Nonlinear long-wave models for imperfectly bonded layered waveguides

We propose a composite lattice model for describing nonlinear waves in a two-layer waveguide with adhesive bonding. We first consider waves in an anharmonic chain of oscillating dipoles and show that the corresponding asymptotic long-wave model for longitudinal waves coincides with the Boussinesq-type equation previously derived for a macroscopic waveguide using the nonlinear elasticity approach. We also show that in this model, there is no simple analogy between long longitudinal and long flexural waves. Then, for a composite lattice, we derive two new model systems of coupled Boussinesq-type equations for long nonlinear longitudinal waves and conjecture that a similar description exists in the framework of dynamic nonlinear elasticity.