Waves in a Plane Rectangular Lattice of Thin Elastic Waveguides

Аннотация

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 μπ²h⁻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.