Higher Order Continuum Wave Equation Calibrated on Lattice Dynamics

TL;DR

This paper extends the classical lattice dynamics continuum approach by incorporating higher order and nonlinear effects, deriving more accurate wave equations that match lattice responses in one-dimensional systems.

Contribution

It introduces higher order and nonlinear terms into continuum wave equations, calibrated on lattice dynamics, to improve modeling accuracy for non-local and non-linear effects.

Findings

Derived higher order dispersion coefficients.

Compared continuum solutions with lattice response.

Validated accuracy in one-dimensional systems.

Abstract

The classical approach to linking lattice dynamics properties to continuum equations of motion, the "method of long waves," is extended to include higher order terms. The additional terms account for non-local and non-linear effects. In the first part of the article, the derivation is made within the harmonic approximation for the perfect lattice response. Higher order terms are included in the continuum equation of motion to account for non-linear dispersion effects. Wave propagation coefficients as well as fourth order dispersion coefficients are obtained. In the second part, the lattice anharmonicity is considered and nonlinear macroscopic equations of motion are obtained within the local approximation. Both continuum solutions are particularized to the one-dimensional case and are compared with the lattice response in order to establish the accuracy of the approximation.