Wave-number analysis of the measurement of finite absorberswith a microphone array

Brandão, E. Acoustical Engineering & Civil Engineering Graduate Program, Federal University of Santa Maria (UFSM), Santa Maria-RS, 97050-140, Brazil, eric.brandao@eac.ufsm.br

Fernandez-Grande, E. Department of Electrical Engineering, Technical University of Denmark (DTU), Ørsteds Plads, Building 352, 2800, Kgs. Lyngby, Denmark, efg@elektro.dtu.dk

Abstract

This paper aims at analyzing the effect of edge diffraction on the in situ impedance measurement with an array of microphones. An analytical solution for the problem is presented, based on a two-dimensional Fourier transform of the sound field above a rectangular sample. A simplified Boundary Element Method (BEM) is used to emulate the measurement of finite samples with a microphone array, and the wave-number spectrum is estimated by a regularized solution of the inverse problem. The properties of the observed wave-number spectrum are associated either with the specular reflection or with the diffracted components, caused by the interaction of the sound wave with the finite absorber. The absorption coefficient is retrieved from the regularized solution and the influence of the size and shape of the samples is also investigated.

Keywords:

impedance measurement

diffraction

microphone array

wave-number domain.

References

[1] ISO 10534-2: Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes-Part 2: Transfer-function method. Technical report, International Organization for Standardization, 2001.

[2] ISO 354: Measurement of sound absorption in a reverberation room. Technical report, International Organization for Standardization, 2003.

[3] Thomasson, S. I. On the absorption coefficient. Acustica, 44:265–273, 1980.

[4] Bruijn, A. A mathematical analysis concerning the edge effect of sound absorbing materials. Acustica, 28:33–44, 1973.

[5] Pereira, M; Mareze, PH; Godinho, L; Amado- Mendes, P e Ramis, J. Proposal of numerical models to predict the diffuse field sound absorption of finite sized porous materials–bem and fem approaches. Applied Acoustics, 180:1–13, 2021.

[6] Brandão, E.; Lenzi, A. e Paul, S. A review of the in situ impedance and sound absorption measurement techniques. Acta Acustica united with Acustica, 101: 443 – 463, 2015.

[7] Kimura, K. e Yamamoto, K. The required sample size in measuring oblique incidence absorption coefficient experimental study. Applied Acoustics, 63:567–578, 2002.

[8] Hirosawa, K.; Takashima, K.; Nakagawa, H.; Kon, M.; Yamamoto, A. e Lauriks, W. Comparison of three measurement techniques for the normal absorption coefficient of sound absorbing materials in the free field. The Journal of the Acoustical Society of America, 126 (6):3020–3027, 2009.

[9] Brandão, E.; Lenzi, A. e Cordioli, J. Estimation and minimization of errors caused by sample size effect in the measurement of the normal absorption coefficient of a locally reactive surface. Applied Acoustics, 73: 543 – 556, 2012.

[10] Tamura, M. Spatial Fourier transform method of measuring reflection coefficients at oblique incidence. I: Theory and numerical examples. The Journal of the Acoustical Society of America, 88(5):2259 – 2264, 1990.

[11] Tamura, M.; Allard, J.F. e Lafarge, D. Spatial Fouriertransform method for measuring reflection coefficients at oblique incidence. II: Experimental results. The Journal of the Acoustical Society of America, 97 (4):2255 – 2262, 1995.

[12] Richard, A. e Fernandez-Grande, E. Comparison of two microphone array geometries for surface impedance estimation. The Journal of the Acoustical Society of America, 146(1):501 – 504, 2019.

[13] Nolan, M. Estimation of angle-dependent absorption coefficients from spatially distributed in situ measurements. The Journal of the Acoustical Society of America, 147(2):EL119–EL124, 2020.

[14] Richard, A.; Fernandez-Grande, E.; Brunskog, J. e Jeong, C-H. Estimation of surface impedance at oblique incidence based on sparse array processing. The Journal of the Acoustical Society of America, 141 (6):4115 – 4125, 2017.

[15] Hald, J.; Song,W.; Haddad, K.; Jeong, C-H e Richard, A. In-situ impedance and absorption coefficient measurements using a double-layer microphone array. Applied Acoustics, 143:74 – 83, 2019.

[16] Ottink, M.; Brunskog, J.; Jeong, C-H; Fernandez- Grande, E.; Trojgaard, P. e Tiana-Roig, E. In situ measurements of the oblique incidence sound absorption coefficient for finite sized absorbers. The Journal of the Acoustical Society of America, 139(1):41 – 52, 2016.

[17] Richard, A.; Fernández Comesaña, D.; Brunskog, J.; Jeong, C-H e Fernandez-Grande, E. Characterization of sound scattering using near-field pressure and particle velocity measurements. The Journal of the Acoustical Society of America, 146(4):2404 – 2414, 2019.

[18] Brandão, Eric e Fernandez-Grande, Efren. Analysis of the sound field above finite absorbers in the wavenumber domain. The Journal of the Acoustical Society of America, 151(5):3019–3030, 2022.

[19] Brunskog, Jonas. The forced sound transmission of finite single leaf walls using a variational technique. The Journal of the Acoustical Society of America, 132 (3):1482–1493, 2012.

[20] Maynard, Julian D; Williams, Earl G e Lee, Y. Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH. The Journal of the Acoustical Society of America, 78(4): 1395–1413, 1985.

[21] Hald, J. Basic theory and properties of statistically optimized near-field acoustical holography. The Journal of the Acoustical Society of America, 125(4):2105– 2120, 2009.

[22] Jacobsen, Finn e Jaud, Virginie. Statistically optimized near field acoustic holography using an array of pressure-velocity probes. The Journal of the Acoustical Society of America, 121(3):1550–1558, 2007.

[23] Nolan, Mélanie; Verburg, Samuel A; Brunskog, Jonas e Fernandez-Grande, Efren. Experimental characterization of the sound field in a reverberation room. The Journal of the Acoustical Society of America, 145(4): 2237–2246, 2019.

[24] Hansen, Per Christian. Discrete inverse problems: insight and algorithms. 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA: Siam, 2010. volume 7. Sec. 5.3.

[25] Miki, Yasushi. Acoustical properties of porous materials-Modifications of Delany-Bazley models. Journal of the Acoustical Society of Japan (E), 11 (1):19–24, 1990.

[26] Allard, J.F. e Atalla, N. Propagation of sound in porous media: modeling sound absorbing materials. West Sussex, PO19 8SQ, United Kingdom: Wiley, 2009. Cap. 11.

[27] Williams, Earl G e Maynard, Julian D. Numerical evaluation of the rayleigh integral for planar radiators using the fft. The Journal of the Acoustical Society of America, 72(6):2020–2030, 1982.

[28] Jacobsen, Finn e Liu, Yang. Near field acoustic holography with particle velocity transducers. The Journal of the Acoustical Society of America, 118(5):3139– 3144, 2005.