.0579

    Localized defect modes generated by a finite line defect composed of several masses, embedded in an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenization, are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.

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