Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

Abstract

This article explores the deep connections that exist between the mathematical representations of dynamic phenomena in functionally graded waveguides and those in periodic media. These connections are at their most obvious for low-frequency and long-wave asymptotics where well established theories hold. However, there is also a complementary limit of high-frequency long-wave asymptotics corresponding to various features that arise near cut-off frequencies in waveguides, including trapped modes. Simultaneously, periodic media exhibits standing wave frequencies, and the long-wave asymptotics near these frequencies characterise localised defect modes along with other high-frequency phenomena. The physics associated with waveguides and periodic media are, at first sight, apparently quite different, however the final equations that distill the essential physics are virtually identical. The connection is illustrated by the comparative study of a periodic string and a functionally graded acoustic waveguide.

Dispersion of elastic waves in a strongly inhomogeneous three-layered plate

“…In addition, we cite Cherdantsev & Cherednichenko (2012), Smyshlyaev (2009), Figotin & Kuchment (1998, and Kaplunov & Nobili (2016) devoted to homogenization of high-contrast periodic composites. Similarity of the asymptotic procedures underlying multi-layered plate theories and homoge-40 nization for periodic media has been recently reported in Craster et al (2014).…”

Composite dynamic models for periodically heterogeneous media

“…In such a case the energy of deformation is localised within the soft component, while the stiff component undergoes a nearly rigid-body motion. This reveals a fundamental analogy to the wave propagation in thin-walled structures: the local solution on the unit cell may correspond to the solution within the transverse cross section of a thin rod, plate, or shell (see Craster et al, 2014;Kaplunov, 2017).…”

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

“…It should be noted that macroscopic dynamic equations obtained by the AHM are valid only in the long-wave case, when l≪ L . Recently, Craster et al (2014) have shown a subtle analogue between the long-wave asymptotic procedures underlying approximate formulations for periodic media and for functionally graded waveguides. The conventional AHM seems to be a counterpart of the classical theories for thin plates, shells and rods.…”