Parametric excitation of traveling waves in a circular non-dispersive medium

This paper discusses a special type of propagating waves created by parametric excitation in a circular taught string. The string, being a non-dispersive medium propagates deformations in a similar manner to electromagnetic waves in vacuum, both have simple wavelength-frequency relationship that play an important role here. Nonlinear equations are derived under the assumption of finite deformations, whose solution produces a square-wave like, limited-amplitude, traveling wave. Closed-form expressions are obtained for the parametric excitation characteristics of the nonlinear system and the steady-state traveling waves are described by a generalized eigenvalue problem. The latter relates the nonlinear elongation of the neutral axis to the participating wavelengths forming the propagating wave. Detailed numerical simulations are provided to validate the solution and to illustrate graphically the waveforms. It is shown that propagating sinusoidal parametric excitation gives rise to various square-wave like deformation shapes which a unique phenomenon is arising in non-dispersive media.