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Waves in a plane rectangular lattice of thin elastic waveguides. / Nazarov, S. A. .
в: Journal of Mathematical Sciences, Том 242, № 2, 2019, стр. 227-279.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Waves in a plane rectangular lattice of thin elastic waveguides
AU - Nazarov, S. A.
N1 - Nazarov, S.A. Waves in a Plane Rectangular Lattice of Thin Elastic Waveguides. J Math Sci 242, 227–279 (2019). https://doi.org/10.1007/s10958-019-04476-7
PY - 2019
Y1 - 2019
N2 - We study the spectrum of a thin (with the relative width h ≪ 1) rectangular lattice of elastic isotropic (with the Lamé constants ⋋ ≥ 0 and μ > 0) plane waveguides simulating joining seams of a doubly periodic system of identical absolutely rigid tiles. We establish that the low-frequency range of the essential spectrum contains two spectral bands (passing ones for waves) of length O(e −δ/(2h)), δ > 0. Above these bands there is a gap of width O(h −2) (stopping zones for waves) and then, in the mid-frequency range, above the cut-off value μπ 2h −2 of the continuous spectrum of the infinite cross-shaped waveguide, there is a family of spectral bands of length O(h); moreover, between some of these bands there are opened up gaps of width O(1). The character of the wave propagation depends on whether the frequencies are below or above the cut-off value. In the first case, the oscillations are strictly concentrated near the lattice nodes and the edges are practically immovable. In the second case, the oscillations are localized on the lattice edges, i.e., the nodes are left at relative rest. We show that single perturbations of nodes or edges can cause the appearance of points of the discrete spectrum under the essential spectrum or inside the gaps; moreover, an infinite collection of identical perturbations of nodes can also change the essential spectrum. Bibliography: 78 titles. Illustrations: 5 figures.
AB - We study the spectrum of a thin (with the relative width h ≪ 1) rectangular lattice of elastic isotropic (with the Lamé constants ⋋ ≥ 0 and μ > 0) plane waveguides simulating joining seams of a doubly periodic system of identical absolutely rigid tiles. We establish that the low-frequency range of the essential spectrum contains two spectral bands (passing ones for waves) of length O(e −δ/(2h)), δ > 0. Above these bands there is a gap of width O(h −2) (stopping zones for waves) and then, in the mid-frequency range, above the cut-off value μπ 2h −2 of the continuous spectrum of the infinite cross-shaped waveguide, there is a family of spectral bands of length O(h); moreover, between some of these bands there are opened up gaps of width O(1). The character of the wave propagation depends on whether the frequencies are below or above the cut-off value. In the first case, the oscillations are strictly concentrated near the lattice nodes and the edges are practically immovable. In the second case, the oscillations are localized on the lattice edges, i.e., the nodes are left at relative rest. We show that single perturbations of nodes or edges can cause the appearance of points of the discrete spectrum under the essential spectrum or inside the gaps; moreover, an infinite collection of identical perturbations of nodes can also change the essential spectrum. Bibliography: 78 titles. Illustrations: 5 figures.
UR - http://www.scopus.com/inward/record.url?scp=85071441403&partnerID=8YFLogxK
U2 - 10.1007/s10958-019-04476-7
DO - 10.1007/s10958-019-04476-7
M3 - Article
VL - 242
SP - 227
EP - 279
JO - Journal of Mathematical Sciences
JF - Journal of Mathematical Sciences
SN - 1072-3374
IS - 2
ER -
ID: 45422742