Weakly nonlinear simple waves in Hertzian chains

The discrete system of equations for a granular chain consisting of a large number of spheres interacting via the hertz force is cast as an effective medium. In the long wavelength limit, the second order equation of motion for the effective medium possess a subset of simple waves obeying a first order equation of reduced nonlinear index. Simple waves are those in which knowledge of one dependent variable determines all the rest. For a given initial strain, the simple wave solution prescribes initial mass velocity. Strain and velocity profiles from the first order equation are used as initial conditions in simulations for the second order discrete system. Results for viscous and inviscid shock formation compare very well between the second order system and the reduced first order equation. Second order simulation of colliding waves reveals the ability of waves to pass through each other, with a phase advance accruing during the collision process. Results may be related to explosions in granular media. [Work supported by the Office of Naval Research.]