Scattering of Waves by Impurities in Precompressed Granular Chains

TL;DR

This paper analyzes wave scattering in precompressed granular chains with impurities, deriving explicit formulas for reflection and transmission, identifying resonance phenomena, and validating findings through experiments and simulations.

Contribution

It provides a closed-form analytical framework for wave scattering in granular chains with impurities, including resonance tuning and experimental validation.

Findings

High-frequency waves are strongly attenuated by single impurities.

Double impurities can produce a resonance enabling full transmission.

RT resonance can be tuned across the pass band and leads to reflectionless modes.

Abstract

We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the reflection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber $k\rightarrow \pm\pi$), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance --- enabling full transmission at a particular frequency --- in a manner that is analogous to the Ramsauer--Townsend (RT) resonance from quantum physics. We also demonstrate that one can tune the frequency of the RT resonance to any value in the pass band of the host chain. We corroborate our theoretical predictions both numerically and experimentally, and we directly observe complete transmission for frequencies close to the RT resonance frequency. Finally, we show how this RT resonance can lead to the existence of reflectionless modes even in granular chains (including disordered ones) with multiple double impurities.