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Analysis of guided waves with a nonlinear boundary condition caused by internal resonance using the method of multiple scales
Affiliation:1. Department of Physics, Ferdowsi University of Mashhad, 91775-1436 Mashhad, Iran;2. Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran;3. Institute for Metals Superplasticity Problems RAS, Khalturin Street 39, 450001 Ufa, Russia;4. National Research Tomsk State University, Lenin Ave. 36, 634050 Tomsk, Russia;1. Laboratoire Ondes et Milieux Complexes (LOMC) UMR CNRS 6294, Université Le Havre Normandie, France;2. Département de Mathématiques, Université de Lomé, Togo;1. Communications and Computer Research Center, Ferdowsi University of Mashhad, Mashhad, Iran;2. Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
Abstract:We theoretically investigated the cumulative nonlinear guided waves caused by internal resonance, using the method of multiple scales (MMS), which can construct better approximations to the solutions of perturbation problems. In this study, we consider nonlinearity only on the boundary instead of material nonlinearity or geometric nonlinearity. We showed nonlinear effects on the amplitudes of a lower mode and a higher mode depending on the propagation length. Also, we examined effects of wavenumber detuning from a phase matching condition of the two modes. If the wavenumber detuning is exactly equal to zero, the mechanical energy of the lower mode is transferred through nonlinear coupling to the energy of the higher mode, unilaterally. However, if a wavenumber detuning is not equal to zero, amplitude of the two modes change in a cyclic fashion during wave propagation. The amount of this amplitude variation and its cycle length are determined by the eigenfunctions of the two modes, the nonlinear parameter and the wavenumber detuning.
Keywords:Nonlinear guided wave  Lamb wave  Internal resonance  The method of multiple scales
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