Multimode long-wave approximation for a viscoelastic coating subject to antiplane shear

A general asymptotic approach involving multimode long-wave approximations is illustrated by a 2D time-harmonic scalar problem for the dynamic antiplane shear of a viscoelastic coating. For the first time, a 1D equation of motion with the coefficients depending on frequency parameters is derived. The associated dispersion relation also seems to be a fresh development approximating its exact counterpart near the vicinities of all the cut-off frequencies. As might be expected, the developed formulation is not valid for short wavelength patterns. At the same time, as it is shown for a delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-type loading, it proves to be robust for various scenarios dominated by long-wave response.