Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions

Abstract

We consider an acoustic waveguide (the Neumann problem for the Helmholtz equation) shaped like a periodic family of identical beads on a thin cylinder rod. Under minor restrictions on the bead and rod geometry, we use asymptotic analysis to establish the opening of spectral gaps and find their geometric characteristics. The main technical difficulties lie in the justification of asymptotic formulas for the eigenvalues of the model problem on the periodicity cell due to its arbitrary shape.

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

“…(1. 15) Taking (1.15) into account, we apply the partial Gelfand transform [9], see also [10] and [11, §3.4],…”

“…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].…”

Bands in the spectrum of a periodic elastic waveguide

“…Lemma on near eigenvalues and eigenvectors. We shall need in Section 4.3 the following operator theoretic result in the form given in [1]; see also [4] or [33] for more simple formulations corresponding to n = 1, γ = 0 and t = τ . Lemma 4.2.…”

Surface waves in a channel with thin tunnels and wells at the bottom: Non-reflecting underwater topography

“…1.2, b), formulations with different methods and goals, see e.g. [11,22,30,31,40,12,34,16,17,18,3]. In particular, complete asymptotic expansions of eigenvalues and eigenfunctions were constructed in [12] by the methods of matched asymptotic expansions.…”

“…[30,31]). Besides, the derivation of asymptotically sharp estimates becomes much more complicated in the curved case, see [40,3] for details. However, the ligaments connecting massive domains are always assumed to be straight.…”