Revisiting Nesterenko’s solitary wave in the precompressed granular alignment held between fixed ends

Abstract

The response of a granular alignment held between fixed end walls to an impulse was introduced in the works of Nesterenko between 1983 and the years that followed. He showed analytically and experimentally that a granular chain admits a propagating solitary wave. In his analytic work, under small precompression compared to the local strain, he showed that one finds a propagating solitary wave. The solitary wave was also seen experimentally but at zero and vanishingly small precompressions. Under stronger precompression a possible Korteweg–de Vries (KdV) solitary wave was suggested though never observed. Later, others confirmed the solitary wave result at zero loading. Sen and Manciu reported seeing the solitary wave behavior in numerical simulations and in 2001 proposed an accurate solution which obtained the solitary wave at zero precompression as seen in some experiments and in numerics. Simulations showed an oscillatory tail following the solitary wave at small precompressions. In an experimental study in 1997, Costé, Falcon and Fauve and later Nesterenko et al. reported seeing propagation of a wave with an oscillatory tail. The oscillatory tail eventually consumed the solitary wave with increasing precompression. How can one reconcile Nesterenko’s solitary wave for the weakly precompressed system with Sen and Manciu’s solitary wave solution for zero precompression? Here we show that there is a separate solitary wave phase at a certain weak but finite loading regime which is distinct from Sen and Manciu’s work and this may be the reason why Nesterenko’s analytic theory seems to admit a solitary wave at finite loadings. We also offer insights into why the KdV solution is not seen.