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.


  1. Klein–Gordon equation
  2. nonsmooth systems
  3. periodic traveling waves

MSC codes

  1. 35B10
  2. 74J30

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1 - 24
ISSN (online): 1095-712X


Submitted: 15 June 2022
Accepted: 5 October 2022
Published online: 25 January 2023


  1. Klein–Gordon equation
  2. nonsmooth systems
  3. periodic traveling waves

MSC codes

  1. 35B10
  2. 74J30



DIISM, Università Politecnica delle Marche, Ancona, Italy.
DICEA, Università Politecnica delle Marche, Ancona, Italy.

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