Effects of Rayleigh Waves on the Essential Spectrum in Perturbed Doubly Periodic Elliptic Problems

Abstract

We give an example of a scalar second order differential operator in the plane with double periodic coefficients and describe its modification, which causes an additional spectral band in the essential spectrum. The modified operator is obtained by applying to the coefficients a mirror reflection with respect to a vertical or horizontal line. This change gives rise to Rayleigh type waves localized near the line. The results are proven using asymptotic analysis, and they are based on high contrast of the coefficient functions.

The band-gap structure of the spectrum in a periodic medium of masonry type

“…Since the right-hand-side of (29) includes the small coefficient h α A , the passing to the limit h → 0 turns Σ h into Σ 0 = ∂Ω 0 (see (1)) and leads to the limit problem…”

“…The spectral bands (18) may overlap, but between them there can also exist gaps, i.e., nonempty open intervals γ h m =]β h m , β h m+1 [ which are free of the spectrum. In the paper [20] it was proved that the spectrum of the problem (4) has at least one open gap of width O (1). In what follows, we will describe the asymptotic structure of the low-frequency range {λ h ∈ σ h : λ h const} of the spectrum (17).…”

“…A similar problem was considered in [28], and there it was in addition shown that the number of open gaps grows unboundedly when ε → +0. The subdomain ω is fixed in the definition (25), but the papers [20,1] deal with the situation described in Section 1.1: the subdomain (1) of the high contrast covers the whole periodicity cell Ω h in the limit h → +0.…”

“…In the present paper we employ an asymptotic method which differs quite significantly from the analysis used in [6,7,28,22,2] and [20,1].…”

The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions

“…In [14] examples of arbitrarily many non-empty spectral bands for the Dirichlet Laplacian in a double-periodic perforated plane are given. Investigation of the spectral bands for other geometries of double-periodic two-dimensional structures are performed in [15] and [16].…”