Localized Modes in a 1D Harmonic Crystal with a Mass-Spring Inclusion

Abstract

The spectral problem concerning the existence of localized modes of oscillation in 1D harmonic crystal with a single mass-spring inclusion is investigated. A crystal is an infinite harmonic chain of particles with nearest-neighbor interaction. The bond stiffnesses are referred to as “springs”. Two types of inclusion are considered, namely, a symmetric and an asymmetric ones. The symmetric inclusion consists of the particle of an alternated mass with two springs of alternated stiffnesses attached. The asymmetric inclusion consists of the particle of an alternated mass with one alternated spring attached. Outside the inclusion the chain is assumed to be uniform. For both types of a mass-spring inclusion, the necessary and sufficient conditions for the existence of localized modes, as well as the corresponding frequencies of localized oscillation, are found.

Appendix

Here we discuss the dispersion relation and the Green function in the frequency domain for the uniform chain. Assuming the solution to be in the form of Eqs. (25.4), (25.6) we get the dispersion relation for a uniform chain corresponding to the one described by Eq. (25.1):

$$\begin{aligned} \Omega ^2=4\sin ^2\frac{q}{2}\equiv 2(1-\cos q), \end{aligned}$$

(25.76)

where $\Omega \in \mathbb {R}$ is the frequency, q is the wave-number. The detailed analysis of the dispersion relation for a uniform chain is given, for example, in Shishkina and Gavrilov (2023).

The whole frequency band $\Omega \in \mathbb R$ can be divided to the pass-band, where

$$\begin{aligned} &\qquad \Omega ^2 < \Omega _{*}^{2}\equiv 4, \end{aligned}$$

(25.77)

$$\begin{aligned} q&=\pm \arccos \frac{2-\Omega ^2}{2}, \end{aligned}$$

(25.78)

i.e., the corresponding wave-numbers $q(\Omega )$ are reals, and the stop-band, where

$$\begin{aligned} \Omega ^{2}&>\Omega _*^2\equiv 4 ,\end{aligned}$$

(25.79)

$$\begin{aligned} q=\pi \pm {\text {i}} {\text {arccosh}}\frac{1}{2}(\Omega ^2-2) = \pi & \pm \textrm{i}\ln \left( \frac{1}{2}(\Omega ^2-2) +\sqrt{\frac{1}{4}(\Omega ^2-2)^2-1} \right) , \end{aligned}$$

(25.80)

i.e., the corresponding wave-numbers are imaginary. Here

$$\begin{aligned} \Omega _*{\mathop {=}\limits ^{\text{ def }}}2 \end{aligned}$$

(25.81)

is the cut-off (or boundary) frequency, which separates the bands.

The Green function in the frequency domain for the corresponding uniform chain in the stop-band is Montroll and Potts (1955), Shishkina and Gavrilov (2023)

$$\begin{aligned} G_n(\Omega )=\frac{(-1)^{|n|}2^{|n|}}{\Phi ^{|n|-1}(\Omega )((-\Omega ^2+2)\Phi (\Omega )+4)}, \end{aligned}$$

(25.82)

where

$$\begin{aligned} \Phi (\Omega ){\mathop {=}\limits ^{\text{ def }}}\Omega ^2-2+|\Omega |\sqrt{\Omega ^2-4}. \end{aligned}$$

(25.83)