Broadband noise insulation of windows using coiled-up silencers consisting of coupled tubes

It has been demonstrated that a staggered window achieves better noise reduction performance than a traditional single glazing one at middle to high frequencies while maintaining a degree of natural ventilation. There is, however, little improvement in the low frequency range. In contrast, this work proposes to apply coiled-up silencers consisting of coupled tubes on the side walls of staggered windows to obtain noise attenuation in a broad band, especially in the low frequency range. Each element in the silencer consists of two coupled tubes with different cross sections so that noise at more frequencies can be attenuated than that with a uniform cross section. The simulation results show that 8.8 dB overall insertion loss can be obtained between 100 and 500 Hz after applying a combination of silencers designed at 7 different frequencies, and the insertion loss of the staggered window is increased from 6.7 to 15.6 dBA between 100 and 2000 Hz for normal incident traffic noise with the proposed silencers installed. The design is validated by the experiments with a 1:4 scale down model.


Scientific Reports
| (2021) 11:6292 | https://doi.org/10.1038/s41598-021-85796-0 www.nature.com/scientificreports/ increases the effective band width to 1900 Hz. Another improvement is that we carried out experiments with a real scale down model of a staggered window, instead of in an impedance tube with normal incident sound, thus demonstrated the feasibility of the silencer to be used in practical applications.

Results
A staggered window can be used with a traditional single glazing window, as shown in Fig. 1(a), to provide sufficient natural ventilation while the single glazing window is closed. The silencers we proposed are attached on the two opposite side walls of the ventilation duct of the staggered window, as shown in Fig. 1(b). The outside noise transmits through the inlet into the duct and radiates to the internal room through the outlet. The size of the ventilation duct under investigation is 1.5 m × 0.32 m × 0.25 m (l x × l y × l z ), and the sizes of the inlet and outlet for ventilation are both 0.2 m × 0.2 m. For this specific window, the silencer on both sides of the duct consists of 7 elements of different sizes. Figure 1(c) shows a cross-sectional sketch of one element in the silencer. It consists of two tubes, with a cross section of d 1 and d 2 , respectively. Both tubes are folded up once in space, and the total size of the silencer is a × p (width × thickness). In the numerical simulations, the primary sound transmits into the staggered window through the inlet and radiates outside to a semi-infinite space through the outlet. The SPL at an evaluation point outside the outlet 0.1 m away from its center is used to evaluate the performance of the silencers. The noise reduction caused by the silencers is defined as the difference between the SPLs at the evaluation point without and with them installed, and it is also the insertion loss (IL) of the silencers. By using two tubes with different cross sections coupled together and adjusting the lengths and cross sections of the two tubes, the proposed structure works at more frequencies in the low frequency range than that with a uniform cross section. The reduction of the sound power radiated to the downstream when a side branch tube is installed on an infinitely long duct can be predicted theoretically (See "Methods" for more details). As shown in Fig. 2, when the side branch tube has a uniform cross section, and the length is l = 0.72 m, the sound power reduction reaches the highest at 119 Hz because a quarter the wavelength of 119 Hz is approximately 0.72 m. There is another sound power reduction peak at 357 Hz, which is the 3rd harmonic of 119 Hz. However, if the side branch tube consists of two parts with different cross sections, it works at 4 frequencies below 500 Hz. Apart from 119 Hz, the reduction at 177 Hz, 415 Hz and 473 Hz are all above 20 dB, indicating that a silencer consisting of two coupled tubes with different cross sections can reduce noise over a wider frequency band. A silencer with coupled tubes might occupy less space than a uniform one in some cases with careful design (for example, in this work a reduction to 0.0159 m 3 from 0.0180 m 3 is achieved). www.nature.com/scientificreports/ To reduce the silencer size, the tubes are coiled up in space, as shown in Fig. 1(c). In order to achieve effective broadband noise reduction, silencers with different parameters are arranged in an array along the side walls of the ventilation duct. They are designed to operate at 7 different frequencies, and the parameters are listed in Table. 1. The 7 elements are installed on two opposite sides of the duct, as shown in Fig. 1(b). When the primary source is a normally incident plane wave with a strength of 1 Pa, the SPLs at the evaluation point with and without the silencers installed are shown in Figs. 3(a), and 3(b) shows the insertion loss caused by the silencers. Figure 3 shows that noise is reduced at all frequencies between 100 and 500 Hz except at 410 Hz, where the SPL is slightly increased by 1.8 dB. The overall insertion loss caused by the silencers can be used to evaluate the noise reduction performance in a broad band. It is calculated with where M is the number of frequencies of interest, and SPL i_without and SPL i_with are the SPLs at the ith frequency without and with the silencers installed, respectively. It is the reduction of the overall SPL at all the frequencies of interest and the number is 8.8 dB between 100 and 500 Hz.
The silencers also work at higher frequencies, and Fig. 4(a) shows the SPLs between 100 and 2000 Hz, when the primary sound is a normal incident plane wave with an amplitude of 1 Pa. The blue and red curves correspond to the SPLs outside the staggered window without and with silencers installed, and the SPLs with silencers are lower at 84.3% of the frequencies between 100 and 2000 Hz. The overall insertion loss is 7.2 dB between 100 and 2000 Hz.
The SPLs at the evaluation point in the free field and with an opened single glazing window are also included in Fig. 4(a) for comparison. The single glazing window is embedded in an infinitely large rigid wall. The window size is 0.2 m × 0.2 m, which is the same as the inlet and outlet of the staggered window, and it is kept open to ensure a similar amount of air circulation. The sound field behind the window is different from that in a free field in the low frequency range when the wavelength is comparable to the window size due to the reflections and scattering by the window frame.    www.nature.com/scientificreports/ It can be observed from Fig. 4(a) and (b) that the staggered window provides extra noise reduction from 500 to 2000 Hz compared with the opened single glazing one, demonstrating the advantage of a staggered window in the middle to high frequency range. The insertion loss of the opened single glazing window is 3.6 dB between 100 and 2000 Hz, and the staggered window increases it to 7.5 dB. By applying silencers on the staggered window, the insertion loss is further increased to 14.7 dB. Figure 4(c) shows the A-weighted SPLs at the evaluation point when the primary sound is normal incident traffic noise, and Fig. 4(d) shows the corresponding insertion loss. The insertion loss of the staggered window with silencers is 15.6 dBA between 100 and 2000 Hz, in comparison to only 6.7 dBA without the silencers.
The experiments were carried out in an anechoic room with a 1:4 scale down model, as shown in Fig. 5(a) and Fig. 5(b) is a detailed view of the staggered window model with the silencers installed. The size of the staggered window model was 375 mm × 80 mm × 67.5 mm, and the size of both the inlet and outlet was 50 mm × 50 mm. The staggered window model was installed at the top of a 1.20 m × 1.10 m × 1.00 m anechoic box, which is a 1:4 scale down model of a rectangular room. The noise source was a loudspeaker inside the box and the silencers were installed on the two opposite side walls of the staggered window model to reduce the noise transmitted through it to the outside.
The silencers were designed for the frequency band from 400 to 2000 Hz for the 1:4 scale model, and the parameters are listed in Table 2. The silencers were built with 3D printing using acrylonitrile-butadience-styrene (ABS) plastic, as shown in Fig. 5(c). Since the density of ABS plastic is 1100 kg/m 3 , the thickness of the walls of the structure was chosen as w = 3.00 mm to ensure a transmission loss of at least 20.4 dB at frequencies no less than 400 Hz according to the mass law 32 , thus the boundaries of the structure can be considered as rigid in the experiments at frequencies of interest. The SPL at the point 2 cm above the center of the outlet was measured to evaluate the noise reduction performance of the silencers.
The loudspeaker inside the anechoic box generated white noise, and the frequency range of interest was 400 Hz-8000 Hz, corresponding to 100 Hz-2000 Hz for the full scale model. The measured SPLs at the evaluation point with and without the silencers installed are shown in Fig. 6(a), and the insertion loss of the silencers is shown in Fig. 6(b). It is clear that the measured SPL is reduced at most frequencies with the silencers installed, and the overall insertion loss caused by the silencers is 12.0 dB between 400 and 2000 Hz.
The numerical simulation results obtained from the commercial finite element model software COMSOL Multiphysics 5.5 for the same silencer are included in Fig. 6(a) and (b) for comparison. The overall insertion loss caused by the silencers is 8.0 dB within this frequency range in the numerical simulations. The trends of the numerical simulation and experimental results are similar, and one of the reasons for the difference is the errors in 3D printing. Further numerical simulation results (not shown here) indicate that a 0.5 mm difference in the tube width or length can cause a 20 Hz difference in the working frequency, which demonstrates the significant effect of 3D printing errors on the noise reduction performance of the silencer.
Other reasons for the difference between numerical simulation and experimental results include: (1) The model created in COMSOL was not exactly the same as that in the experiments; (2) The primary source in the  www.nature.com/scientificreports/ numerical simulations was a point monopole, while the primary loudspeaker was not omni-directional in the experiments, especially in the high frequency range; (3) The primary loudspeaker was put near a corner of the anechoic box, instead of the position in the numerical simulations which was hard to be installed at. Although there exists difference between numerical simulation and experimental results, the measured results have demonstrated the effectiveness of the proposed silencer. The silencers can also reduce noise at higher frequencies, which is shown in Fig. 7(a), contributing to a total insertion loss of 12.6 dB at the evaluation point across the whole frequency range from 400 to 8000 Hz, and the insertion loss spectrum is shown in Fig. 7(b). The experimental results demonstrate that the broadband noise reduction can be achieved by introducing specially designed silencers.

Discussion
In this paper, we propose applying an array of thin silencers on a staggered window to constitute a natural ventilation window that attenuates noise in a broad frequency band. Each element of the silencer consists of two coupled tubes with different cross sections that are coiled up once in space; this coupled tube operates effectively at two more frequencies than tubes with a uniform cross section. With a hybrid silencer designed at 7 different frequencies, the SPL can be reduced at all frequencies between 100 and 500 Hz, except at 410 Hz, where a slight SPL increase of 1.8 dB is observed. The overall insertion loss of the silencer is 8.8 dB within this frequency range. The thickness of the whole structure is only 1/10 the wavelength of 100 Hz. The silencer also reduces noise at higher frequencies, and the insertion loss of the staggered window between 100 and 2000 Hz can be increased from 6.7 to 15.6 dBA for normal incident traffic noise with the silencers installed. The experimental results

Methods
Theory. The reduction of the sound power radiated to the downstream of an infinitely long duct can be predicted theoretically. Figure 8 shows the sketches of a duct with two kinds of side branch tubes. A plane wave is assumed to propagate in the infinitely long duct with a cross section of S = l y × l z , and a side branch tube is attached on the side wall. The acoustic impedance of the side branch tube in Fig. 8(b) is 32 where S b1 = d 1 × l y is the cross section of the side branch tube entrance, k is the wavenumber, ρ 0 and c 0 are the air density and sound speed in the air, respectively, and is the acoustic impedance of the tube with length l 2 in Fig. 8(b). The transmission coefficient of the sound power to the downstream is where R b and X b are the real and imaginary parts of Z b , respectively. The reduction of the sound power is defined as For the side branch tube with a uniform cross section shown in Fig. 8(a), Eq. (2) can be simplified as where S b = d × l y is the cross section of the side branch tube entrance. Equation (5) is also used to calculate the sound power reduction in this case.
For coiled-up tubes investigated in this paper, the l 1 and l 2 are the equivalent lengths of the two parts, which are approximately 2a − d 1 and 2a − d 2 , respectively. Applying symmetric silencers on two opposite sides of the ventilation duct increases the noise reduction, but has limited effect on the working frequencies for the case we investigated.
Numerical simulations. The results in Figs. 3 and 4 for a normal incident plane wave are obtained using LMS Virtual.Lab 13.10. The results in Fig. 6 with a point monopole as the primary sound source are obtained using COMSOL Multiphysics 5.5. The air density is 1.225 kg/m 3 and the sound speed in the air is 343.202 m/s. All the boundaries of the models used in COMSOL and Virtual.Lab are set as rigid.