Speed of Sound in Periodic Elastic Composites

Abstract

We consider the low-frequency limit (homogenization) for propagation of sound waves in periodic elastic medium (phononic crystals). Exact analytical formulas for the speed of sound propagating in a three-dimensional periodic arrangement of liquid and gas or in a two-dimensional arrangement of solids are derived. We apply our formulas to the well-known phenomenon of the drop of the speed of sound in mixtures. For air bubbles in water we obtain a perfect agreement with the recent results of coherent potential approximation obtained by M. Kafesaki, R. S. Penciu, and E. N. Economou [Phys. Rev. Lett. 84, 6050 (2000)] if the filling of air bubbles is far from close packing. When air spheres almost touch each other, the approximation gives 10 times lower speed of sound than the exact theory does. DOI: 10.1103/PhysRevLett.91.264302 PACS numbers: 43.20.+g, 43.58.+z, 43.90.+v Acoustic waves play an important role in our life, being a universal carrier of information in vivid nature. In most cases, the media where the acoustic waves propagate are inhomogeneous. A common example is a mixture of water and air. During the past century the problem of propagation of sound in heterogeneous media has been extensively studied [1]. In the last few years artificial periodic elastic structures, phononic crystals, have been successfully fabricated [2]. Because of the presence of acoustic gaps -regions of frequencies where sound does not propagate [3] -phononic crystals can be used as soundless background for many technological devices. Phononic crystal with point, linear, and surface defects permit manipulation of sound: they guide the acoustic waves, split, and bend them [4]. At low frequencies phononic crystals possess a property to focus a sound beam [5] and thus may find numerous applications in acoustic surgery [6]. To design an acoustic lens one needs to know the refractive index of the material or the effective speed of sound. At low frequencies (well below the band gap) the dispersion relation is linear, ! c eff k, since one wavelength covers many periods of the structure, thus averaging the inhomogeneous medium. Calculation of the effective parameters of the uniform medium (effective speed of sound, c eff , and the effective elastic moduli) is a long-standing problem of the mathematical theory of homogenization [7]. Although the theory itself is well developed and predicts that different media homogenize at low frequencies, there are no explicit formulas that can be used for calculation of the effective parameters (see recent review [8] on homogenization for different types of the wave equations). There are also approximate methods that allow calculations of the effective parameters for particular structures [9][10][11]. Sometimes the effective elastic moduli can be evaluated from the exact upper and lower bounds [12].In this Letter we develop an exact analytical theory of homogenization of periodic elastic structures. Our approach is based on the plane wave method that has been successfully used for homogenization of pe...

Metafluid with anisotropic dynamic mass

“…air bubbles in water) was calculated in Ref. 28. This result is equally valid for 2D phononic crystal of solid cylinders embedded in fluid, since the transverse mode is suppressed.…”

Sound focusing by gradient index sonic lenses

“…1 However, in the range of low frequencies ͑homogenization limit͒ they behave like homogeneous media whose effective acoustic parameters, dynamical mass density, and bulk modulus, basically depend on the lattice filling fraction. [2][3][4] The homogenization properties of SCs have been employed to design refractive devices like, for example, acoustic lenses whose focusing properties are based on their external curved surfaces [5][6][7] or Fabry-Perot type acoustic interferometers. 8 Gradient index ͑GRIN͒ sonic lenses based on homogenized 2D SCs have been proposed.…”

Wave focusing using symmetry matching in axisymmetric acoustic gradient index lenses

“…In the long wavelength regime, the minor radius of each scatterer can be selected to fix the filling fraction, f (r) = πR(r) 2 /a 2 , at a position r from the center of the lens 7 , and the index of refraction, n(r), can be written in terms of f as 7,8 n(r) = c host c ef f Then, by a gradual change of the filling fraction we can design a refraction index profile in the vertical plane of the lens, perpendicular to the axial z-direction. In this work we use the hyperbolic secant profile that has been proved to reduce the aberration of the focal spot 23 , defined as…”