Sound power reflection coefficients in main duct with anechoic termination. 

Contexts in source publication

Context 1

... the case for the sidebranch resonance, the end condition of the region between the side-branches in the main duct is 1/4 flanged and the others are flat. It is believed that ␧ is likely to exceed 1.7 times the equivalent radius of the main duct cross section. This is because one can observe from Ref. 3 that 1.7 is the factor for a flanged open-end tube and 1.2 is that for the unflanged case, suggesting an increase in this factor as higher restriction is imposed to the radiation direc- tion at the tube ends. However, a formula or data for deter- mining ␧ is not known to the authors. The results of Redmore and Mulholland 8 with a single sidebranch do not facilitate the estimation of such factor. Since this factor is increased by 0.5 upon a four-side 50% reduction in the radiation surface, a rough extrapolation here for the factor is 1.7 ϩ (1.7 Ϫ 1.2)/4 ϫ 3 ϭ 2.1. Equivalent radius of the main duct cross section here is 90.9 mm and thus ␧ is estimated to be 188.6 mm. For L / w ϭ 1.59, 4.62, and 7.49, a drop of 3 to 4.3 dB in the TL prediction can result with such correction ( ϭ 188.6 mm). This seems able to explain the observation of ‘‘missing peak’’ in Figs. 9 ͑ c ͒ and ͑ d ͒ . One should note that the evanescent wave scattered out from the entrance of the upstream sidebranch is still relatively strong at that of the downstream sidebranch at L / w ϭ 1.59 ͑ only a ϳ 2 dB drop in magnitude ͒ . The difference in TL drops to ϳ 2 dB after the inclusion of ␧ . However, such correction does not work when L / w ϭ 2.72 and 8.50. Indeed, the plane wave theory predicts reasonably the experimental results for these lengths ͓ Figs. 9 ͑ b ͒ and ͑ e ͔͒ implying no separation correction is required. It is found that these L ’s are shorter than the nearest integer multiples of half the wavelength at this frequency, while the opposite are observed for the other lengths. This behavior tends to imply that resonance inside the main duct is relatively more difficult to occur at L / w ϭ 2.72 and 8.5. The resonant condition inside the main duct appears also important for ␧ . An investigation on the diffraction correction at junction is left to further study. It is believed that the effect of ␧ still exists in the high pass filter cases, but the broadband gradual variation of the acoustic impedance makes it much less distinguishable. At f / f c ϳ 0.51, Z d / Z b Ϸ 1, suggesting a vanishing reactance in Z b . The plane wave theory without allowance for ␧ results in more than 3 dB overestimation in TL. Such overestimation is reduced to below 2.6 dB for all L ’s if a ␧ of 188.6 mm is included. The overestimation drops below 2.3 dB if ␧ ϭ 163 mm is taken. This tends to suggest that ␧ depends on frequency as diffraction does. It may also depend on the branch impedance. Further investigation is essential. It also shows that the measurement error in length, which is at most 2 mm in the present study, is not going to affect the results significantly. The high of TL at f / f c ϳ 0.61 after the introduction of a second sidebranch predicted by the plane-wave theory is not confirmed by the experiment, especially for large L ͓ Figs. 9 ͑ d ͒ and ͑ e ͔͒ . The introduction of a correction ␧ of 163 mm to 188.6 mm as before gives prediction 1–3 dB closer to the experimental results for L / w ϭ 2.72, 4.62, and 7.49, but about 1.5 dB more derivation for L / w ϭ 8.50. The case for L / w ϭ 1.59 is exceptional as the evanescent wave is not substantially attenuated in the region bounded by the two sidebranches. One can notice that at this frequency and for L / w ϭ 1.59 and 8.50, the nominal branch separation L is shorter than the nearest integer multiple of half the wavelength, while the opposite is observed for other L ’s. This further suggests that the correction ␧ will depend on the resonant condition along the main duct. The predictions from plane-wave theory without correction in L agree basically with the experimental results at f / f c ϳ 0.63. The plane-wave theory again overestimates TL at frequencies of strong resonance and thus weak Z b / Z d ͑ Fig. 10 ͒ . At f / f c ϭ 0.32, an overestimation of about 10 dB is found at L / w ϭ 1.59 ͓ Fig. 10 ͑ a ͔͒ , while for L / w у 2.72 ͓ Figs. 10 ͑ b ͒ – ͑ e ͔͒ , such difference is capped under 6 dB. Similar situation can be found at f / f c ϳ 0.16 ͑ not shown here due to less sat- isfactory anechoic termination performance, cf. Fig. 2 ͒ . For short L , the presence of evanescent wave of significant magnitude close to the proximity of the second branch certainly makes the plane-wave assumption less valid ͑ 8 dB magnitude drop at L / w ϭ 1.59, Ͼ 23 dB at L / w у 2.72). It is again found that the introduction of ␧ of 188.6 mm into Eq. ͑ 9 ͒ reduces considerably the discrepancy between predictions and the experimental results even at L / w ϭ 1.59. The correction results in only 3 dB TL overestimation except at L / w ϭ 7.49, but it still gives 1.5 dB better prediction than that without this length correction. In fact, neither a reduction nor an increase of ␧ from 188.6 mm can produce better results. It is noted that the introduction of ␧ brings about a 1.5 dB increase of the TL overestimation at L / w ϭ 2.72. One can observe that for L / w ϭ 2.72 and 8.50, L is relatively far away from the nearest integer multiple of half the wavelength at f / f c ϭ 0.32 than for other values of L / w , suggesting again the resonant condition along the main duct can affect the correction or even the prediction. This is again left to further investigation. The TL peak at f / f c ϭ 0.50 for the single sidebranch of l / w ϭ 5.23 again disappears upon the introduction of a second sidebranch for all L ’s investigated in the present study ͑ Fig. 10 ͒ . However, a small ‘‘kick’’ can be seen at around this frequency on the experimental TL curves for L / w у 2.72 ͓ Figs. 10 ͑ b ͒ – ͑ e ͔͒ . This large discrepancy in TL of about 6 dB cannot be explained even by including a correction ␧ of 188.6 mm to L , though such correction improves the prediction by 0.5–3 dB except for L / w ϭ 1.59 and 7.49 where the TL overestimations are increased by 1.6 and 0.6 dB, respectively. However, this correction still gives the best prediction overall. Anyway, Z b / Z d ϳ 0.56 ϩ 0.04 j here, which suggests that this is not a frequency of strong branch resonance. The TL at f / f c ϭ 0.48 is amplified by the introduction of a second sidebranch for all L ’s as shown in Figs. 10 ͑ a ͒ – ͑ e ͒ . This TL is due to the noise breakout of the sidebranches, which is dissipative to sound in the main duct. This is also the only frequency that the plane-wave theory clearly under- estimates TL. Such underestimation reaches 5 dB at L / w ϭ 1.59 ͓ Fig. 10 ͑ a ͔͒ , drops to about 2 dB at L / w ϭ 2.72 ͓ Fig. 10 ͑ b ͔͒ , and then gradually increases to about 4 dB at L / w ϭ 8.50 ͓ Figs. 10 ͑ c ͒ – ͑ e ͔͒ . The impedance of the branch at this frequency Z b / Z d ϭ 1 Ϫ 0.02 j , which is close to that of an infinitely long sidebranch ͑ϳ 1.06 in the present study theo- retically ͒ . Though this impedance is nearly the same as that at f / f c ϳ 0.51 in the weak resonance case, which is due to a longitudinal resonance in the branches, it is believed that the mechanisms of sound reflection and transmission across the branches in these two cases are different. The sound pressure level drop related to this noise breakout alone accounts for as much as ϳ 30 dB m along this sidebranch ͑ not shown here ͒ . One therefore does not expect much reflection from the opened branch end, resulting in an impedance close to that of an infinitely long branch. Some theoretical considerations on the effects of flexible duct walls on TL can be found in Huang. 14 The vibration of branch walls at the breakout frequency is strong and will probably lead to much more vigorous air motion ͑ low sound pressure ͒ at the entrances of the branches than that in the weak resonance case. Higher dissipation of energy by damping at the edges at the entrance of each sidebranch is then expected. One should also note that the noise broken out from one branch will affect the wall vibration of the other branch. 15 This will lead to some kinds of coordinated wall vibrations, which interferes with the air movement inside the branches and along the main duct. Such interference, which is not considered by the plane-wave theory, can be substantial for short L , due to the direct bombardment of breakout sound waves onto the branch walls, and for L which is close to any integer multiple of half the wavelength at f / f c ϭ 0.48. The separation L / w ϭ 8.50 corresponds to 4.08 times this half wavelength. The higher TLs at f / f c ϳ 0.48 shown in Figs. 10 ͑ a ͒ ( L / w ϭ 1.59) and Fig. 10 ͑ e ͒ ( L / w ϭ 8.50) than in Figs. 10 ͑ b ͒ – ͑ d ͒ tend to support the present conjecture. However, further investigations are required to clarify the present observed sound transmission phenomena near the break-out frequency of the sidebranch ͑ es ͒ . Other results, not presented here, show that the TLs of the single or double sidebranch arrangements are not affected very much by in-duct airflow up to a speed of 7 m/s, except that one can find rougher experimental TL curves in the presence of the flow. In the presence of the flow, the loudspeaker was mounted on a wall of the main duct and the anechoic termination determined by Neise et al. 16 was used. Though this ...

Context 2

... propagation inside an air duct and its attenuation is one of the most important issues in building noise control. The problem is more serious at low frequencies because of the poor performance of the dissipative silencer in this frequency range. 1 The low frequency sound attenuation performance of various duct elements has been investigated exten- sively during the past few decades. They include the expansion chambers and mufflers, 2 the Helmholtz resonators, 3 and different forms of waveguides ͑ for instance, Ref. 4 ͒ . Formulas for the prediction of low frequency sound transmission loss of some of these conventional passive duct elements are currently available for engineering design purposes. 5 The use of active control for low frequency attenuation in ducts is also possible nowadays. 6 When engineers calculate the required sound attenuation for a particular air distribution path in a duct system, the transmission losses produced by all the duct elements involved have to be taken into account. Sidebranches, except those specifically installed as a reactive silencer ͑ such as the Helmholtz resonator or simply a cavity 7 ͒ , are indispensable as they are important for conveying fresh air from the main air supply duct into different parts in the interior of a building. Similar branches can also be found in the air exhaust system. Though these sidebranches are not designed to provide sound attenuation, they do have contributions in the noise control as they produce a change in the acoustic impedance along a duct. 8 The termination of a sidebranch, its length, and its area relative to that of the main duct cross section affect the overall sound power transmission loss across the branch. For a double sidebranch configuration, the coupling between the two sidebranches and the air mass in between the branches inside the main duct have crucial effects on the overall sound power transmission loss. Optimal design for sound attenuation therefore requires a better knowledge of the acoustical effects associated with branching. Since it is the low frequency noise that is of concern, the one-dimensional wave theory has been applied to predict the sound power transmission loss, TL, due to sidebranches ͑ for instance, Reynolds 9 ͒ . For an infinitely long sidebranch, this approach suggests that the corresponding TL depends solely on the cross-sectional area ratio between the sidebranch and the main duct. 9 For a double sidebranch, the current engineering practice usually ignores the contribution of the air mass inside the main duct bounded by the two sidebranches, 5 though it is included in the theoretical consideration. 9 De- spite the importance of the acoustical behaviors of multiple sidebranches on sound propagation inside a duct, a detailed comparison between the predictions from plane-wave theory and experimental results, at least to the knowledge of the authors, is not well documented. In the present investigation, the experimentally determined sound power transmission loss resulting from double sidebranches is compared with that predicted from the plane- wave theory in detail. The impedance of each sidebranch is estimated from the corresponding complex sound pressure ratios using the plane-wave theory. Different characteristics of the sidebranch impedance are included and their effects discussed. It is hoped that the present results can clarify the extent to which the plane-wave theory can be used to predict the sound power transmission losses of the double sidebranches and provide useful information for building noise control and further modeling. Figure 1 illustrates the schematic of the experimental set rig assembly for the present study. The main duct was made of 12.7-mm-thick Perspex and had a rectangular cross sec- tion of dimension 173 mm by 150 mm giving a first cut-off frequency, f c , of 991 Hz, which was also the highest frequency concerned in the present study. A loudspeaker was mounted at one end of the main duct, while the other end was fitted to an anechoic termination. Owing to the size of the loudspeaker, sound of frequency lower than 100 Hz could not be effectively generated. The sound power reflection coefficient, measured by the two microphone method ͑ discussed later ͒ , associated with the anechoic termination shows that this termination basically absorbed 98% of the incident sound energy for frequency at around 180 Hz ( f / f c ϭ 0.18, Fig. 2 ͒ . In fact, for f / f c Ͼ 0.22, the sound power reflection coefficient was less than 0.6%, indicating an anechoic condition was established. The sidebranches had a cross section of 163 mm by 150 mm. They were made of 5.8 mm Perspex and each of their open ends contained a small flange. Branch length l could be varied and the ends of the sidebranches were either left opened or fitted with anechoic terminations. This was to vary the acoustic impedance of the sidebranch and will be discussed later. The small thickness of the sidebranch allowed a possible noise breakout, which is due to the vibration of the duct walls, somewhere around 450 Hz ( f / f c 0.45)—a fea- ture specifically introduced for the present investigation to complicate the acoustic impedance of the sidebranches. The exact noise breakout frequency depends on the branch length as well as on the structure of the branch. However, this phenomenon was only important when the branch length was sufficiently long. This will be discussed later. The corresponding noise breakout frequency of the main duct was higher than 1 kHz. One sidebranch was removed during the measurements of single side-branch impedances and the corresponding sound power transmission losses. Sound pressure fluctuations were measured by four Br ̈ el & Kj , r 4935 4 1 in. microphones with the appropriate signal conditioners. One microphone pair was located on each side of the duct branching and the microphones in each pair were separated by a distance of 20 mm. The locations of measurements were more than two duct widths from the entrance or exit of the branching in order to allow for the decay of all the nonplanar modes. 10 These positions were also more than 0.5 m from the loudspeaker or the anechoic termination. In a preliminary trial test, sound pressure spectra at various points along the centerlines of the main duct cross section were measured by a probe microphone. The variation of spectral densities from 180 to 991 Hz was less than 0.6 dB, confirming that only plane waves were propagating along the main duct within the experimental frequency range. The sound pressure fluctuations from the microphones were si- multaneously recorded onto tapes using a SONY PC208Ax digital recorded for later analyses. Sampling rate per channel was 24 kHz and each recording lasted for 30 s. The experimental investigation was carried out in a large laboratory with a height of around 12 m and a floor area roughly over 100 m 2 . There were plenty of scattering objects and the sound absorption was good. The acoustical interaction between the laboratory and the test rig should be insig- nificant. The laboratory was air-conditioned with air tem- perature and relative humidity maintained at 23 °C and 60%, respectively. The two-microphone transfer function method was em- ployed to calculate the sound transmission loss and the impedance of a single sidebranch. Since the gain and phase responses of the microphones are in general not identical, the sensor-switching technique was used. Though main details of this method for measuring complex sound reflection coefficient can be found in Chung and Blaser, 11 a brief account of it is given in the following for completeness and also as a reference for later derivation of the acoustic impedance measurement and the sound power transmission loss. The subscripts 1, 2, 3, and 4 denote hereinafter quanti- ties related to microphones 1, 2, 3 and 4, respectively. During each measurement, four complex pressure signals, p i , can be ...