Periodic traveling waves in a taut cable on a bilinear elastic substrate

TL;DR

This paper derives closed-form solutions for periodic traveling waves on a taut cable over a bilinear elastic substrate, analyzing their dependence on substrate stiffness and stability through analytical and numerical methods.

Contribution

It provides explicit solutions for wave phase velocity and form in a bilinear substrate system, highlighting the effects of substrate stiffness ratios and rigidity on wave propagation.

Findings

Wave solutions depend solely on the ratio of soil stiffnesses.

No wave propagation occurs if one substrate side is rigid.

Numerical simulations confirm analytical results and stability analysis.

Abstract

The wave propagation problem on a taut cable resting on a bilinear substrate is investigated, without and with a distribute transversal load. The piecewise nature of the problem offers a sufficiently simple kind of nonlinearity as to permit a closed form solution both for the wave phase velocity and the wave form. We show that the solution depends only on the ratio between the two soil stiffnesses, and that no waves propagate if one side of the substrate is rigid. Some numerical simulations, based on a finite difference method, are performed to confirm the analytical findings. The stability of the proposed waves is discussed analytically and numerically.