Noise reduction through periodic serial side branches containing two defects

. Noise pollution is a challenging issue requiring the development of acoustic systems that can reduce noise. Besides the environmental impacts of noise pollution, it can cause serious health problems. This work originally discusses noise reduction using a new configuration of side branches grafted in series with different terminations. The combination of two serial resonators shows acoustic coupling effects. This effect is used to increase localized modes of two serial defective resonators located in the middle of the structure. The transmission of acoustic waves can be controlled by the dimensions of the introduced defects as well as their boundary conditions. We also show that defects can totally filter the transmission of the pass-bands. The proposed structure can then play the role of an acoustical filter.


Introduction
Noise pollution is not only an environmental issue but also can have important impacts on health. According to the WHO, traffic-related noise accounts for over 1 million healthy years of life lost annually leading to illness, disability, or early death in the western European countries [1]. An important study shows that sleep disturbance can occur if there are a level of 50 dB(A) indoors or more in 50 noise events per night [2]. The use of periodic these filters are based on resonators. In a piping system, acoustic filters are discontinuities in the structure inducing significant reflections. These discontinuities can be in form of an open pipe termination or a closed branch [3][4][5]. The acoustic response of such filters has been widely discussed [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. At low frequencies, the electrical diagram equivalent to a side-branch can be made up of two inductors in series [20][21], these last represent the effect of antisymmetrical modes on the side-branch. The electrical diagram can also consist of an impedance, representing the effect of the plane mode, in series with an inductance representing the effect of the higher modes. The equivalent electric circuit of a symmetrical lateral branch is obtained by Dubos using the modal decomposition [22]. However, the circular geometry presented difficulties. Several tests like the influence of the distance between the side branches, the resonator's length, the pressure and the angle between the branches, have been carried out to study the effect of the flow and geometrical parameters on the acoustic response of two side-branches [23]. The resonance of the two resonators is found to be more important when compared with a single branch under similar test conditions. The difference is due to the acoustic coupling effect between the two side branches which drastically reduces the acoustic radiation into the main duct.
Side branches have an important applications in aeronautical and marine systems [24][25][26][27][28]. Series arrangements of a closed termination resonators were used to reduce synchronous turbo noise. It has been shown that the presented onedimensional model (1D) gives acceptable results, where the two resonant frequencies and their maximum values were analysed by Allam [29]. However, there is even offset at high frequency and in presence of flow due to the complexity of the chamber of the resonators which brings a 3D effects.
The objective of this work is the study of the propagation of acoustic waves through a new association of serial resonators based on side branches with different terminations. This structure is called the combination of closed and open resonators (CCOR). The parallel configuration of the CCOR system, in which the two different side branches are connected one in front of the other is discussed [30]. The presence of defects inside the periodic parallel resonators shows the influence of acoustic coupling modes of defects.
In this study, we extend our research on CCOR to examine another configuration in which the two side branches are connected in series. The acoustic performance of this new configuration is originally presented using the interface response theory [31]. The boundary conditions of the serial side branches are different. In addition, we focus on the existence of the localized modes inside the forbidden bands due to the presence of two serial defects inside the structure as the effect of one defective resonator is already studied [32]. The coupling effect given by defect modes is examined.
The localized mode's transmission are controlled by the modification of the sections of the defects. The analytical and numerical results are validated by those obtained by the transfer matrix method.

General case of a periodic structure of series of different side branches
In this section, we theoretically study the attenuation of sound through the CCOR system in serial configuration. Side branches are often used in industrial and building extraction and supply networks, in order to prevent the noise generated by acoustic sources from reaching the duct outlet. The behaviour of the acoustic wave in the latter depends essentially on the dimensional property of the guide as well as the boundary conditions of its termination. The ducts behave like acoustic waveguides. When the wavelength is significantly large, the wave motion can be considered as a plane wave, which offers great simplifications. The plane wave plays an important role in the understanding of the involved physical phenomena.
The proposed configuration is given by two serial resonators (or side branches) along a rectangular waveguide ( Figure. where ‫ܨ‬ = −j߱/‫ݖ‬ (j = √−1, i = 1 − 3), ‫ܥ‬ = cosh (݇ ݀ ) ܵ = sinh (݇ ݀ ), , ݇ = ߱/ܿ ߱ is the angular frequency of the incident acoustic wave, c0 the speed of sound speed, and zi being the characteristic acoustic impedance of the medium i. g ஶ ିଵ (M, M) of the infinite periodic structure is given by the following matrix: The dispersion relation of the infinite serial resonators is given by: Where K is the Block wave number.
Assuming that the acoustic structure is constructed with the same material, the dispersion relation is given by: This equation is similar to the equation obtained in [32] using the transfer matrix.
Using this later, the transmission loss of the serial side branches is given by where yop and yc are the acoustic impedances of the open side resonator and the closed resonator, respectively. Figure 2 represents the variation of the transmission loss coefficient through serial closed and open-side branches connected along a rectangular waveguide. One can clearly observe the two resonance frequencies corresponding to these resonators. Since the open resonator is a high-pass filter, the first peak corresponds to the latter. The behaviour of high-pass filter of the open resonator dominates. To identify the origin of the second resonance, we varied the length of the closed resonator (d2) for a fixed value of d3. The second resonance (TL = 60) located at frequency 1317Hz moves towards lower frequencies as d2 increases. While the first peak remains the same. This confirms that this resonance corresponds to that of the closed resonator.

Transmission loss coefficient of serial side branches
To take advantage of the configuration of the serial resonators, an arrangement of closed resonators have been experimentally examined by Allam [29]. The numerical results show that the transmission loss of the overall system exhibits a combined behaviour that superimposes the two peaks provided by the gated resonators in a non-overlapping frequency range. This result shows that cascading different resonators in the resonator design can achieve more attenuation performance, provided the resonators are properly designed and optimized to be effective in different frequency bands.

Transmission of the periodic structure of serial side branches containing two dimensional defects
Here, we study the appearance of localized modes inside the structure which contains the compound defects of the closed and open resonator in a series configuration (Figure 3). The resonators and the main tube are filled with air. We note that the geometries of all the grafted side branches, including the defects, consist of cylindrical sections, while the main tube has a square cross-section (50 × 50mm 2 ). The transmission coefficient of the defective structure is calculated using the transfer matrix [32]. The simulation results are based on experimental data obtained from literature.
As a unit cell is composed of two serial side-branches, the defect can be one of the two resonators corresponding to the central cell located. Therefore, the defect of the resonator can have a closed, an open end or both.

Effect of the presence of one defective resonator inside the periodic side branches
We should consider the effect of the presence of one defective resonator that is already discussed [32] which can make the interpretation of the coupling effect of the two defects modes more clear. The other resonator, located in the same cell, is similar to the common closed side branches. The transmission coefficient of the perfect structure is presented in Figure  4(c). A localized mode appears in the fifth band gap. As the cross-section of the defective open resonator increases, the peak shifts to higher frequencies. Its reaches the middle of the fifth band gap while keeping low transmission below 0.50. Inside the sixth band gap, notably the limit of the fifth pass-band, a localized mode of amplitude T = 0.69 appears at a fixed position. This peak is located on frequency 2724Hz and is not affected by the variation of s03. This type of localized mode is also observed in the parallel configuration resonators [30].
The fourth pass band is affected by the insertion of the defective open resonator. For s03/s3 > 1, the transmission inside the first pass-band is also decreased ( Figure  4 (Figure 4(j)-(f)). These remarks bring us back to the conclusion that this structure can play the role of an acoustic filter rather than a selective filter since the localized mode inside the structure has a very weak transmission. However, the high-pass character of the defective open resonator shifts the localized mode towards high frequencies, this is not the case for the parallel configuration of the CCOR system which contains a defective open resonator [30].   (Figure 5(a)) to the limit of the third pass-band ( Figure 5(f)). This peak moves through the entire fourth band gap. The transmission decreases when the section closes to that of common open resonators. This peak becomes more important as the section becomes larger. The direction of defect mode shift in this case towards low frequencies. This is the opposite of what is observed in the case of defective open resonator (Figure 4). This behaviour is due to the fact that the closed resonator works as a low pass filter. Hence, the defective closed resonator acts as a selective filter that can shift the mode towards low frequencies.
A localized mode located at the frequency 1537Hz appears inside the third band gap with high transmission (T = 0.90). This peak is not affected by the variation of the defective section. Also, a localized mode on the frequency 2308Hz appears inside the fifth forbidden band for s02/s2 > 1.4. A significant influence of the defective closed resonator appears inside the fifth pass-band in which the entire pass-band is depressed giving rise to a large band gap. This new forbidden band is caused by the Eigen-frequency of the defect which falls on the pass-band. The pass-bands are also affected by the presence of the defect, in particular the fourth pass band ( Figure  5(f)).

Effect of the presence of two serial defects inside the CCOR
After presenting the case of a one defect inside the CCOR system, we present the study of the effect of inserting two defects with two different terminations inside the periodic side branches which is the main objective of this investigation. The two serial resonators corresponding to the central cell are dimensional defects (closed and open resonator in series) with two different cross-sections. The identification of the localized modes of the two defects requires, for instant, the length of one of the two defects to be fixed.  Figure 5) shows no localized mode inside the second gap, and considering the fact that the position of this mode is located on the frequency 676Hz, we can conclude that this mode comes from the fixed defective closed resonator. Another remark confirms our conclusion is the fact that the position of the localized mode frequency in the case of a single defective closed resonator ( Figure 5(a)) is 685Hz for s02/s2 = 0.4, this frequency is very close to the position of the localized mode observed for two defects (Figure 6(a)). This confirms a coupling effect between the two defect modes. Moreover, the decrease of the localized mode transmission for s03/s3 > 1 is due to the variation of s03. The latter is also decreases the transmission of the first pass-band. This behaviour is similar to that of a single defective open resonator (Figure 4(d)-(f)). On the other hand, the two defects contribute to the attenuation of transmission waves inside the fourth passband. The defect of the open-side branch has no effect on the reduction of the transmission of the third pass-band. This is mainly caused by the defect of the closed side-branch.
In the third forbidden band, a localized mode appears, its transmission 0.90 at a fixed frequency equal to 1537Hz. It is exactly that of the defect of the closed resonator ( Figure 5). Moreover, this peak does not appear in the absence of the defective open resonator (Figure 6(c)) which confirm our remark. The defect mode due to the section variation of the defective closed resonator also appears in the fourth forbidden band, its transmission increases as the section s02 increases. This transmission remains fixed for s02/s2 > 1.
The defect mode corresponds to the defective open resonator appears in the middle of the fifth forbidden band ( Figure  6(b)). Its frequency is close to that obtained by a single defective open resonator ( Figure 4) with very low transmission (T = 0.35). This case is the only one where the localized mode of this defect is observed. On the other hand, the localized mode of the defective closed resonator appears to the left of that of the defective open resonator (Figure 6(b)). Its transmission remains fixed at the position 2296Hz which is very close to that observed in Figure 5. Half of the transmission of this peak is attenuated due to the variation of the defective section of the open resonator. If we compare this transmission with that illustrated in Figure 5(e), we can remark that the transmission of the defective open resonator goes from 0.69 to 0.22 ( Figure  6(e)).
The defect of the closed resonator has destroyed the entire transmission at the level of the fifth pass-band, this is a behaviour already described in Figure 5 as a result of the coupling effect of defect modes. The series configuration of the CCOR system gives very useful results. In fact, this configuration shows two characters of selection of the defect modes obtained by different filters in particular that of low and high frequencies. The selection of modes in two different

Conclusion
We theoretically presented an original study of a 1D periodic CCOR composed of two different side branches grafted along a rectangular acoustic duct. The infinite periodic structure has wide acoustic band gaps that can be used in noise reduction, these band gaps come from both closed and open-side branches. The band structure of the perfect CCOR system shows that the combination of the filters has the character of a highpass filter, this behaviour is mainly imposed by the open-side branches.
The behaviour of the localized modes is analysed by varying the cross-section of the defective closed resonator in the case where the length of the defective open resonator is fixed. We show that both defects contribute to give rise to the localized modes. These results are very important and show the coupling effect of defects. In particular, when they are caused by two different filters. Moreover, the transmission of pass-bands is also affected by the presence of defects. In fact, by introducing a defective open-side resonator, we are able to attenuate the transmission inside pass-bands.
In the case of a single defective open resonator, the direction of defect mode displacement is towards high frequencies, which is the opposite of that obtained by a single defective closed resonator. The low/high filtering of frequencies appears clearly in the CCOR of series configuration. The presence of the defects therefore makes it possible to exploit the forbidden frequency regions. We showed that defect modes can change their movement direction depending on the boundary condition of the defective resonator. Thus, the new configurations proposed in this study can play the role of a selective filter of low or high frequencies.
The transmission of acoustic waves can be controlled by the dimensions of the introduced defects as well as their boundary conditions.