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Calculation of band structures of a phononic crystal within a waveguide in 3D with cubic inclusions using a Periodic Green’s Function Method

The Phononic Crystals have generated a growing scientific interest as a means to control the dispersion of waves in various technological applications such as telecommunications. In particular, Phononic Crystal Waveguides are composed of periodic distributions of dispersers immersed in a propagation medium and, designed by an arrangement with dimensions and periods comparable to the wavelength. These crystals have properties that give them the ability to guide acoustic waves efficiently. In this paper, we present a numerical Boundary Element Method, which requires the use of a Periodic Green’s Function. This method allows to calculate the band structure of phononic crystals in two- and three-dimensions. In particular, the band structure is calculated for a waveguide formed by two flat, and parallel plates that involve a two-dimensional periodic arrangement of cubic inclusions. All surfaces involved are considered acoustic hard surfaces. The system considered, in addition to being a waveguide is in itself a phononic crystal, so that this type of systems present an alternative to manufacture to phononic crystal that can act as a phononic crystal and as an acoustic waveguide. These properties present some interest from a technological point of view.


Introduction
The Phononic Crystals (PnCs) are artificial structures composed of at least two materials with different mass density and elastic properties. These are constructed in such a way that the elastic constants and/or the mass density of this structured compound are periodic functions in space. The material type of the PnCs can be formed can be solid, liquid or gaseous. The PnCs are designed for the control of the propagation of acoustic waves. These allow covering a large part of the continuous sonic spectrum, from infrasound (hertz) to thermal waves (terahertz), passing through audible sounds (kilohertz), ultrasound (megahertz) and hipersound (gigahertz) [1].
The dimensions that intervene in the PnCs range from a few meters to a hundred nanometers. On this scale, matter appears as continuous, so that the laws of classical mechanics are applicable and are used as a general rule. PnCs open the possibility to develop novel materials with advantageous properties that arise from the ability to tailor the propagation of acoustic and elastic waves and with it its effective properties [2], so condensed matter physicists are beginning to take interest in them.
In literature, one can find a lot of works on PnCs; however, most of the works are limited to twodimensional PnCs. So far, so both theoretically [2,3] and experimentally [4,5], only some examples of completely three-dimensional (3D) phononic crystals have been investigated. In most calculations the plane-wave expansion (PWE) method [6,7] has been used. However, PWE method has shown to be inadequate for to produce accurate results for modeling periodic composites consisting of solid  [8], fast multipole boundary element method (FMBEM) [9] based on the Burton-Miller formulation, etc.
In particular, the Boundary Element Methods (BEMs) have been widely used to determine the scattered fields in two-dimensional (2D) and three-dimensional (3D) systems in optics, propagation of seismic waves, and acoustics [10,11]. BEM has shown various advantages such as ease of coding and computational efficiency, so that has been proposed as a versatile method to determine the functions of the field and its normal derivative directly, in contrast to other methods that use the expansion of them into a functional basis [12]. It is possible that the most important feature of BEM is its versatility for solving resonant modes in cavities and scattering problems [13].
Among the possible applications of the PnC systems we can mention, of course, the acoustic isolation [14], the waveguides [15,16] and another less intuitive is the use of these structures to focus the waves of the sea towards a plant with the purpose of converting mechanical energy into electrical energy [17].
In the present paper, we show a numerical study for a Phononic Crystal Waveguides (PnCWs) formed by 3D acoustic rectangular waveguides with internal structures, composed by an array of cubic inclusions periodically in two directions (3DPnCW). Surfaces considered are hard acoustic surfaces that correspond to Neumann boundary condition. This paper is organized as follows. Sec. 2 introduces a numerical integral method to try the physical system considered, based in the use of the Periodic Green's Function (PGF) method and BEM [18]. In Sec. 3, we show results of phononic band structures for a 3DPnCW with two different sizes of cubic inclusions. In this section, we also discuss the numerical results. Finally, we present our conclusions in Sec. 4.

Periodic Green's Function Method applied to calculation of the band structure of a phononic crystal within a waveguide in 3D with cubic inclusions
In this section, we will introduce a numerical method to calculate the band structure of a 3DPnCW formed by two flat plates that involve a periodic 2D array of cubic inclusions as shown in figure 1. It is considered that all surfaces involved are acoustic hard surfaces and the medium between the plates is air. The unit cell of the system is shown in figure 2.
where , ∈ ℤ, c is the longitudinal speed of the wave in the acoustic medium and r is the observer's position. The subscripts l and m represent the position of a cell in the direction x and y within the arrangement with two-dimensional periodicity, respectively. Our problem is to calculate the band structure of a periodic system, for this we propose to use an integral numerical method that can be formulated from a PGF and BEM to solve the equation of Helmholtz under Neumann boundary conditions (acoustic hard surfaces). Periodicity in the x and y-direction of the 3DPnCW is considered, thus the Bloch theorem can be applied. This implies the following four relations of periodicity: , ( , + , , ) = ,0 ( , , ) where = ( , ) is the two-dimensional Bloch vector. In order to determine the band structure, the dispersion relation must be calculated. To do this, let us consider a Green's function for the 3D geometry given by where r and r´ are observation and integration points, respectively. The Green's function is a solution to Considering the geometry of the system shown in figure 1, and applying the three-dimensional Green's second identity for a pressure wave , ( ) and the Green`s function G in the air region, we obtain the expression ∑ ∑ the numerical values of the pressure, 0,0 (r´) and 0,0 (´)⁄ its normal derivative, evaluated at the n-th central point, denoted by , of the small surface elements ∆ , , respectively. In this way, equation (9) can be represented numerically in terms of a homogeneous system of algebraic equations as The matrix elements ℒ ( , ) and ( , ), that depend of the Bloch vector and the frequency , are given by and where is the maximum value of the indices l y m, numerically considered. This positive integer number represents a simple form to truncate the series, if is large enough, almost accurate results are obtained. Information about how the numerical values of ( ) and ( ) are obtained as outlined in Ref. [13].
Acoustic hard surfaces for the system are considered in this work. So that the normal derivative of the acoustic pressure is zero (Neumann boundary condition) on all the surfaces. This leads to = 0 for n=1,2,…,N, thereby obtaining the expression Equation (13) Numerically, this function presents local minima points that will give us the numerical dispersion relation = ( ). In this way, the band structure of the 3DPnCW can be calculated. The complexity of the system is enough to not allow us from obtaining analytical solutions. However, to validate the application of the PGF method to calculate of the band structure of a 3DPnCW, let us consider a limiting case that presents analytical solutions. A simple 3D waveguide formed by two flat, parallel and acoustic hard surfaces (without inclusions) has the analytical dispersion relation where , = 0, ±1, ±2, ±3, … ; = 0, 1,2,3, …, and the Bloch vector = � , � within the first Brillouin zone. In the band structure it is common to introduce dimensionless quantities, so our results will be expressed in terms of a reduced frequency given by = 2 ⁄ and a reduced Bloch vector given by = 2 ⁄ . A comparison between the analytical and the corresponding numerical results for a 3D waveguide with acoustic hard surfaces is shown in figure 3, where the distance between the flat surfaces of = 2 was considered, and = 15. Analytical results are illustrated by the blue dotted curves and numerical results by the red dotted curves. Figure 3 shows that the phononic band structures calculated by both methods are overlap, showing an excellent concordance.
The excellent agreement between both band structures that is shown in figure 3 gives reliability to PGM to obtain accurate numerical results for more complex systems.

Phononic Band Structures of a 3DPnCW
The numerical results shown in figure 3 allows us to now consider a waveguide formed by two flat, parallel and acoustic hard surfaces that have a two-dimensional arrangement of cubic inclusions as shown in figure 1. Figure 4 shows the unit cell when the separation between flat surfaces is = 2 ,