Propagation of mechanical waves in a one-dimensional nonlinear disordered lattice

Abstract

The propagation of transverse waves along a string loaded by masses, each of them being fixed to a spring with a quadratic nonlinearity, is studied. After presenting the nonlinear model and stating the equation of propagation into a lattice with discrete nonlinearities and disorder, we propose a perturbation approach to wave propagation in a nonlinear lattice using the Green's function formalism. We show how the nonlinearity acts on the propagation into a disordered lattice. In the low-frequency approximation, an analytical expression of the boundary between the propagative regime and the evanescent one is found. Numerical results are compared to the analytical results and phase diagrams are proposed in the ordered and disordered cases. A behavior of the transmission coefficient is found, on an empirical basis, as a function of the length of the lattice and the localization length in the nonlinear case. Finally, a dynamic approach is developed and the ordered and disordered cases are addressed. This method is based on a finite difference equation and allows the construction of the Poincaré section describing the propagation of the wave into the lattice. This approach distinguishes between the properties of propagation in the lattice in a propagative regime and in an evanescent one.

Group velocity and energy transport velocity near the band edge of a disordered coupled cavity waveguide: an analytical approach

“…In the first case we expect an increase in v E (super-diffusive regime), whereas in the second case v E should decrease (sub-diffusive regime). We would like to stress that nonlinear effects profoundly modify the role played by the disorder as, for instance, simulated in [38] with a mechanical model based on a linear chain of masses linked by springs with a quadratic nonlinearity. In particular the nonlinearity can lead to an effective smoothing of the disordered dielectric potential, in analogy to the transition from Bose-glass phase to superfluid phase as predicted in the Bose gas [39].…”

Low-frequency band gaps in chains with attached non-linear oscillators

“…For given values of the amplitudes B 1,j and A 1,j of mass j and the corresponding attached oscillator, the value of µ u can be calculated using (14).…”

“…Localisation phenomena can also be observed in perfectly periodic non-linear structures [8][9][10]14], where spring-mass chains are studied and the non-linear behaviour is introduced either in the spring between the two neighbour masses, or by adding non-linear springs between the ground and the masses. The applications of these filtering phenomena are mainly in the high frequency range, as the distance between the inclusions has to be comparable with the wave length.…”

Green's function formulation of multiple nonlinear Dirac δ-function potential in one dimension

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