The anti-localization of non-stationary linear waves and its relation to the localization. The simplest illustrative problem

TL;DR

This paper introduces anti-localization of non-stationary linear waves as asymptotic weakening of the propagating field near an inclusion due to destructive interference of pass-band harmonics. It analyzes a simple 1D model—a taut string on a Winkler foundation with a discrete mass–spring oscillator—solving via a Fourier integral $u(x,t)=\frac{1}{2\pi}\int \mathscr G(x,\Omega) e^{-i\Omega t}\,d\Omega$ and separating the stop- and pass-band contributions. Stationary-phase analysis yields $I^{\mathrm{pass}}+\mathrm{c.c.}=-\frac{A(w)}{\sqrt{t}}\cos(\sqrt{1-w^2}\,t+\frac{\pi}{4}+\psi)+O(t^{-3/2})$ with $w=|x|/t$, and crucially $A(0)=0$ for $K\neq M$, establishing anti-localization; a trapped-mode pole at $\Omega_0$ appears when $K<M$, producing localization in the stop-band. The paper clarifies the relation between anti-localization and the localization domain boundary $K=M$, discusses coexistence scenarios, and suggests applications to acoustic isolation and seismic protection in ordered media.

Abstract

We introduce a new wave phenomenon, which can be observed in continuum and discrete systems, where a trapped mode exists under certain conditions, namely, the anti-localization of non-stationary linear waves. This is zeroing of the non-localized propagating component of the wave-field in a neighbourhood of an inclusion. In other words, it is a tendency for non-stationary waves to propagate avoiding a neighbourhood of an inclusion. The anti-localization is caused by a destructive interference of the harmonics involved into the representation of the solution in the form of a Fourier integral. The anti-localization is associated with the waves from the pass-band, whereas the localization related with a trapped mode is due to poles inside the stop-band. In the framework of a simple illustrative problem considered in the paper, we have demonstrated that the anti-localization exists for all cases excepting the boundary of the domain in the parameter space where the wave localization occurs. Thus, the anti-localization can be observed in the absence of the localization as well as together with the localization. We also investigate the influence of the anti-localization on the wave-field in whole.

Figures (5)

• Figure 1: Comparing of the asymptotic solution for $u$ given by Eqs. \ref{['sum-I_1+I_2-single-wave']}--\ref{['psi']}, wherein $w=|x|/t$, and the corresponding numerical solution of Eq. \ref{['OSC-maineq-SPRING']} in the case $K>M$. One can observe the anti-localization near $x=0$.
• Figure 2: Comparing of the asymptotic solution for $u$ given by Eqs. \ref{['OSC-fr-root']}--\ref{['psi']}, wherein $w=|x|/t$, and the corresponding numerical solution of Eq. \ref{['OSC-maineq-SPRING']} in the case $K<M$. The anti-localization near $x=0$ co-exists with the localization, see plots for the propagating component \ref{['sum-I_1+I_2-single-wave']} and the localized one \ref{['S-MS-sol']}.
• Figure 3: The amplitude $A(w)$ defined by Eq. \ref{['sum-I_1+I_2-single-wave-amp']} for various system parameters (note that $A(1)\to\infty$ if $M=0$ and $A(1)=0$ otherwise.) In the case $K\neq M$ (see the solid lines) one can observe the anti-localization near $x=0$.
• Figure 4: Comparing the asymptotic solution for $u$ just at the inclusion in the form of the right-hand side of Eq. \ref{['S-MS-int2-fin']}, and the corresponding numerical solution of Eq. \ref{['OSC-maineq-SPRING']} in the case $K>M$.
• Figure 5: Comparing of the asymptotic solution for $u$ given by Eqs. \ref{['sum-I_1+I_2-single-wave']}--\ref{['psi']}, wherein $w=|x|/t$, and the corresponding numerical solution of Eq. \ref{['OSC-maineq-SPRING']} in the case $K=M\neq0$. There is no anti-localization near $x=0$.

Theorems & Definitions (2)

• Remark 1
• Theorem 1