Eur. Phys. J. B 3, 155–161 (1998) T HE EUROPEAN P HYSICAL JOURNAL B c  EDP Sciences Springer-Verlag 1998 Sonic stop-bands for cubic arrays of rigid inclusions in air M.S. Kushwaha a , B. Djafari-Rouhani, L. Dobrzynski, and J.O. Vasseur Laboratoire de Dynamique et Structures des Mat´ eriaux Mol´ eculaires b , U.F.R. de Physique, Universit´ e de Lille I, 59655 Villeneuve d’Ascq Cedex, France Received: 24 October 1997 / Revised: 1 December 1997 and 26 January 1998 / Accepted: 6 March 1998 Abstract. Extensive band structures have been computed for cubic arrays of rigid spheres and cubes in air. Complete stop bands are obtained for the face-centered-cubic (fcc) structure; however, there is no gap for the body-centered-cubic (bcc) and simple-cubic (sc) structures. These gaps start opening up for a filling fraction of ≈ 54% (27%) for spherical (cubic) inclusions and tend to increase with the filling fraction, exhibiting a maximum at the close-packing. We also propose a tandem structure that allows the achievement of an ultrawideband filter for environmental or industrial noise in the desired frequency range. This work is motivated by the recent experimental measurement of sound attenuation on the sculpture, by Eusebio Sempere, exhibited at the Juan March Foundation in Madrid (Nature 378, 241 (1995)) and complements the corresponding theoretical work (Appl. Phys. Lett. 70, 3218 (1997)). PACS. 43.40.+s Structural acoustics and vibration – 63.20.-e Phonons in crystal lattices – 42.45.Fx Diffraction and scattering 1 Introduction An architectural proposal of Yablonovitch [1] and a con- ceptual hypothesis of Sajeev John [2] triggered the pri- mary interest in photonic crystals. These are periodic dielectric structures that exhibit a band gap, by anal- ogy with the electronic band gap in semiconductor crys- tals. Within these photonic band gaps, the atoms are de- nied spontaneously absorbing and re-emitting light; this has signalled practical interest to produce highly efficient lasers. From the fundamental point of view, the existence of complete or pseudo gaps in a weakly disordered sys- tem is paramount in determining the transport properties and realizing the Anderson localization of light. Since the prominent phenomena emerging from the physics of pho- tonic crystals are all consequences of the existence of a photonic band gap, much of the research effort has been dedicated to the search for such photonic crystals [3]. Within a few years of the emergence of photonic crystals, a growing interest in the analogous studies on “phononic crystals” has been seen [4–17]. These are the two- and three-dimensional periodic elastic/acoustic com- posites that exhibit spectral gaps in the band structure. In analogy to the photonic crystals, the prime interest of the band theorists has been the occurrence of complete elas- tic/acoustic band gaps (or stop bands). The term com- plete refers to the gaps which exist independent of the a Permanent address: Institute of Physics, University of Puebla, Box J–48, Puebla 72570, Mexico. e-mail: manvir@sirio.ifuap.buap.mx b URA CNRS No. 801 polarization of the wave as well as its direction of propaga- tion. Within these gaps the sound, vibrations and phonons are all forbidden. This is of interest for applications such as elastic/acoustic filters, improvements in the design of transducers, and noise control; as well as for pure physics concerned with the Anderson localization of sound and vi- brations [18,19]. Piezoelectric and piezomagnetic compos- ites are already known to have long standing applications as medical ultrasonics and naval transducers; as well as for related tasks in medical imaging [20,21]. Such com- posites were initially fabricated for sonar applications and are now widely used for ultrasonic transducers. It is interesting to remark that in all artificial peri- odic structures –dielectric composites, elastic composites, magnetic composites, etc.– the existence of complete gaps is attributed to the joint effect of the Bragg diffraction and the Mie scattering. The destructive interference due to Bragg diffraction accompanied by the Mie resonances due to strong scattering from individual spheres is the conceptual base of the complete gaps. The latter becomes effective when the diameter of the sphere is close to an integer multiple of wavelength [3]. In the quest for achieving complete gaps one must resort to the band structure calculations. These have been performed for several geometries of periodic elas- tic composites and for various types of waves [4–17]. One-dimensional (1D) periodic systems (superlattices, for example) allow longitudinal, transverse, and mixed modes. Two-dimensional (2D) composites permit the propagation of pure transverse and mixed modes indepen- dently; no longitudinal modes are possible, however. In