Anderson Localization in Disordered Vibrating Rods

TL;DR

This study investigates Anderson localization in disordered elastic rods through experiments and simulations, revealing exponential wave localization, frequency-dependent localization length, and connections to Random Matrix Theory.

Contribution

It provides the first combined experimental and numerical analysis of Anderson localization in torsional waves of disordered rods, linking localization length to spectral statistics.

Findings

Wave amplitudes are exponentially localized.

Localization length decreases with frequency.

Level statistics show level repulsion varying with frequency.

Abstract

We study, both experimentally and numerically, the Anderson localization phenomenon in torsional waves of a disordered elastic rod, which consists of a cylinder with randomly spaced notches. We find that the normal-mode wave amplitudes are exponentially localized as occurs in disordered solids. The localization length is measured using these wave amplitudes and it is shown to decrease as a function of frequency. The normal-mode spectrum is also measured as well as computed, so its level statistics can be analyzed. Fitting the nearest-neighbor spacing distribution a level repulsion parameter is defined that also varies with frequency. The localization length can then be expressed as a function of the repulsion parameter. There exists a range in which the localization length is a linear function of the repulsion parameter, which is consistent with Random Matrix Theory. However, at low values of the repulsion parameter the linear dependence does not hold.