**1. Introduction**

Even though the propagation of elastic/acoustic waves in inhomogeneous and layered media has been an active research topic for many decades already, new problems and challenges continue to be posed even up to now. In fact, during the last few years, renewed interests have been witnessed by researchers in the various fields of acoustics, such as acoustic mirrors, filters, resonators, waveguides, and other kinds of acoustic devices, in relation to wave propagation in periodic elastic media. In acoustics and applied mechanics, these developments have been triggered by the need for new acoustic devices in order to obtain quality control of elastic/acoustic waves.

What sort of material can allow us to have complete control over the elastic/acoustic wave's propagation? We would like to discuss and answer this question in this chapter. It is well known that the successful applications of photonic band-gap materials have hastened the related researches on phononic band-gap materials. *Analysis of Acoustic Wave in Homogeneous and Inhomogeneous Media Using Finite Element Method* explores the theoretical road leading to the possible applications of phononic band gaps. It should quickly bring the elastic/acoustic professionals and engineers up to speed in this field of study where elastic/acoustic waves and solid-state physics meet. It will also provide an excellent overview to any course in elastic/acoustic media.

Previous research on photonic crystals (Johnson & Joannopoulos, 2001, 2003; Joannopoulos et al., 1995; Leung & Liu, 1990) has sparked rapidly growing interest in the analogous acoustic effects of phononic crystals and periodic elastic structures. The various techniques for band structure calculations were introduced (Hussein, 2009). There are many wellknown methods of calculating the band structures of photonic and phononic crystals in addition to the reduced Bloch mode expansion method: the plane-wave expansion (PWE) method (Huang & Wu, 2005; Kushwaha et al., 1993; Laude et al., 2005; Tanaka & Tamura, 1998; Wu et al., 2004 ; Wu & Huang, 2004), the multiple-scattering theory (MST) (Leung & Qiu, 1993; Kafesaki & Economou, 1999; Psarobas & Stefanou, 2000; Wang et al., 1993), the finite-difference (FD) method (Garica-Pabloset et al., 2000; Sun & Wu, 2005; Yang, 1996), the transfer matrix method (Pendry & MacKinnon, 1992), the meshless method (Jun et al., 2003), the multiple multipole method (Moreno et al., 2002), the wavelet method (Checoury & Lourtioz, 2006; Yan & Wang, 2006), the pseudospectral method (Chiang et al., 2007), the finite element method (FEM) (Axmann & Kuchment, 1999; Dobson, 1999; Huang & Chen,

Analysis of Acoustic Wave

**2.1 Real space and k space** 

are determined by (Kittel, 1996)

where *ijk* ε

2 *<sup>i</sup> <sup>j</sup>*

πδ*ij* **b a**⋅ = , where *ij*

δ

Fig. 1. Primitive unit cell in real space

in Homogeneous and Inhomogeneous Media Using Finite Element Method 5

It is well-known that the analysis of wave motion in infinite periodic structures is difficult in real space. For dealing with the periodic structures, the Fourier series and Bloch's theorem are used to expand the periodic parameters such as the density, material constants, displacement fields, or potential. Regarding to the transformation of the real space and **k** space, the reciprocal lattice vectors (RLVs) are adopted from the solid-state physics. In general, we consider a three-dimensional phononic crystal with primitive lattice vectors **<sup>1</sup> a** , <sup>2</sup> **a** , and 3 **a** . The complete set of lattice vectors is written as {**RR a a a** | =++ *lll* <sup>123</sup> **<sup>123</sup>**} , where *l*1, *l*2, and *l*3 are integers. The associated primitive reciprocal lattice vectors **b1** , **b2** , and **b3**

> 2 , ( ) *ijk j k*

**aaa**

**<sup>123</sup>** ⋅ × **a a**

is the three-dimensional Levi-Civita completely antisymmetric symbol. The

**1 a**

**2 a**

III

**b2**

**b1**

is the kronecker symbol. Note that the associated primitive

(1)

ε

complete set of reciprocal lattice vectors is written as {**GG b b b** | =++ *NNN* <sup>123</sup> **<sup>123</sup>**} , where *N*1, *N*2, and *N*3 are integers. Figure 1 shows the primitive unit cell in two-dimensional real space while the Fig. 2 shows the relationship between the real space and **k** space. A property between the primitive lattice vectors and associated primitive reciprocal lattice vectors is

reciprocal lattice vectors are constructed as **k** space from the concept of crystal diffraction.

**1 a**

Real space **k** space

We will find that, in following sections, the discrete translational symmetry of a phononic crystal allows us to classify the elastic/acoustic waves with a wave vector **k**. The

<sup>×</sup> <sup>=</sup>

π

*i*

**b**

**2 a**

**1 a**

II

I

**2 a**

**2 a**

Fig. 2. Relationship between the real space and **k** space

**1 a**

2011; Wu et al., 2008), the mass-in-mass lattice model (Huang & Sun, 2010), and the micropolar continuous modeling (Salehian & Inman, 2010).

Many studies on phononic band structures from the past decade use the PWE, MST, and FD methods to analyze the frequency band gaps of bulk acoustic waves (BAW) in composite materials or phononic band structures. Studies adopting the PWE method investigate the dispersion relations and the frequency band-gap feathers of the BAW and surface acoustic wave (SAW) modes. Other studies use the layered MST to study the frequency band gaps of bulk acoustic waves in three-dimensional periodic acoustic composites and the band structures of phononic crystals consisting of complex and frequency-dependent Lame′ coefficients. Other researchers applied the finite-difference time-domain method to predict the precise transmission properties of slabs of phononic crystals and analyze the mode coupling in joined parallel phononic crystal waveguides.

The techniques for tuning frequency band gaps of elastic/acoustic waves in phononic crystals are very important, and remain exciting research topics in the physics community. The filling fraction, rotation of noncircular rods, different cuts of anisotropic materials, and the temperature effect all produce large frequency band gaps in the BAW and SAW modes of periodic structures. A previous review paper (Burger et al., 2004) discusses the technique used to optimize the unit cell material distribution, achieving the largest possible band gap in photonic crystals for a given cell symmetry. Studies over the past decade focus on the theoretical and numerical analysis of phononic structures based on circular or square cylinders embedded in background materials. In this case, the PWE method can easily calculate the dispersion relations by constructing the structural functions with Bessel or Sinc functions. However, research on the more complicated problem of waves in the reticular and other special periodic band structures has not started until recently.

This chapter uses the 2D and 3D finite element methods to discuss the wave velocities of isotropic and anisotropic materials in homogeneous media. It also considers the tunable band gaps of acoustic waves in two-dimensional phononic crystals with reticular geometric structures (Huang & Chen, 2011). The concept of adopting a reticular geometric structure comes from the variations of similar geometry in bio-structural reticular formation and fibers. The PWE method used to calculate the structural functions of densities and elastic constants cannot numerically analyze the Gibbs phenomenon. Therefore, this chapter adopts the FEM to discuss this special periodic band structure. Changing the filling fraction, scale parameters, and rotating angles of reticular geometric structures can tune the frequency band gaps of mixed polarization modes. This technique is suitable for analyzing the phenomenon of frequency band gaps in special band structures.

### **2. Theory**

In this chapter, based on the theorems of solid-state physics and the finite element method with Bloch calculations, equation of motion of the acoustic modes in two-dimensional inhomogeneous media, phononic band structures, are derived and discussed in detail. In the beginning, the concepts of the real space and **k** space are introduced while the Brillouin zone is also addressed in the text. Generalized techniques of Bloch calculations in finite element method are used to analyze the acoustic modes in two-dimensional homogeneous and inhomogeneous media, phononic band structures, consisting of materials with general anisotropy. The mixed and transverse polarization modes and quasi-polarization modes are investigated in the text.

### **2.1 Real space and k space**

4 Acoustic Waves – From Microdevices to Helioseismology

2011; Wu et al., 2008), the mass-in-mass lattice model (Huang & Sun, 2010), and the

Many studies on phononic band structures from the past decade use the PWE, MST, and FD methods to analyze the frequency band gaps of bulk acoustic waves (BAW) in composite materials or phononic band structures. Studies adopting the PWE method investigate the dispersion relations and the frequency band-gap feathers of the BAW and surface acoustic wave (SAW) modes. Other studies use the layered MST to study the frequency band gaps of bulk acoustic waves in three-dimensional periodic acoustic composites and the band structures of phononic crystals consisting of complex and frequency-dependent Lame′ coefficients. Other researchers applied the finite-difference time-domain method to predict the precise transmission properties of slabs of phononic crystals and analyze the mode

The techniques for tuning frequency band gaps of elastic/acoustic waves in phononic crystals are very important, and remain exciting research topics in the physics community. The filling fraction, rotation of noncircular rods, different cuts of anisotropic materials, and the temperature effect all produce large frequency band gaps in the BAW and SAW modes of periodic structures. A previous review paper (Burger et al., 2004) discusses the technique used to optimize the unit cell material distribution, achieving the largest possible band gap in photonic crystals for a given cell symmetry. Studies over the past decade focus on the theoretical and numerical analysis of phononic structures based on circular or square cylinders embedded in background materials. In this case, the PWE method can easily calculate the dispersion relations by constructing the structural functions with Bessel or Sinc functions. However, research on the more complicated problem of waves in the reticular

This chapter uses the 2D and 3D finite element methods to discuss the wave velocities of isotropic and anisotropic materials in homogeneous media. It also considers the tunable band gaps of acoustic waves in two-dimensional phononic crystals with reticular geometric structures (Huang & Chen, 2011). The concept of adopting a reticular geometric structure comes from the variations of similar geometry in bio-structural reticular formation and fibers. The PWE method used to calculate the structural functions of densities and elastic constants cannot numerically analyze the Gibbs phenomenon. Therefore, this chapter adopts the FEM to discuss this special periodic band structure. Changing the filling fraction, scale parameters, and rotating angles of reticular geometric structures can tune the frequency band gaps of mixed polarization modes. This technique is suitable for analyzing the

In this chapter, based on the theorems of solid-state physics and the finite element method with Bloch calculations, equation of motion of the acoustic modes in two-dimensional inhomogeneous media, phononic band structures, are derived and discussed in detail. In the beginning, the concepts of the real space and **k** space are introduced while the Brillouin zone is also addressed in the text. Generalized techniques of Bloch calculations in finite element method are used to analyze the acoustic modes in two-dimensional homogeneous and inhomogeneous media, phononic band structures, consisting of materials with general anisotropy. The mixed and transverse polarization modes and quasi-polarization modes are

micropolar continuous modeling (Salehian & Inman, 2010).

coupling in joined parallel phononic crystal waveguides.

and other special periodic band structures has not started until recently.

phenomenon of frequency band gaps in special band structures.

**2. Theory** 

investigated in the text.

It is well-known that the analysis of wave motion in infinite periodic structures is difficult in real space. For dealing with the periodic structures, the Fourier series and Bloch's theorem are used to expand the periodic parameters such as the density, material constants, displacement fields, or potential. Regarding to the transformation of the real space and **k** space, the reciprocal lattice vectors (RLVs) are adopted from the solid-state physics. In general, we consider a three-dimensional phononic crystal with primitive lattice vectors **<sup>1</sup> a** , <sup>2</sup> **a** , and 3 **a** . The complete set of lattice vectors is written as {**RR a a a** | =++ *lll* <sup>123</sup> **<sup>123</sup>**} , where *l*1, *l*2, and *l*3 are integers. The associated primitive reciprocal lattice vectors **b1** , **b2** , and **b3** are determined by (Kittel, 1996)

$$\mathbf{b}\_{i} = 2\pi \frac{\mathbf{c}\_{ijk}\mathbf{a}\_{j} \times \mathbf{a}\_{k}}{\mathbf{a}\_{1} \cdot (\mathbf{a}\_{2} \times \mathbf{a}\_{3})},\tag{1}$$

where *ijk* ε is the three-dimensional Levi-Civita completely antisymmetric symbol. The complete set of reciprocal lattice vectors is written as {**GG b b b** | =++ *NNN* <sup>123</sup> **<sup>123</sup>**} , where *N*1, *N*2, and *N*3 are integers. Figure 1 shows the primitive unit cell in two-dimensional real space while the Fig. 2 shows the relationship between the real space and **k** space. A property between the primitive lattice vectors and associated primitive reciprocal lattice vectors is 2 *<sup>i</sup> <sup>j</sup>* πδ *ij* **b a**⋅ = , where *ij* δ is the kronecker symbol. Note that the associated primitive reciprocal lattice vectors are constructed as **k** space from the concept of crystal diffraction.

Fig. 1. Primitive unit cell in real space

Fig. 2. Relationship between the real space and **k** space

We will find that, in following sections, the discrete translational symmetry of a phononic crystal allows us to classify the elastic/acoustic waves with a wave vector **k**. The

Analysis of Acoustic Wave

π *a* to π

**2.2 Equation of motion** 

the displacement vector **u r**(,)*t* can be written as

 ( ) ( , ) [ ( ) ( , )], ρ

between −

is recovered.

x

y

to coincide, the homogeneous case is recovered

components in the periodic structure are expressed as follows:

( , ) ( , ), *<sup>i</sup> u t eU t i i*

A B

in Homogeneous and Inhomogeneous Media Using Finite Element Method 7

By the periodicity of the reciprocal lattice, any reciprocal lattice point which represents a wave vector **k** outside the first Brillouin zone can be found a corresponding point in the first Brillouin zone. Therefore, the wave vectors **k** can always be confined in the first Brillouin zone. In the square lattice, only the wave vectors **k** in the region of the first Brillouin zone

*first*, *second*, and *third Brillouin zones*. For more details, it is best to consult the first few chapters of a solid-state physics text, such as Kittel, 1996, or consult the appendix of popular

This section provides a brief introduction of the theory of analyzing acoustic wave propagation in inhomogeneous media like as phononic band structures. The theory in this chapter can also be used to discuss acoustic wave propagation in homogeneous media

In an inhomogeneous linear elastic medium with no body force, the equation of motion of

are the position-dependent mass density and elastic stiffness tensor, respectively. The following discussion considers a periodic structure consisting of a two-dimensional periodic array (x-y plane) of material A embedded in a background material B shown in Fig. 5. It is noted that when the properties of materials A and B tend to coincide, the homogeneous case

z

0

Half space r0

Fig. 5. Periodic structures with square lattice. When the properties of materials A and B tend

To calculate the dispersion diagrams of periodic structures, this study uses COMSOL Multiphysics software to apply the Bloch boundary condition to the unit cell domain in the FEM method. Based on the periodicity of phononic crystals, the displacement and stress

a

x

y

photonic text like Joannopoulos et al. 1995 and Johnson & Joannopoulos, 2001, 2003.

because a homogeneous medium is symmetric with respect to any periodicity.

where **r x** = = (,) (,,) *z xyz* is the position vector, *t* is the time variable, and

*a* (the lattice constant is *a*) need to be considered. The Fig. 4 shows the

*ut C u t <sup>i</sup>* =∂ ∂ *j ijmn n m* **rr r r** (3)

B A

<sup>⋅</sup> = **k x x x** (4)

ρ

( )**r** and ( ) *Cijmn* **r**

propagating modes can be written in "Bloch form," consisting of a plane wave modulated by a function that shares the periodicity of the lattice (Joannopoulos et al., 1995):

$$\mathbf{P\_{k}(r)} = e^{i\mathbf{k}\cdot\mathbf{r}}\mathbf{u\_{k}(r)} = e^{i\mathbf{k}\cdot\mathbf{r}}\mathbf{u\_{k}(r+R)}.\tag{2}$$

The important feature of the Bloch states is that different values of **k** do not necessarily lead to different modes. It is clear that a mode with wave vector **k** and a mode with wave vector **k+G** are the same mode, where **G** is a reciprocal lattice vector. The wave vector **k** serves to specify the phase relationship between the various cells that are described by **u**. If **k** is increased by **G**, then the phase between cells is increased by **G**⋅**R**, which we know is 2π*n* (*n= l*1*N*1*+l*2*N*2*+ l*3*N*3 is an integer) and not really a phase difference at all. So incrementing **k** by **G** results in the same physical mode. This means that we can restrict our attention to a finite zone in reciprocal space in which we *cannot* get from one part of the volume to another by adding any **G**. All values of **k** that lie outside of this zone, by definition, can be reached from within the zone by adding **G**, and are therefore redundant labels shown in Fig. 3. This zone is the so-called *Brillouin zone*.

Fig. 3. All values of **k** that lie outside of this zone, by definition, can be reached from within the zone by adding **G**

Fig. 4. Brillouin zones in a square lattice

By the periodicity of the reciprocal lattice, any reciprocal lattice point which represents a wave vector **k** outside the first Brillouin zone can be found a corresponding point in the first Brillouin zone. Therefore, the wave vectors **k** can always be confined in the first Brillouin zone. In the square lattice, only the wave vectors **k** in the region of the first Brillouin zone between −π *a* to π *a* (the lattice constant is *a*) need to be considered. The Fig. 4 shows the *first*, *second*, and *third Brillouin zones*. For more details, it is best to consult the first few chapters of a solid-state physics text, such as Kittel, 1996, or consult the appendix of popular photonic text like Joannopoulos et al. 1995 and Johnson & Joannopoulos, 2001, 2003.
