https://doi.org/10.1140/epjb/e2012-30521-4
Regular Article
From the inverted pendulum to the periodic interface modes
1 Centre d’Elaboration de Matériaux et d’Etudes Structurales, CNRS UPR 8011, 29 rue J. Marvig, BP 94347, 31055 Toulouse cedex 4, France
2 Université de Toulouse, UPS, 31055 Toulouse, France
a
e-mail: combe@cemes.fr
Received: 27 June 2012
Received in final form: 24 October 2012
Published online: 28 January 2013
The physics of the spatial propagation of monochromatic waves in periodic media is related to the temporal evolution of the parametric oscillators. We transpose the possibility that a parametric pendulum oscillates in the vicinity of its unstable equilibrium position to the case of monochromatic waves in a lossless unidimensional periodic medium. We develop this concept, that can formally applies to any kind of waves, to the case of longitudinal elastic wave. Our analysis yields us to study the propagation of monochromatic waves in a periodic structure involving two main periods. We evidence a class of phonons we refer to as periodic interface modes that propagate in these structures. These modes are similar to the optical Tamm states exhibited in photonic crystals. Our analysis is based on both a formal and an analytical approach. The application of the concept to the case of phonons in an experimentally realizable structure is given. We finally show how to control the frequencies of these phonons from the engineering of the periodic structure.
Key words: Solid State and Materials
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2013