Simple waves in Hertzian chains

The discrete system of equations for a chain consisting of a large number of spheres interacting via the Hertz force of index 3/2 in strain is examined in the very long wavelength limit, yielding an effective medium description. The resulting continuum second-order equation of motion possesses a subset of simple waves obeying a first-order equation of reduced index 5/4. These simple waves appear not to have examined before. For a given initial strain, the simple wave solution prescribes initial sphere centroid velocities. Together the initial strain and velocities are used in the second-order discrete system. Results for shock wave development compare very well between the second-order discrete system (minus physically valid oscillations) and the reduced first-order equation. A second-order simulation of colliding waves examines the ability of waves to pass through each other, with a phase advance accruing during the collision process. An arbitrary initial condition is shown to evolve toward a universal similarity solution proportional to (x/t)(4). A closed-form solution is given including the complete history of the waveform, shock location, and amplitude.