Perturbation solution for growth of nonlinear shock waves in a lattice

Abstract

A perturbation solution is obtained for the propagation of longitudinal shock waves in a one-dimensional lattice with a velocity step applied to the first mass. The nonlinear part of the elastic interaction force is assumed to be of parabolic form. The growth of particle velocity at the head of the wave and the noticeable high-frequency contributions travelling behind, that arise due to nonlinearity, have been observed previously in numerical studies and are found also in the perturbation solution. Within the range of validity of the solution, the nonlinear growth of maximum particle velocity at the head of the shock wave is shown to increase with distance into the lattice as the two-thirds power.