# 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

- ISO 10534-1; Acoustics—Determination of Sound Absorption Coefficient and Impedance in Impedance Tubes. Part 1: Method Using Standing Wave Ratio. ISO: Geneva, Switzerland, 1996.
- ISO 10534-2; Acoustics—Determination of Sound Absorption Coefficient and Impedance in Impedances Tubes. Part 2: Transfer-function Method. ISO: Geneva, Switzerland, 1998.
- Chung, J.Y.; Blaser, D.A. Transfer function method of measuring in-duct acoustic properties. I. Theory. J. Acoust. Soc. Am.
**1980**, 68, 907–913. [Google Scholar] [CrossRef] - Chung, J.Y.; Blaser, D.A. Transfer function method of measuring in-duct acoustic properties. II. Experiment. J. Acoust. Soc. Am.
**1980**, 68, 914–921. [Google Scholar] [CrossRef] - Watson, W.R. A New Method for Determining Acoustic-Liner Admittance in a Rectangular Duct with Grazing Flow from Experimental Data; Technical Report No. TP-2310; NASA: Hampton, VA, USA, 1984.
- Watson, W.R. A New Method for Determining Acoustic-Liner Admittance in Ducts with Sheared Flow in Two Cross-Sectional Directions; Technical Report No. TP-2518; NASA: Hampton, VA, USA, 1985.
- Parrot, T.L.; Watson, W.R.; Jones, M.G. Experimental Validation of a Two-Dimensional Shear-Flow Model for Determining Acoustic Impedance; Technical Report No. TP-2679; NASA: Hampton, VA, USA, 1987.
- Watson, W.R.; Jones, M.G.; Tanner, S.E.; Parrot, T.L. A Finite Element Propagation Model for Extracting Normal Incidence Impedance in Nonprogressive Acoustic Wave Fields; Technical Report No. TM-110160; NASA: Hampton, VA, USA, 1995.
- Jing, X.; Peng, S.; Sun, X. A straightforward method for wall impedance eduction in a flow duct. J. Acoust. Soc. Am.
**2008**, 124, 227–234. [Google Scholar] [CrossRef] [PubMed] - Watson, W.R.; Jones, M.G. A comparative study of four impedance eduction methodologies using several test liners. In Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference, Berlin, Germany, 27–29 May 2013. [Google Scholar]
- Armstrong, D.L.; Beckemeyer, R.; Olsen, R.F. Impedance measurements of acoustic duct liners with grazing flow. J. Acoust. Soc. Am.
**1974**, 55, S59. [Google Scholar] [CrossRef] - Aurégan, Y.; Leroux, M.; Pagneux, V. Measurement of Liner Impedance with Flow by an Inverse Method. In Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference, Manchester, UK, 10–12 May 2004. [Google Scholar]
- Sobolev, A.F. Determination of impedance of liners on installation “Interferometer with the flow”. In Proceedings of the 1-st All-Russian Acoustical Conference, Moscow, Russian, 6–9 October 2014. [Google Scholar]
- Sobolev, A.F.; Ostrikov, N.N.; Anoshkin, A.N.; Palchikovskiy, V.V.; Burdakov, R.V.; Ipatov, M.S.; Ostroumov, M.N.; Yakovets, M.A. Comparison of liner impedance derived from the results of measurements at two different setups using a small number of microphones. PNRPU Aerosp. Eng. Bull.
**2016**, 45, 89–113. [Google Scholar] [CrossRef] - Elnady, T.; Boden, H. An inverse analytical method for extracting liner impedance from pressure measurements. In Proceedings of the 10th AIAA/CEAS Aeroacoustics Conference, Manchester, UK, 10–12 May 2004. [Google Scholar]
- Ostrikov, N.N.; Yakovets, M.A.; Ipatov, M.S. Experimental Confirmation of an Analytical Model of the Sound Propagation in a Rectangular Duct in the Presence of Impedance Transitions and Development of an Impedance Eduction Method Based on it. Acoust. Phys.
**2020**, 66, 105–122. [Google Scholar] [CrossRef] - Dean, P.D. An in-situ method of wall acoustic impedance measurement in flow duct. J. Sound Vib.
**1974**, 34, 97–130. [Google Scholar] [CrossRef] - Gaeta, R.J.; Mendoza, J.M.; Jones, M.G. Implementation of in-situ impedance techniques on a full-scale aero-engine system. In Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), Rome, Italy, 21–23 May 2007. [Google Scholar]
- Khramtsov, I.; Kustov, O.; Palchikovskiy, V.; Ershov, V. Investigation of the reason for the difference in the acoustic liner impedance determined by the transfer function method and Dean’s method. Akustika
**2021**, 39, 224–229. [Google Scholar] [CrossRef] - Anoshkin, A.N.; Zakharov, A.G.; Gorodkova, N.A.; Chursin, V.A. Computational and experimental studies of resonance sound-absorbing multilayer structures. PNRPU Mech. Bull.
**2015**, 5–20. [Google Scholar] [CrossRef] - Kustov, O.Y.; Khramtsov, I.V.; Palchikovskiy, V.V.; Bulbovich, R.V. Comparison of acoustic characteristics of resonant liner samples at normal incidence of waves based on semiempirical model, natural experiment and numerical simulation. AIP Conf. Proc.
**2021**, 2351, 1–9. [Google Scholar]

**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