We present detailed numerical studies on the motion of an initial perturbation in a chain of spheres which are characterized by Hertzian contacts. We consider the propagation of the perturbation in the presence and in the absence of gravitational compaction in the chain [see R. S. Sinkovits and S. Sen, Phys. Rev. Lett. 74, 2686 (1995)]. Our results show that robust solitonlike pulses carry the initial perturbation from the surface to the depths of the Hertzian chains for various magnitudes of the initial perturbation at finite loadings of the column. In addition, we probe the structural characteristics of the solitonlike pulse as a function of the magnitude of the initial impact, of loading, and of the gravitational field. Our results suggest that the solitonlike features, while altered, persist at finite gravitational fields when the compaction of grains with increasing depth becomes important. In the presence of light mass impurities in the Hertzian chain, we present evidence of backscattering of the solitonlike pulse and suggest that acoustic backscattering can be a possible probe of buried light mass impurities in granular beds. In closing, we discuss the propagation of perturbations in finite Hertzian chains when the surface grain is coupled to a transducer which vibrates with very low amplitude and frequency, and we report the formation of metastable standing-wave-like patterns in finite Hertzian chains.

The equation of motion for a bead in a chain of uncompressed elastic beads in contact that interact via the potential $V\left(\delta \right)\sim {\delta }^n,$ $n>\mathrm{}2,$ δ being overlap, supports solitary waves and does not accommodate sound propagation [V. Nesterenko, J. Appl. Mech. Tech. Phys. 5, 733 (1983)]. We present an iteratively exact solution to describe the solitary wave as a function of material parameters and a universal, infinite set of coefficients, which depend only on n. We compute any arbitrary number of coefficients to desired accuracy and show that only the first few coefficients of our solution significantly improves upon Nesterenko’s solution. The improved solution is a necessary step to develop a theoretical understanding of the formation of secondary solitary waves [M. Manciu, et al., Phys. Rev. E 63, 011614 (2001)].