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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.

At present there is a great interest in fabricating artificial materials, having extraordinary properties, which considerably extend those of natural materials. Such a new class of materials is known as metamaterials. From the beginning, the photonic metamaterials, possessing negative index of refraction, have intensively been investigated. It was established that such an unusual optical property is found in periodic structures (photonic crystals) whose dielectric function is spatially modulated. A peculiarity of the photonic metamaterials is the high dielectric contrast between the components. The negative refraction has been observed in double negative metamaterials with simultaneously negative effective permittivity and permeability [

Several homogenization theories, which are valid when the acoustic wavelength is much longer than the lattice constant of the phononic crystal, have been proposed [

The work is divided as follows: in Section

Let us consider a one-dimensional phononic crystal (or elastic superlattice) composed of alternating layers of isotropic elastic materials, A and B (Figure

Scheme of the binary superlattice.

In this case, the second Newton’s law for the displacement vector

Due to the periodicity of the functions

According to our geometry (see Figures

Unit cell of the binary superlattice.

For a periodic system, the solution of (

Substituting (

The inverse

The condition (

In the case when the wave length of sound is much larger than the lattice constant of the phononic crystal, this artificial periodic heterostructure can be modeled as a homogeneous medium with effective acoustic parameters. In this section, we shall calculate the effective sound velocities for both transverse and longitudinal vibrational modes, propagating along the growth direction of a superlattice like that considered in the previous sections.

From the Bloch theorem, given by (

Let us rewrite (

Let us apply the derived formula (^{3} and ^{3}, transverse sound velocities

Figures

Effective sound velocity for transverse vibrational modes propagating along the growth direction of a Polyethylene-Si superlattice. Here, solid line was obtained by using formula (

Effective sound velocity for longitudinal vibrational modes propagating along the growth direction of a Polyethylene-Si superlattice. Here, solid line was obtained by using formula (

Figures ^{3} and ^{3}, transverse sound velocities

Effective sound velocity for transverse vibrational modes propagating along the growth direction of a W-Al superlattice. Here, solid line was obtained by using formula (

Effective sound velocity for longitudinal vibrational modes propagating along the growth direction of a W-Al superlattice. Here, solid line was obtained by using formula (

In order to verify our numerical results in Figures

Figure

Graphs of the effective mass density

Otherwise, formula (^{3} and longitudinal sound velocity

Dependence of the effective longitudinal sound velocity for a homogenized 1D phononic crystal of aluminium embedded in water upon the aluminium filling fraction

In the developed method in this work, the homogenization is achieved in the limit

We have derived explicit formulas for the calculation of the effective sound velocities in a 1D phononic crystal in the long-wavelength limit. The formulas were applied for analyzing the dependence of the effective, transverse and longitudinal, sound velocities upon the inclusion filling fraction for binary superlattices composed of Polyethylene-Si and W-Al. In the latter case, the contrast of material parameters is relatively larger and, as a result, at Al filling fractions

Although the homogenization theory developed here is valid only for phononic crystals with one-dimensional periodicity and isotropic inclusions, it shows the usefulness of the plane wave expansion method to obtain explicit expressions for theoretical results of the effective sound velocity. The generalization of this approach to 2D and 3D periodic elastic structures with anisotropic inclusions and arbitrary contrast of the materials parameters will be presented elsewhere.

The authors declare that they have no competing interests.

This work was partially supported by VIEP-BUAP (Grant FOMJ-ING16-I). The authors also thank the support of PRODEP program (Grant BUAP-PTC-384, Agreement no. DSA/103.5/15/7449).