Two-scale series expansions for travelling wave packets in one-dimensional periodic media

TL;DR

This paper develops a two-scale asymptotic expansion method to analyze wave packets in one-dimensional periodic media, capturing both microscale oscillations and macroscale variations, with applications demonstrated through elastic pulse propagation.

Contribution

It introduces a novel two-scale series expansion approach for wave packets in periodic media, integrating geometric optics and energy transport descriptions.

Findings

Derived recurrence relations for wave packet propagation

Validated theory with elastic pulse example in contrasting materials

Provided insights into microscale and macroscale wave interactions

Abstract

Starting from the wave equation for a medium with material properties that vary periodically, we study a system of recurrence relations that describe propagation of wave packets that oscillate on the microscale (i.e. on lengths of the order of the period of the medium) and vary slowly on the macroscale (i.e. on lengths that contain a large number of periods). The resulting equations contain a version of the geometric optics and the overall energy transport description for periodic media. We illustrate the developed asymptotic theory using the example of a point pulse propagating through a periodic arrangement of two materials with highly contrasting elastic moduli.