Asymptotic Analysis of Periodic Structures

Abstract

This chapter discusses the asymptotic problems in periodic structures involving wave propagation when the typical wavelength of the motion is comparable to the period of the structure and both are small. The chapter begins with a review on some notions and conventions regarding scaling. It presents a general introduction to the WKB or geometrical optics methods. The chapter examines diffraction effects in a very indirect way. It also presents some results on the asymptotic simplification (order reduction) of hyperbolic systems. The chapter analyzes the spectral properties of differential operators with periodic coefficients and presents the so called Bloch expansion theorem. The Bloch expansion is used directly to solve several problems and analyze their asymptotic behavior.