**4.1 Setup of experiment**

*Noise and Vibration Control - From Theory to Practice*

position of the flow fluctuation velocity in the tube bank. As represented in **Figure 20**, the excitation flow fluctuation is not generated in the entire tube bank under the condition where the self-sustained tone is not generated. On the other hand, **Figure 20** represents that it is generated in the entire tube bank under the condition of the self-sustained tone being generated. Therefore, the two parameters particle velocity and the excitation flow fluctuation are controlled by inserting the baffle plate. Ishihara et al. [12] thought that it is the suppression mechanism of the self-sustained tone to decrease the sound power by controlling

*The fluctuation velocity of flow and the sound pressure level at observation point* ③ *with the baffle plate* 

*Fluctuation velocity of flow on tube bank and the measurement position ([12]).*

**68**

**Figure 20.**

**Figure 19.**

*(Pattern of baffle plate position is "0") ([26]).*

these two parameters.

In this section, we describe the experiments performed by Ishihara and Nakaoka [13] and Ishihara [28]. They carried out some experiments to examine the suppression effect of the perforated plates and cavities installed, and confirmed the suppression effect. They defined the aperture ratio *ϕ* of the perforated plate and investigated how the change in the value of the aperture ratio *ϕ* of the perforated plate affects the self-sustained tone in the duct. They varied the value of the aperture ratio *ϕ* from 1 to 32%. The setup of the experiment is shown in **Figure 21**. The duct is made of acrylic plates that have a thickness of 1 cm. The tube bank consists of an array of bronze tubes whose diameter is *D* = 6 mm. The array geometry is represented in **Figure 3(b)**, where the spacings *T*/*D* and *L*/*D* are 2.0. In the tube bank, there are 9 rows of tubes in the flow direction and 19 tubes in the width direction, which is perpendicular to the flow, and the length in the flow direction is 102 mm. The sound pressure signal is measured using the microphone set near the duct outlet as shown in **Figure 3(a)**, and converted to frequency domain with FFT analyzer.

The perforated plate is made of iron, and has a length of 400 mm, a height of 250 mm, and a thickness of 2.3 mm. A hole with a diameter of 3 mm was opened in a staggered arrangement on a plate. As shown in **Figure 22(a)**, perforated plates can be mounted from the slit (shown in green), and the duct has two cavities with a depth of *Lc* = 100, 66 and 33 mm. In this experiment, in order to examine the influence of the aperture ratio of the perforated plate on the self-sustained tone suppressing effect, as shown in **Figure 22(b)**, assuming a hole diameter of 3 mm, Ishihara and Nakaoka [13] made six patterns (1, 2, 4, 8, 16, and 32%) of the perforated plates. Here, the aperture ratio *ϕ* is the ratio of the area of the holes to the total area of the perforated plate and is defined by Eq. (5).

$$
\phi\_- = \frac{n\_h \left(\frac{ml^2}{4}\right)}{S\_p} \tag{5}
$$

Here, *nh* is the number of the hole, *Sp* = 200 × 420 mm is the total area of the perforated plate, and d is the hole diameter (3 mm). Even at the same aperture ratio, if the hole diameter is different, the influence on self-sustained tone may be different. However, in this study, it is assumed that the hole diameter is constant

**Figure 21.** *Setup of experiment [29].*

**Figure 22.**

*Tube bank part with perforated plates ([29]). (a) Detail of tube bank part with perforated plates and cavities. (b) Perforated plate.*

(3 mm) [13]. Since the case that the aperture ratio is 0% (holeless plate) is defined as the standard of this experiment, *ϕ* = 0% is also a parameter of the aperture ratio.

#### **4.2 Results of experiments**

An effect of the aperture ratio on sound pressure level spectra at the gap velocity *Vg* = 21.3 m/s is represented in **Figure 23**. The sound pressure decreases more than 30 dB in cases where the perforated plates and cavities are installed (*ϕ* ≥ 0%), and the sound pressure level decreases in the high-frequency region as the aperture ratio increases. Hence, the perforated plate is assumed to have a sound-absorbing property at high frequencies. The experimental results of the relation between overall sound pressure level and the gap velocity in cases of aperture ratios *ϕ* = 1, 2, 4, 8, 16, and 32% are represented in **Figure 24**. Sound following the 5th power law is the ordinary aerodynamic sound, as mentioned in Section 2.3. On the other hand, as shown by the blue circle in **Figure 24**, the noise which does not follow the power law and becomes extremely large is referred to as the self-sustained tone. As shown in **Figure 24**, the self-sustained tone is generated only in the duct with the plate of the aperture ratio *ϕ* = 0%, which is the normal duct without holes. When the perforated plate with aperture ratios of *ϕ* = 1–32% is applied on the duct wall surfaces, the overall sound pressure level rises as per the 5th power law, and the self-sustained tone is not generated.

**71**

**Figure 24.**

*8, 16, and 32%, and Lc = 100 mm.*

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct*

Ishihara [28] studied experimentally the effect of a cavity volume which is used with perforated plates on the SPL. He concluded that the effect of a cavity volume on the SPL is a little. **Figure 25** shows the effect of a cavity volume or depth on the sound pressure level in the case of the aperture ratio *ϕ* = 4% as the typical example of the aperture ratio. Further, Ishihara [28] clarified experimentally the critical and optimum aperture ratios. The results show that the critical aperture ratio is about

Mori et al. [29] performed compressible CFD simulations and acoustic simulations, and compared the simulation results with the measurements [13] to numerically verify the effect of the aperture ratio of the perforated plate on the self-sustained tone and acoustic resonance frequencies. The numerical method for unsteady CFD simulations is described in Section 2.4. The CFD model for the normal duct without holes, which corresponds to the duct with the perforated plates of the aperture ratio *ϕ* = 0%, is shown in **Figure 6**. As an example of the CFD model for the duct with the perforated plates, **Figure 26(a)** represents the CFD model

*Relation between overall sound pressure level and the gap velocity in cases of aperture ratios ϕ(AR) = 0, 1, 2, 4,* 

0.25%, and the optimum aperture ratio is concluded to be 4%.

*Effect of aperture ratio on sound pressure level spectra at Vg = 21.3 m/s.*

**4.3 Unsteady CFD simulations**

**Figure23.**

*DOI: http://dx.doi.org/10.5772/intechopen.86039*

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct DOI: http://dx.doi.org/10.5772/intechopen.86039*

**Figure23.** *Effect of aperture ratio on sound pressure level spectra at Vg = 21.3 m/s.*

Ishihara [28] studied experimentally the effect of a cavity volume which is used with perforated plates on the SPL. He concluded that the effect of a cavity volume on the SPL is a little. **Figure 25** shows the effect of a cavity volume or depth on the sound pressure level in the case of the aperture ratio *ϕ* = 4% as the typical example of the aperture ratio. Further, Ishihara [28] clarified experimentally the critical and optimum aperture ratios. The results show that the critical aperture ratio is about 0.25%, and the optimum aperture ratio is concluded to be 4%.

#### **4.3 Unsteady CFD simulations**

*Noise and Vibration Control - From Theory to Practice*

(3 mm) [13]. Since the case that the aperture ratio is 0% (holeless plate) is defined as the standard of this experiment, *ϕ* = 0% is also a parameter of the aperture ratio.

*Tube bank part with perforated plates ([29]). (a) Detail of tube bank part with perforated plates and cavities.* 

An effect of the aperture ratio on sound pressure level spectra at the gap velocity *Vg* = 21.3 m/s is represented in **Figure 23**. The sound pressure decreases more than 30 dB in cases where the perforated plates and cavities are installed (*ϕ* ≥ 0%), and the sound pressure level decreases in the high-frequency region as the aperture ratio increases. Hence, the perforated plate is assumed to have a sound-absorbing property at high frequencies. The experimental results of the relation between overall sound pressure level and the gap velocity in cases of aperture ratios *ϕ* = 1, 2, 4, 8, 16, and 32% are represented in **Figure 24**. Sound following the 5th power law is the ordinary aerodynamic sound, as mentioned in Section 2.3. On the other hand, as shown by the blue circle in **Figure 24**, the noise which does not follow the power law and becomes extremely large is referred to as the self-sustained tone. As shown in **Figure 24**, the self-sustained tone is generated only in the duct with the plate of the aperture ratio *ϕ* = 0%, which is the normal duct without holes. When the perforated plate with aperture ratios of *ϕ* = 1–32% is applied on the duct wall surfaces, the overall sound pressure

level rises as per the 5th power law, and the self-sustained tone is not generated.

**70**

**4.2 Results of experiments**

**Figure 22.**

*(b) Perforated plate.*

Mori et al. [29] performed compressible CFD simulations and acoustic simulations, and compared the simulation results with the measurements [13] to numerically verify the effect of the aperture ratio of the perforated plate on the self-sustained tone and acoustic resonance frequencies. The numerical method for unsteady CFD simulations is described in Section 2.4. The CFD model for the normal duct without holes, which corresponds to the duct with the perforated plates of the aperture ratio *ϕ* = 0%, is shown in **Figure 6**. As an example of the CFD model for the duct with the perforated plates, **Figure 26(a)** represents the CFD model

#### **Figure 24.**

*Relation between overall sound pressure level and the gap velocity in cases of aperture ratios ϕ(AR) = 0, 1, 2, 4, 8, 16, and 32%, and Lc = 100 mm.*

#### **Figure 25.**

*Relation between overall sound pressure level and the gap velocity in cases of aperture ratios ϕ = 4% and Lc = 100, 66, and 33 mm [29].*

for the duct with the perforated plates of the aperture ratio *ϕ* = 2%. **Figure 26(b)** shows a part of the CFD model near the perforated plates and cavities in the case of the aperture ratio *ϕ* = 2%. The CFD domain contains 10,727,088 cells and 6,612,429 nodes for the case of the aperture ratio *ϕ* = 1%; 12,053,432 cells and 7,308,716 nodes for the case of the aperture ratio *ϕ* = 2%; and 13,646,420 cells and 8,004,142 nodes for the case of the aperture ratio *ϕ* = 4%, respectively.

**Figure 27** shows instantaneous snapshots of the fluctuation pressure field. Comparing the cases of *ϕ* = 0% with *ϕ* = 1%, the fluctuation pressure in the case of *ϕ* = 0% is much larger than that in the case of *ϕ* = 1%. In the cases of *ϕ* = 1%, the pressure fluctuation does not represent the resonance mode in the duct width direction which appears in the case of *ϕ* = 0% in **Figure 27(a)**.

The effect of the aperture ratio on the frequency spectra of SPL monitored on the wall of the duct near the outflow boundary is represented in **Figure 28(a)**. For comparison, both the simulated and measured data are represented, and the frequency spectra of SPL in the case of *ϕ* = 0% is also displayed in **Figure 28(a)**. In both simulations and experiments, the self-sustained tone is generated in the case of *ϕ* = 0%; however, the self-sustained tone is not generated in the cases of *ϕ* = 1%. The effect of the aperture ratio on the overall sound pressure level is represented in

#### **Figure 26.**

*CFD model for duct with perforated plates and cavities with a depth of Lc=100 mm. (a) Duct with the perforated plates of the aperture ratio ϕ = 2%. (b) Duct near the perforated plates and cavities in the case of the aperture ratio ϕ = 2% [29].*

**73**

**Figure 28.**

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct*

**Figure 28(b)**. The SPL decreases with an increase of the aperture ratio, and rapidly

*Fluctuation pressure fields in the cases of aperture ratio at Vg = 21.3 m/s. (a) ϕ =0%. (b) ϕ = 1%.*

*Effect of aperture ratio on SPL at Vg = 21.3 m/s. (a) ϕ = 1%. (b) Overall SPL (200–2000 Hz).*

**Figure 29** shows the SPL on the wall of the duct in frequency domain at the peak frequency, 740 Hz. In the case of *ϕ* = 0%, the SPL on the wall of the duct clearly represents the acoustic mode in the duct width direction as represented in **Figure 27(a)**. On the other hand, in the cases of *ϕ* = 1%, **Figure 29(b)** shows that

decreases when the aperture ratio is between 0 and 1%.

*DOI: http://dx.doi.org/10.5772/intechopen.86039*

**Figure 27.**

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct DOI: http://dx.doi.org/10.5772/intechopen.86039*

**Figure 27.**

*Noise and Vibration Control - From Theory to Practice*

for the duct with the perforated plates of the aperture ratio *ϕ* = 2%. **Figure 26(b)** shows a part of the CFD model near the perforated plates and cavities in the case of the aperture ratio *ϕ* = 2%. The CFD domain contains 10,727,088 cells and 6,612,429 nodes for the case of the aperture ratio *ϕ* = 1%; 12,053,432 cells and 7,308,716 nodes for the case of the aperture ratio *ϕ* = 2%; and 13,646,420 cells and 8,004,142 nodes

*Relation between overall sound pressure level and the gap velocity in cases of aperture ratios ϕ = 4% and* 

**Figure 27** shows instantaneous snapshots of the fluctuation pressure field. Comparing the cases of *ϕ* = 0% with *ϕ* = 1%, the fluctuation pressure in the case of *ϕ* = 0% is much larger than that in the case of *ϕ* = 1%. In the cases of *ϕ* = 1%, the pressure fluctuation does not represent the resonance mode in the duct width direc-

The effect of the aperture ratio on the frequency spectra of SPL monitored on the wall of the duct near the outflow boundary is represented in **Figure 28(a)**. For comparison, both the simulated and measured data are represented, and the frequency spectra of SPL in the case of *ϕ* = 0% is also displayed in **Figure 28(a)**. In both simulations and experiments, the self-sustained tone is generated in the case of *ϕ* = 0%; however, the self-sustained tone is not generated in the cases of *ϕ* = 1%. The effect of the aperture ratio on the overall sound pressure level is represented in

*CFD model for duct with perforated plates and cavities with a depth of Lc=100 mm. (a) Duct with the perforated plates of the aperture ratio ϕ = 2%. (b) Duct near the perforated plates and cavities in the case of* 

for the case of the aperture ratio *ϕ* = 4%, respectively.

tion which appears in the case of *ϕ* = 0% in **Figure 27(a)**.

**72**

**Figure 26.**

**Figure 25.**

*Lc = 100, 66, and 33 mm [29].*

*the aperture ratio ϕ = 2% [29].*

*Fluctuation pressure fields in the cases of aperture ratio at Vg = 21.3 m/s. (a) ϕ =0%. (b) ϕ = 1%.*

**Figure 28(b)**. The SPL decreases with an increase of the aperture ratio, and rapidly decreases when the aperture ratio is between 0 and 1%.

**Figure 29** shows the SPL on the wall of the duct in frequency domain at the peak frequency, 740 Hz. In the case of *ϕ* = 0%, the SPL on the wall of the duct clearly represents the acoustic mode in the duct width direction as represented in **Figure 27(a)**. On the other hand, in the cases of *ϕ* = 1%, **Figure 29(b)** shows that

**Figure 28.** *Effect of aperture ratio on SPL at Vg = 21.3 m/s. (a) ϕ = 1%. (b) Overall SPL (200–2000 Hz).*

**Figure 29.** *SPL on the wall of the duct at Vg = 21.3 m/s [29]. (a) ϕ = 0%. (b) ϕ = 1%.*

the acoustic mode is not clearly represented and the value of SPL on the duct wall is close to 40 dB smaller than that in the case of *ϕ* = 0%.

The relation between overall sound pressure level and the gap velocity in the cases of *ϕ* = 0 and 1% is represented in **Figure 30**. As represented in **Figure 30**, the self-sustained tone is generated in the case of *ϕ* = 0%, and the sound pressure level does not follow the 5th power law when the gap velocity is high, as mentioned in Section 2.4. However, in the case of *ϕ* = 1%, the self-sustained tone is not generated, and regardless of the gap velocity, the overall sound pressure level rises along the 5th power law in both simulations and experiments. Therefore, when the perforated plate is installed on the duct wall surfaces, the self-sustained tone is not generated in the CFD simulations, as in the experiments [13].

## **4.4 Acoustic simulations and suppression mechanism of self-sustained tone by perforated plates**

The acoustic characteristics of the duct with the perforated plates and cavities without the flow were calculated by means of BEM (the commercial code, WAON) [30]. Boundary element models for the cases of the aperture ratio *ϕ* = 0 and *ϕ* = 2% are shown in **Figure 31(a)** and **(b)**, respectively, and in the case of *ϕ* = 2%, the duct

#### **Figure 30.**

*Relation between overall sound pressure level (200–2000 Hz) and gap velocity in cases of aperture ratios ϕ = 0% and ϕ = 1%.*

**75**

**Figure 32.**

*Acoustic frequency responses.*

**Figure 31.**

*(d) ϕ = 0%. (e) ϕ = 2%.*

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct*

with the perforated plates and cavities is modeled as shown in **Figure 31(c)**. There are 80,530 boundary elements for the case of the aperture ratio *ϕ* = 0%; 167,361 boundary elements for the case of the aperture ratio *ϕ* = 1%; 226,079 boundary elements for the case of the aperture ratio *ϕ* = 2%; and 307,747 boundary elements for the case of the aperture ratio *ϕ* = 4%, respectively. The maximum element size is 0.013 m. An impedance boundary condition is imposed at the inflow boundary to consider acoustic waves moving from the inflow boundary to the outside and the value of the impedance is *ρc*. The outflow boundary is the surface connecting the duct inside and outside. At the outflow boundary, the interface boundary, where the particle velocity and acoustic pressure of the internal and external sound fields of the duct are coupled, is imposed. At other wall boundaries except the holes, weak absorption boundaries, whose absorption coefficient is 0.02, are imposed. The absorption coefficient "0.02"

*Boundary element model and position of monopole point source [29]. (a) ϕ = 0%. (b) ϕ = 2%. (c) ϕ = 2%.* 

*DOI: http://dx.doi.org/10.5772/intechopen.86039*

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct DOI: http://dx.doi.org/10.5772/intechopen.86039*

**Figure 31.**

*Noise and Vibration Control - From Theory to Practice*

**74**

**Figure 30.**

*ϕ = 0% and ϕ = 1%.*

*Relation between overall sound pressure level (200–2000 Hz) and gap velocity in cases of aperture ratios* 

the acoustic mode is not clearly represented and the value of SPL on the duct wall is

The relation between overall sound pressure level and the gap velocity in the cases of *ϕ* = 0 and 1% is represented in **Figure 30**. As represented in **Figure 30**, the self-sustained tone is generated in the case of *ϕ* = 0%, and the sound pressure level does not follow the 5th power law when the gap velocity is high, as mentioned in Section 2.4. However, in the case of *ϕ* = 1%, the self-sustained tone is not generated, and regardless of the gap velocity, the overall sound pressure level rises along the 5th power law in both simulations and experiments. Therefore, when the perforated plate is installed on the duct wall surfaces, the self-sustained tone is not generated in

**4.4 Acoustic simulations and suppression mechanism of self-sustained tone by** 

The acoustic characteristics of the duct with the perforated plates and cavities without the flow were calculated by means of BEM (the commercial code, WAON) [30]. Boundary element models for the cases of the aperture ratio *ϕ* = 0 and *ϕ* = 2% are shown in **Figure 31(a)** and **(b)**, respectively, and in the case of *ϕ* = 2%, the duct

close to 40 dB smaller than that in the case of *ϕ* = 0%.

*SPL on the wall of the duct at Vg = 21.3 m/s [29]. (a) ϕ = 0%. (b) ϕ = 1%.*

the CFD simulations, as in the experiments [13].

**perforated plates**

**Figure 29.**

*Boundary element model and position of monopole point source [29]. (a) ϕ = 0%. (b) ϕ = 2%. (c) ϕ = 2%. (d) ϕ = 0%. (e) ϕ = 2%.*

with the perforated plates and cavities is modeled as shown in **Figure 31(c)**. There are 80,530 boundary elements for the case of the aperture ratio *ϕ* = 0%; 167,361 boundary elements for the case of the aperture ratio *ϕ* = 1%; 226,079 boundary elements for the case of the aperture ratio *ϕ* = 2%; and 307,747 boundary elements for the case of the aperture ratio *ϕ* = 4%, respectively. The maximum element size is 0.013 m. An impedance boundary condition is imposed at the inflow boundary to consider acoustic waves moving from the inflow boundary to the outside and the value of the impedance is *ρc*. The outflow boundary is the surface connecting the duct inside and outside. At the outflow boundary, the interface boundary, where the particle velocity and acoustic pressure of the internal and external sound fields of the duct are coupled, is imposed. At other wall boundaries except the holes, weak absorption boundaries, whose absorption coefficient is 0.02, are imposed. The absorption coefficient "0.02"

**Figure 32.** *Acoustic frequency responses.*


#### **Table 2.**

*Peak frequency in each case of the aperture ratio [29].*

corresponds to that of the general acrylic surface. To clarify the acoustic characteristics of the tube bank duct, the acoustic frequency responses have been calculated using the monopole point sources (without the flow) that are supposed to be vortices generated on the downstream side behind the tube as shown in **Figure 31(d)** and **(e)**. The magnitude of the point sources is 1 Pa at all frequencies. The point source is located at (0.25*H, −*3.86*H,* 0) downstream of the tube bank, and *H* is 200 mm in the height of the tube bank duct. In order to excite the resonance mode in the duct width direction, the sound source was arranged asymmetrically with respect to the YZ plane.

**Figure 32** represents the acoustic frequency responses that have been calculated using the monopole point source (without the flow). The monitor point is located

**Figure 33.** *Acoustic modes at each peak frequency [29]. (a) ϕ = 0%. (b) ϕ = 1%. (c) ϕ = 2%. (d) ϕ = 4%.*

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*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct*

at (0.585*H*, −4.76*H*, 0) downstream of the tube bank. In the case of *ϕ* = 0%, the peak frequency surrounded by the green circle is 725 Hz, and close to the resonance frequency in the duct width direction, as mentioned in Section 2.3. On the other hand, in the cases of *ϕ* = 1, 2, and 4%, peak frequencies surrounded by blue, red, and purple circles are close to the resonance frequency in the duct width direction, and change to higher frequencies than that in the case of *ϕ* = 0% with an increase of the aperture ratio. **Table 2** represents the peak frequency in each case of the aperture ratio. It has been also confirmed that the resonance frequency increases as the aperture ratio increases in the previous study [31] in which an acoustic resonance frequency has been experimentally and analytically verified by assuming a one-dimensional sound field in a duct partitioned by a perforated plate, and similar results are obtained in this study. **Figure 33** shows the acoustic mode at each peak frequency in the cases of *ϕ* = 0, 1, 2, and 4%. In the case of *ϕ* = 0%, the acoustic mode in the width direction strongly appears at 725 Hz. However, in the cases of *ϕ* = 1, 2, and 4%, the acoustic mode in the duct width direction does not clearly appear, and the value of SPL decreases as the aperture ratio increases. Therefore, it

was assumed that the perforated plate has an effect of attenuating sound.

upstream and when it is inserted downstream.

1.The sound pressure level rises with an increase of the gap flow velocity by following the 5th power law when the gap velocity is low. However, when the gap velocity is high, the self-sustained tone is generated not by following the

2.Insertion of baffle plates in the tube bank decreases the natural frequency of the duct and increases the onset gap velocity of the self-sustained tone. Hence, the natural frequency of the duct does not seem to be related with the suppression of the self-sustained tone when the baffle plate is installed in the tube bank. The self-sustained tone is the most effectively suppressed by inserting the baffle in the entire tube bank because the baffle plate decreases the particle velocity and vorticity. Furthermore, there is a difference in the onset gap velocity of the self-sustained tone between when the baffle plate is inserted

3.The perforated plates installed on the duct walls suppress the self-sustained tone and increase the resonance frequency in the duct width direction. Consequently, if the perforated plates are installed on the duct walls, the self-sustained tone is assumed to be suppressed by an increase of the resonant frequency in the duct

width direction and sound-absorbing effect of the perforated plates.

*DOI: http://dx.doi.org/10.5772/intechopen.86039*

**5. Conclusions**

5th power law.

*Countermeasure for High Level Sound Generated from Boiler Tube Bank Duct DOI: http://dx.doi.org/10.5772/intechopen.86039*

at (0.585*H*, −4.76*H*, 0) downstream of the tube bank. In the case of *ϕ* = 0%, the peak frequency surrounded by the green circle is 725 Hz, and close to the resonance frequency in the duct width direction, as mentioned in Section 2.3. On the other hand, in the cases of *ϕ* = 1, 2, and 4%, peak frequencies surrounded by blue, red, and purple circles are close to the resonance frequency in the duct width direction, and change to higher frequencies than that in the case of *ϕ* = 0% with an increase of the aperture ratio. **Table 2** represents the peak frequency in each case of the aperture ratio. It has been also confirmed that the resonance frequency increases as the aperture ratio increases in the previous study [31] in which an acoustic resonance frequency has been experimentally and analytically verified by assuming a one-dimensional sound field in a duct partitioned by a perforated plate, and similar results are obtained in this study. **Figure 33** shows the acoustic mode at each peak frequency in the cases of *ϕ* = 0, 1, 2, and 4%. In the case of *ϕ* = 0%, the acoustic mode in the width direction strongly appears at 725 Hz. However, in the cases of *ϕ* = 1, 2, and 4%, the acoustic mode in the duct width direction does not clearly appear, and the value of SPL decreases as the aperture ratio increases. Therefore, it was assumed that the perforated plate has an effect of attenuating sound.
