**3. Acoustic wave in homogeneous media**

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 are also observed and discussed.

### **3.1 Isotropic medium**

8 Acoustic Waves – From Microdevices to Helioseismology

( , ) ( , ), *<sup>i</sup> ij ij*

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

( , ) ( , ), *T tT t ij* + = *ij* **xR x** (7)

where **R** is a lattice translation vector with components of *R*1 and *R*2 in the x and y

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

( ) ( , ) ( , ) ( , ) ( , ). *<sup>i</sup> i i <sup>i</sup> ij ij ij ij*

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

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

⋅ + ⋅ ⋅ <sup>⋅</sup> += += = **kxR kR kx k R x R x R x x** (8)

*t e T t e eT t e t* ⋅ + ⋅ ⋅ <sup>⋅</sup> += += = **kxR kR kx k R x R x R x x** (9)

directions. The relationships between the original variables ( ,) *u t <sup>i</sup>* **x** , ( , ) *ij*

**x R**+ *t* about the Bloch boundary conditions are characterized as:

*t eT t* <sup>⋅</sup> = **k x x x** (5)

( , ) ( , ), *U tU t i i* **xR x** + = (6)

σ

 σ **x** *t* , ( ,) *u t <sup>i</sup>* **x R**+ ,

σ

functions that satisfy the following relation (Tanaka et al., 2000):

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

and ( , ) *ij* σ

following sections.

σ

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 reduced wave vector \* *k kR* = /π . Here, *k* is the wave vector along the Brillouin zone. The Young's modulus E, Poisson's ratio ν , and density ρ of the material Al utilized in this example are E=70 GPa, ν =0.33, and ρ=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*, and *ux* respectively. It is noted that wave velocity , , / 2\* *S L S L c d dk R m* = = ω , so the slopes of 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 can be obtain from

Analysis of Acoustic Wave

**3.2 Anisotropic medium** 

reduced wave vector.

piezoelectric and anisotropic material. The density

Table 1. The elastic constants of quartz in GPa unit

Table 3. The relative permittivity of quartz

Table 2. The piezoelectric constants of quartz in C/m2 unit

( Γ −*Y* section) are 3922, 4311, and 6009 m/s respectively.

from the truly longitudinal and truly transverse waves of Fig. 8.

in Homogeneous and Inhomogeneous Media Using Finite Element Method 11

Similarly, the method in this chapter is used to discuss the wave velocities of acoustic modes in an anisotropic material. Consider a periodic structure consisting of quartz circular cylinders embedded in a background material of quartz forming a two-dimensional square lattice with lattice spacing R. This is also a homogeneous medium. The quartz is a

piezoelectric constants, and relative permittivity of quartz utilized in this example are shown in Tables 1-3. The piezoelectric material, quartz, is a complete structural-electrical material, and thus all piezoelectric material properties were defined and entered into the FEM model. Figure 9 shows the dispersion relations along the boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ− −Γ Y due to the material anisotropy. In the calculations, the x-y plane is parallel to the (001) plane and the x axis is along the [100] direction of quartz. The vertical axis is the frequency in Hz unit and the horizontal axis is the

86.7362 6.98527 11.9104 17.9081 0 0 6.98527 86.7362 11.9104 -17.9081 0 0 11.9104 11.9104 107.194 0 0 0 17.9081 -17.9081 0 57.9428 0 0 0 0 0 0 57.9492 17.9224 0 0 0 0 17.9224 39.9073


> 4.4093 0 0 0 4.4092 0 0 0 4.68

Shown in Γ−Χ section of Fig. 9, the cross symbols represent the quasi shear horizontal (quasi-SH) mode. The square symbols represent the quasi shear vertical (quasi-SV) mode and the open circle symbols represent the quasi longitudinal (quasi-L) mode. The wave velocities of quasi-SH, quasi-SV, and quasi-L modes along x axis are 3306, 5116, and 5741 m/s. Similarly, The wave velocities of quasi-SH, quasi-SV, and quasi-L modes along y axis

Figure 10 also shows the vibration mode shapes of unit cell for quasi-SH, quasi-SV, and quasi-L modes in X point. The arrows shown in Fig. 10 are the polarizations. In this example, the quasi-longitudinal and quasi-transverse waves are almost indistinguishable

ρ

=2651 kg/m3. The elastic constants,

$$\mathbf{c}\_{S} = \sqrt{\frac{E}{\rho} \frac{1}{2(1+\nu)}} = \mathfrak{J}122 \quad m \text{ / s.} \tag{10}$$

$$\mathcal{L}\_{\perp} = \sqrt{\frac{E}{\rho} \frac{(1-\nu)}{(1+\nu)(1-2\nu)}} = 6031 \quad m \text{ / s.} \tag{11}$$

Note that the FEM method can easily describe the mode characteristics. Figure 8 shows the vibration mode shapes of unit cell for shear and longuitudinal modes in X point. In this example, Fig. 8(a) is a shear horizontal mode with mode vibrating displacement along the y direction when the wave propagates along the x direction ( Γ−Χ direction). Also, Fig. 8(b) is a shear vertical mode with mode vibrating displacement along z direction, and Fig. 8(c) is a longitudinal mode with mode vibrating displacement along x direction. The arrows shown in Fig. 8 are the polarizations.

Fig. 7. The dispersion relations of homogeneous and isotropic material Al along the boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ−Γ

Fig. 8. (a) shear horizontal mode (b) shear vertical mode, (c) longitudinal mode in the Al

### **3.2 Anisotropic medium**

10 Acoustic Waves – From Microdevices to Helioseismology

<sup>1</sup> 3122 / , 2(1 ) *<sup>S</sup>*

(1 ) 6031 / . (1 )(1 2 ) *<sup>L</sup>*

Note that the FEM method can easily describe the mode characteristics. Figure 8 shows the vibration mode shapes of unit cell for shear and longuitudinal modes in X point. In this example, Fig. 8(a) is a shear horizontal mode with mode vibrating displacement along the y direction when the wave propagates along the x direction ( Γ−Χ direction). Also, Fig. 8(b) is a shear vertical mode with mode vibrating displacement along z direction, and Fig. 8(c) is a longitudinal mode with mode vibrating displacement along x direction. The arrows shown

Fig. 7. The dispersion relations of homogeneous and isotropic material Al along the

(a) (b) (c)

Fig. 8. (a) shear horizontal mode (b) shear vertical mode, (c) longitudinal mode in the Al

boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ−Γ

 ν

*c m s* ν

= = <sup>+</sup> (10)

<sup>−</sup> = = + − (11)

*c m s*

 ν

*E*

ρ

ρν

*E*

in Fig. 8 are the polarizations.

Similarly, the method in this chapter is used to discuss the wave velocities of acoustic modes in an anisotropic material. Consider a periodic structure consisting of quartz circular cylinders embedded in a background material of quartz forming a two-dimensional square lattice with lattice spacing R. This is also a homogeneous medium. The quartz is a piezoelectric and anisotropic material. The density ρ =2651 kg/m3. The elastic constants, piezoelectric constants, and relative permittivity of quartz utilized in this example are shown in Tables 1-3. The piezoelectric material, quartz, is a complete structural-electrical material, and thus all piezoelectric material properties were defined and entered into the FEM model. Figure 9 shows the dispersion relations along the boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ− −Γ Y due to the material anisotropy. In the calculations, the x-y plane is parallel to the (001) plane and the x axis is along the [100] direction of quartz. The vertical axis is the frequency in Hz unit and the horizontal axis is the reduced wave vector.


Table 1. The elastic constants of quartz in GPa unit


Table 2. The piezoelectric constants of quartz in C/m2 unit


Table 3. The relative permittivity of quartz

Shown in Γ−Χ section of Fig. 9, the cross symbols represent the quasi shear horizontal (quasi-SH) mode. The square symbols represent the quasi shear vertical (quasi-SV) mode and the open circle symbols represent the quasi longitudinal (quasi-L) mode. The wave velocities of quasi-SH, quasi-SV, and quasi-L modes along x axis are 3306, 5116, and 5741 m/s. Similarly, The wave velocities of quasi-SH, quasi-SV, and quasi-L modes along y axis ( Γ −*Y* section) are 3922, 4311, and 6009 m/s respectively.

Figure 10 also shows the vibration mode shapes of unit cell for quasi-SH, quasi-SV, and quasi-L modes in X point. The arrows shown in Fig. 10 are the polarizations. In this example, the quasi-longitudinal and quasi-transverse waves are almost indistinguishable from the truly longitudinal and truly transverse waves of Fig. 8.

Analysis of Acoustic Wave

polarization modes.

**4.1 Periodic structure with two media** 

represents the normalized frequency \*

reduced wave number \* *k kR* = /

ρ

and density

CPU time.

=8905 kg/m3.

ρ

in Homogeneous and Inhomogeneous Media Using Finite Element Method 13

calculates and discusses the band gap variations of the bulk modes due to different sizes of reticular geometric structures. Results show that adjusting the orientation of the reticular geometric structures can increase or decrease the total elastic band gaps for mixed

It is necessary and worthy to provide evidence supporting the FEM method's (COMSOL Multiphysics) ability to perform Bloch calculations with two media. This chapter compares the dispersion relations of Al/Ni band structure using the PWE method with the results of using the FEM method. Consider a phononic structure consisting of Al circular cylinders embedded in a background material of Ni to form a two-dimensional square lattice with lattice spacing R. Figure 11 shows the dispersion relations along the boundaries of the irreducible part of the Brillouin zone in Fig. 6(a) with filling ratio 0.6. The vertical axis

= *R C*/ *<sup>t</sup>* and the horizontal axis represents the

ν,

=0.336, and

ν

. Here, *Ct* and *k* are the shear velocity of Ni and the wave

ω ω

Fig. 11. Comparison of Bloch calculations between the PWE and FEM methods

vector along the Brillouin zone, respectively. The Young's modulus E, Poisson's ratio

of the material Ni utilized in this example are E=214 GPa,

The diamond symbols represent the dispersion relations of the transverse polarization modes (shear vertical modes), and the cross symbols represent the mixed polarization modes (shear horizontal mode coupled with longitudinal mode) in the PWE method. The open circles represent the dispersion relations of all modes in the FEM method with a 3D model. The results of the FEM method match well with those of the PWE method. In the similar cases, when the differences of mass densities and elastic constants between the two periodic materials are larger, the convergence of the PWE method is slower and costs more

π

Fig. 9. The dispersion relations of homogeneous material quartz along the boundaries of the irreducible part of the Brillouin zone Γ−Χ−Μ− −Γ Y

Fig. 10. (a) quasi shear horizontal mode (b) quasi shear vertical mode, (c) quasi longitudinal mode in the quartz

From the discussion, it shows that the method adopted in this chapter can be used to discuss the wave propagations in isotropic and anisotropic media.

### **4. Acoustic wave in inhomogeneous media**

Previous studies on photonic crystals raise the exciting topic of phononic crystals. This section presents the results of acoustic waves in inhomogeneous media, Al/Ni periodic structures and phononic crystals with reticular geometric structures. It also discusses the tunable band gaps in the acoustic waves of two-dimensional phononic crystals with reticular geometric structures using the 2D and 3D finite element methods. This section calculates and discusses the band gap variations of the bulk modes due to different sizes of reticular geometric structures. Results show that adjusting the orientation of the reticular geometric structures can increase or decrease the total elastic band gaps for mixed polarization modes.

## **4.1 Periodic structure with two media**

12 Acoustic Waves – From Microdevices to Helioseismology

Fig. 9. The dispersion relations of homogeneous material quartz along the boundaries of the

(a) (b) (c) Fig. 10. (a) quasi shear horizontal mode (b) quasi shear vertical mode, (c) quasi longitudinal

From the discussion, it shows that the method adopted in this chapter can be used to discuss

Previous studies on photonic crystals raise the exciting topic of phononic crystals. This section presents the results of acoustic waves in inhomogeneous media, Al/Ni periodic structures and phononic crystals with reticular geometric structures. It also discusses the tunable band gaps in the acoustic waves of two-dimensional phononic crystals with reticular geometric structures using the 2D and 3D finite element methods. This section

irreducible part of the Brillouin zone Γ−Χ−Μ− −Γ Y

the wave propagations in isotropic and anisotropic media.

**4. Acoustic wave in inhomogeneous media** 

mode in the quartz

It is necessary and worthy to provide evidence supporting the FEM method's (COMSOL Multiphysics) ability to perform Bloch calculations with two media. This chapter compares the dispersion relations of Al/Ni band structure using the PWE method with the results of using the FEM method. Consider a phononic structure consisting of Al circular cylinders embedded in a background material of Ni to form a two-dimensional square lattice with lattice spacing R. Figure 11 shows the dispersion relations along the boundaries of the irreducible part of the Brillouin zone in Fig. 6(a) with filling ratio 0.6. The vertical axis represents the normalized frequency \* ω ω= *R C*/ *<sup>t</sup>* and the horizontal axis represents the reduced wave number \* *k kR* = /π . Here, *Ct* and *k* are the shear velocity of Ni and the wave vector along the Brillouin zone, respectively. The Young's modulus E, Poisson's ratio ν , and density ρ of the material Ni utilized in this example are E=214 GPa, ν =0.336, and ρ=8905 kg/m3.

The diamond symbols represent the dispersion relations of the transverse polarization modes (shear vertical modes), and the cross symbols represent the mixed polarization modes (shear horizontal mode coupled with longitudinal mode) in the PWE method. The open circles represent the dispersion relations of all modes in the FEM method with a 3D model. The results of the FEM method match well with those of the PWE method. In the similar cases, when the differences of mass densities and elastic constants between the two periodic materials are larger, the convergence of the PWE method is slower and costs more CPU time.

Fig. 11. Comparison of Bloch calculations between the PWE and FEM methods

Analysis of Acoustic Wave

a=1.2

modes indicated in Fig. 13

couple with the mixed polarization modes.

in Homogeneous and Inhomogeneous Media Using Finite Element Method 15

relations of mixed polarization modes in the 2D FEM model, while solid circles represent the results of all bulk modes in the 3D FEM model. Figure 14 shows the eigenmode shapes with 4×4 supercell of total displacements for M1 and M2 modes indicated in Fig. 13. These figures clearly show the phenomena of wave localizations in this reticular geometric structure. Note that the FEM method can easily describe the mode characteristics. In this chapter, M1 is a shear horizontal mode with mode vibrating displacement along the y direction when the wave propagates along the x direction ( Γ−Χ direction). Also, M2 is a shear vertical mode with mode vibrating displacement along z direction, and it does not

Fig. 13. The dispersion relations of the mixed and transverse polarization modes along the boundaries of the irreducible part of the Brillouin zone with the scales R=h=1, c=0.8, and

Fig. 14. The eigenmode shapes with 4×4 supercell of total displacements for M1 and M2

M1 M2

As the elastic waves propagate along the *x* axis, the nonvanishing displacement fields of the shear horizontal mode, shear vertical mode, and longitudinal mode are *uy*, *uz*, and *ux* respectively. For the sequence modes appear, the modes are always the same. When representing the whole wave vector space by the first Brillouin zone alone, they appear as further branches from higher Brillouin zones. In this example, the phase velocities of the SV0 mode (diamond symbols) are larger than those of the SH0 mode. The boundary of the Brillouin zone X-M of Fig. 11 represents the dispersion of the bulk waves with propagating direction varied 0 deg~ 45 deg counterclockwise away from the *x* direction.

### **4.2 Periodic structure with single medium**

Figure 12(a) depicts a two-dimensional phononic crystal with the reticular geometric structures of square lattice. These reticular structures are parallel to the z-axis. In a perfect two-dimensional phononic crystal, the periodic structure is constant in the z direction and the size of the structure is infinite in the x and y directions. To analyze the dispersion relations of all bulk acoustic modes in this band structure, the FEM should consider the 3D model in Fig. 12(c). The dimensions of the unit cell in Fig. 12(a) are c=d=0.8R and R=h=1 in the calculations.

Fig. 12. (a) square lattice with lattice spacing R and (b) rectangular lattice with lattice spacing aR along x-axis and R along y-axis, (c) a unit cell with reticular structures in a 3D FEM model

The material of the reticular structures in the unit cell in this chapter is aluminum. Figure 12(c) shows a diagram of the unit square lattice in a 3D FEM model. The periodicity of phononic crystals along the z direction is used to calculate the dispersion relations of the mixed and transverse polarization modes. The types of finite elements used for the 2D and 3D cases are the default element types, Lagrange-Quadratic, in COMSOL Multiphysics. Figure 13 shows the dispersion relations of the mixed and transverse polarization modes along the boundaries of the irreducible part of the Brillouin zone in Fig. 6(b) with the scales R=h=1, c=0.8, and a=1.2. The horizontal axis represents the reduced wave number along Γ−Χ−Μ− −Γ Y and the vertical axis represents the frequency (Hz). Note that this band structure shows no full band gap of the mixed and transverse polarization modes. Adopting the 2D FEM model to discuss the mixed polarization modes in this kind of band structure shows that there is only one full frequency band gap in Fig. 13, located at 3311 ~ 3400 Hz. Figure 13 compares the 3D and 2D FEM models. Open circles represent the dispersion

As the elastic waves propagate along the *x* axis, the nonvanishing displacement fields of the shear horizontal mode, shear vertical mode, and longitudinal mode are *uy*, *uz*, and *ux* respectively. For the sequence modes appear, the modes are always the same. When representing the whole wave vector space by the first Brillouin zone alone, they appear as further branches from higher Brillouin zones. In this example, the phase velocities of the SV0 mode (diamond symbols) are larger than those of the SH0 mode. The boundary of the Brillouin zone X-M of Fig. 11 represents the dispersion of the bulk waves with propagating

Figure 12(a) depicts a two-dimensional phononic crystal with the reticular geometric structures of square lattice. These reticular structures are parallel to the z-axis. In a perfect two-dimensional phononic crystal, the periodic structure is constant in the z direction and the size of the structure is infinite in the x and y directions. To analyze the dispersion relations of all bulk acoustic modes in this band structure, the FEM should consider the 3D model in Fig. 12(c). The dimensions of the unit cell in Fig. 12(a) are c=d=0.8R and R=h=1 in

Fig. 12. (a) square lattice with lattice spacing R and (b) rectangular lattice with lattice spacing aR along x-axis and R along y-axis, (c) a unit cell with reticular structures in a 3D FEM

The material of the reticular structures in the unit cell in this chapter is aluminum. Figure 12(c) shows a diagram of the unit square lattice in a 3D FEM model. The periodicity of phononic crystals along the z direction is used to calculate the dispersion relations of the mixed and transverse polarization modes. The types of finite elements used for the 2D and 3D cases are the default element types, Lagrange-Quadratic, in COMSOL Multiphysics. Figure 13 shows the dispersion relations of the mixed and transverse polarization modes along the boundaries of the irreducible part of the Brillouin zone in Fig. 6(b) with the scales R=h=1, c=0.8, and a=1.2. The horizontal axis represents the reduced wave number along Γ−Χ−Μ− −Γ Y and the vertical axis represents the frequency (Hz). Note that this band structure shows no full band gap of the mixed and transverse polarization modes. Adopting the 2D FEM model to discuss the mixed polarization modes in this kind of band structure shows that there is only one full frequency band gap in Fig. 13, located at 3311 ~ 3400 Hz. Figure 13 compares the 3D and 2D FEM models. Open circles represent the dispersion

direction varied 0 deg~ 45 deg counterclockwise away from the *x* direction.

**4.2 Periodic structure with single medium** 

the calculations.

model

relations of mixed polarization modes in the 2D FEM model, while solid circles represent the results of all bulk modes in the 3D FEM model. Figure 14 shows the eigenmode shapes with 4×4 supercell of total displacements for M1 and M2 modes indicated in Fig. 13. These figures clearly show the phenomena of wave localizations in this reticular geometric structure. Note that the FEM method can easily describe the mode characteristics. In this chapter, M1 is a shear horizontal mode with mode vibrating displacement along the y direction when the wave propagates along the x direction ( Γ−Χ direction). Also, M2 is a shear vertical mode with mode vibrating displacement along z direction, and it does not couple with the mixed polarization modes.

Fig. 13. The dispersion relations of the mixed and transverse polarization modes along the boundaries of the irreducible part of the Brillouin zone with the scales R=h=1, c=0.8, and a=1.2

Fig. 14. The eigenmode shapes with 4×4 supercell of total displacements for M1 and M2 modes indicated in Fig. 13

Analysis of Acoustic Wave

deg to 65 deg.

deg, 75 deg, and 90 deg

in Homogeneous and Inhomogeneous Media Using Finite Element Method 17

Finally, the rotating angles of reticular geometric structures were changed to analyze the distribution of total band gaps. Figure 17 shows the 2D diagrams of unit rectangular lattices in different rotating angles D=30 deg, 45 deg, 75 deg, and 90 deg. In these cases, the widths of aluminum remain constant, 0.14R, in the reticular geometric structures with different rotating angles in the calculations. Figure 18 shows the band gap widths of rectangular lattices with different rotating angles of reticular geometric structures. Based on the symmetry of the geometry, the different angles in the Bloch calculations were adopted from 15 deg ~ 90 deg. In the calculated results, no band gap is detected from D=5

Fig. 17. 2D diagrams of unit rectangular lattices in different rotating angles D=30 deg, 45

Fig. 16. The band gap widths with the scale a from 0.1 to 2.0

The following discussion addresses several parameters of the reticular geometric in this chapter. First, the effect of filling fraction is discussed when the parameters c=d varied from 0.1 to 0.9 in Fig. 12(a). Figure 15 shows the distribution of the total band gaps of mixed polarization modes, in which only one total band gap appears at approximately 3560 ~ 3736 Hz in c=d=0.8. The horizontal axis represents the parameter c, and the vertical axis represents frequency (Hz). Figure 15 also shows the 2D diagrams of the reticular geometric structures with c=d=0.1, 0.5, and 0.8.

Fig. 15. The band gap width with parameters c=d varying from 0.1 to 0.9 when the vertical range is selected from 3500 to 4500 Hz

On the other hand, the scale a in Fig. 12(b) varies from 0.1 to 2.0 along the x direction and the width of the unit cell along y direction remains 1.0 in the Bloch calculations. Changing the scale a from 0.1 to 2.0 can tune the full frequency band gaps of mixed polarization modes. Using detailed calculations of dispersion relations of reticular geometric structures with scale a=0.1 to 2.0, Fig. 16 shows the band gap widths with the scale a from 0.1 to 2.0 when the vertical range ranges from 2400 to 5200 Hz. The horizontal axis ranges from 0 to 2.0, and the vertical axis represents frequency (Hz). No full frequency band gap exists when the scale a are 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 1.5, 1.6, and 1.7. These results clearly show that changing the scale a can increase or decrease the full frequency band gap.

It is noted that the unit cells with *a*=0.5 and 2.0 are the same in the Bloch calculations. However, the dispersion phenomena is similar except for the scalar of the eigenmode frequencies in the vertical axis of dispersion relations. In both cases, there is only one total band gap of the mixed polarization modes. The location of the band gap ranges from approximately 5009 to 5017.4 Hz with *a*=0.5, while that for *a*=2.0 ranges from approximately 2504.5 to 2508.7 Hz.

The following discussion addresses several parameters of the reticular geometric in this chapter. First, the effect of filling fraction is discussed when the parameters c=d varied from 0.1 to 0.9 in Fig. 12(a). Figure 15 shows the distribution of the total band gaps of mixed polarization modes, in which only one total band gap appears at approximately 3560 ~ 3736 Hz in c=d=0.8. The horizontal axis represents the parameter c, and the vertical axis represents frequency (Hz). Figure 15 also shows the 2D diagrams of the reticular geometric

Fig. 15. The band gap width with parameters c=d varying from 0.1 to 0.9 when the vertical

On the other hand, the scale a in Fig. 12(b) varies from 0.1 to 2.0 along the x direction and the width of the unit cell along y direction remains 1.0 in the Bloch calculations. Changing the scale a from 0.1 to 2.0 can tune the full frequency band gaps of mixed polarization modes. Using detailed calculations of dispersion relations of reticular geometric structures with scale a=0.1 to 2.0, Fig. 16 shows the band gap widths with the scale a from 0.1 to 2.0 when the vertical range ranges from 2400 to 5200 Hz. The horizontal axis ranges from 0 to 2.0, and the vertical axis represents frequency (Hz). No full frequency band gap exists when the scale a are 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 1.5, 1.6, and 1.7. These results clearly show that

It is noted that the unit cells with *a*=0.5 and 2.0 are the same in the Bloch calculations. However, the dispersion phenomena is similar except for the scalar of the eigenmode frequencies in the vertical axis of dispersion relations. In both cases, there is only one total band gap of the mixed polarization modes. The location of the band gap ranges from approximately 5009 to 5017.4 Hz with *a*=0.5, while that for *a*=2.0 ranges from approximately

changing the scale a can increase or decrease the full frequency band gap.

structures with c=d=0.1, 0.5, and 0.8.

range is selected from 3500 to 4500 Hz

2504.5 to 2508.7 Hz.

Fig. 16. The band gap widths with the scale a from 0.1 to 2.0

Finally, the rotating angles of reticular geometric structures were changed to analyze the distribution of total band gaps. Figure 17 shows the 2D diagrams of unit rectangular lattices in different rotating angles D=30 deg, 45 deg, 75 deg, and 90 deg. In these cases, the widths of aluminum remain constant, 0.14R, in the reticular geometric structures with different rotating angles in the calculations. Figure 18 shows the band gap widths of rectangular lattices with different rotating angles of reticular geometric structures. Based on the symmetry of the geometry, the different angles in the Bloch calculations were adopted from 15 deg ~ 90 deg. In the calculated results, no band gap is detected from D=5 deg to 65 deg.

Fig. 17. 2D diagrams of unit rectangular lattices in different rotating angles D=30 deg, 45 deg, 75 deg, and 90 deg

Analysis of Acoustic Wave

ISSN 1539-3755

ISSN 1048-9002

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in Homogeneous and Inhomogeneous Media Using Finite Element Method 19

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Fig. 18. The band gap widths of the rectangular lattices with different rotating angles of reticular geometric structures
