Bands in the spectrum of a periodic elastic waveguide

TL;DR

This paper analyzes the spectral properties of a periodic elastic waveguide with thin connections, deriving asymptotic formulas for spectral bands and gaps as the ligament diameter approaches zero.

Contribution

It provides new asymptotic formulas for the spectral bands and gaps of a periodic elastic waveguide with thin ligaments as their diameter tends to zero.

Findings

Asymptotic formulas for spectral band positions

Asymptotic formulas for spectral gap positions

Characterization of the band-gap structure in the limit h → 0

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 $h \to 0$.