Power-Law Behaviors in Nonlinearly Coupled Granular Chain under Gravity

We find power-law behaviors of grain velocity in both propagation and backscattering in a gravitationally compacted granular chain with nonlinear contact force. We focus on the leading peak of the velocity signal which decreases in a power-law $d^{-\alpha}$, where $d$ is the location of the peak, as the signal goes down. The ratio of backscattered to incident leading velocity also follows a power-law $d_i^{-\beta}$, where $d_i$ is the depth of impurity. The up-going backscattered signal is nearly solitary. Therefore, the overall change of the leading velocity peak is given by the power-law $d_i^{-(\alpha+\beta)}$. We find $\alpha=0.250$ and $\beta=0.167$ for the Hertzian contact force.