Shock propagation in a granular chain

The study of granular matter is of interest for fundamental physics as well as for technological applications. The intensive studies of liquids and solids gave physicists powerful tools for investigating these states of matter. These tools are, however, very difficult to apply to granular matter @1#. Little work has been done on sound propagation in real granular media. Experiments have been performed by Liu and Nagel @2–4# who investigated the propagation of low amplitude vibrations in a box of spherical beads. It was concluded that nonlinearity and disorder make wave propagation behave in very unexpected ways: the fragile structure of contacts between grains makes them sensitive to rearrangements, dramatically changing the amplitude response of the receptor to the source. Besides, disorder gives rise to important interference effects that can lead to localization @5#. Numerical simulations have been made by Melin @6#, who studied the depth dependence of sound speed in granular media. The results obtained were different from the power law behavior between sound speed and pressure predicted by Goddard @7# in an effective medium calculation. Three-dimensional models seem to be very difficult to investigate directly and we need to begin with simplified models in order to isolate specific features of real granular media. In this paper we try to understand the effect of nonlinearity on wave propagation in one dimension. A previous study was done by Nesterenko @8# on a chain of spherical beads obeying the elastic Hertz law of contact. It was shown both analytically and numerically that the chain submitted to compression pulses involves solitary wave propagation. This has been confirmed by the experiments of Lazaridi and Nesterenko @9# and of Coste, Falcon, and Fauve @10#. Here, we want to study the case of a chain of beads initially in contact and submitted to a shock perturbation. The beads interact via an elastic contact law only when they are compressed. The problem is therefore highly nonlinear because as soon as there are broken contacts the chain behaves as an ensemble of independent elastic systems. We can look at this problem even in terms of a linear chain of points connected by springs with completely asymmetric elastic