Power-law in depth-dependence of signal speed in vertical granular chain

The signal generated by an initial perturbation dispersively propagates in the vertical granular chain under gravity. For the power-law-type contact f orce, the signal speed follows power-law with the depth. When the perturbat ion is very weak, the exponent is 1/2(1 - 1/p). When the perturbation is ve ry strong, the exponent approaches zero. The transition of the exponent fro m oscillatory regime with weak nonlinearity to quasi-solitary regime with s trong nonlinearity is smooth.