On Boussinesq-type models for long longitudinal waves in elastic rods

TL;DR

This paper derives and compares two Boussinesq-type models for long nonlinear longitudinal waves in elastic rods, incorporating effects like pre-stretch and boundary loading, supported by experimental insights.

Contribution

The paper introduces two new forced Boussinesq-type models for elastic rods, extending previous derivations to include boundary effects and pre-stretch conditions.

Findings

Comparison of linear dispersion curves for the models

Analysis of solitary wave solutions

Potential application to laboratory experiments

Abstract

In this paper we revisit the derivations of model equations describing long nonlinear longitudinal bulk strain waves in elastic rods within the scope of the Murnaghan model in order to derive a Boussinesq-type model, and extend these derivations to include axially symmetric loading on the lateral boundary surface, and longitudinal pre-stretch. We systematically derive two forced Boussinesq-type models from the full equations of motion and non-zero surface boundary conditions, utilising the presence of two small parameters characterising the smallness of the wave amplitude and the long wavelength compared to the radius of the waveguide. We compare the basic dynamical properties of both models (linear dispersion curves and solitary wave solutions). We also briefly describe the laboratory experiments on generation of bulk strain solitary waves in the Ioffe Institute, and suggest that this generation process can be modelled using the derived equations.