# On the Influence of Certain Geometric Characteristics of the Resonator on the Impedance Determined by the Dean’s Method

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## Abstract

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## 1. Introduction

- evaluation of the effect of the orifice arrangement in the resonator cover at low height and low perforation degree of the resonator on the impedance determined by the Dean’s Formula (1);
- elimination of the inaccuracy in determining the impedance associated with a more complex modal structure of the sound field at the resonator backing.

## 2. Numerical Simulation of the Physical Process in a Resonator under the Incidence of Plane Waves

## 3. Calculation of Resonator Impedance Using the Normal Dean’s Formula

## 4. Impedance Determination by the Modified Dean’s Formula

## 5. Discussion

- the sound field at the resonator backing is non-uniform for different orifice arrangements;
- the sound pressure distribution at the resonator backing is different for different orifice arrangements.

- the amplitude coefficients of the zeroth order mode have been determined;
- the amplitude coefficients of the zeroth order mode turn out to be almost the same for different orifice arrangements.

- impedance does not depend on the orifice arrangement in the resonator cover;
- with a uniform orifice arrangement, the impedance is the same as the impedance determined by the normal Dean’s formula.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Triple-layer locally reacting liner: (

**a**) Overall view; (

**b**) Geometry of layers; p is perforation degree; h is height of resonant cavity.

**Figure 2.**Installation of microphones into the resonator cell for determination of the impedance by Dean’s method.

**Figure 5.**Distribution of acoustic pressure along the symmetry axis of the geometric model at a frequency of 3000 Hz: the red line is the real part; the blue line is the imaginary part; coordinate z = 0 is on the resonator surface.

**Figure 7.**Normalized impedance determined by the Formula (1): (

**a**) Real part; (

**b**) Imaginary part; Variant 1 of the orifice arrangement; Variant 2 of the orifice arrangement; Variant 3 of the orifice arrangement.

**Figure 8.**Array of points at the resonator backing for recording the acoustic pressure in numerical simulation.

**Figure 9.**Normalized impedance determined by Formula (4): (

**a**) Real part; (

**b**) Imaginary part; Variant 1 of the orifice arrangement; Variant 2 of the orifice arrangement; Variant 3 of the orifice arrangement.

Characteristic | Symbol | Value |
---|---|---|

Inner diameter of the resonator, mm | D | 30 |

Height of the resonant cavity, mm | h | 5 |

Perforated plate thickness, mm | t | 1 |

Orifice diameter, mm | d | 2 |

Number of the orifices | n | 7 |

Perforation degree, % | p | 3.1 |

Characteristic | Symbol | Value |
---|---|---|

Static pressure, Pa | ${p}_{0}$ | 101,325 |

Static temperature, C | ${T}_{0}$ | 20 |

Maximum element size, mm | ${\delta}_{\mathrm{max}}$ | 2.8 |

Minimum element size, mm | ${\delta}_{\mathrm{min}}$ | 0.028 |

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**MDPI and ACS Style**

Palchikovskiy, V.; Khramtsov, I.; Kustov, O. On the Influence of Certain Geometric Characteristics of the Resonator on the Impedance Determined by the Dean’s Method. *Acoustics* **2022**, *4*, 382-393.
https://doi.org/10.3390/acoustics4020023

**AMA Style**

Palchikovskiy V, Khramtsov I, Kustov O. On the Influence of Certain Geometric Characteristics of the Resonator on the Impedance Determined by the Dean’s Method. *Acoustics*. 2022; 4(2):382-393.
https://doi.org/10.3390/acoustics4020023

**Chicago/Turabian Style**

Palchikovskiy, Vadim, Igor Khramtsov, and Oleg Kustov. 2022. "On the Influence of Certain Geometric Characteristics of the Resonator on the Impedance Determined by the Dean’s Method" *Acoustics* 4, no. 2: 382-393.
https://doi.org/10.3390/acoustics4020023