On the oscillatory and mean motions due to waves in a thin viscoelastic layer

A perturbation analysis based on equations of motion in Lagrangian form is presented for the oscillatory and mean motions induced by a forced periodic wave propagating in a thin layer of viscoelastic material. The material is assumed to be a Voigt body, for which the constitutive relation is a linear combination of viscous and elastic parts. In this work, the elastic part of the stress is assumed to be a linear function of the Lagrangian deformation tensor. The aim is to show a proper approach of carrying out the analysis to the second order in order to determine the mean deformation undergone by the material. This approach is in sharp contrast to the previous studies, which have mistakenly applied the complex viscoelastic parameter to the second order and assumed the mean motion to be a steady Lagrangian drift. It is shown here that for a sufficiently soft and viscous material the mean motion is actually a creeping motion that slowly dies out as a limit of finite deformation is approached. Numerical results are also generated to illustrate the combined effects due to viscous damping and elasticity on the first-order oscillatory and the second-order mean displacements of particles in the material.