Burgers's-Korteweg-De Vries equation for viscoelastic rods and plates

Abstract

A perturbation scheme is developed to study the far-field structure of waves in cylinders and plates. There exist four small parameters in the problem: the first gives waves of small but finite amplitude; the second is the reciprocal of a Reynolds number and represents viscous effects; the third is the ratio of the radius of the cylinder to an axial length, or of the thickness of the plate to an in-plane length, and represents dispersion; the last parameter is used to stretch time so that the description sought is an asymptotic one. The highest order problem leads to known longitudinal waves in rods and extensional waves in plates. The terms next in order lead to the equation describing the structure of these waves, its generality depending on the similarity hypothesis made. The most general result is a generalization to any number of dimensions of the B.K.D.V. equation, obtained by giving equal weightage to the four small parameters. Some self-similar solutions of these are discussed which exhibit the structure of these waves.