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

The wave propagation problem in a second-order nonlinear PDE, with specific reference to a mechanical system consisting of a taut cable, or string, resting on a piecewise linear foundation 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 theoretically and numerically, also by using return maps in phase space. (C) 2022 Elsevier Inc. All rights reserved.