Steady Shock Profile in a One-Dimensional Lattice

Abstract

The equations of motion of a one-dimensional lattice of mass points connected by nonlinear springs are set forth and compared with the equations of the corresponding continuum. A permanent regime for the damped lattice is obtained by series approximation and shown to agree with that of the continuum. A higher approximation leads to a permanent regime profile for the undamped lattice which oscillates steadily after shock arrival. This is shown to be in qualitative accord with the results of numerical integrations of the transient problem. However, comparison of periods of steady oscillation with those obtained in the transient problem indicate that the series approximation to the permanent regime is quantitatively unsatisfactory, though qualitatively correct. Scaling of the problem with a parameter u1α is noted, where u1 is steady particle velocity behind the shock and α is a parameter of nonlinearity.