Wave Propagation in a String Resting on a General Nonlinear Substrate

Traveling waves propagating on a taut cable resting on an elastic substrate are investigated by an equivalent mechanical model based on the classical Klein–Gordon equation. The formulation is devised for an elastic response of generally arbitrary shape, and permits one to compute the propagation wave velocity without solving the equation of motion, thus providing a unified theoretical framework for a large class of response functions, linear or nonlinear, smooth or nonsmooth. The general solution is then applied to the cases of a general polynomial substrate, a bilinear substrate, a bilinear substrate with a cubic correction, and a negative linear stiffness substrate, all of them falling within the realm of nonsmooth systems when a piecewise continuous stiffness is chosen; in the second one, we recover results present in the literature and obtained by employing a method based on matching conditions, which are spared in the approach used in this paper. Finally, the application to the sine-Gordon equation is considered.