6. Acoustical simulation of straight and side inlet/outlet rectangular plenums using the FEM method
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1 Research Signpost 37/661 (2), Fort P.O. Trivandrum Kerala, India Noise Control: Theory, Application and Optimization in Engineering, 2014: ISBN: Editors: Min-Chie Chiu and J. H. Huang 6. Acoustical simulation of straight and side inlet/outlet rectangular plenums using the FEM method Ying-Chun Chang 1, Min-Chie Chiu 2 and Yi-HSin Chou 1 1 Department of Mechanical Engineering, Tatung University; 2 Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology Abstract. Noise can cause psychological and physiological ailments in industrial plants. Because of this, noise control must be seriously addressed. In order to reduce low frequency noise emitted from venting systems, plenum systems are customarily used. To find better mechanisms for plenums within a constrained space, eleven side inlet/outlet rectangular plenums (plenums A~K) and a straight rectangular plenum (plenum L) are proposed. These will be simulated using a finite element method (FEM) run on COMSOL, which is a commercial package. Before the FEM simulation is performed, the accuracy of the FEM is verified using analytical data from Ih. Consequently, the acoustical simulation of the plenum system using a FEM method proposed in this study is an efficient method for predicting the acoustical performance of the plenums with complicated mechanisms. Correspondence/Reprint request: Dr. Min-Chie Chiu, Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Taiwan 104, R.O.C.. minchie.chiu@msa.hinet.net
2 120 Ying-Chun Chang et al. 1. Introduction Noise causing psychological and physiological ailments is often found in industrial plants. And so, noise control is crucial [1, 2]. Various acoustical elements used in reducing sound energy have been widely assessed. In 1974, Cummings [3] investigated the acoustical performance of a curved duct having a rectangular section and a circular section. Later, Rostafinski [4] proposed a mathematical model for a sound wave that propagated along a curved duct. In 1999, Kim and Ih [5] predicted the acoustical efficiency of a curved expanded chamber using a four-pole matrix. In addition, much research has been directed toward the flow velocity and temperature gradient as they relate to the reactive muffler [6, 7, 8]. Kim and Ih [9] have also developed a sound-reduction mathematical model used in a sound energy transported along a curved and expanded duct that uses a four-pole transfer matrix. In order to depress the venting noise emitted from an industrial plant, a plenum system is customarily employed [10]. Munjal [11] analyzed the acoustical performance of a one-chamber side inlet/outlet plenum using the plane wave theory. Li and Hansen [12], in 2005, estimated the acoustical performance of a rectangular plenum using theoretical and experimental data. Later, in 2009, Liu and Herrin [13] evaluated the influence of the plenum s acoustical performance with respect to the perforated hole s distance using a finite element method. However, the plenum designs mentioned above focused on a one-chamber plenum in which the acoustical mechanism was a simple plenum without baffles. Therefore, the acoustical performance of the above plenums was limited. In order to enhance the acoustical performance of the plenum system, eleven side inlet/outlet rectangular plenums (plenums A~K) and a straight rectangular plenum (plenum L) within a fixed space will be introduced and simulated by using a finite element method (FEM) which is run on COMSOL. 2. Mathematical Model of the FEM (Run on the COMSOL Package) The boundary condition for the acoustical field (a solid boundary) of the plenum used in the acoustical model on the COMSOL package is 1 n { ( pt q)} = 0 (1) ρ c
3 FEM simulation on side inlet/outlet rectangular plenums 121 where q (a dipole sound source) is set at zero, c (the sound speed) is set at 343 (m/s), ρ (air density) is set at ( 3 kg m ). Assuming that a plane wave with a sound pressure (p) of 1 (Pa) propagates into the plenum s inlet, the governing equation of the sound wave yields 2 1 keq pt ( pt q ) Q ρ ρ = (2a) where t c b c p = p+ p (2b) k ω = ( ) (2c) 2 2 eq cc cc ρc = c (2d) = ρ (2e) Here, pb is the background pressure field. The Sound Transmission Loss (TL) is calculated as Win TL = 10log (3) W where out W is the inlet sound power and in W is the outlet sound power. out 3. Model check Before performing the acoustical simulation on plenums, an accuracy check of the FEM mathematical model on the fundamental elements of a straight type one-chamber plenum and a side inlet/outlet one-chamber plenum shown in Fig. 1 and Fig. 2 was performed using analytical data from Ih [14]. As revealed in Fig. 1 and Fig. 2, the FEM (run on the COMSOL package) and the analytical data for the two plenums are in agreement. Consequently, the simulation of eleven side inlet/outlet rectangular plenums (plenums A~K) and a straight rectangular plenum (plenum L) within a fixed space simulated by using the finite element method (FEM) is carried out in the following section.
4 122 Ying-Chun Chang et al. Figure 1. Accuracy check of a straight plenum compared to the Ih theory [14] and COMSOL. Figure 2. Accuracy check of a side inlet/outlet plenum compared to the Ih theory [14] and COMSOL. 4. Case studies In this paper, eleven side inlet/outlet rectangular plenums (plenums A~K) and a straight rectangular plenum (plenum L) within a fixed space will be analyzed using the finite element method (FEM).
5 FEM simulation on side inlet/outlet rectangular plenums Mechanism of a side inlet/outlet plenum (Plenum A) As indicated in Fig. 3, a side inlet/outlet plenum with a fixed length of 0.7 m is proposed for acoustical simulation where the section of the plenum is 0.12 m in length and 0.12 m in width (L=0.12 m). In addition, the dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width. Figure 3. Mechanism of a side inlet/outlet plenum (plenum A) Mechanism of a side inlet/outlet plenum hybridized with a center-hole baffle (Plenum B) As indicated in Fig. 4, a side inlet/outlet plenum with a fixed length of 0.7 m is introduced for acoustical simulation. The section of the plenum is 0.12 m in length and 0.12 m in width (L=0.12 m). Also, the dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width. A baffle with a rectangular hole of L1*L1 is located inside the plenum. The STL with respect to L1/L of 0.25 and 1.0 is assessed.
6 124 Ying-Chun Chang et al. Figure 4. Mechanism of a side inlet/outlet plenum hybridized with a center-hole baffle (plenum B) Mechanism of a side inlet/outlet plenum hybridized with two center-hole baffles (Plenum C) As indicated in Fig. 5, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is adopted for acoustical simulation. The dimensions of the inlet and outlet duct are set at 0.04 m in length and 0.04 m in width. Two baffles with rectangular holes of L1*L1 are inserted inside the plenum. The influence of the acoustical performance with respect to the ratio of L1/L and the simulation of STL at L1/L of 0.25 and 1.0 is assessed.
7 FEM simulation on side inlet/outlet rectangular plenums 125 Figure 5. Mechanism of a side inlet/outlet plenum hybridized with two center-hole baffles (plenum C). 4.4 Mechanism of a side inlet/outlet plenum hybridized with a center-hole and an extruding baffle (Plenum D) As indicated in Fig. 6, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is used for acoustical simulation. The dimensions of the inlet and outlet duct that are 0.04 m in length and 0.04 m in width are fixed. A baffle is connected with a rectangular extruding tube that is L2 in length, L1 in width, and L1 in height. To analyze the influence of the acoustical performance with respect to L2 at a fixed ratio of L1/L (=0.25), the simulation of STL at L2 of 0.2 and 0.4 is assessed.
8 126 Ying-Chun Chang et al. Figure 6. Mechanism of a side inlet/outlet plenum hybridized with a center-hole extruding baffle (plenum D) Mechanism of a side inlet/outlet plenum hybridized with two center holes and two extruding baffles (Plenum E) As indicated in Fig. 7, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is analyzed for acoustical simulation. Here, the dimensions of the inlet and outlet duct that are 0.04 m in length and 0.04 m in width are preset. Two baffles are connected with rectangular extruding tubes that are L2 in length, L1 in width, and L1 in height. To understand the influence of the acoustical performance with respect to the rectangular extruding tubes, the simulation of STL at L2 of 0.2 and L1/L with and without adding the rectangular extruding tubes is assessed.
9 FEM simulation on side inlet/outlet rectangular plenums 127 Figure 7. Mechanism of a side inlet/outlet plenum hybridized with two center-hole extruding baffles (plenum E) Mechanism of an extended side inlet/outlet plenum (Plenum F) As indicated in Fig. 8, an extended side inlet/outlet plenum with a fixed length of 0.7 m is proposed for acoustical simulation that is 0.12 m in length and 0.12 m in width (L=0.12 m). In addition, the dimensions of the inlet and outlet duct are set at 0.04 m in length and 0.04 m in width. Here, the extended length at the side inlet and outlet is 0.06 m.
10 128 Ying-Chun Chang et al. Figure 8. Mechanism of an extended side inlet/outlet plenum (plenum F) Mechanism of a side inlet/outlet plenum hybridized with a center-hole and a tilted extruding baffle (Plenum G) As indicated in Fig. 9, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is used for acoustical simulation. The dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width and are preset. A baffle is connected with a tilted rectangular extruding tube that is L2 in length, L1 in width, and L1 in height. To analyze the influence of the acoustical performance with respect to θ(the tilted angle) at a fixed ratio of L1/L (=0.25) and a fixed L2 (=0.2), the simulation of STL atθ of (-15 o, 0 o, and 15 o ) is assessed.
11 FEM simulation on side inlet/outlet rectangular plenums 129 Figure 9. Mechanism of a side inlet/outlet plenum hybridized with a center-hole and a tilted extruding baffle (plenum G) Mechanism of a side inlet/outlet plenum hybridized with a centerhole and contracted/expanded extruding baffle (Plenum H-1 and Plenum H-2) As indicated in Figs. 10 and 11, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is introduced for acoustical simulation. The dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width and are preset. Two baffles are connected with two contracted/expanded rectangular extruding tubes (with an angle of 15 o ) that are L2 in length, L1 in width, and L1 in height. To investigate the influence of the acoustical performance with respect to the contraction/expansion effect on the extruding tubes fixed at L2 (=0.2), a simulation of the STL for two plenums shown in Figs. 10 and 11 is assessed.
12 130 Ying-Chun Chang et al. Figure 10. Mechanism of a side inlet/outlet plenum hybridized with a center-hole and an contracted extruding baffle (plenum H-1). Figure 11. Mechanism of a side inlet/outlet plenum hybridized with a center-hole and an expanded extruding baffle (plenum H-2).
13 FEM simulation on side inlet/outlet rectangular plenums Mechanism of a side inlet/outlet plenum hybridized with a center-splitter baffle (Plenum I) As indicated in Fig. 12, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is adopted for acoustical simulation. The dimensions of the inlet and outlet duct are also fixed with 0.04 m in length and 0.04 m in width. A baffle with a rectangular opening (splitter shape) is inserted inside the plenum. To analyze the influence of the acoustical performance with respect to the ratio L1/L, the simulation of the STL with respect to L1/L (0, 0.2, 0.4) is assessed. Figure 12. Mechanism of a side inlet/outlet plenum hybridized with a center-splitter baffle (plenum I) Mechanism of a side inlet/outlet plenum hybridized with two center-splitter baffles (Plenum J) As indicated in Fig. 13, a side inlet/outlet plenum with a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m is proposed for acoustical simulation. The dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width. Two baffles with two rectangular openings
14 132 Ying-Chun Chang et al. Figure 13. Mechanism of a side inlet/outlet plenum hybridized with two centersplitter baffles (plenum J). (splitter shapes) are located within the plenum. To understand the influence of the acoustical performance with respect to the ratio L1/L, the simulation of the STL with respect to L1/L (0, 0.2, 0.4) is assessed Mechanism of a side inlet/outlet plenum hybridized with four crossed baffles (Plenum K) As indicated in Fig. 14, a side inlet/outlet plenum has a fixed length of 0.7 m, a fixed width of 0.12 m, and a fixed height of 0.12 m. The dimensions of the inlet and outlet duct are 0.04 m in length and 0.04 m in width. Four baffles are crossed within the plenum. To understand the influence of the acoustical performance with respect to the ratio L1/L, the simulation of STL with respect to L1/L (0, 0.2, 0.4) is assessed.
15 FEM simulation on side inlet/outlet rectangular plenums 133 Figure 14. Mechanism of a side inlet/outlet plenum hybridized with four crossed baffles (plenum K) Mechanisms of multi-chamber straight plenums (Plenum L-1, Plenum L-2, Plenum L-4, Plenum L-8, and Plenum L-16) As indicated in Figs. 15 and 16, straight plenums hybridized with one, two, four, eight, and sixteen simple expansion chambers have lengths of 0.7 m. To understand the influence of the acoustical performance with respect to the number of chambers at L of 0.12 m, the simulation of the STL with respect to the number of chambers (1, 2, and 4) and (1, 8, and 16) is assessed. Moreover, in order to evaluate the acoustical effect of the section s length, the simulation of the STL for a four-chamber straight plenum and a eight-chamber straight plenum with respect to an L of 0.12, 0.16, and 0.2 m is assessed.
16 134 Ying-Chun Chang et al. Figure 15. Mechanisms of one-chamber, two-chamber, and four-chamber straight plenums (plenums L-1, L-2, and L-4 ). Figure 16. Mechanisms of one-chamber, eight-chamber, and sixteen-chamber straight plenums (plenums L-1, L-8, and L-16).
17 FEM simulation on side inlet/outlet rectangular plenums Results and discussion 5.1 Results By using the FEM in the acoustical simulation of plenums A~L, the results of the STL profiles with respect to the key parameters are shown in Figs , 31, and 33. The STL with respect to the ratio L1/L for plenum B is depicted in Fig. 17. In addition, the STL with respect to the ratio L1/L for plenum C is shown in Fig. 18. To evaluate the baffle effect, a comparison of STLs for plenums equipped with no baffle, one baffle, and two baffles (plenums A, B, and C) is illustrated in Fig. 19. Similarly, the STL with respect to L2 for plenum D is plotted in Fig. 20. The STL with respect to L2 for plenum E is depicted in Fig. 21. To understand the extended effect at the inlet and the outlet of the plenum, the simulated result of the STL with respect to L2 is shown in Fig. 22. Likewise, to understand the influence of the tilted angle (θ) for the baffle s extruding duct, the tilted angle effect for the acoustic performance is simulated and plotted in Fig. 23. Subsequently, the comparison of STLs for plenums equipped with no extruding tube, a contracted extruding tube, and an expanded extruding tube (plenums D, H-1, and H-2) is shown in Fig. 24. And, in order to realize the influence of the acoustical performance with respect to a baffle hybridized with a center-splitter baffle, the STL with respect to the ratio L1/L for plenum I is assessed and shown in Fig. 25. Moreover, to realize the influence of the acoustical performance with respect to two baffles hybridized with two center-splitter baffles, the STL with respect to the ratio L1/L for plenum J is assessed and shown in Fig. 26. Furthermore, for plenum K, the investigation of the STL with respect to the ratio L1/L for a plenum hybridized with four crossed baffles is also carried out and illustrated in Fig. 27. Equally, to initiate the influence of the acoustical performance with respect to the number of plenum chambers (plenum L), the STLs with respect to one chamber, two chambers, and four chambers are simulated and plotted in Fig. 28. Also, the STLs with respect to one chamber, eight chambers, and sixteen chambers are simulated and plotted in Fig. 29. Moreover, in order to analyze the influence of the acoustical performance with respect to the length of the section (L) in plenum L-4 (shown in Fig. 30), the STL with respect to L is simulated and plotted in Fig. 31. Consequently, the STL with respect to L for plenum L-16 (shown in Fig. 32) is simulated and depicted in Fig Discussion As indicated in Fig. 17, for plenum B with one center-hole baffle, the broadband STL will be improved if the ratio L1/L decreases. This means that
18 136 Ying-Chun Chang et al. the acoustical performance will greatly increase if the opening of the baffle decreases. Similarly, as indicated in Fig. 18, the broadband STL for plenum C with two center-hole baffles will increase if the ratio L1/L (baffle s opening ratio) decreases. Also, Fig. 19 indicates that plenum C with more center-hole baffles is superior to plenum B and plenum A. And, plenum A without a center-hole baffle is not as good as the other two plenums. Therefore, the smaller opening ratio of the center-hole baffle will result in a larger STL. As indicated in Fig. 20, for plenum D with a center-hole and an extruding baffle, it is obvious that the STLs for a plenum having one centerhole and extruding baffles is better than that of plenum B which is without a center-hole and an extruding baffle. Similarly, Fig. 21 indicates that the STLs for plenum E with two center-holes and an extruding baffle is better than a plenum without a center-hole and an extruding baffle (plenum B). Therefore, the acoustical effect of the center-hole and extruding baffle is superior. As seen in Fig. 22, for plenum F, the STL will greatly increase if the internally extended length (L2) of the side inlet/ outlet is added. Therefore, the acoustical effect for the internally extended length (L2) of the side inlet/ outlet is essential. As shown in Fig. 23, the influence of the acoustical performance for plenum G with respect to the tilted angle θ is trivial. Therefore, the variation of tilted angle for a baffle hybridized with an extruding duct is not important. Fig. 24 indicates that STLs for plenum H equipped with no extruding tube, a contracted extruding tube, and an expanded extruding tube (plenums D, H-1, and H-2) are similar. The effect of contraction and expansion for an extruding tube linked with a baffle is small. Equally, as indicated in Fig. 25, for plenum I (equipped with one centersplitter baffle), the ratio L1/L for the center-splitter baffle will influence the STL at higher frequencies (beyond 1250 Hz). The acoustical effect for the ratio L1/L will be invalid for low frequencies. Similarly, Fig. 26 reveals that the influence of STL with respect to the ratio L1/L for plenum J (equipped with two center-splitter baffles) is trivial in lower frequencies and effective in higher frequencies. Therefore, the changes of the ratio L1/L will somewhat improve the STL at higher frequencies. Also, as indicated in Fig. 27, the influence of the acoustical performance with respect to the ratio L1/L of crossed baffles in plenum K is obvious in higher frequencies; however, the effect of the ratio L1/L will be trivial for low frequencies. Furthermore, Figs. 28 and 29 illustrates that the STL is proportional to the number of chambers inside plenum L. In addition, the STL curve will be
19 FEM simulation on side inlet/outlet rectangular plenums 137 broadened when the chambers increase. Therefore, the increment of the number of straight plenums will largely increase the broadband STL. Moreover, as indicated in Fig. 31, for a straight plenum with four chambers, the STL will increase and the profile will be broadened when the L of plenum L-4 increases. Consequently, as indicated in Fig. 33, for a straight plenum with eight chambers, the STL will shift to the left (low frequency region) when the L of plenum L-8 increases. This means that the acoustical performance in lower frequencies will increase when the length of the section increases. Figure 17. STL with respect to the L1/L ratio for plenum B. Figure 18. STL with respect to the L1/L ratio for plenum C.
20 138 Ying-Chun Chang et al. Figure 19. Comparison of STLs for plenums equipped with no baffles, one baffle, and two baffles (plenums A, B, and C). Figure 20. STL with respect to L2 for plenum D. Figure 21. STL with respect to L2 for plenum E.
21 FEM simulation on side inlet/outlet rectangular plenums 139 Figure 22. STL with respect to L2 for plenum F. Figure 23. STL with respect to θ for plenum G. Figure 24. Comparison of STLs for plenums equipped with no extruding tubes, a contracted extruding tube, and an expanded extruding tube (plenums D, H-1, and H-2).
22 140 Ying-Chun Chang et al. Figure 25 STL with respect to the L1/L ratio for plenum I. Figure 26 STL with respect to the L1/L ratio for plenum J. Figure 27. STL with respect to the L1/L ratio for plenum K.
23 FEM simulation on side inlet/outlet rectangular plenums 141 Figure 28. Comparison of STLs for plenums equipped with 1 chamber, 2 chambers, and 4 chambers (plenums L-1, L-2, and L-4). Figure 29. Comparison of STLs for plenums equipped with 1 chamber, 8 chambers, and 16 chambers (plenums L-1, L-8, and L-16). Figure 30. Mechanism of a straight plenum (plenum L-4).
24 142 Ying-Chun Chang et al. Figure 31. STL with respect to L for plenum L-4. Figure 32. Mechanism of a straight plenum (plenum L-8). Figure 33 STL with respect to L for plenum L-8.
25 FEM simulation on side inlet/outlet rectangular plenums Conclusion To find a better mechanism in plenum design, eleven side inlet/outlet rectangular plenums (plenums A~K) and a straight rectangular plenum (plenum L) within a fixed space are introduced and simulated by using a finite element method (FEM) which is run on COMSOL, a commercial package. Simulated results indicate that for a plenum with one center-hole baffle, the smaller opening ratio of the center-hole baffle will result in a larger STL. In addition, for a plenum with center-hole and extruding baffles, the STLs for a plenum with a center-hole and an extruding baffles (plenums D and E) is better than a plenum without a center-hole and an extruding baffle (plenum B). Also, for plenum F, the acoustical effect for the internally extended length (L2) of the side inlet/ outlet is essential. And, for a plenum with center-splitter baffles, the variety of the ratios L1/L will somewhat improve the STL at higher frequencies. Furthermore, the influence of STL with respect to the ratio L1/L of the crossed baffles in plenum K is effective in higher frequencies and trivial in lower frequencies. Moreover, the increment of the number of the straight plenums will increase the broadband STL. Additionally, for a straight plenum, the STL will increase and the profile will be broadened when L of the plenum L-4 increases. Furthermore, the STL in lower frequencies will increase when the size (L) of the plenum L-8 s section increases. Consequently, the acoustical simulation of a plenum system using the FEM method proposed in this study provides an efficient way to predict the acoustical performance of the plenums with complicated mechanisms. References 1. Ising, H., Kruppa, B., Health Effects Caused by Noise: Evidence in the Literature from the Past 25 Years, Noise and Health, 2004, 6, Willich, S. N., Wegscheider, K., Stallmann, M., Keil, T., Noise Burden and the Risk of Myocardial Infarction, European Heart Journal, 2006, 27, Cummings, A., Sound Transmission in Curved Ducts Bends, Journal of Sound and Vibration, 1974, 35, Rostafinski, W., Transmission of Wave Energy in Curved Ducts, Journal of the Acoustical Society of America, 1974, 56, Kim, J. T., Ih, J. G.., Transfer Matrix of Curved Duct Bends and Sound Attenuation in Curved Expansion Chambers, Applied Acoustics, 1999, 56, Prasad, M. G., Crocker, M. J., Studies of Acoustical Performance of a Multicylinder Engine Exhaust Muffler System, Journal of Sound and Vibration, 1983, 90(4),
26 144 Ying-Chun Chang et al. 7. Prasad, M. G., A Note on Acoustic Plane Waves in a Uniform Pipe With Mean Flow, Journal of Sound and Vibration, 1984, 95(2), Munjal, M. L., Acoustics of Ducts and Mufflers with Application to Exhaust and Ventilation, John Wiley and Sons, New York, Kim, J. T., Ih, J. G., Transfer Matrix of Curved Duct Bends and Sound Attenuation in Curved Expansion Chambers, Applied Acoustics, 1999, 56, Magrab, E. B., Environmental Noise Control; John Wiley & Sons, New York, Munjal, M. L., Plane Wave Analysis of Side Inlet/outlet Chamber Mufflers with Mean Flow, Applied Acoustics, 1997, 52, Li, X., Hansen, C. H., Comparison of Models for Predicting the Transmission Loss of Plenum Chambers, Applied Acoustics, 2005, 66(7), Liu, J., Herrin, D. W., Enhancing Micro-perforated Panel Attenuation by Partitioning the Adjoining Cavity, Applied Acoustics, 2010, 71, Ih, J. G., The Reactive Attenuation of Rectangular Plenum Chambers, Journal of Sound and Vibration, 1992, 157(1),
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