Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Abstract

Transverse elastic waves behave differently in nonlinear isotropic and anisotropic media. While in the former the quadratically nonlinear coupling in the evolution equations for wave amplitudes is not possible, such a coupling may occur for certain directions in anisotropic materials. We identify the expression responsible for the coupling and we derive coupled canonical evolution equations for transverse wave amplitudes in the case of the two-fold and three-fold symmetry acoustic axes. We illustrate our considerations by examples for a cubic crystal.

Degenerate weakly non-linear elastic plane waves

“…We illustrate these general statements with some examples of particular elastic materials. Some of these results appeared previously in an abbreviated form (Domanski and Norris, 2008).…”

The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

“…Our aim in this work is to derive equations similar to those in [2] but by a different method. Here, unlike the presentation in [2] (see also [3][4][5][6][7][8]), where the method of weakly nonlinear geometric optics was used, we will apply a double-scale expansion. Instead of introducing a new fast variable and applying geometric optics-type asymptotics, which leads to evolution equations with three independent variables, we introduce here just two new independent variables: a slow time variable τ and a characteristic variable θ .…”