Far-field analysis of nonlinear shock waves in a lattice

Abstract

An analysis is made of the far-field behavior of a longitudinal shock wave propagating in a one-dimensional semi-infinite elastic 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. Using the method of matched asymptotic expansions and stretched coordinates, for small nonlinearity, the original lattice equations of motion are replaced by a partial differential equation that governs the slow evolution of the wave with distance. A numerical solution of the partial differential equation shows the transition as the wave propagates from an oscillatory shock profile in the near field to a sequence of solitary waves in the far field of the lattice. The individual solitary waves at the head of the pulse eventually achieve essentially the same steady maximum amplitude and the same steady form, while continuing to spread apart from each other. The maximum particle velocity for the far-field solitary waves is found to be approximately 1.76 times as great as the velocity applied to the first mass. For the first-order nonlinear theory that is derived, the existence of a maximum amplitude of particle velocity that is found to be steady with distance as the wave propagates also implies that amplitude is independent of the magnitude of the nonlinearity, so long as the nonlinearity is small, but nonzero.