Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Voume 9, 23 ICA 23 Montrea Montrea, Canada 2-7 June 23 Architectura Acoustics Session 4pAAa: Room Acoustics Computer Simuation II 4pAAa9. Numerica modes for predicting absorption/insuation performance of acoustic eements Naohisa Inoue* and Tetsuya Sakuma *Corresponding author's address: Univ. of Tokyo, Kashiwa, , Chiba, Japan, WIth a great improvement oomputer resource avaiabiity, numerica anaysis is widey used to investigate acoustic characteristics of various materias. A further expectation wi be to predict absorption/insuation performance of acoustic eements used for buidings, automobie and so on, which can be the aternative to the actua measurements. This paper presents genera numerica modes for predicting the absorption coefficient and the transmission oss of acoustic eements with arbitrary shape and materia composition. The features of the modes are :) A test sampe is mounted in the cavity or aperture on a thick rigid baffe :2) FEM is empoyed for the materias and the air in the cavity, and couped with sound fieds out of the baffe by BEM :3) Acoustica indices are cacuated from the incidence power and the absorption/transmission power on the interfaces. Numerica simuation demonstrates the obique incidence absorption coefficients and transmission osses of singe ayered materias. The infuence of the area and depth of the cavity/aperture are discussed in comparison with theoretica vaues for the infinite area. Pubished by the Acoustica Society of America through the American Institute of Physics 23 Acoustica Society of America [DOI: /.48278] Received 22 Jan 23; pubished 2 Jun 23 Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page

2 INTRODUCTION With a great improvement oomputer resource avaiabiity, numerica anaysis is widey used to investigate acoustic characteristics of various materias. A further expectation wi be to predict absorption/insuation performance of acoustic eements used for buidings, automobie and so on, which can be the aternative to the actua measurements. Severa researchers have reported the numerica modes and methods [-3] to provide the obique incidence acoustic indices for the finite materia. Amost a works buid the particuar modes for absorption and/or transmission probem in order to simpify the probems. However, It is desirabe to use a coherent mode for the evauation of the absorption and insuation performances of same materia. This paper presents genera numerica modes for predicting the absorption coefficient and the transmission oss of acoustic eements with arbitrary shape and materia composition as shown in Figure. The features of the modes are: ) A test sampe is mounted in the cavity or aperture on a thick rigid baffe: 2) FEM is empoyed for the materias and the air in the cavity, and couped with sound fieds out of the baffe by BEM on the imaginary interface(s): 3) Acoustica indices are cacuated from the incidence power and the absorption/transmission power on the imaginary interface(s). Numerica anayses by the proposed modes are carried out on two probems; one is about the area effect in the absorption coefficient anaysis, and the other is about niche effect in the transmission oss anaysis. In order to vaidate the proposed modes, these resuts are compared with theoretica vaue, and with experimenta vaue in absorption probem. infinite rigid baffe (a) (b) Ω (c) Semi-infinite air space: i y θ y Imaginary boundary: Γi materia Air space: ΩA x x Impedance boundary: Γ A z Δθ φ θ z incident pane wave φ θ z incident pane wave Porous materia: Ω P Pate: Γ pt Imaginary boundary: Γt Semi-infinite air space: Ω t x θ φ Δφ y FIGURE. Schematic of the cacuation modes. (a) Three-dimensiona absorption and transmission mode (b) Domain notation in this artice (c) discrete direction mode for random incidence from π/2 space. THEORETICAL DESCRIPTION The discrete systems of the proposed mode are formuated for air space inside and outside the thick baffe. On the boundaries and interfaces, norma directions are defined as the outward to each domain, and partice veocities v f are defined as the inward to each domain. Formuation of the Acoustic Pressure Fied Outside of the Thick Baffe On the fat imaginary interfaces Γ i and Γ t, discretizing the Kirchhoff-Hemhotz integra equation taken into account the mirror image gives the foowing matrix equations: Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 2

3 () where, p B is sound pressure, v f B is partice veocity, [I] is unit matrix, and [G] is the infuence matrix and its components are given as foow: (2) where, ρ is air density and k is wave number in air. Externa force to the imaginary interface Γ i is given as the pane wave incidence from each direction that is given by discretizing the π/2 space into approximatey equisoid anges as shown in Fig. (c). Assuming the unit-ampitude veocity potentia, components of externa force matrix [D] is given as foow where, k is wave number vector representing propagation direction of pane wave. Inside of the Thick Baffe In the acoustic fied of air space Ω a in cavity/aperture, foowing matrix equation is given by discretized Hemhotz equation in a weak form. (3) (4) where, p F is sound pressure, v f F is partice veocity and [K a ]/[M a ]/[C a ] are eastance/inertance/dissipative matrices for acoustic fied, given as foow. Couping Conditions Vibro-acoustic system within the thick baffe is couped with semi-infinite acoustic fieds out of the baffe through the air space at the imaginary interfaces Γ i and Γ t. There are two approaches to coupe acoustic fieds formuated by BEM and FEM according to the interpoation scheme of sound pressure in BEM eements. One is interpoation eement formuation. Couping condition in this approach is attained by the eimination of degree of freedom in common on the interface boundary. The other approach is constant eement formuation. In this approach, unknowns of sound pressure are defined at different points in BEM and FEM, thus the eimination of noda unknowns cannot be appied. Aternativey, foowing couping conditions are imposed to each eement. Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 3

4 First equation (a) represents the continuity of the forces across an interface eement. Foowing matrix equation can be obtained by assembing equations to a interface eements. (a) (b) (6) where, Δ e is area of a couping eement. Second equation (b) represents the continuity of the partice veocity in a couping eement. Partice veocity on the interface is assumed to be constant in the eement both in BEM and FEM. Thus the conformity matrix [Q a ] is obtained by (7) Finay, the couped system is composed of above reations as foow: (8) Cacuation of Acoustic Indices First, supposing the incident sound pressure given as eq. (3), incidence power at an incidence ange θ is obtained by. (9) where, S is the area of an interface on incidence side and c is sound speed. Second, absorption and transmission power at the incidence ange is obtained by. () where I n is net active intensity passing interfaces, and represented by. () Subsequenty, absorption and transmission coefficients at the incidence ange are obtained respectivey by. (2) Finay, statistica acoustic indices are obtained by the average with weighting in accordance with incident ange as foow.,, (3) Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 4

5 NUMERICAL EXAMPLES AND DISCUSSION Foowing cacuations have been carried out at every /24-octave bands center frequencies. In case of transmission oss anaysis, /3-octave bands vaues are converted from the resuts, and shown with thicker ines. Incidence anges are given by discretizing the soid ange π/2 into 87 directions. Investigation on Absorption Coefficient The method presented in this artice invoves area effect in absorption coefficient. Accordingy, the cavity and materia geometries, such as area, depth and so on, greaty affect the resut, which is examined in the cacuation of the statistica absorption coefficient for singe ayer porous materia set on the cubic cavity. In the porous materia domain Ω P, the effective density ρ eff and the effective soun d speed c eff are cacuated by using Kato s mode [], and the absorption eement proposed by Craggs [4] is used. Tabe. shows the properties of hair fet as a test specimen. For reference, marking the theoretica vaue for the infinite materia by transfer matrix method (TMM) [6] and aso the measured vaues by impedance tube and reverberation room method. Effect of Cavity Area FIGURE.2 shows norma, random and fied incidence absorption coefficients cacuated for the different cavity and materia area. Materia thickness is 2 mm and cavity depth is mm. It is seen that the genera tendency of norma incidence absorption coefficient obeys the theory for infinite materia. However, the discrepancy is aso recognized in mid-frequency range. This is because of the resonance for the cavity depth and the effect of diffraction. In random and fied incidence conditions, absorption coefficients remarkaby exceed the theoretica vaue for smaer specimens. And at high frequency range, absorption coefficients approach the theoretica vaue. These tendencies are we-known behaviors of area effect. The remarkabe discrepancy is observed between the cacuation and measured vaues at mid- and highfrequency ranges. It may be caused by the difference between the height of the fence surrounding materia side in the measurement and the cavity depth set in the cacuation. Effect of Cavity Depth FIGURE.3 shows norma, random and fied incidence absorption coefficients cacuated for the different cavity depth. Cavity and materia have a 2 m square in common. In genera, tendencies resembe those mentioned in the effect oavity area. However, in case of random and fied incidence, remarkabe peaks appear in the resuts for materia set in deeper cavity. They are considered to be the resonance to opposed faces of the cavity. As the cavity become deeper, absorption coefficients remarkaby approach the theoretica vaue at mid-frequency range, which support the vaidity of deep-we approach in absorption coefficient measurement by reverberation room method. TABLE. Physica properties and dimensions of a hair fet for the cacuation Property Name and Symbo Vaue Dimension Materia Density: ρ s 86 [kg/m 3 ] Buk Density: ρ [kg/m 3 ] Fiber Diameter: D 2 µm Thickness: t 2 mm Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page

6 Absorption Coefficient (a) Norma (b) Random (c) Fied Meas. m. 2 Area a m 4. : a 2 m 2 9. m 2 a m TMM Meas. (Imp. Tube) FIGURE 2. Absorption coefficients for the porous materia of different materia area, set on the bottom oavity with mm depth, (a) norma incidence, (b) random incidence and (c) fied incidence conditions.. Absorption Coefficient. (a) Norma d mm 2 4 mm TMM Meas. (Imp. Tube) 2 mm. (b) Random Meas. m 2. (c) Fied FIGURE 3. Absorption coefficients for the porous materia set on the bottom oavity with different depth, (a) norma incidence, (b) random incidence and (c) fied incidence conditions. Investigation on Transmission Loss Proposed method invoves niche effect as for the transmission probem because of the difference between materia and baffe thicknesses. Thus, the effect of the niches is examined by the cacuation of transmission oss of a pate mounted in a cubic aperture. Appying the FEM (with non-conforming trianguar eements [7] ) to the pate vibration equation, and the couped finite eement system is composed of air space and pate vibration in the aperture. Aperture and a pate have a m square, and the pate is supported simpy. Tabe.2 shows the properties of a gass pane as a test pate. For reference, marking the theoretica vaues for infinite pate by TMM and mass aw, and aso for finite pate by Sewe s expression [8]. Effect of Aperture Depth FIGURE.4 shows norma, random and fied incidence transmission osses cacuated for the different baffe thicknesses. The specimen is paced at the center of the thick baffe. Norma incidence transmission osses approximatey foow mass aw in the non-resonance frequency. As the baffe is thicker, transmission oss is owered at ow-frequency range, and the baffe thickness is reated to the sharpness of the dips caused by resonances at mid-frequency. In random and fied incidence condition, transmission osses obey Sewe s expression except for resonance frequencies. However, fied incidence transmission oss is generay,2 db ower compared to random incidence vaue. Under the critica frequency, specimens set in thicker baffe show ower transmission oss at non-resonance frequencies. Effect of Materia Position FIGURE. shows norma, random and fied incidence transmission osses cacuated for the specimen mounted on the different position in the aperture. The Specimen position is set such that the depth of the two sides are oppositey varied with the constant tota depth, + 2 = mm. Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 6

7 Norma incidence transmission oss foows mass aw in the non-resonance frequency. In random and fied incidence condition, reciprocity of the niche depths of the incident and transmitted sides is observed in the transmission oss. Transmission oss tends to be owered at mid-frequency range when the specimen is set nearby the center of aperture. Transmission Loss [db] TABLE.2 Physica properties and dimensions of a gass pane for the cacuation Property Name and Symbo Vaue Dimension Materia Density: ρ s kg/m 3 Young Moduus: E 7. N/m 3 Poisson s Ratio: ν 2 Loss Factor: η.2 Thickness: t. m FIGURE 4. Transmission oss for the pate set on the different thickness aperture: (a) norma incidence: (b) random incidence: (c) fied incidence conditions. Transmission Loss [db] (a) Norma (a) Norma inc. side mm /2 / (b) Random 2 (b) Random [mm] FIGURE. Transmission oss for the pate set on the different position of the aperture with mm thickness: (a) norma incidence: (b) random incidence: (c) fied incidence conditions. CONCLUSION This artice showed the genera numerica modes for predicting the absorption coefficient and the transmission oss of acoustic eements with arbitrary shape and materia composition in arbitrary incidence conditions. Through the case studies, foowing tendencies were observed for the absorption coefficients. Area effect is greater when the cavity is sma and shaow. As the cavity become deeper, the notabe effect of the resonances to the opposite faces oavity appears. There is remarkabe difference between random and fied incidence absorption coefficients. And for the transmission osses, foowing tendencies were observed. Niche effect is remarkabe for the specimen mounted in thicker baffe, and additionay the maximum effect appears by equa depth of niches. The reciprocity of niche depths is observed in statistica transmission oss. There is reativey minor difference between random and fied incidence transmission osses. These tendencies support that the proposa method gives vaid resuts judging from a theoretica point of view. - [mm] (c) Fied (c) Fied Mass Law Sewe Mass Law Sewe TMM TMM Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 7

8 ACKNOWLEDGMENTS This work is supported by HOWA TEXTILE INDUSTRY CO., LTD. REFERENCES. K. Hirosawa, Estimation of the absorption coefficient for pane porous materia with arbitrary construction based on wave theory, (in Japanese) PhD thesis, Kyushu University, (24) 2. Y. Kosaka and T.SakumaL. Numerica examination on the scattering coefficients of architectura surfaces using the boundary eement method, Acoust. Sci. Tech. 26, (). 3. T. Sakuma and T. Oshima, Appication of a vibro-acoustic method to prediction of sound insuation performance of buiding eement, Proc. INTER-NOISE 2, No.8 (2). 4. A. Craggs, Couping of finite eement acoustic absorption modes, J. Sound Vib., 66(4), 6-347, D. Kato, Predictive mode of sound propagation in porous materia: Extension of appicabiity in Kato mode, (in Japanese) J. Acoust. Sc. Jpn., 64, , (28) 6. J.F.Aard and N.Ataa, Propagation of Sound in Porous Media, 2nd Edition, WILEY, O. C. Zienkiewicz and R. L. Tayor, The finite eement method for soid and structura mechanics, Esevier,. 8. E. C. Sewe, Transmission of reverberant sound through a singe-eaf partition surrounded by an infinite rigid baffe, J. Sound. Vib. 2, 2-32, 97. Proceedings of Meetings on Acoustics, Vo. 9, 9 (23) Page 8