*Article* **The Influence of Floor Layering on Airborne Sound Insulation and Impact Noise Reduction: A Study on Cross Laminated Timber (CLT) Structures**

**Federica Bettarello 1, Andrea Gasparella <sup>2</sup> and Marco Caniato 2,\***


**Abstract:** The use of timber constructions recently increased. In particular, Cross Laminated Timber floors are often used in multi-story buildings. The development of standardization processes, product testing, design of details and joints, the speed of construction, and the advantages of eco-sustainability are the main reasons why these structures play a paramount role on the international building scene. However, for further developments, it is essential to investigate sound insulation properties, in order to meet the requirements of indoor comfort and comply with current building regulations. This work presents the results obtained by in field measurements developed using different sound sources (tapping machine, impact rubber ball, and airborne dodecahedral speaker) on Cross Laminated Timber floors, changing different sound insulation layering (suspended ceiling and floating floors). Results clearly show that the influence on noise reduction caused by different layering stimulated by diverse noise source is not constant and furthermore that no available analytical model is able to correctly predict Cross Laminated Timber floors acoustic performances.

**Keywords:** cross laminated timber; impact noise; rubber ball; sustainable; sound insulation; timber

#### **1. Introduction**

At present, the need of sustainable buildings is rising all over Europe and thus their construction is growing quickly [1]. Therefore, high-rise wooden edifices are more and more requested in the market [2–4]. These edifices are composed using different elements. Often, timber frame is used for the construction of walls [5,6] and Cross Laminated Timber (CLT) for floors [7,8]. In this light, CLT horizontal partitions have to fulfil many requisites like structural integrity etc., but recently sound insulation and impact noise reduction as well as indoor acoustic comfort are becoming important issues to manage. Anyway, bare horizontal partitions do not easily fulfill acoustic law requirements [9] and thus many other layers have to be added.

In order to solve these problems, many works were developed in years to study the acoustic behavior of this type of timber element. In a recent review, Di Bella and Mitrovic [10] focused on bare structures elucidating their properties and construction phases.


When focusing on layered CLT elements, Pérez and Fuente [11] presented a dedicated study using laboratory and field measurements of sound insulation and impact noise reduction of some CLT components. Anyway, no parametric study related to the influence

**Citation:** Bettarello, F.; Gasparella, A.; Caniato, M. The Influence of Floor Layering on Airborne Sound Insulation and Impact Noise Reduction: A Study on Cross Laminated Timber (CLT) Structures. *Appl. Sci.* **2021**, *11*, 5938. https://doi.org/10.3390/ app11135938

Academic Editors: Edoardo Piana, Paolo Bonfiglio and Monika Rychtarikova

Received: 7 June 2021 Accepted: 23 June 2021 Published: 25 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the single layer is included. The same consideration could be applied to many other works [12–15].

Due to the reduced weight of CLT elements, their poor acoustic performances could lead to problems in the lower frequency range when in presence of impact noises such as children or adults walking or running. In order to reduce impact noise or to increase sound insulation, some other technologies have to be coupled to the bare timber floor. Two of the most used are (i) floating floor and (ii) suspended ceiling.

Kim et al. [16] found that a floating floor addition using resilient materials ensures good performance against lightweight impact noise, but has rather negative effects due to the resonance on the heavy weight impact noise caused by the falling of heavy objects or children walking/running.

It is thus evident how there is a lack of parametric studies, discussing the influence of single noise reduction action on CLT floors and, in addition, their overall contribution when laid together on the same horizontal partition. Furthermore, there is a lack of studies comprising different acoustic excitation techniques on such floors [17], providing single configuration influence on final sound reduction.

Timber constructions are relatively new in the European market. People usually live in traditional heavyweight ones made of masonry and/or concrete. When moving on new sustainable edifices, people feel new indoor environment and new noises, which were not present in traditional houses. Thus, subjective evaluations are now part of the research order to understand if timber buildings could provide a suitable environment from the point of view of acoustic comfort [18]. In this view, the rubber impact ball was demonstrated to be the noise source most associated with subjective reactions [19,20]. However, to the authors' knowledge, no parametric research comparing the influence of noise reduction technologies on CLT floors is available in literature, using such a source.

For these reasons, this research presents the results of acoustic measurements using rubber ball and tapping machine for impact noise and dodecahedral source for airborne noise. Tests were carried out in situ in a timber building featuring CLT floors. The measurements were made step-by-step during the construction phase, firstly considering the bare CLT floor and after all the various layers. The aim of this research is to parametrically determine the influence of different layers on impact and airborne noise reduction as well as to understand if available analytical models could predict the measured values.

#### **2. Materials and Methods**

The test-building where the in situ measurements (airborne and impact noise tests) were performed consists of five CLT floors (Figure 1), featuring timber frame walls, as depicted in Figure 2.

**Figure 1.** Realization of the test-building.

**Figure 2.** Pictures of the internal partition.

In these conditions, thanks to (i) the rock wool included within the timber studs of the vertical partitions, (ii) the coupling of massive (CLT) and lightweight (timber frame) partitions and (iii) the point connections, flanking transmissions are very limited (up to 1 dB overall) [21–24].

#### *2.1. Investigated Structures*

Different configurations of floor structures were built and tested, in order to understand their influence on sound insulation and impact noise reduction, as follows:


The first configuration deals with the characterization of the bare floor. This structure features a thickness of 180 mm of Cross Laminated Timber as reported in Figure 3.

**Figure 3.** Configuration 1: Bare floor.

The second studied configuration presents a common solution used in timber buildings: suspended ceiling (configuration 2). A suspended ceiling is used for the following most frequent reasons: (i) including HVAC systems and thermos-hygrometric indoor conditions [25], (ii) including air or fluids pipes or ducts [26,27], (iii) protecting timber structures from fire [28]. Less frequently, it is intended to be used as a sound insulation layer or impact noise reduction technology. In Figure 4, the configuration 2 is depicted.

This solution does not interfere in vibration transmission but does on the airborne noise one. Accordingly, the suspended ceiling acts as an added layering, namely another and different impedance from the bare floor. Thus, it constitutes a sound insulating element laid between the source (vibrating floor) and the receiver (room). Subsequently, its influence is related to the airborne noise more than the structure borne one.

**Figure 4.** Configuration 2: Bare floor coupled with suspended ceiling.

The third configuration features a well-established technology for impact noise reduction: a floating floor. Using this decoupling approach, vibration transmission is decreased by means of the mass-spring effect [29,30]. It is known that this technology diminishes the transmitted noise by decoupling the covering heavyweight screed from the bare floor. In this way, vibrations are reduced and thus the transmission to the other room will be significantly reduced. Configuration 3 is represented in Figure 5.

**Figure 5.** Configuration 3: Bare floor coupled with floating floor.

In order to investigate also the coupled effect of both suspended ceiling and floating floor on cross laminated timber, a further configuration (configuration 4, -Figure 6) was considered.

**Figure 6.** Configuration 4: Bare floor coupled with floating floor and suspended ceiling.

In Table 1, different layers used and tested are reported, describing their thickness, density, and elasticity.


**Table 1.** Floor elements description.

#### *2.2. Experimental Structures Characterization*

In order to investigate the influence of different layers on bare CLT, three different noise excitation sources were used: dodecahedral speaker for airborne noise generation, ISO tapping machine for heavyweight impact noise generation, and rubber ball for lightweight noise generation (Figure 7). Four different floors were tested for each configuration. For the sake of brevity, only average results are presented and discussed.

**Figure 7.** Tapping machine and rubber ball used to test the impact noise of CLT floors.

The measurement methods of airborne and impact sound insulation were conducted in accordance with international standards ISO 16283 part 1 (airborne noise) [31] and part 2 (impact noise) [32]. In particular, part 2 of the standard has recently introduced the use of the rubber ball also at an international level, associating it to subjective perception evaluation. The indices are calculated in accordance with the procedures indicated in the ISO 717 standards part 1 (airborne noise) and part 2 (impact noise).

The used tapping machine features the following characteristics:


The rubber ball generates the impact force exposure level LFE in each octave band shown in Table 2, when it is dropped vertically in a free fall from the height of 100 cm ± 1 cm, measured from the bottom of the rubber ball to the surface of the floor under test. The used rubber ball features the following characteristics:

(a) hollow ball of 180 mm in diameter with 30 mm thickness;


The impact force exposure level, LFE, is expressed by Equation (1):

$$\mathbf{L}\_{\rm FE} = 10 \lg \left[ \frac{1}{\mathbf{T}\_{\rm ref}} \int\_{\mathbf{t}\_1}^{\mathbf{t}\_2} \frac{\mathbf{F}^2(\mathbf{t})}{\mathbf{F}\_0^2} \mathbf{dt} \right] (\mathbf{dB}) \tag{1}$$

where F(t) is the instantaneous force acted on the floor under test when the rubber ball is dropped on the floor [N], F0 = 1 N is the reference force, t2 − t1 is the time range of the impact force [s], and Tref = 1 s is the reference time interval. In Table 2, the standard rubber ball force is depicted.



The dodecahedral source features 12 speaker units. All speaker units in the same cabinet radiate in phase. The directivity of loudspeakers is approximately uniform and omnidirectional.

#### *2.3. Acoustic Parameters*

The apparent sound reduction index R is calculated in accordance with Equation (2):

$$\mathbf{R}' = \mathbf{L}\_1 - \mathbf{L}\_2 + 10 \log \frac{\mathbf{S}}{\mathbf{A}} \text{ (dB)}\tag{2}$$

L1 is the energy-average sound pressure level in the source room (dB);

L2 is the energy-average sound pressure level in the receiving room (dB);

S is area of the common partition [m2];

A is the equivalent absorption area in the receiving room [m2];

The normalized impact sound pressure level generated by standard tapping machine is calculated using Equation (3):

$$\mathbf{L}'\_{\rm n} = \mathbf{L}\_{\rm i} + 10 \lg \frac{\mathbf{A}}{\mathbf{A}\_0} \text{ (dB)}\tag{3}$$

where A0 = 10 m<sup>2</sup> is the reference equivalent absorption area.

The maximum impact sound pressure level measured with rubber ball L i,Fmax is the maximum sound pressure level, tested using the "fast" time constant.

From the values measured in 1/3 octave bands it is possible to derive the evaluation indices R w, L n,w, and L iA,Fmax according to ISO 717 part 1 and 2 standard. R w and L n,w are evaluated in the frequency range 100–3150 Hz, while L iA,Fmax is the A-weighted sound pressure level evaluated both in the frequency range 50–630 Hz and 20–2500 Hz. This last extended range was performed in order to consider low frequency comfort according to Späh et al. [20].

#### *2.4. Acoustic Models*

In order to verify if available traditional models are suitable for acoustic performance predictions of Cross Laminated Timber floors, in the following, for the four presented configurations, analytical equations retrieved from literature and standard are presented. It has to be highlighted here that, at present, for the impact rubber ball, no analytical model is available for the noise prediction in the receiving room.

For the bare floor, the traditional model is the ISO 12354-2 [32]. In this view, analytic expression is reported in Equation (4):

$$\mathrm{L\_n} = 155 - \left[ \left( 30 \log \mathrm{m}^{\prime}\_{\mathrm{floc}} \right) + \left( 10 \log \mathrm{T\_s} \right) + \left( 10 \log \sigma \right) + \left( 10 \log \frac{\mathrm{f}}{\mathrm{f\_{ref}}} \right) \right] \text{ (dB)} \tag{4}$$

where m floor is the mass per square meter [kg/m2] of the bare floor, Ts is the structural reverberation time, σ is the radiation efficiency, f is the excitation frequency, and fref is the reference frequency at 1000 Hz.

The structural reverberation time is calculated according to Equation (5):

$$\mathbf{T\_s} = \frac{2.2}{\mathbf{f}\,\eta} \,\mathrm{[s]}\tag{5}$$

where η is the overall damping.

The radiation efficiency is calculated according to Equation (6), using the Waterhouse correction [33]:

$$\sigma = \frac{\frac{\mathbf{p}^2}{4\rho\_0 c\_0} \mathbf{A} \left(1 + \frac{\mathbf{S}\_\Gamma \lambda\_0}{8\mathbf{V}}\right)}{\rho\_0 c\_0 \mathbf{S}\_\Gamma \mathbf{v}^2} \tag{6}$$

where *ρ*0c0 is the air impedance, A is the absorption area retrieved from the reverberation time [m2], ST is the floor area [m2], V is the volume of the receiving room [V], and v is the vibration velocity [m/s].

For the floating floor, the Cremer's equation is available [34], according to Equation (7):

$$
\Delta L'\_{\text{n,w,flosing,Cramer}} = 30 \log \frac{\text{f}}{\text{f}\_0} \text{ (dB)}\tag{7}
$$

where f0 is the resonance frequency [Hz] of the floating floor composed by the resilient layer and the screed and expressed by Equation (8):

$$\mathbf{f}\_0 = \frac{1}{2\pi} \sqrt{\frac{\mathbf{s'}}{\mathbf{m'}\_{\text{scored}}}} \begin{bmatrix} \mathbb{H}\mathbf{z} \end{bmatrix} \tag{8}$$

where s is the dynamic stiffness of resilient layer [MN/m3] and m screed is the mass per square meter [kg/m2] of the screed.

Finally, some models are present for sound insulation prediction of lightweight partitions. Anyway, most of them are related only to the weighted index and do not provide a frequency trend. The only available approach could be the Sharp's one [35,36], providing a frequency domain formulation, reported in Equation (9):

$$\text{AR} = 20 \log \left( \text{m}'\_{\text{partition}} \,\text{f} \right) - 47.2 \text{ (dB)} \tag{9}$$

where m partition is the mass per square meter [kg/m2] of the wall or floor.

#### **3. Results and Discussion**

#### *3.1. Impact Noise–Tapping Machine*

The impact sound pressure level results in 1/3 octave bands for bare CLT floors are shown in Figure 8.

Interestingly, the impact sound pressure levels of the weighted index measured using the tapping machine as the generator provide very similar results for all the different horizontal partitions. This is very important since it demonstrates that all the further studies and noise reduction actions will have very similar influence on all bare floors. The retrieved differences depend on floor dimensions, receiving room shapes and volumes [37]. Accordingly, in the low frequency range, some differences are evidenced. In the middle

frequencies (1000–2000 Hz), they tend to offer very similar results, while at higher ranges, again, some diversities are present.

**Figure 8.** In situ measurements of normalized impact sound pressure level (tapping machine) for four bare CLT floors (configuration 1).

In Table 3, the 1/3 octave band average trend and the standard deviation are reported. It is evident how for most 1/3 octave frequency bands, the standard deviation falls within the ±3 dB range. This permit to consider the average data reliable and thus a reference.

**Table 3.** Average trend and standard deviation of averaged impact sound pressure levels (tapping machine).


When using Equation (4), the predicted trend is not similar to the measured one, as depicted in Figure 9.

It is possible to highlight that for middle-low frequencies, the model more or less fits the measured values. In Table 4, the difference between measured (average) and calculated values are reported. It could be noticed that low and middle frequencies (125–800 Hz) present very good agreement. Accordingly, the prediction falls within a range of ±3 dB. In contrast, from 1000 Hz on, the model is not able to correctly fit middle-high and high frequencies, mostly because the measurement of the structural reverberation time is measured using hard-surface sources (hammer). When the head of the hammer impinges the wood, it tends to present a resilient behavior (compared to concrete) on middle and high frequencies. For this reason, the measurement could not accurately determine this parameter, thus affecting acoustic performance predictions.


**Table 4.** Average trend and difference between measured and calculated values (Δ).

When considering a suspended ceiling addition, no analytical model is present in literature or in standards.

In Figure 10, the results are reported when the exciting source is the tapping machine. It can be seen how different configurations can act on the impact noise reduction when compared to the average impact noise of the bare CLT floor (configuration 1). Accordingly, when at the bare floor a suspended ceiling is added (configuration 2), a significant noise reduction both in frequency and weighted index is verified. In particular, this reduction follows the trend of the average bare floor almost constantly.

**Figure 10.** Impact sound pressure level for CLT floors (configurations 1, 2, 3, and 4). Source: tapping machine.

When considering the range of 250–3150 Hz, we can also easily derive a regressive equation (Equation (10)):

$$
\Delta L'\_{\text{n,w,celling}} = -0.028 \text{ (f)} + 23.8 \text{ (dB)} \tag{10}
$$

where f is the frequency [Hz].

The results could be fitted, with a regression coefficient of R<sup>2</sup> = 0.89. By means of this equation, it could be possible to estimate the effect of this kind of suspended ceiling on a generic CLT floor of 20 cm thickness.

Moving onto configuration 3 (only floating floor and bare CLT floor), it could be acknowledged that a similar reduction is proposed, compared to configuration 2. Here, low frequencies (100–200 Hz) are reduced more efficiently, as well as high frequencies (1600–4000 Hz). However, this technology could not work properly, because its performance requires a heavy mass as bare floor. In this case, a cross laminated plate could not represent this element, because of its lightweight structure.

When applying Equation (7) to configuration 3, a different resulting trend is produced. In Figure 11, the frequency tendencies of the two impact noise reductions are depicted. Clearly, Cremer's model cannot be applied to this kind of wooden partitions, since it fails by a large amount.

The main reason is that Cremer's model considers the bare floor as completely rigid, featuring an ideally infinite mass in comparison to the floating floor. In the case of CLT floors, this does not happen. Accordingly, the density of a CLT floor is 90 kg/m2, very similar to the floating floor. It is evident that the bare timber floor cannot be considered neither more rigid nor more massive than the floating floor, thus significantly affecting the application of Cremer's model.

**Figure 11.** Comparison between Cremer's and measured ΔL n,w for configuration 3.

When investigating the frequency influence on the measured noise reduction index reported in Figure 11, in the range 250–3150 Hz, a relation can be found, as expressed in Equation (11), with a regression coefficient of R2 = 0.70. However, a poor influence of frequency on impact noise reduction is highlighted:

$$
\Delta L'\_{\text{n,w,flosing,CLT model}} = -0.012 \text{ f} + 22.9 \text{ (dB)}\tag{11}
$$

when combining the two technologies in the bare floor (configuration 4), a significant reduction is verified, in comparison to configuration 1 (bare CLT floor) and to both configuration 2 and 3. Anyway, when combining Equations (10) and (11), the obtained result is not reliable (Figure 12).

**Figure 12.** Measured vs. calculated ΔL n,w.

The retrieved equation from measured values is reported below, with a regression coefficient of R<sup>2</sup> = 0.81 (Equation (12)).

$$
\Delta L'\_{\text{n,w,overall}} = -1.9 \text{ f} + 40.1 \text{ (dB)} \tag{12}
$$

Here, we can see that frequency affects more significantly impact sound reduction, in comparison to configuration 2 and 3. It is thus evident that we have to avoid the combination of the two equations related to single actions, as the merging would lead to a significant underestimation of the final results.

#### *3.2. Impact Noise–Rubber Ball*

When using the impact ball as noise source, we have to consider that the excitation is different from the traditional tapping machine. As reported above, this methodology injects into the structures an impulse which is poor of middle-high frequencies (800–5000 Hz) and focuses its action in the range 100–630 Hz. The results of the standardized maximum impact sound pressure relate to bare and lined CLT floors are shown in Figure 13. As demonstrated above, bare CLT floors mostly present the same frequency trends when excited. Therefore, for the sake of brevity, only average bare floor trends are presented hereafter.

**Figure 13.** Impact sound pressure level for CLT floors (configurations 1, 2, 3, and 4). Source: rubber ball.

From Figure 13, it is evident that configuration 2 acts efficiently in noise reduction especially at low (20–100 Hz) and high frequencies (1600–5000 Hz). The presence of the fibrous material within the ceiling air gap could influence the impact noise propagation. However, when the resonance frequency is overcome (over 80 Hz), its efficiency significantly decreases. Since the measured trend is composed by three different zones, showing three different behaviors, it is not worthy to infer a regressive equation.

When only floating floor is considered (configuration 3), it is evident how the provided noise reduction is significantly lower. This is mainly due to the fact that the floating floor works at a different frequency range. It was previously demonstrated [38] that this technology reduces significantly the transmission on middle-high frequency ranges (1000–5000 Hz). When using the rubber ball, this range is not injected in the floor, as reported in Table 2. For this reason, a smaller reduction is found, compared to configuration 2. In this case, the trend presents a homogeneous behavior and therefore a regressive approach may be used. The result is presented in Equation (13), with a regression coefficient of R2 = 0.97:

$$
\Delta L'\_{\text{iA,Fmax,flating}} = 35.1 \log(\text{f}) - 49.5 \text{ (dB)}\tag{13}
$$

It is interesting to note that, in comparison to tapping machine excitation, the equation is not linear anymore, but it follows a logarithmic trend, based on the exciting frequency.

When considering suspended ceiling and floating floor together (configuration 4), a significant overall reduction is verified. Accordingly, in some frequencies, the background noise could have influenced the results, since they are comparable to it.

In this case too, the trend presents a homogeneous behavior and therefore a regressive approach can be used. The result is presented in Equation (14), with a regression coefficient of R2 = 0.74.

$$
\Delta L'\_{\text{fA,Fmax,flocating}} = 10 \log(\text{f}) - 38.1 \text{ (dB)}\tag{14}
$$

In this case, the frequency contribution is less significant than the configuration 3 represented by Equation (13) and a logarithmic trend is evidenced.

#### *3.3. Airborne Sound Insulation*

The investigation of the sound insulation to airborne noise is useful in order to understand if the actions of the floating floor and of the suspended ceiling, applied for impact noise reduction, can influence also soundwaves propagation in air. For this reason, in Figure 14, the measured trends are reported.

**DĞĂƐƵƌĞĚĂŝƌďŽƌŶĞƐŽƵŶĚŝŶƐƵůĂƚŝŽŶŽĨĚŝĨĨĞƌĞŶƚ>dĨůŽŽƌƐĐŽŶĨŝŐƵƌĂƚŝŽŶƐ**

**Figure 14.** Airborne sound insulation for CLT floors (configurations 1, 2, 3, and 4).

It is evident how the bare floor provides very poor frequency performances, while the addition of a suspended ceiling (configuration 2) positively affects sound insulation. In this case, also floating floor (configuration 3) positively affects sound insulation, showing a very similar trend compared to configuration 2. When combining the two solutions, some significant improvements can be seen at middle-low frequencies, until 1600 Hz.

In Table 5, the results of the calculated frequency sound insulation are reported. In addition, the difference Δ<sup>n</sup> between the measured and the calculated values (using Equation (9) are included after each configuration. It is interesting to notice that configuration 1 (bare floor) is not fitted robustly. Accordingly, differences up to −19.8 dB are verified, with a mean value of −13.9 dB. When considering the suspended ceiling addition

(configuration 2), we can verify a significant improvement of the sound insulation prediction. A maximum of 9.8 dB difference is provided, featuring a mean value of −1.6 dB. It is worthy to highlight that from 100 to 2000 Hz, the prediction falls within a range of ± 3 dB, with the exception of 315 Hz (3.7 dB) and 400 Hz (3.5 dB). This fact demonstrates how low, middle, and middle-high frequencies can be successfully calculated using Sharp's theory.


**Table 5.** Calculated sound insulation values, using Sharp's theory. Δ<sup>n</sup> represents the difference between the measured and calculated results.

When considering only floating floor addition to the bare floor (configuration 3), the prediction is not to be considered robust and reliable. Differences are significant (up to −12.4 dB) with a mean value of −6.3 dB. A different trend is verified for the complete layering (configuration 4). For low frequencies (100 Hz–500 Hz), a good prediction is provided, falling within the ± 3 dB range, with the exception of 125 Hz (4.2 dB) and 160 Hz (3.2 dB). From 630 Hz, a constant increase in the difference is verified, reaching its maximum at 5000 Hz (−15.0 dB) and a mean value of −5 dB.

As an overall result, we can say that the Sharp's model provides a reliable prediction only for configuration 2 up to middle frequencies, while for the other configurations, no consistent trends are found. For all configurations, the Sharp's model tends to overestimate the sound insulation and the worst range is represented by high frequencies (>1600 Hz).

#### *3.4. Weighted Indexes Results*

Even if weighted indexes do not provide frequency information, thus eliminating some of the interesting results, it is evident how it is easier to compare measurements using just the single index instead of 18 different values (1/3 octave bands). For this reason, results in terms of weighted index are reported in Table 6.

It is possible to notice that, for impact noise reduction using the tapping machine, the significant variation is caused by the use of whether a suspend ceiling or a floating floor is similar. Anyway, when applying the second configuration, it is possible to notice that the reduction is different if measured with the rubber ball or with the tapping machine. According to the standard, L iA,Fmax,50–630 is the parameter representing the subjective evaluation of the noise disturbance produced by floor impact sound. Conversely, according to literature [20], L iA,Fmax,20–2500 is more accurate. In this case, we can see no significant difference comparing the two parameters.


**Table 6.** Impact noise and airborne sound insulation weighted index results for each configuration.

Considering configuration 3, we can see that almost the same impact reduction is measured with both tapping machine and rubber ball and that no significant difference is found between L iA,Fmax,50–630 and L iA,Fmax,20–2500.

Moving to configuration 4, a significant difference can be reported between tapping machine and rubber ball test, evidencing how, for the latter, a significant improvement in performance is assessed in comparison with the bare floor.

From the sound insulation point of view, when adding a technology of impact noise reduction, we can see a substantial increment of the performances. Anyway, (i) changing the configurations or (ii) merging them do not vary importantly final results.

Another overall finding is that configuration 3 (floating floor) and 4, where the combination of the two technologies ensures a more insulated partition, are capable to respect the law requirements of most European countries [39].

#### **4. Conclusions**

In this work, airborne and impact noise insulation measurements were carried out on a building featuring Cross Laminated Timber floors. In particular, the apparent sound reduction index and the impact sound pressure level measured with a normalized generator and rubber ball impulses were measured step-by-step during the construction phase. In particular, the application of floating floor, suspended ceiling, and the merging of these two technologies applied to the bare Cross Laminated Timber floor was investigated.

Findings highlighted how this lightweight sustainable timber structures do not present the same performances of heavyweight ones. We can then resume our main conclusions as follows:


**Author Contributions:** F.B. and M.C. developed the research. F.B. and M.C. defined the methods and comparisons. F.B and M.C. studied and analyzed typical wooden structure. F.B. and M.C. performed acoustic measurements. M.C. and F.B. collected data and analyses results. M.C. and A.G. overviewed and supervised the research. M.C. and F.B. wrote the paper. All authors edited and proofread the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was developed within the Interreg Project BIGWOOD, Interreg V-A Italy-Austria 2014–2020 (code ITAT 1081 CUP: I54I18000300006), which is gratefully acknowledged. This work was supported by the Open Access Publishing Fund of the Free University of Bozen-Bolzano.

**Acknowledgments:** Authors would like to really thank Ater Trieste for their precious help in this research and specifically Andrea Zeriali.

**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

#### **References**


## *Article* **A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption**

**Xin Li, Bilong Liu \* and Chong Qin**

School of Mechanical & Automobile Engineering, Qingdao University of Technology, No. 777 Jialingjiang Road, Qingdao 266520, China; jz03-4lx@163.com (X.L.); qinchong95@163.com (C.Q.) **\*** Correspondence: liubilong@qut.edu.cn

**Abstract:** A perforated plate with stepwise apertures (PPSA) is proposed to improve sound absorption for low frequencies. In contrast with an ordinary perforated plate with insufficient acoustic resistance and small acoustic mass, the perforated plate with stepped holes could match the acoustic resistance of air characteristic impedance and also moderately increase acoustic mass especially at low frequencies. Prototypes made by 3D printing technology are tested in an impedance tube. The measured results agree well with that of prediction through theoretical and numerical models. In addition, an absorber array of perforated plates with stepwise apertures is presented to extend the sound absorption bandwidth due to the introduced multiple local resonances.

**Keywords:** perforated plate; stepwise apertures; sound absorption; low frequency

#### **1. Introduction**

Porous materials and resonant structures are widely used for sound absorption [1,2]. Typical resonant structures for sound absorption are perforated plates, micro-perforated plates, Helmholtz resonators and thin plate resonators. One of the conditions for effective sound absorption in resonant structures is that their acoustic resistance should match the characteristic impedance of the air. Usually, the acoustic resistances of an ordinary perforated plate with apertures in the range of a few millimeters to centimeters are insufficient and therefore the absorption coefficients are very small. In building acoustics, perforated plates with large perforation ratio are often used as protective plates over porous layers for sound absorption [3–5]. To replace porous materials, Maa proposed a well-known micro-perforated plate (MPP) for sound absorption in the 1970s [6]. For MPP, the apertures are reduced to submillimeter and thus sufficient acoustic resistances can be provided when the perforation ratio is specified. Additionally, in contrast with Helmholtz resonators, when the frequency is away from the resonance frequency, the acoustical reactance of MPP increases slowly to ensure the value is smaller than that of the acoustical resistance in a wide bandwidth, and this characteristic guarantees the broadband absorption of MPP.

When the perforation ratio is constant, MPP with smaller aperture has better sound absorption performance, while in the meantime, the number of holes is increased and the thickness shall be thinner [7]. The increase of hole numbers will increase the cost for the perforation and thinner plates may result in insufficient strength in application. In contrast, a thick MPP with small apertures will lead to excessive acoustic resistance and a decrease in sound absorption performance. To reduce the acoustic resistance of a thick MPP, MPP with variable section have been proposed in recent years [8–11]. Randeberg [8] proposed a micro-horn shaped MPP and the numerical results showed that micro-horn perforation has the potential to improve the sound absorption bandwidth. Sakagami et al. [9] conducted a pilot study to improve the absorption of a 10 mm thick MPP using a tapered perforation, and the measured results exhibited the shift of the resonant frequency to lower frequencies, but the absorption peak decreases. Lu et al. [10] studied the acoustic properties of MPP with variable cross-section, and showed that the absorption performance of such MPP

**Citation:** Li, X.; Liu, B.; Qin, C. A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption. *Appl. Sci.* **2021**, *11*, 6180. https://doi.org/10.3390/app11136180

Academic Editor: Dimitrios G. Aggelis

Received: 9 May 2021 Accepted: 29 June 2021 Published: 2 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

mainly depends on the part of small holes, and the part of large holes is only to increase the plate thickness. He et al. [11] experimentally analyzed the effects of tapered and stepped holes on the sound absorption performance of thick MPP, and the results showed that large tapered holes can broaden the absorption bandwidth in the higher frequency domain. Based on the MPP model, Ma [12] performed an equivalent simulation of the experimental results for a tapered MPP, and the equivalent aperture obtained is between the large part and the small part of the tapered hole. Qian et al. [13] developed a numerical model of MPP with a tapered hole in the acoustic module of COMSOL Multiphysics, and the simulation results showed that the absorption performance of MPP with tapered holes was mainly influenced by the inlet diameter and outlet diameter. In addition, Jiang et al. [14] gave an empirical impedance correction model related to the cross-sectional ratio based on 176 sets of numerical simulations. These aforementioned literatures concern with reducing the excess acoustic resistance of large-thick MPPs by replacing straight-through holes with sub-millimeter variable cross-section holes.

In comparison with MPP, perforated plates with large apertures have the advantage of less perforation holes if the acoustic resistance can be improved sufficiently through some ways. Recently, various perforated structures for low to medium sound absorption, such as perforated panel with extended tubes (PPET) [15], composite honeycomb sandwich panels (CHSPs) [16], perforated composite Helmholtz-resonator (PCHR) [17], coiled space resonators (CSRs) [18], parallel-arranged perforated panel absorbers (PPAs) [19], panel containing coiled Helmholtz resonators [20] and inhomogeneous multi-layer Helmholtz resonators with extended necks (HRENs) [21], have been investigated. However, perforated plates with stepwise apertures for low frequency sound absorption have not been reported. For this motivation, a perforated plate with stepwise apertures (PPSA) larger than 1.5 mm is proposed for low-frequency (100–300 Hz) sound absorption in a compact space.

Additionally, the array structures consisting of multiple sub-absorbers in parallel arrangement have been studied to improve the sound absorption. Cha et al. [22] designed a MPP absorber array with two different cavities and gave the measured normal absorption coefficients by impedance tube. Wang et al. [23] established a numerical model to study the sound absorption mechanism of a MPP absorbers array with different cavity depths in detail. Uenishi et al. [24,25] studied a permeable membranes absorber array (PMAR) numerically and experimentally, and the results showed that PMAR is an effective absorbing structure due to the influence of multiple locally reacting air cavities at different depths. Furthermore, Wu et al. [26] proposed a profiled structure using perforated plates in some wells, thus adjusting the depths of the wells to improve low frequency absorption. In this paper, based on the local resonance effect, without changing the structure parameters, a simple PPSA absorbers array is initially designed to extend the sound absorption bandwidth only by inserting one rigid partition plate in the air cavity.

This paper is organized as follows. In Section 2, theoretical and numerical models are developed to predict the absorption coefficients of PPSA sound absorbers, and the predictions are verified by impedance tube measurements. In Section 3, the sound absorption performances of PPSA absorber and single perforated panel (PP) absorber are compared and discussed, and a simple array structure of two PPSA absorbers in parallel is presented to improve low frequency sound absorption. Finally, conclusions are drawn in Section 4.

#### **2. Models and Methods**

#### *2.1. Theoretical Calculation*

Figure 1a shows the structure of a PPSA absorber, which consists of a PPSA and an air cavity supported by a rigid wall. The PPSA is perforated with a series of stepped holes, which are composed of two unequal circular apertures in the coaxial line. For each circular aperture, the ratio of diameter to depth is greater than 1 and the diameter is not less than 1.5 mm. The diameter, thickness and perforation ratio of the small aperture are *d*1, *t*1, *σ*1, those of the large aperture are *d*2, *t*2, *σ*2, respectively, and the air cavity depth is *D*. Structurally, PPSA can be viewed as a serial combination of two perforated panels PP1 and PP2 without spacing.

**Figure 1.** A perforated plate with stepwise apertures (PPSA) absorber. (**a**) Structure diagram; (**b**) Onedimensional acoustical system.

Based on the transfer matrix method [27], for the one-dimensional acoustical system element of PPSA absorber, as shown in Figure 1b, the sound pressure *p*<sup>1</sup> and particle velocity *u*<sup>1</sup> on the left side and the sound pressure *p*<sup>3</sup> and particle velocity *u*<sup>3</sup> on the right side can be expressed as:

$$
\begin{bmatrix} p\_1 \\ \mu\_1 \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} p\_3 \\ \mu\_3 \end{bmatrix} \tag{1}
$$

where *T*11, *T*12, *T*<sup>21</sup> and *T*<sup>22</sup> are the four pole parameters of the total transfer matrix [*T*]. The total transfer matrix [*T*] is obtained by multiplying the unit transfer matrix of PP1, PP2 and air cavity, written as:

$$[T] = [T\_{\rm PP1}][T\_{\rm PP2}][T\_{Air}]\_\prime \tag{2}$$

For PP1 and PP2, the transfer matrix [*Tpp*] can be given as:

$$\begin{bmatrix} \left[ T\_{\text{PP1},2} \right] \end{bmatrix} = \begin{bmatrix} 1 & Z\_{P1,2} \\ 0 & 1 \end{bmatrix} \prime \tag{3}$$

The acoustic energy loss of the perforated panel mainly includes the air viscous dissipation inside the hole and the end correction caused by viscous friction and sound radiation. Therefore, the acoustic impedance of perforated panel is the sum of the acoustic impedance inside the hole and the end correction outside the hole. According to the viscous motion theory in the tube derived by Rayleigh and simplified by Crandall [28], the acoustic impedance in the circular hole is expressed as:

$$z\_{\rm hole} = j\omega\rho\_0 t \left[ 1 - \frac{2}{k\sqrt{-j}} \frac{f\_1\left(k\sqrt{-j}\right)}{\ln\left(k\sqrt{-j}\right)} \right]^{-1},\tag{4}$$

According to Ingard and Rayleigh theory [29], the corrections of acoustic resistance and reactance at both ends of the hole are 2*ωρ*0*η*/2 and 0.85*d*, respectively. Due to the continuity of the surfaces of PP1 and PP2, the corrections of acoustic impedance at one end is considered. Thus, the acoustic impedance of PP1 and PP2 is written as:

For the air cavity, the transfer matrix [*TAir*] can be written as:

$$Z\_{P1,2} = \frac{1}{\sigma\_{1,2}} \left( j\omega\rho\_0 t\_{1,2} \left( 1 - \frac{2}{k\_{1,2}\sqrt{-j}} \frac{l\_1(k\_{1,2}\sqrt{-j})}{l\_0(k\_{1,2}\sqrt{-j})} \right)^{-1} + j\omega 0.425 d\_{1,2} + \frac{\sqrt{2\omega\rho\_0\eta}}{4} \right), \tag{5}$$

$$\begin{bmatrix} T\_{Air} \end{bmatrix} = \begin{bmatrix} \cos(k\_0 D) & j\rho\_0 c\_0 \sin(k\_0 D) \\\ j\sin(k\_0 D)/\rho\_0 c\_0 & \cos(k\_0 D) \end{bmatrix} \tag{6}$$

where *ω* = 2*πf* is the angular frequency, *η* is the dynamic viscosity coefficient of the air, *k* = *d* √*ωρ*0*η*/2 is the ratio of the inner radius to the viscous boundary layer thickness inside the tube, for square arrangement, perforation ratio *σ* = 0.785*d*2/*b*2, *d* and *b* are the diameter of holes and the spacing between holes, *J*<sup>0</sup> and *J*<sup>1</sup> are the Bessel functions of the zero order and the first order, respectively. *ρ<sup>0</sup>* is the air density, *c*<sup>0</sup> is the sound speed in the air, *k*<sup>0</sup> = *ω*/*c*<sup>0</sup> is the air wave number.

Since the air cavity is supported by a rigid wall, the particle velocity on the rigid wall is *u*<sup>3</sup> = 0. Therefore, it can be obtained from Equation (1):

$$p\_1 = T\_{11} p\_{3\prime} \,\mu\_1 = T\_{21} p\_{3\prime} \,\tag{7}$$

Then, the surface impedance of the PPSA absorber at normal incidence is expressed as:

$$Z = \frac{p\_1}{\mu\_1} = \frac{T\_{11}}{T\_{21}}\tag{8}$$

Therefore, the surface reflection coefficient of the PPSA absorber is given by:

$$R = \frac{Z - \rho\_0 c\_0}{Z + \rho\_0 c\_0} \tag{9}$$

Thus, the absorption coefficient of the PPSA absorber for normal incident wave is written as:

$$\alpha = 1 - |R|^2 = \frac{4\text{Re}(Z/\rho\_0 c\_0)}{\left[1 + \text{Re}(Z/\rho\_0 c\_0)\right]^2 + \left[\text{Im}(Z/\rho\_0 c\_0)\right]^2} \tag{10}$$

#### *2.2. FEM Simulation*

In the thermo-viscous and pressure acoustics frequency domain interface of COMSOL Multi-physics, 3D finite element models are performed to simulate the acoustic behavior of the PPSA absorber under normal incidence. Considering viscosity and thermal loss, the acoustic field of single PPSA is built up in the thermo-viscous-acoustics, frequency domain interface, as shown in Figure 2a. Due to the stepped hole is a symmetric structure, to reduce the simulation time, 1/4 unit of the stepped hole is constructed. This numerical element model is composed of a stepped hole domain, two air domains and two perfectly matched layers (PML) at the ends. Moreover, the background pressure field is defined in an air domain, and PML acts as an infinite air field. The domains marked by the yellow and red line are the inlet and outlet surfaces of acoustic wave, respectively. In the simulation, the maximum frequency is chosen as 1000 Hz, the calculated frequency range is 100–500 Hz and the frequency interval is 5 Hz. For the stepped hole domain, the maximum element size of free tetrahedral meshes is set as the viscous boundary layer thickness size, and for the background sound field and air domain, the maximum element size of free tetrahedral meshes are equal to 1/3 of the small hole radius. In addition, the distribution of mesh elements for PML is at least six layer using the swept node. The mechanical and thermal boundary conditions for the sidewalls of the steppe hole are no slip and isothermal, respectively. For a stepped hole with *d*<sup>1</sup> = 1.5 mm, *d*<sup>2</sup> = 4 mm, *t* = 2 mm and hole spacing of 5 mm, the 3D finite model is meshed with a total of 72,269 tetrahedral meshes and the maximum element for the stepped hole domain is 0.07 mm.

The transfer impedance *Ztrans* of the PPSA is defined as [6]:

$$Z\_{trans} = \frac{\Delta p}{\overline{u}}\tag{11}$$

where Δ*p* is the pressure drop between the inlet and outlet surfaces, and *u* is the mean velocity in the stepped hole. Then, define the numerical value of *Ztrans* as a global variable using derived values.

**Figure 2.** Finite element simulation. (**a**) Thermos-viscous model of PPSA; (**b**) Pressure acoustic model of PPSA absorber.

After that, a finite element model of the PPSA absorber is established in the pressure acoustics frequency domain interface. The upper PML acts as an infinite air domain, the sound incident is placed in background pressure field and the air cavity behind the PPSA is set as air domain, as shown in Figure 2b. The values of the transfer impedance *Ztrans* are then imported using the Interpolation function command in the Definitions toolbar, thus the acoustic impedance of the PPSA can be defined by the built-in impedance. Thus, the sound pressure reflection coefficient *R* is expressed as:

$$R = \frac{p\_{sc}}{p\_{in}}\tag{12}$$

where *psc* and *pin* are the scattered and incident sound pressure, respectively. Finally, the sound absorption coefficient of the PPSA absorber can be obtained from Equation (10).

Figure 3 plots the theoretical and simulated sound absorption of PPSA absorbers with different perforation ratio and cavity depth. The parameters for three PPSA absorbers are listed in Table 1, for PPSA1 and PPSA2 absorbers, the theoretical sound absorption coefficient curves are the same as that of the numerical simulation. In addition, for PPSA3 there is an offset of about 2 Hz between the theoretical and the numerical resonances. In addition, in Figure 3b, there exists a difference between the theoretical and simulated acoustic resistance at frequencies far from resonance, which may be further modified by model mesh refinement and considering the thermal effects of air in the calculations, but its acoustic reactance is almost identical. Overall, the finite element simulation is in agreement with the theoretical calculation.

**Figure 3.** The theoretical and simulated results for PPSA absorbers. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.

**Table 1.** Parameters for three PPSA absorbers in the theoretical and simulated comparison.


#### *2.3. Experiment Measurements*

To further verify the feasibility of the theoretical and numerical models, the sound absorption coefficients of PPSA absorber at normal incidence are measured by SW422 impedance tube with diameter of 100 mm, as shown in Figure 4a, and PPSA samples are made of resin materials by 3D printing using SLA (light curing molding) equipment. The outer diameter of the specimen is about 99 mm, so the sample is placed in the specimen tube with sealing strips or tape to ensure the tightness.

The sound absorption measurement in impedance tube follows the two-microphone transfer function method [30], and the principle is described below. The sound pressure *p*<sup>1</sup> and *p*<sup>2</sup> of two microphones located at position 1 and position 2 are expressed as:

$$p\_1 = P\_1 e^{j\mathbf{k}\_0 x\_1} + P\_R e^{-j\mathbf{k}\_0 x\_1}, \\ p\_2 = P\_1 e^{j\mathbf{k}\_0 x\_2} + P\_R e^{-j\mathbf{k}\_0 x\_2} \tag{13}$$

The transfer function of the incident wave and the reflected wave are written as:

$$H\_I = \frac{p\_{2I}}{p\_{1I}} = e^{-j\mathbf{k}\_0(\mathbf{x}\_1 - \mathbf{x}\_2)} = e^{-j\mathbf{k}\_0 \mathbf{s}},\\ H\_R = \frac{p\_{2R}}{p\_{1R}} = e^{j\mathbf{k}\_0(\mathbf{x}\_1 - \mathbf{x}\_2)} = e^{j\mathbf{k}\_0 \mathbf{s}}\tag{14}$$

According to Equation (13), the transfer function of the total sound field is denoted as:

$$H\_{12} = \frac{p\_2}{p\_1} = \frac{e^{jk\_0x\_2} + re^{-jk\_0x\_2}}{e^{jk\_0x\_1} + re^{-jk\_0x\_1}}\tag{15}$$

Substituting Equation (14) into Equation (15), the reflection coefficient is expressed as:

$$r = \frac{H\_{12} - H\_I}{H\_R - H\_{12}} e^{j2k\_0 x\_1} \tag{16}$$

where *x*<sup>1</sup> and *x*<sup>2</sup> are the distances between microphone position 1 or position 2 and the front surface of the test sample, respectively, and *s* is the distance between two microphone position 1 and position 2. Once the reflection coefficient is determined, the sound absorption coefficient of the test sample can be calculated.

The sound absorption coefficients of a PPSA absorber in the frequency range of 100–300 Hz from theoretical, numerical and experimental results are plotted in Figure 4b. The parameters of the PPSA are *d*<sup>1</sup> =1.5 mm, *d*<sup>1</sup> =3 mm, *t*<sup>1</sup> = 1 mm, *t*<sup>2</sup> = 1 mm, *p*<sup>1</sup> = 0.24%, *p*<sup>2</sup> = 1.08% and the air cavity depth is *D* = 80 mm. The peak frequencies of three absorption coefficient curves are almost at 190 Hz, and the peaks are 0.999, 0.999 and 0.992, and the frequency ranges of absorption coefficients greater than 0.6 are 152–244 Hz, 153–242 Hz and 154–239 Hz, respectively. The results showed that there are tiny errors between the measured and predicted results. These unavoidable deviations may be due to factors such as machining accuracy and inaccurate mounting. In conclusion, the proposed theoretical and numerical models are feasible for predicting the sound absorption of the PPSA absorber. Meanwhile, it implies that the expected sound absorption is also relied on the accuracy of the sample processing.

**Figure 4.** Experimental setup and results. (**a**) Impedance tube system and test sample; (**b**) Absorption coefficients of PPSA absorber from three models: theoretical, FEM and experimental.

#### **3. Results and Discuss**

#### *3.1. Sound Absorption of PPSA and Single Perforated Panle(PP) Absorber*

To explain the feasibility of the proposed PPSA structure for low frequency sound absorption, comparisons of the sound absorption of PPSA absorber with that of single PP absorber are given here, with a depth of 80 mm for each air cavity. Figure 5 displays the theoretical sound absorption coefficients and acoustic impedance of PPSA and single PP absorbers with the same perforation ratio. The relevant parameters of PPSA and PP are listed in Table 2. In Figure 5a, the resonance peaks of PPSA, PP1 and PP2 absorber are located at 168 Hz, 195 Hz and 219 Hz, and the bandwidth of the sound absorption coefficient well above 0.6 are 76 Hz, 40 Hz and 110 Hz, respectively. In Figure 5b, PPSA has a relatively matched acoustic resistance from 1 to 1.41 in the range of 100–300 Hz. In contrast, the acoustic resistance of PP is extremely inadequate, especially the maximum acoustic resistance of PP1 with a large aperture is below 0.5. In addition, compared to the PP absorbers, the zero acoustic reactance of the PPSA absorber is shifted to a lower frequency. Even though, PP2 has a relatively large absorption bandwidth, its resonant frequency occurs at higher frequency, and a large number of perforations will increase the manufacturing costs. Thus, PPSA is more suitable for low-frequency sound absorption.

**Figure 5.** Sound absorption of PPSA and single PP absorbers with the same perforation ratio. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.

**Table 2.** Parameters for PPSA and single PP absorbers with the same perforation ratio.


Figure 6 illustrated the theoretical sound absorption coefficients and acoustic impedance of PPSA and single PP absorbers with the same hole spacing. The parameters of PPSA and PP are given in Table 3. First, comparing PPSA and PP1, their resonant frequencies are close to 153 Hz and their sound reactance curves are almost overlapping as shown in Figure 6b. For PPSA with *d*<sup>1</sup> = 2 mm and *d*<sup>2</sup> = 4 mm, its thickness is *t* = 2 mm, while for PP1 with *d* = 4 mm, its thickness is 6.5 mm, thus the panel thickness of PPSA is about 3.25 times that of PP1. That is, for the same cavity depth and hole spacing, when PPSA and PP absorbers are resonating at the same frequency, PP with the large hole diameter requires a larger thickness. Meanwhile, the resonance peak of PPSA is close to 1, and that of PP1 is 0.88. The sound absorption coefficient of PPSA is greater than 0.6 from 135 Hz–196 Hz, and that of PP1 is greater than 0.6 from 143 Hz–182 Hz. Obviously, the sound absorption of PPSA is better than that of PP1. Furthermore, for PP2 of the thickness *t* = 2 mm, its resonance occurs at a higher frequency of 226 Hz with a peak of only 0.6. This is due to its inadequate acoustic resistance and high acoustic reactance. Therefore, the introduction of stepped holes not only effectively increase the acoustic resistance, but also shift the zero acoustic reactance of the PPSA absorber to lower frequencies.

**Figure 6.** Sound absorption of PPSA and single PP absorbers with the same hole spacing. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.


**Table 3.** Parameters for PPSA and single PP absorbers with the same hole spacing.

#### *3.2. Sound Absorption of PPSA Absorber with Varied Hole Spacing, Aperture and Plate Thickness*

The theoretical absorption coefficients of the single PPSA absorber with varied hole spacing, aperture and plate thickness are marked with the color bar in Figure 7a–d. The parameters of PPSA are listed in Table 2 and hole spacing b = 30 mm, air cavity D = 80 mm. In Figure 7a, when other parameters remain unchanged, with the increase of hole spacing b, the effective sound absorption is distributed in the lower frequency range, and the maximum sound absorption coefficient reaches 1 and then decreases. In Figure 7b, when aperture *d*<sup>1</sup> increases, the resonance shifts to higher frequencies and becomes weak, and the maximum sound absorption coefficient is significantly reduced. That is, when the perforation ratio is constant, the PPSA with a smaller aperture *d*<sup>1</sup> is more conducive to sound absorption in the low frequency range. The effect of aperture *d*<sup>2</sup> on the sound absorption is similar to that of aperture *d*1, so no explanation is given here. In Figure 7c, as the thickness *t*<sup>1</sup> increases, the resonance occurs in the lower frequency, but the maximum sound absorption coefficient becomes lower. This is because the small aperture with a large thickness causes excessive damping. In Figure 7d, when the thickness *t*<sup>2</sup> increases, the resonance also moves to lower frequencies, however, the maximum absorption coefficient of almost 1 is perfect in the whole range of *t*2. This is because for the large aperture, the sound mass changes obviously as the thickness increases, while the sound resistance changes relatively slightly. Therefore, proper parameters should be considered to obtain an effective PPSA absorber for low frequency sound absorption.

**Figure 7.** The sound absorption coefficients of PPSA absorber with varied structure parameters. (**a**) Hole spacing b; (**b**) Hole aperture *d*1; (**c**) Plate thickness *t*<sup>1</sup> of the small aperture; (**d**) Plate thickness *t*<sup>2</sup> of the large aperture.

#### *3.3. Sound Absorption of PPSA Absorber Array*

Due to its Helmholtz resonance nature, PPSA absorber exhibits good sound absorption in the vicinity of the resonance. To further improve the sound absorption of PPSA absorber, only one rigid plate is inserted to separate the air cavities, thus introducing additional resonances to expand the absorption bandwidth, while the structural parameters of the PPSA remain unchanged.

Without loss of simplicity, a PPSA absorber array consisting of two sub-PPSA absorbers arranged in parallel is shown in Figure 8a. According to the electroacoustic analogy method, the surface impedance of the PPSA absorber array is expressed as:

$$Z = \left(\frac{\Phi\_1}{Z\_1} + \frac{\Phi\_2}{Z\_2}\right)^{-1} \tag{17}$$

where *ϕ*<sup>1</sup> and *ϕ*<sup>2</sup> are the area ratios of sub-PPSA1 and sub-PPSA2 to the PPSA absorber array, *Z*<sup>1</sup> and *Z*<sup>2</sup> are the surface impedance of sub-PPSA1 and sub-PPSA2 absorbers, respectively.

**Figure 8.** PPSA absorber array. (**a**) Structure diagram; (**b**) Sound absorption coefficients; (**c**) The acoustic pressure field (color map) and velocity distribution (yellow arrow).

Next, we will discuss the sound absorption performance of PPSA absorber array. As an example, a PPSA absorber array with panel thickness *t* = 2 mm, air cavity *D* = 100 mm, diameters of stepped hole *d*<sup>1</sup> =1.5 mm and *d*<sup>2</sup> = 4 mm, and its relevant parameters are listed in Table 4. The sound absorption coefficients of PPSA absorber array, sub-PPSA absorber and single PPSA absorber are demonstrated in Figure 8b. It can be observed that, the two peak of the PPSA absorber array are located at 158 Hz and 208 Hz, and the peaks of subPPSA1 and sub-PPSA2 absorber are positioned at 154 Hz and 213 Hz, respectively. That is, the two peaks of the PPSA absorber array are corresponding to those of sub-PPSA1 and sub-PPSA2 absorber, and the apparent frequency shift can be attributed to the interaction of sub-PPSA1 and sub-PPSA2. In addition, the PPSA absorber array has a sound absorption coefficient greater than 0.6 from 129 Hz to 257 Hz, and its sound absorption bandwidth is about 128 Hz. In contrast, the single PPSA absorber has a sound absorption coefficient greater than 0.6 from 150 Hz to 237 Hz, and its sound absorption bandwidth is about 87 Hz. Consequently, an array of PPSA absorbers can expand absorption bandwidth for low frequency under the common coupling effect of multi-local resonances.


**Table 4.** Parameters for PPSA absorber array and single PPSA absorber.

Intuitively, Figure 8c shows the distribution of the normalized sound pressure and particle velocity at two absorption peak frequencies *f* = 158 Hz and *f* = 208 Hz. As observed, at *f* = 158 Hz, the sound pressure in the air cavity of sub-PPSA1 is approximately four times higher than that of the incident sound field, and most of the particle velocity flow is distributed in sub-PPSA1 absorber. Similarly, at *f* = 208 Hz, due to strong local resonance, the maximum sound pressure and particle velocity flow are mostly concentrated in the sub-PPSA2 absorber, and even the sound pressure does not change in the Sub-PPSA1. Thus, it is confirmed that acoustic energy is mainly dissipated by the local resonances of the PPSA absorber array.

For PPSA absorber array and single PPSA absorber, the measured and theoretical sound absorption of sample 1 and sample 2 are shown in Figures 9 and 10, respectively. The air cavity depth of two samples is 80 mm, and the approximate parameters for samples are given in Table 5. There are some differences between the measured and the theoretical absorption coefficients, as well as the acoustic impedance, but their sound absorption characteristics are almost identical. For PPSA absorber array1, as shown in Figure 9a, the measured sound absorption coefficient is higher than 0.5 from 141 Hz to 280 Hz, and its effective sound absorption bandwidth is about 105 Hz. While for single PPSA absorber1, the measured sound absorption coefficient is above 0.5 from 153 Hz to 247 Hz, and its effective sound absorption bandwidth is about 59 Hz. Moreover, compared to a single PPSA absorber without partition, as observed in Figure 9b, the specific reactance of the PPSA absorber array changes relatively slowly and tends to 0, and its specific acoustic resistance is close to 1 except for the non-strong coupling domain of the sub-PPSA absorber.

**Figure 9.** The sound absorption of Sample 1. (**a**) The measured and theoretical sound absorption coefficients; (**b**) The measured and theoretical specific acoustic impedance.

**Figure 10.** The sound absorption of Sample 2. (**a**) The measured and theoretical sound absorption coefficients; (**b**) The measured and theoretical specific acoustic impedance.


**Table 5.** Parameters of Samples of PPSA absorber array and single PPSA absorber.

Similarly, for PPSA absorber array2, as shown in Figure 10a, the measured sound absorption coefficient is higher than 0.5 from 165 Hz to 334 Hz, and its effective sound absorption bandwidth is about 129 Hz. While for single PPSA absorber2, the measured sound absorption coefficient is above 0.5 from 186 Hz to 302 Hz, and its effective sound absorption bandwidth is about 72 Hz. In addition, Figure 10b exhibits that the acoustic impedance characteristics of PPSA absorber array2 are the same as that of PPSA absorber array1. In summary, it is also demonstrated that the PPSA absorber array has the potential to enhance sound absorption compared to a single PPSA absorber. In addition, the overall dimensions of the fabricated specimens are 82 mm, which are only about 1/30th and 1/27th of the maximum wavelength of the semi-absorption coefficient, indicating that the PPSA absorber array is a compact structure for low frequency sound absorption.

#### **4. Conclusions**

In this paper, a PPSA absorber is studied to improve sound absorption for low frequencies. The advantages of the PPSA are that it matches acoustic resistance of air characteristic impedance and moderately increase acoustic mass by introducing stepped hole. The theoretical and numerical predictions agree well with the experimental results. In addition, an array consisting of two PPSA absorbers arranged in parallel is explored, and PPSA absorber array exhibits an effective broadband sound absorption due to the coupling effect of local resonances. The measured results also show that PPSA absorber array as a compact structure has the potential to control large wavelength noise in a limit space. Consequently, the proposed PPSA provides a meaningful method for the application of perforated plates in low frequency sound absorption. Moreover, in the subsequent work, we will investigate the sound absorption characteristics of the array structure of multiple inhomogeneous PPSA absorbers with different cavities.

**Author Contributions:** Conceptualization, X.L. and B.L.; methodology, data curation and investigation, X.L.; validation and software, C.Q.; resources and supervision, B.L.; writing—original draft preparation, X.L.; writing—review and editing, X.L. and C.Q. All authors have read and agreed to the published version of the manuscript.

**Funding:** The financial support given by NSFC with Grant No. 11874034 and Taishan Scholar Program of Shandong (No. ts201712054) are highly appreciated for this research.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

