Modification of Dean's Method for Determining Impedance with an Inhomogeneous Sound Field in a Resonator

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Abstract

A modification of Dean’s method is proposed for determining the impedance in the case of a nonuniform sound field on the front and bottom surfaces of a resonator. Instead of acoustic pressures in Dean’s formula, the modification uses the coefficients of eigenfunctions, which correspond to a uniform acoustic pressure distribution on the front and bottom surfaces of the resonator. The eigenproblem is solved by the finite element method; the coefficients of the eigenfunctions are found by the least squares method. At the current stage of research, the full-scale experiment has been replaced by numerical simulation in a linear formulation of sound propagation in an impedance tube with normal wave incidence with a honeycomb resonator attached to it. The inhomogeneity of the pressure field over the cross section of the resonator is created from the different positions of holes in the resonator face plate. The study is done for a different number of acoustic pressure measurement points at the bottom of the resonator. Calculations show that the proposed method is efficient and provides good agreement with the straight method for determining impedance. However, the possibilities of using modification of Dean’s method in full-scale measurements are limited, because accurate resonator impedance determination requires a large number of measurement points.

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About the authors

V. V. Palchikovsky

Perm National Research Polytechnic University

Author for correspondence.
Email: vvpal@pstu.ru
Russian Federation, Perm

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Supplementary files

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2. Fig. 1. Examples of some of the resonator shapes used in sound absorbing cladding: (a) - cylindrical resonator, (b) - triangular resonator, (c) - square/rectangular resonator, (d) - honeycomb resonator

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3. Fig. 2. (a) - Fragment of the geometry of the computational domain and (b) - finite element mesh

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4. Fig. 3. Variants of location of perforation holes and points of acoustic pressure reading on the front surface of the resonator (shown in red): (a) - perforation 1, (b) - perforation 2, (c) - perforation 3

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5. Fig. 4. Amplitudes of acoustic pressure at the bottom of the resonator and their difference for different variants of perforation location: (a) - 2000 Hz, (b) - 4000 Hz, (c) - 6000 Hz

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6. Fig. 5. Reading points of acoustic pressures at the bottom of the honeycomb resonator: (a) - array 1, (b) - array 2, (c) - array 3, (d) - array 4

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7. Fig. 6. Normalised impedance obtained by Dean's method using different formulas: (a) - array 1, (b) - array 2, (c) - array 3; (d) - array 4. ------- perforation 1, f. (1); ------- perforation 2, f. (1); ------- perforation 3, f. (1); - - perforation 1, f. (3); - - perforation 2, f. (3); - perforation 3, f. (3); -- perforation 1, f. (6); -- perforation 2, f. (6); -- perforation 3, f. (6)

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8. Fig. 7. Comparison of impedance obtained by the direct method and the classical Dean method: -- perforation 1, f. (7); -- perforation 2, f. (7); -- perforation 3, f. (7); ------- perforation 1, f. (1); ------- perforation 2, f. (1); ------- perforation 3, f. (1)

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9. Fig. 8. Comparison of the impedance obtained by formulae (10) and (11): -- perforation 1, f. (11); -- perforation 2, f. (11); -- perforation 3, f. (11); ------- perforation 1, f. (10); ------- perforation 2, f. (10); ------- perforation 3, f. (10)

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