# Anti-plane surface waves in media with surface structure: discrete vs. continuum model

## TL;DR

This paper compares continuum and discrete models of anti-plane surface waves in elastic media with surface structures, highlighting their similarities and differences in dispersion relations and phase velocity behavior.

## Contribution

It introduces a discrete lattice model with surface particles and compares its dispersion relations to the continuum Gurtin-Murdoch model for surface elasticity.

## Key findings

- Both models support anti-plane surface waves.
- Phase velocity behavior can characterize surface shear modulus.
- Discrete and continuum models show notable similarities.

## Abstract

We present a comparison of the dispersion relations derived for anti-plane surface waves using the two distinct approaches of the surface elasticity vis-a-vis the lattice dynamics. We consider an elastic half-space with surface stresses described within the Gurtin-Murdoch model, and present a formulation of its discrete counterpart that is a square lattice half-plane with surface row of particles having mass and elastic bonds different from the ones in the bulk. As both models possess anti-plane surface waves we discuss similarities between the continuum and discrete viewpoint. In particular, in the context of the behaviour of phase velocity, we discuss the possible characterization of the surface shear modulus through the parameters involved in lattice formulation.