The Use of Generalized Zakharov Systems in Elastic Surface Waves

Abstract

In recent works^{1, 2} the possible propagation of envelope solitons having the nature of shear-horizontal (SH) elastic surface waves was unequivocally proved mathematically for an elastic structure likely to be experimentally tested. This phenomenon may occur along a composite structure made of a nonlinear elastic isotropic substrate coated with a thin “slow”linear elastic film. Couples of materials for which this is indeed realizable were also determined. However, a strong decoupling hypothesis layed dormant in that approach. Namely, it was assumed that the SH wave in question remains decoupled from the so-called Rayleigh component, i. e., that vectorial elastic component that is polarized parallel to the sagittal plane (plane spanned by the direction of propagation X₁ and the normal to the limiting surface N̰). This was considered in order to simplify the analysis, but it does not hold true in all rigor. A simple way to realize this fact is to recall what happens for bulk waves in nonlinear isotropic (a fortiori anisotropic) elasticity (See Ref. 3, pp. 36–37). In that theory a longitudinal motion necessarilly accompanies a transverse motion; e. g., one can write

$$ \left. {\begin{array}{*{20}{c}} {{\square _T}\;v = 0,} \\ {{\square _L}\;u = \gamma {v_x}{v_{xx}},} \end{array}} \right\} $$

((1.1))