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Advances in Materials Science and Engineering
Volume 2016, Article ID 3017835, 9 pages
Research Article

Plane Wave-Perturbative Method for Evaluating the Effective Speed of Sound in 1D Phononic Crystals

1Benemérita Universidad Autónoma de Puebla, Ciudad Universitaria, Blvd. Valsequillo y Esquina, Av. San Claudio s/n, Col. San Manuel, 72570 Puebla, PUE, Mexico
2Instituto Tecnológico de Puebla, División de Estudios de Posgrado e Investigación, Av. Tecnológico No. 420, Maravillas, 72220 Puebla, PUE, Mexico
3CONACYT, Dirección Adjunta al Desarrollo Científico, Dirección de Cátedras, Insurgentes Sur 1582, Crédito Constructor, Benito Juárez, 03940 Ciudad de México, Mexico

Received 6 July 2016; Revised 1 October 2016; Accepted 3 November 2016

Academic Editor: Tiejun Liu

Copyright © 2016 J. Flores Méndez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A method for calculating the effective sound velocities for a 1D phononic crystal is presented; it is valid when the lattice constant is much smaller than the acoustic wave length; therefore, the periodic medium could be regarded as a homogeneous one. The method is based on the expansion of the displacements field into plane waves, satisfying the Bloch theorem. The expansion allows us to obtain a wave equation for the amplitude of the macroscopic displacements field. From the form of this equation we identify the effective parameters, namely, the effective sound velocities for the transverse and longitudinal macroscopic displacements in the homogenized 1D phononic crystal. As a result, the explicit expressions for the effective sound velocities in terms of the parameters of isotropic inclusions in the unit cell are obtained: mass density and elastic moduli. These expressions are used for studying the dependence of the effective, transverse and longitudinal, sound velocities for a binary 1D phononic crystal upon the inclusion filling fraction. A particular case is presented for 1D phononic crystals composed of W-Al and Polyethylene-Si, extending for a case solid-fluid.