# Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids

## TL;DR

This paper generalizes the Zabolotskaya equation to all incompressible isotropic elastic solids, revealing the impossibility of linear polarization in nonlinear shear waves and analyzing harmonic generation in Gaussian beams.

## Contribution

It derives a new coupled nonlinear equation system for shear waves in all incompressible isotropic elastic solids, extending the Zabolotskaya equation and exploring polarization and harmonic phenomena.

## Key findings

- Coupling between anti-plane and in-plane motions cannot be removed in general.
- Pure shear beams generate only odd harmonics.
- Slight in-plane noise induces second harmonic generation.

## Abstract

We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible non-linear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarised) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarisation.