Power Laws in Nonlinear Granular Chain under Gravity

The signal generated by a weak impulse propagates dispersively in a gravitationally compacted granular chain. For a power-law-type contact force, we show analytically that the type of dispersion follows power laws in depth. The power law for a grain displacement signal is given by h^-(1/4)[1-(1/p)] where h and p denote depth and the exponent of the contact force, respectively, and the power law for the grain velocity is h^-(1/4)[(1/3)+(1/p)]. Other depth-dependent power laws for oscillation frequency, wavelength, and period are given by combining the above two expressions and the phase velocity power law h^(1/2)[1-(1/p)]. We verify these power laws by comparing with the data obtained by numerical simulations.