**2.2 Equation of motion**

6 Acoustic Waves – From Microdevices to Helioseismology

propagating modes can be written in "Bloch form," consisting of a plane wave modulated

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

ky

π a


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

**G**

**K**


1

0

 

4 5

2 3

6 7 8

2 /a π

Fig. 4. Brillouin zones in a square lattice

**k'**

π a

. *i i*

*e e* ⋅ ⋅ = =+ **k r k r P (r) u (r) u (r R) kkk** (2)

kx

1 Brillouin zone st

3 Brillouin zone rd

2 Brillouin zone nd

by a function that shares the periodicity of the lattice (Joannopoulos et al., 1995):

is the so-called *Brillouin zone*.

the zone by adding **G**

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

In an inhomogeneous linear elastic medium with no body force, the equation of motion of the displacement vector **u r**(,)*t* can be written as

$$\rho(\mathbf{r})\ddot{u}\_i(\mathbf{r},t) = \partial\_j \left[\mathbf{C}\_{i|mn}(\mathbf{r})\partial\_n u\_m(\mathbf{r},t)\right],\tag{3}$$

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

Fig. 5. Periodic structures with square lattice. When the properties of materials A and B tend to coincide, the homogeneous case is recovered

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 components in the periodic structure are expressed as follows:

$$
\mu\_i(\mathbf{x}, t) = e^{i\mathbf{k} \cdot \mathbf{x}} \mathcal{U}\_i(\mathbf{x}, t), \tag{4}
$$

Analysis of Acoustic Wave

and transverse polarization modes.

are also observed and discussed.

reduced wave vector \* *k kR* = /

example are E=70 GPa,

can be obtain from

Young's modulus E, Poisson's ratio

ν

π

=0.33, and

ν

ρ

and *ux* respectively. It is noted that wave velocity , , / 2\* *S L S L c d dk R m* = =

**3.1 Isotropic medium** 

**3. Acoustic wave in homogeneous media** 

in Homogeneous and Inhomogeneous Media Using Finite Element Method 9

the irreducible part of the Brillouin zone, which is a triangle with vertexes Γ , Χ , and Μ . Similarly, Fig. 6(b) shows the irreducible part of the Brillouin zone of a rectangular lattice due to the geometric anisotropy, which is a rectangle with vertexes Γ , Χ , Μ , and Y , and

The finite element method divides a unit cell with a three-dimensional model into finite elements connected by nodes. The FEM obtains the eigen-solutions and contours of a mode shape. To simulate the dispersion diagrams, the wave vectors are condensed inside the first Brillouin zone in the square and rectangular lattices. Using the theories above, the following section presents the results of dispersion relations in a band structure for the Γ−Χ−Μ−Γ square lattice or isotropic materials, and Γ−Χ−Μ− −Γ Y rectangular lattice or anisotropic materials. Note that the 2D FEM model calculates the dispersion relations of mixed polarization modes, while the 3D FEM model describes the dispersion relations of mixed

It can be noted that a homogeneous medium is symmetric with respect to any periodicity, and it can be shown that the results for an infinite homogeneous medium can be cast in the form appropriate for a periodic medium. In this section, we introduce the mixed polarization modes and transverse polarization modes in a homogeneous medium. Displacement fields (polarizations) are also investigated and used to distinguish the different modes in the dispersion relations. The aluminum and quartz are adopted for examples and discussed in the section. The wave velocities of different propogating modes

In Fig. 5, when the properties of materials A and B tend to coincide, the homogeneous case is recovered. Consider a periodic structure consisting of aluminum (Al) circular cylinders embedded in a background material of Al forming a two-dimensional square lattice with lattice spacing R. It means this is a homogeneous medium in a 3D FEM model. Figure 7 shows the dispersion relations along the boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ−Γ . The vertical axis is the frequency (Hz) and the horizontal axis is the

, and density

 =2700 kg/m3. As the elastic waves propagate along the *x* axis, the nonvanishing displacement fields of the shear horizontal mode (SH), shear vertical mode (SV), and longitudinal mode (L) are *uy*, *uz*,

dispersion curves in the Γ−Χ section of Fig. 7 are exactly the straight lines and can be explained as the wave velocities of shear (S) and longitudinal (L) modes. Here, *m*S,L are the slopes of shear and longitudinal modes in Fig. 7. It is noted that the wave velocities of shear horizontal mode and shear vertical mode are the same in an isotropic material. From the results in Fig. 7, the wave velocities of shear and longitudinal modes are 3119 and 6174 m/s. As we know, the wave velocities of shear and longitudinal modes in an isotropic material

. Here, *k* is the wave vector along the Brillouin zone. The

ω

of the material Al utilized in this

, so the slopes of

ρ

the same as discussing the material anisotropy (Wu et al., 2004).

$$
\sigma\_{\vec{\boldsymbol{\eta}}}(\mathbf{x},t) = e^{i\mathbf{k}\cdot\mathbf{x}} T\_{\vec{\boldsymbol{\eta}}}(\mathbf{x},t),\tag{5}
$$

where 1 2 **k** = (,) *k k* is the Bloch wave vector, and *i* = −1 ; ( ,) *U t <sup>i</sup>* **x** and ( , ) *T t ij* **x** are periodic functions that satisfy the following relation (Tanaka et al., 2000):

$$\mathcal{U}I\_i(\mathbf{x} + \mathbf{R}, t) = \mathcal{U}\_i(\mathbf{x}, t), \tag{6}$$

$$T\_{ij}(\mathbf{x} + \mathbf{R}\_{\prime}t) = T\_{ij}(\mathbf{x}, t)\_{\prime} \tag{7}$$

where **R** is a lattice translation vector with components of *R*1 and *R*2 in the x and y directions. The relationships between the original variables ( ,) *u t <sup>i</sup>* **x** , ( , ) *ij* σ **x** *t* , ( ,) *u t <sup>i</sup>* **x R**+ , and ( , ) *ij* σ**x R**+ *t* about the Bloch boundary conditions are characterized as:

$$
\mu\_i(\mathbf{x} + \mathbf{R}, t) = e^{i\mathbf{k} \cdot (\mathbf{x} + \mathbf{R})} L\_i(\mathbf{x} + \mathbf{R}, t) = e^{i\mathbf{k} \cdot \mathbf{R}} e^{i\mathbf{k} \cdot \mathbf{x}} L\_i(\mathbf{x}, t) = e^{i\mathbf{k} \cdot \mathbf{R}} \mu\_i(\mathbf{x}, t), \tag{8}
$$

$$e\sigma\_{\vec{\boldsymbol{\eta}}}(\mathbf{x}+\mathbf{R},t) = e^{i\mathbf{k}\cdot(\mathbf{x}+\mathbf{R})}T\_{\vec{\boldsymbol{\eta}}}(\mathbf{x}+\mathbf{R},t) = e^{i\mathbf{k}\cdot\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{x}}T\_{\vec{\boldsymbol{\eta}}}(\mathbf{x},t) = e^{i\mathbf{k}\cdot\mathbf{R}}\sigma\_{\vec{\boldsymbol{\eta}}}(\mathbf{x},t). \tag{9}$$

The Bloch calculations in this study record the variation of the displacements, stress fields, and eigen-frequencies as the wave vector increases. By using the FEM, the unit cell is meshed and divided into finite elements which connect by nodes, and is used to obtain the eigen-solutions and mechanical displacements. The types of finite elements used in this chapter are the default element types, Lagrange-quadratic, in COMSOL Multiphysics. In order to simulate the dispersion diagrams, the wave vectors are condensed inside the first Brillouin zone in the square lattice. According to the above theories, the results of dispersion relations in a band structure along the Γ−Χ−Μ−Γ are characterized and presented in the following sections.

Fig. 6. Brillouin regions of the square and rectangular lattices

This chapter considers a periodic homogeneous medium with square lattice and phononic structures with square and rectangular lattices. These lattices consist of periodic structures that form two-dimensional lattices with lattice spacing R (square lattice) and lattice spacing aR (rectangular lattice). The term a is a scale from 0.1 to 2.0 in this chapter. The periodic structures are parallel to the z-axis. Figures 6(a) and 6(b) illustrate the Brillouin regions of the square lattice and rectangular lattice, respectively. In the square lattice, Fig. 6(a) shows the irreducible part of the Brillouin zone, which is a triangle with vertexes Γ , Χ , and Μ . Similarly, Fig. 6(b) shows the irreducible part of the Brillouin zone of a rectangular lattice due to the geometric anisotropy, which is a rectangle with vertexes Γ , Χ , Μ , and Y , and the same as discussing the material anisotropy (Wu et al., 2004).

The finite element method divides a unit cell with a three-dimensional model into finite elements connected by nodes. The FEM obtains the eigen-solutions and contours of a mode shape. To simulate the dispersion diagrams, the wave vectors are condensed inside the first Brillouin zone in the square and rectangular lattices. Using the theories above, the following section presents the results of dispersion relations in a band structure for the Γ−Χ−Μ−Γ square lattice or isotropic materials, and Γ−Χ−Μ− −Γ Y rectangular lattice or anisotropic materials. Note that the 2D FEM model calculates the dispersion relations of mixed polarization modes, while the 3D FEM model describes the dispersion relations of mixed and transverse polarization modes.
