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.
Whitham’s variational method for nonlinear dispersive wave propagation is used to obtain the nonlinear dispersion relation between frequency, wavenumber, and amplitude in a monatomic chain of particles. The interactive force in the chain is of third degree in the relative displacement of nearest neighbours. The frequency is shown to be larger than its infinitesimal amplitude value at low wavenumbers and less at high wavenumbers, changing sign at a critical frequency which depends on the force constants of the lattice. The periodic dependence of frequency upon wavenumber is preserved, but the cutoff frequency diminishes with increasing amplitude. The dilatation or expansion of the lattice is found always to be positive and it increases with wavenumber and amplitude. The Gruneisen ratio, i. e. the relative rate of change of frequency with respect to dilatation, depends on wavenumber and force constants. The formula reproduces the value given by quasilinear theory at the edge of the Brillouin zone, but predicts a sharp decrease with decreasing wavenumber, leading even to negative ratios under some circumstances. Finally, the space-time evolution of modulation perturbations on the nonlinear uniform wave is analyzed on the basis of Karpman’s and Krushkal’s theories. The linear chain is found to be mechanically unstable for wavenumbers less than a wavenumber
K
0
defined by the wavenumber
k
0
of the uniform wave and the force constants, but the instability may or may not be severe, since the amplitude of the perturbation is self-limiting.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.