Z. Angew. Math. Phys. (2017) 68:102
2017 Springer International Publishing AG
published online August 16, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Bands in the spectrum of a periodic elastic waveguide
F. L. Bakharev and J. Taskinen
Abstract. We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence
of identical bounded cells connected by thin ligaments of diameter of order h>0. The essential spectrum of the problem
is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as
h → 0.
Mathematics Subject Classiﬁcation. Primary 35J57; Secondary 35P99, 35Q74, 74J05.
Keywords. Elliptic system, Linearized elasticity problem, Essential spectrum, Spectral band, Spectral gap, Asymptotic
analysis, Floquet–Bloch theory.
We study the essential spectrum of the linearized elasticity system with traction-free boundary conditions
in unbounded periodic waveguides denoted by Π
. The waveguide (see Fig. 1) consists of inﬁnitely many
identical, translated bounded cells connected with small cylindrical ligaments, the length and radius
of cross section of which are both proportional to a small parameter h>0 so that the volume of
the ligament is O(h
). A number of papers (e.g., [3,33,34]) has been devoted to geometrically similar
waveguides consisting of arrays of macroscopic cells connected with thin structures, and, using rigorous
perturbation arguments, the existence of gaps in the essential spectra has been detected. This has been
done for elliptic boundary problems in elasticity, linear water wave theory, piezoelectricity etc.
In this geometric setting, the emerging of gaps is explained by that for small h, the problem can be
seen as a perturbation of a “limit spectral problem” (h = 0) on a bounded domain, which consists of a
. The spectrum of such a problem is in general a sequence (λ
of eigenvalues, and the
spectral bands of the original problem are situated “close” to the eigenvalues λ
. To analyze this closeness
and its dependence on h becomes a mathematical challenge: If that can be done accurately enough, one
ﬁnds that disjoint eigenvalues correspond to spectral bands with a gap in between. In fact, this scheme
can work only for a ﬁnite number of the lowest eigenvalues in the sense that for each ﬁxed small h,at
most a ﬁnite number of gaps can be found.
The purpose of this work is to reﬁne the existing results by proving more accurate estimates than
before for the end-points of spectral bands and gaps. For example, in , it was shown that for small
h and also k,thekth spectral band is situated within a distance Ch from λ
,thekth eigenvalue of
the limit problem, where C>0 is a constant depending on the shape of the cells and on the physical
constants of the elastic material, but not on the size of the small ligaments. In comparison, we shall ﬁnd
here an asymptotic formula for the spectral bands Υ
. Namely, by the Floquet–Bloch theory of periodic
problems, the bands are formed by eigenvalues Λ
(η) of the “model problem” depending on the parameter
F. L. Bakharev was supported by the St. Petersburg State University Grant 184.108.40.2062, by the Chebyshev
Laboratory—RF Government Grant 11.G34.31.0026, and by JSC “Gazprom Neft”. J. Taskinen was supported by Grants
from the Magnus Ehrnroot Foundation and the V¨ais¨al¨a Foundation of the Finnish Academy of Sciences and Letters.