Waves in Media with Periodic Fine Structure

Abstract

For nonlinear waves passing through an elastic medium having periodic fine structure, a dynamic, nonlinear generalization of a homogenization procedure is outlined. The propagation equations obtained are nonlinear hyperbolic equations modified by terms involving higher derivatives. The derivation applies for waves having length scale significantly longer than that of the periodic structural cell. It is shown that the constitutive properties of the effective medium are determined from solution of certain canonical nonlinear problems having the same periodicity as the structural cell. Moreover, these canonical problems may be framed as variational problems. An explicit example for one-dimensional waves in a periodically layered medium is given.