"Scattering of Waves by Impurities in Precompressed Granular Chains"

You may have heard of the Ramsaur–Townsend resonance from scattering problems. The simplest version of it (by examing a particle in a square well) is one of the canonical textbook problems in quantum mechanics.

This is one of the important effects illustrating the need for a notion of wave mechanics.

It turns out that one can also get an RT resonance in a macroscopic, classical system.

The discovery of this classical RT effect (in granular crystals) is the subject of a paper by my collaborators and me, out in final form in Physical Review E today. Here are the details of the article.


Title: "Scattering of Waves by Impurities in Precompressed Granular Chains"

Authors: Alejandro J. Martínez, Hiromi Yasuda, Eunho Kim, P. G. Kevrekidis, Mason A. Porter, and Jinkyu Yang

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 → ±π), 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 almost 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 in granular chains (including disordered ones) with multiple double impurities.