Normal Incidence of Sound Transmission Loss of a Double-Leaf Partition Inserted with a Microperforated Panel

A double-leaf partition in engineering structures has been widely applied for its advantages, that is, in terms of its mechanical strength as well as its lightweight property. In noise control, the double-leaf also serves as an effective noise barrier. Unfortunately at low frequency, the sound transmission loss reduces significantly due to the coupling between the panels and the air between them. This paper studies the effect of a microperforated panel (MPP) inserted inside a double-leaf partition on the sound transmission loss performance of the system. The MPP insertion is proposed to provide a hygienic double-leaf noise insulator replacing the classical abrasive porous materials between the panels. It is found that the transmission loss improves at the troublesome mass-airmass resonant frequency if the MPP is located closer to the solid panel. The mathematical model is derived for normal incidence of acoustic loading.


Introduction
A double-leaf structure is a common structural design for many engineering applications.The vehicle body, such as in cars, trains and airplanes, and the walls of a building are some examples of double-leaf partition in practice.From the acoustical engineering point of view, the double leaf is proposed to be a better noise barrier compared to the single-leaf.However, there remains a problem on the double-panel which is the weak sound transmission loss (STL) performance at low frequency due to the "mass-air-mass" resonance.This causes the double leaf to lose its superiority over the single-leaf [1].
Several works have been established to solve this problem.This includes employing absorptive materials inside the gap of a double-leaf, for example, fiberglass [2] and rockwool [3] which can effectively increase the STL due to additional damping to the air layer provided by the absorbent.Mao and Pietrzko [4] proposed a technique by installing the Helmholtz resonators at the air gap.The resonator acts like single degree of freedom system of which its natural frequency depends on its geometry.In order to increase the STL at massair-mass resonance, the Hemholtz resonator is tuned to the same resonant frequency.Li and Cheng [5] used an active control system to control the acoustic modes in the gap by using a sound source and an actuator.The sound source reduces the transmission energy by suppressing certain acoustic modes in the air gap while the actuator reduces energy from the structural path by creating counter forces on the two panels to suppress the vibration.Similarly, Li et al. [6] used long T-shaped resonators embedded along the edge of the double panel.This is also aimed to actively control both acoustics and structural path in the gap.It is found that by varying the location of the resonators, the STL at resonance can be significantly improved.Mahjoob et al. [7] introduced the newtonian fluids to control the acoustic path inside the gap.Air, oil and ferromagnetic nanoparticle fluid were used as a filler between the two panels.Although not practical, this method is also shown to increase the STL at resonance.
However, use of acoustic absorbers such as foam or fibrous type materials inside a double-panel is still the most cheapest and common practice to increase the sound insulation performance [2,3].For noise control application where abrasive and polluting materials cannot be presented, such as in the food industry where hygienic condition is critical to be maintained around the processing machines, conventional synthetic fibrous materials are thus not the solution.Although it is hidden between the panels, a noise barrier panel which is easy to be cleaned handled and is free from hazardous substances to health is therefore necessary.
An alternative fiber-free absorber which has gained more popularity is a microperforated panel (MPP) absorber.MPP is a perforated panel with millimetric size holes backed by air cavity and rigid surface found by Maa in 1975.The hole diameter must be in the range between 0.05 and 1 mm and the perforation ratio between 0.5 and 1.5% for optimum absorption [8].As the MPP can be made from panel, it provides several advantages such as nonfibrous, nonabrasive, nonpolluting, and safer in case of fire hazard.Although the MPP is mainly applied for sound absorber, several works have also been published concerning its sound insulation performance.
Dupont et al. [9] investigated the sound transmission loss of a double-leaf structure where a MPP is backed by a solid panel.Toyoda and Takahashi [10] studied the sound transmission loss of a MPP by subdividing the air cavity behind the MPP to have the sound propagation in normal incidence in the cavity.The transmission loss is found to increase at midfrequencies.Most recently, models of sound transmission loss for a multilayer partition with an MPP are proposed by Mu et al. [11].In their model, the MPP is located at the outer layer of the system.
In this paper, a similar multilayer structure is proposed, but with the MPP inserted between two solid plates.Apart for hygienic purposes, the application can also be found in for example, a multilayer window system where a transparent panel is required to improve the noise insulation.The next section describes the derivation of the mathematical model and presents the simulation results of the effect of the MPP insertion, in terms of its location in the gap as well as its hole size and perforation ratio, on the sound transmission loss.The derivation is conducted only for the sound field with normal incidence.Recent finding suggests that the effect of mass-airmass resonance for an infinite double-panel system subjected to the diffuse field incidence is not correct due to the internal resonance in the cavity in the direction parallel to the panel [12].Numerical modelling technique is required, but this is beyond the scope of this paper.

Governing Equations
2.1.Propagating Acoustic Pressure.A mechanical system of a double-leaf inserted with an MPP (abbreviated here as DL-MPP) under normal incidence of acoustic loading can be seen in Figure 1.The solid panels are separated by distance  and the MPP is located by distance  from the back solid plate.Each of the solid and the MPP panels have mass per unit areas  and , respectively, and they are assumed to be supported on identical mountings having stiffness per unit area  and damping constant per unit area .The incident pressure is expressed as and the reflected pressure is given by where  = / for  represents the acoustic wavenumber,  is the angular velocity, and  is the sound speed in the air.
Here and for the rest of the equations, time dependence   is implicitly assumed.At  = 0, the acoustic pressure acting on the incident side of the front panel can be written as In the same way as in ( 1) and ( 2), the total pressure on the other side of the front panel surface is thus The relation between the average surface particle velocity V and the sound pressure exciting the panel can be obtained by using Euler equation V = −1/(/) [13].For both surfaces of each panel, at  = 0 for the front panel this gives while at  =  −  for the MPP and at  =  for the back panel where V  is the velocity of the panel, V is the mean particle velocity over the MPP surface, and   =  is the acoustic impedance of air with  being the air density.Note that for the solid plate, the mean particle velocity on its surface equals to the velocity of the panel V = V  .This is valid for light fluid such as air and not for heavy medium such as water.
For convenience, the distance between the panel is assumed much smaller compared to the acoustic wavelength ( ≪ 1).The cavity pressures can therefore be assumed to be uniform between each gap Substituting ( 10) into (7) yields Using the same way to the surface pressure on the back solid panel ( = ) gives As the cavity pressure is uniform, ( 5) and ( 11) can be combined to give while for ( 12) and ( 13), it yields

Hole Impedance and Mean Particle Velocity.
As the acoustic pressure impinges on the MPP, the air particles penetrate the holes and also excite the remaining solid surface of the panel.The combination between the panel velocity and particle velocity inside the holes creates the mean particle velocity given by [14] where  is the perforation ratio and V ℎ is the particle velocity inside the holes.The motion of fluid inside the hole depends on the impedance of the hole which according to Maa [8] is given by with where   = (  /2)(/V  ) 1/2 ,   is the hole diameter,  is the plate thickness, and V  is the viscosity of the air, that is, 1.8 × 10 −5 Ns/m 2 .The real part of the impedance  , represents the viscous effect responsible for the friction between the inner solid surface of hole and the air, and the imaginary part  , represents the inertia of the air inside the holes of which the air moves like a piston.From these mechanisms, the net pressure Δ on the surface of the MPP can be expressed as [14] Equation ( 19) can also be rearranged as By substituting this into (16), the mean particle surface velocity can also be expressed as the function of the net pressure given by where  = 1 − ( , /  ) is the complex nondimensional terms.

Sound Transmission Loss.
The equation of motion for the solid back panel is given by where   3 =   1 =  +  − / is the mechanical impedance of the panel.The damping constant can be written as  =    with   = (/) 1/2 being the natural frequency of the system and  the damping loss factor.Substituting ( 9), (15), and (21) into (22), then dividing both sides with V  3 yield the panel velocity ratio The equation of motion for the MPP is expressed as where   2 =  +  − /.Substituting ( 14), (15), and (21) into (24), and again dividing both side with V  3 yield It can be seen that the velocity ratio of the solid panels depends on the location of the MPP inside the gap.From the equation of motion of the front solid panel and using the relation between incident and reflected pressure in (3) and ( 5) it gives By dividing both sides with   =   V  3 , the ratio of the incident and reflected pressure is given by As for plane wave, the sound power  is proportional to the sound intensity  which is simply a ratio of squared magnitude sound pressure to the air impedance,  = | and the transmission loss in dB unit is

Effect of MPP Location, Hole Diameter, and Perforation
Ratio. Figure 2 shows the transmission loss under normal incidence of acoustic loading for double leaf (DL) [1], triple leaf (TL), and double leaf with MPP (DL-MPP) located exactly at the middle of the solid panels ( = 0.5).All three panels have the same thickness of 1 mm made of aluminium (density 2700 kg/m 3 ) with air gap  = 100 mm between the solid plates.Throughout the paper, the stiffness per unit area of the mounting used in the calculation is  = 100 N/m 3 and the damping loss factor is  = 0.01.The graph is plotted from 50 Hz to 1 kHz to have better clarity around the resonance as well as for ease of analysis.The "mass-air-mass" resonance of the DL can be seen to occur around 170 Hz shown by the "drop" value of STL to 0 dB, a well-known phenomenon which occurs when the panels move out of phase.It can also be seen that inserting another solid panel between the double panels (TL) yields the second resonance at 280 Hz corresponding to the gap between the middle and the back panel.This can be considered to worsen the problem although the STL at mid-high frequency significantly increases due to the increase of mass.The insertion of MPP between the DLs (in the middle) overcomes the second resonance.However, the first resonance still occurs which corresponds to the gap between the solid plates.As the aim is to improve the STL of the conventional double leaf at the resonance, Figure 3 shows the results for the DL and DL-MPP for different distance  of the MPP to the solid plate.As in Figure 2, the resonance can be seen at 170 Hz for the DL and also for the DL-MPP with MPP at  the middle of the gap.The presence of the MPP gives no effect to overcome the resonance in this case.
For other locations of the MPP in Figure 3, as the MPP shifts closer to the solid panel, regardless, the front or back solid panel, the STL can be observed to increase at the resonance.The additional damping due to the viscous force in the MPP holes influences the air layer in front of the solid plate which breaks the coupling between the solid panels and the air.It can also be seen that the position of the MPP in the gap also affects the STL at mid-to-high frequency in this case above 400 Hz.Contrary to the STL at resonance, the STL above the resonance increases as it moves away from the solid panel within halfway of the gap.The effect of MPP to breach the mass-air-mass resonance is also discussed by Mu et al. [11], where the MPP is located at the outer layer  of the partition system.However, no detailed discussion is presented regarding the gap of the MPP.
Figure 4 shows the effect of hole diameter of MPP on the STL for fixed MPP location,  = 0.1.Around the resonance region up to 400 Hz, decreasing the hole diameter improves the STL as this increases the domination of the real part of the hole impedance which thus provides more viscous force or damping to the MPP.
In Figure 5, the effect of the perforation ratio is investigated.It can be seen that increasing the perforation ratio does not give significant differences to the STL around the resonance.Therefore, to benefit with STL improvement at high frequency due to added mass in the system, the lowest perforation ratio for the MPP, that is,  = 0.5% is preferred.
Increasing the air gap of the solid plate as in Figure 6 can be seen to shift the effect of the resonance to lower frequency.The improvement at the resonance due the MPP is the same.

STL Improvement.
For clarity of analysis, it is of interest to quantify the level of improvement of the STL which is the dB difference after and before inserting the MPP to the double leaf.This is also the same as the ratio of the transmitted sound power (represented by the power transmission coefficient) before (  ) and after (  ) the MPP insertion in dB unit which is given by where STL  is the transmission loss of the DL-MPP and STL  is for the DL. Figure 7 presents the STL improvement, Ω of the DL-MPP system from results in Figures 3, 4, 5, and 6 plotted up to 5 kHz to give clarity at high frequencies.In Figure 7(a), it can be seen that Ω can be achieved up to nearly 10 dB at the resonance for the MPP at  = 0.1 from the solid plate.These results also show that significant improvement of 5 dB or more can be achieved for hole diameter of 2 mm or less.At higher frequency above the resonance, Ω increases rapidly with frequency by more than 20 dB/decade resembling the "mass-law" trend.Figure 7(b) shows that smaller hole is preferred for good Ω.This could add the cost to the system as panel with smaller microholes are more difficult to fabricate.However, this can be compromised with minimum perforation ratio as shown in Figure 7(c), where almost no further improvement is given to Ω around the resonance by varying the perforation ratio.Again the effect can only be seen above the resonance at high frequency (in this case above 70 Hz), where small perforation ratio provides greater Ω.It is also interesting to note the deterioration of Ω just after the resonance (indicated by negative Ω) which can be seen to be greater as the air gap distance is increased.As this is due to the effect of the amount of solid part in the panel, this can be reduced by increasing the perforation ratio as shown in Figure 7(c).In this case, large perforation ratio is chosen if this reduction effect cannot be tolerated in the design.

Experiment
The experiment to measure the transmission loss of the proposed system was conducted using the impedance tube method, where the specimen was located inside the tube and was excited by a sound field from a loudspeaker.The tube has 50 mm diameter.Two GRAS acoustic microphones 1/2 inch type 40AE were placed before the sample and the other two microphones were after the sample.The recorded signal from the microphones was then processed by a spectrum analyzer LDS Photon.The diagram of the measurement setup is shown in Figure 8. Three samples were prepared for the experiment, where a sample consisted of three solid 1 mm thick and round aluminium plates with diameter of also 50 mm to properly fit inside the impedance tube.The samples were fitted in a sample holder.To hold the plate sample in its position, a light tape was used between the plate perimeter and the holder.This was also to ensure that the whole plate surface can have small movement when it was exposed by a plane wave acoustic loading to closely resemble the model in Figure 1.Use of light tape was to minimise additional mass introduced to the plate.One of the plates was then perforated with submillimetric holes having diameters of 0.3 mm, 0.4 mm, and 0.5 mm for each sample.The gap between the solid plates is 70 mm and the MPP was located at 5 mm from the back solid plate.
The tube was fed with white noise up to 800 Hz to only focus the analysis at low frequency range, where the effect of mass-air-mass resonance occurs (at around 200 Hz).In this frequency range, the acoustic loading still has plane waves propagating along the tube.The signal processing technique for the transmission loss employed the wave decomposition method proposed by Salissou and Panneton [15].This method applies two-load technique, which means that it requires two different loadings for the termination conditions for the transmission coefficient formula to be assembled.In this experiment, the loads were made from glass wool and have two different shapes: conical and circular.The former shape is to provide an anechoic termination in the tube.
Figure 9 shows the experimental results of the transmission loss for several hole diameters and perforation ratios.

Advances in Acoustics and Vibration
The measurement data is found to be only valid from 400 Hz.This is due to the conical termination which is difficult to be anechoic at low frequencies.The reflected waves thus affect the recorded signal.This could be overcome by having a longer tube for the downstream part (i.e., the tube at the transmission region) to give the reflected waves more time to arrive at the microphone as in [11].However, above 400 Hz, it can be seen that the measurement data shows reasonably good agreement with the theory.

Conclusions
The sound transmission loss of a double-leaf partition system inserted with MPP under normal incidence of acoustic loading has been reported.It is found that the MPP insertion reduces the effect of mass-air-mass resonance found in the conventional double-leaf partition at low frequency.However, this is only effective when the MPP distance is less than half of the air gap of the solid panels and improves as it approaches the solid plate.Reducing the size of the hole improves the STL at resonance, while varying the perforation ratio gives only small effect.Optimum effect of sound transmission loss improvement can therefore be achieved with small microhole diameter and small perforation ratio.At high frequency above the resonance, for any MPP parameters, the STL of the system increases dramatically due to added mass.The experimental result shows good agreement with the theory at the mass law region, but validation at low frequencies needs to be improved to observe the phenomenon at the mass-air-mass resonance.Employing the MPP for a multilayer structure is thus feasible, particularly for the system exposed with predominantly low frequency noise, for example, a window system of a control room close to a stamping machine where the sound impinges at normal direction.The proposed model can be used as a design guide.

Figure 1 :
Figure 1: A schematic diagram of a DL-MPP system.

Figure 7 (
Figure 7(d) shows the shift of the resonance area because of the change of the air gap distance.Different peak level of Ω in the results is due to different air gap  which also results in different distance  of the MPP to the solid panel.It is also interesting to note the deterioration of Ω just after the resonance (indicated by negative Ω) which can be seen to be greater as the air gap distance is increased.As this is due to the effect of the amount of solid part in the panel, this can be reduced by increasing the perforation ratio as shown in Figure7(c).In this case, large perforation ratio is chosen if this reduction effect cannot be tolerated in the design.

Figure 8 :Figure 9 :
Figure 8: Diagram of the experimental setup for the sound transmission loss measurement.
2|/  .The transmission coefficient is therefore written as