# Mathematical Models and Experiments on the Acoustic Properties of Granular Packing Structures (Measurement of Tortuosity in Hexagonal Close-Packed and Face-Centered Cubic Lattices)

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## Abstract

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## 1. Introduction

## 2. Samples and Measuring Device Used to Measure the Sound Absorption Coefficient

#### 2.1. Transmission Loss Measurement

#### 2.2. Measurement Equipment for Sound Absorption Coefficient

## 3. Measurement Method and Results of Tortuosity

#### 3.1. Overview of Tortuosity

_{∞}is expressed as Equation (1) using the sound velocity c

_{0}in the air and the apparent sound velocity c in the packed structure [11]:

#### 3.2. Tortuosity Measurement

_{∞}at each frequency was measured for each packed structure. Then, the reciprocal of the square root of the frequency used for the measurement was determined as the value of the horizontal axis, and the tortuosity α

_{∞}at each frequency obtained by the measurement was plotted as the value of the vertical axis. The linear approximation of these point clouds leads to a soaring straight line using the least-squares method. When the frequency of the approximate line is set to infinity, the limiting value of the tortuosity, i.e., the y-intercept of the graph, becomes the tortuosity α

_{∞}of the packing structure.

_{∞}of each packing structure. At 40 kHz, the deviation was excluded for the hexagonal structure. The tortuosity of each packed structure was 1.44 for the hexagonal close-packed structure and 1.43 for the face-centered cubic lattice.

## 4. Theoretical Analysis

#### 4.1. Analysis Units and Element Division

_{n}and real volume V

_{n}of the clearance were geometrically calculated and approximated to the thickness of the clearance between two planes b

_{n}, as shown in Figure 7b and Equation (2).

_{h}of the wall of the sample holder was also taken into consideration, and it was approximated to the clearance between two planes. As shown in Figure 8a,b, when the clearance thickness is set to b

_{n}’, S

_{n}, S

_{h}, and V

_{n}are expressed as Equation (3):

#### 4.2. Derivation of the Surface Area of a Sphere in a Divided Element

_{n}of spheres in the divided element for the hexagonal close-packed and face-centered cubic lattices used in Equations (2) and (3).

_{unit}of the dividing element in the x-axis direction, the surface area S

_{n}of the sphere at the dividing element can be obtained by Equation (4):

_{h}of the blue portion of the sample holder wall was considered for the divided element in contact with the sample holder.

#### 4.3. Derivation of the Volume of the Clearance in a Divided Element

_{n}of clearances in the divided element of the hexagonal close-packed and face-centered cubic lattice structures.

_{n}, for the hexagonal close-packed structure, shows the clearance in the range surrounded by the broken line in Figure 5a and is derived geometrically using the dimensions of the radius r of the sphere and the length Z

_{unit}of the divided element in the x-axis direction, as follows:

_{n}, for the face-centered cubic structure, shows the clearance in the range surrounded by the broken line in Figure 5b, which is geometrically derived as follows:

#### 4.4. Propagation Constant and Characteristic Impedance Considering Tortuosity

_{s}and compressibility C

_{s}were obtained from a three-dimensional analysis using Equations (7) and (8) [14], respectively, using the Navier–Stokes equations, the equation of state of gas, the continuity equation, the energy equation, and the dissipation function representing heat transfer, where ρ

_{0}is the density of the air, λ

_{s}is an intermediary variable, b

_{n}is the clearance thickness between two planes, ω is the angular frequency, η is the viscosity of the air, κ is the specific heat ratio of the air, P

_{0}is the atmospheric pressure, and N

_{pr}is the Prandtl number.

_{s}multiplied by the tortuosity α

_{∞}, the propagation constant and characteristic impedance considering the tortuosity can be obtained [16]. Therefore, the propagation constant γ and the characteristic impedance Z

_{c}can be expressed by the following equations [16] in terms of the effective density ρ

_{s}and compressibility C

_{s}when the tortuosity α

_{∞}is considered:

#### 4.5. Transfer Matrix

_{n}; and the four-terminal constants of the acoustic tube element, A to D, can be calculated using Equation (11):

_{1}and u

_{1}at Plane 1 and p

_{2}and u

_{2}at Plane 2, respectively, and the transfer matrix can be expressed as Equation (12):

_{unit}and T

_{top}of the analysis unit and the analysis unit at the upper edge of the sample were obtained by cascading the transfer matrix of each divided element based on the equivalent circuit shown in Figure 11.

#### 4.6. Normal Incident Sound Absorption Coefficient

_{all}obtained in Section 4.5. Since the end of the specimen used in this study was a rigid wall, the particle velocity u

_{2}= 0, Equation (12), can be deformed as shown in Equation (13), yielding Equation (14):

_{0}and u

_{0}, the specific acoustic impedance Z

_{0}from the incident surface of the sample is expressed as follows:

_{0}= p

_{1}, S

_{0}u

_{0}= Su

_{1}, and Equation (15), the specific acoustic impedance Z

_{0}of the sample is expressed as

_{0}is the aperture ratio of the sample shown in Table 1.

_{0}and the reflectance R is expressed by the following equation:

## 5. Comparison between Measured and Theoretical Values

## 6. Conclusions

- In both packing structures, the real area of the granular surface and the real volume of the clearance were obtained geometrically and analyzed theoretically.
- In both packing structures, the peak frequency tended to appear at a higher frequency than the measured value when the tortuosity was not considered.
- In the theoretical sound absorption, the peak value was higher when the tortuosity was considered compared to that without the consideration of the tortuosity (the peak frequency moved to a lower frequency). As a result, the theoretical value was becoming closer to the measured value.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## References

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**Figure 1.**Test samples: (

**a**) Hexagonal close-packed lattice (d = 4 mm); (

**b**) Hexagonal close-packed lattice (d = 8 mm); (

**c**) Face-centered cubic lattice (d = 4 mm); (

**d**) Face-centered cubic lattice (d = 8 mm).

**Figure 2.**Configuration diagram of a two-microphone impedance tube for the absorption coefficient measurement.

**Figure 5.**Analysis unit in each structure: (

**a**) Hexagonal close-packed lattice; (

**b**) Face-centered cubic lattice.

**Figure 6.**Analysis unit at the top of the sample: (

**a**) Hexagonal close-packed lattice; (

**b**) Face-centered cubic lattice.

**Figure 7.**Divided element approximated to the clearance between two planes: (

**a**) Divided element (face-centered cubic); (

**b**) Approximated clearance between two planes.

**Figure 8.**Approximated clearance between two planes at the element in contact with the sample holder: (

**a**) Divided element in contact with the sample holder on two sides; (

**b**) Divided element in contact with the sample holder on one side.

**Figure 11.**Equivalent circuit in the analysis unit (cascade connecting the transfer matrix of each element).

**Figure 13.**Comparison between the experimental and calculated values (considering tortuosity in Section 3.2 and tortuosity in Reference (Lee et al. 2009) [5]) of the peak frequency and the sound absorption coefficient: (

**a**) Hexagonal close-packed lattice (d = 4 mm); (

**b**) Hexagonal close-packed lattice (d = 8 mm); (

**c**) Face-centered cubic lattice (d = 4 mm); and (

**d**) Face-centered cubic lattice (d = 8 mm).

Packing Structure | Diameter [mm] | Length [mm] | Aperture Ratio of Sample Holder | Filling Rate | Measured Tortuosity | Correspondence to Figure |
---|---|---|---|---|---|---|

Hexagonal close-packed | 4 | 27 | 0.57 | 0.74 | 1.44 | 1a |

8 | 27 | 0.67 | 0.74 | 1.44 | 1b | |

Face-centered cubic | 4 | 22 | 0.58 | 0.74 | 1.43 | 1c |

8 | 21 | 0.85 | 0.74 | 1.43 | 1d |

Frequency [kHz] | Inverse of the Square Root of Frequency $[1/\sqrt{\mathbf{Hz}}]$ | Distance between Sensors [mm] | Transmission Time [ms] | Tortuosity | |
---|---|---|---|---|---|

Hexagonal Close-Packed | Face-Centered Cubic | ||||

32.7 | 0.00559 | 395 | 1.229 | 4.78 | 4.48 |

40 | 0.005 | 395 | - | - | 3.35 |

58 | 0.004152 | 395 | 1.199 | 4.31 | 3.98 |

110 | 0.003015 | 345 | 1.044 | 3.59 | 3.36 |

150 | 0.002582 | 204 | 0.626 | 3.11 | 2.84 |

200 | 0.002236 | 204 | 0.618 | 2.56 | 2.67 |

300 | 0.001826 | 204 | 0.615 | 2.56 | 1.93 |

∞ | 0 | - | - | 1.44 | 1.43 |

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**MDPI and ACS Style**

Sakamoto, S.; Suzuki, K.; Toda, K.; Seino, S.
Mathematical Models and Experiments on the Acoustic Properties of Granular Packing Structures (Measurement of Tortuosity in Hexagonal Close-Packed and Face-Centered Cubic Lattices). *Materials* **2022**, *15*, 7393.
https://doi.org/10.3390/ma15207393

**AMA Style**

Sakamoto S, Suzuki K, Toda K, Seino S.
Mathematical Models and Experiments on the Acoustic Properties of Granular Packing Structures (Measurement of Tortuosity in Hexagonal Close-Packed and Face-Centered Cubic Lattices). *Materials*. 2022; 15(20):7393.
https://doi.org/10.3390/ma15207393

**Chicago/Turabian Style**

Sakamoto, Shuichi, Kyosuke Suzuki, Kentaro Toda, and Shotaro Seino.
2022. "Mathematical Models and Experiments on the Acoustic Properties of Granular Packing Structures (Measurement of Tortuosity in Hexagonal Close-Packed and Face-Centered Cubic Lattices)" *Materials* 15, no. 20: 7393.
https://doi.org/10.3390/ma15207393