Nonlinear interaction of co-directional shear horizontal waves in a two-layered elastic medium

The nonlinear interaction of co-directional shear horizontal (SH) waves propagating in a two-layered plate of uniform thickness is considered. It is assumed that the constituent materials of both layers are nonlinear, homogeneous, isotropic and incompressible elastic and having different mechanical properties. It is also assumed that between the linear shear velocities of the top layer, c₁, and the bottom layer, c₂, the inequality c₁< c₂ is valid. For the existence of a SH wave, the phase velocity of the wave, c, must satisfy either the inequality c₁< c₂≤ c or the one c₁< c≤ c₂ and here the problem is examined under the second condition. By employing an asymptotic perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the first order slowly varying amplitudes of interacting waves are governed asymptotically by a coupled nonlinear Schrödinger (CNLS) equations. Then the effects of nonlinearity of the materials and the ratio of layers’ thickness on the linear instabilities of solutions of the CNLS equations and the existence of solitary wave solutions of the CNLS are examined.