Generation of uncertainty envelopes for thin acoustic screens with uncertain parameters

The properties of the materials used for building sound proofing systems are known to exhibit large variations. These may lead to significant differences in the acoustic responses within a given material batch, particularly when resistive screens are used as a surface component for a multi-layered absorbing panel. In such thin films, it is mostly the thickness and the flow resistivity, but in some cases also the porosity, that are difficult to control in the production process. All these potential variations influence the acoustic response of the complete panel. In the present contribution, a method to isolate and evaluate the effect of uncertainties in a film is proposed. Using a transmission line approach, it is shown to be possible to predict the modification of the response induced by the uncertainties. The proposed technique is then adapted to determine uncertainty envelopes of the absorption coefficient, for experimentally acquired responses, that are closer to measured envelopes as compared to those generated using Monte Carlo simulations or simplified approaches. The method is tested both on numerical and experimental cases and shows, in both cases, a very good agreement with the reference solutions. Unlike Monte Carlo approaches, the proposed method does not require a massive computational effort which makes it suitable for real life applications.


Introduction
Reducing noise through dedicated acoustic treatments is essential in modern society to counteract the negative impact on health that is linked to exposure. Many approaches have been proposed over the years, either with a broadband noise reduction objective [1] or targeting a given frequency band [2][3][4]. Recent reports and directives [5][6][7] set evermore ambitious targets in term of noise mitigation for the future. This overall endeavour to reduce noise and its impact requires the development of efficient, versatile strategies to be combined with punctual, more specific solutions to mitigate particular features of the noise content (emerging tones for instance).
A widely used and rather inexpensive approach to noise reduction over a fairly large frequency band uses poroelastic materials (foam, fibre panels, etc.) arranged as a multi-layered laminate. These composite structures usually involve a small number of foam/fibrous layers with carefully chosen properties and thicknesses in order to meet specific absorption objectives [1]. To protect the panel and sometimes fine tune its properties, thin films can be added on the free surface [8,9]. Because of the films' airflow resistivity is often high they are known to play a prominent role in the behaviour of the overall systems [10][11][12]. As their precise characterisation might be challenging [13], it is of primary importance to understand how the uncertainties on the film parameters induce variations of the overall system response. Apart from this objective, another goal of the study is to propose an efficient method to compute the associated response frequency envelope curves both on simulated response and on experimental results. It is shown that the proposed approach allows one to merge statistical data from the characterisation of the screen with a single measurement of the overall system response. The resulting envelopes are shown to convey the same statistical information as Monte Carlo results obtained by measuring a large number of samples.
The paper is organised as follows. In the first section a study model is introduced and used to derive the influence of the film and its uncertain parameters on the response. A technique for combining an existing deterministic response of a panel and statistical information on the film to infer the associated envelope is discussed. In a second section, the proposed developments are tested against Monte Carlo simulations and validated. A third section presents an experimental set-up to measure the absorption coefficient of different laminates. Using a large number of samples from two different films, a statistical characterisation is performed following a procedure described in the literature [13]. Using the same rig, different systems composed of an absorbing backing lined by one of these films are measured for an important number of film samples in order to obtain experimental envelopes. The experimental results are compared to those given by the proposed envelope generation technique. A final section discusses the results. The key idea of this section is to write the system response such that it is possible to isolate the contribution of uncertain film parameters. The chosen configuration is presented in Figure 1 and involves a backing represented by its surface impedance Z B , a resistive but permeable layer (film, screen, etc.) of thickness d and the so-called incident medium in which the excitation propagates.
In the following, a positive time convention e jxt is assumed where x is the pulsation, t is the time variable and j 2 = À1.

Transmission through the film
Throughout this work, the film layer is assumed to be thin, of the order of half a millimetre. It is furthermore assumed that the elastic deformation of the film may be neglected. This allows for a class of simplified models where the film may be regarded as an equivalent fluid [1,14]. In the examples, a limp model was used for this purpose.
Based on the above assumptions, the thin film layer is modelled through an equivalent characteristic impedance: where q f and K f are respectively the equivalent density and dynamic bulk modulus. Different expressions of these quantities exist in the literature [1,15,16] but are out of the scope of this paper and will not be discussed in the present contribution.

Evaluating the surface impedance
In order to determine the response of the multi-layered system to a plane wave excitation, one needs its surface impedance Z s . Using an equivalent fluid model for the film, this quantity can be deduced from the backing surface impedance Z B and the film properties using the impedance translation formula [1]: where, is the wavenumber in the film.
As the thickness of the film is small compared to the wavelength (e ¼ k f d ( 1), a first order Taylor expansion of the tangent gives tanðk f dÞ ¼ k f d þ Oðk f dÞ. After substitution in (2) and simplification, this may be rewritten as: which may be simplified further, through a Taylor expansion of the denominator: Finally, substituting equations (3) and (1), and neglecting the ðk f dÞ 2 term gives: The influence of the film is then entirely captured in Z F which, apart from the angle of incidence and the thickness of the screen, shows a dependency on the backing impedance Z B .

Introducing uncertainties
As indicated in the introduction, the parameters of acoustic screens are seldom known with perfect accuracy. The sources of uncertainty can be either linked to samplewise variations or to the characterisation itself. These films, woven or non-woven, are rather delicate and it is difficult to ensure that cutting and manipulating the samples does not alter their properties. Thus, different factors tend to render these media challenging to characterise even though standardised procedures are available [13,17]. Their influence on the acoustic performance is mainly controlled by two properties, the airflow resistivity r and the thickness d [1,12]. In this section, the corresponding uncertainties induced on the model parameters of the medium are considered in terms of the higher level descriptors: q f and K f . Through these two descriptors, and the thickness d, the resulting deviation may be directly introduced into (6). It should be noted that the hypotheses leading to (6) henceforth are assumed to apply. In order to determine the variation of Z F when the different parameters vary, the total differential of Z F is obtained as: Using this, the variation of Z s as a whole is then calculated through: Thus, in the case of a variation of q f alone, the resulting surface impedance is given by: which is independent of the backing system. In other cases, e.g. when the thickness or the bulk modulus are known to vary, the surface impedance of the backing system is required at the frequency of interest. The cases for which this last quantity is not readily available are discussed in Section 2.3.

Deriving reflection and absorption coefficients
Considering the deviation of the surface impedance Z s ¼ " Z s þ ÁZ s , it is straightforward to compute the reflection coefficient: and factorising " Z s þ Z 0 , the denominator contains a small parameter ÁZ s ð " Z s þ Z 0 Þ À1 and thus may be expanded in a Taylor series, which allows to separate the nominal and deviated part of R as, From (11), one can derive the absorption coefficient as (with RðgÞ denoting the real part of g):

Envelope determination
Having established equation (7), it is now possible to generate an envelope curve using as input a single response of the overall system and information on the uncertainties in the film. This process will be outlined in the following, based on the notation introduced in Figure 2. Here two possibly independent sets of measurements are used. On one hand, a characterisation of the film is required including a rather accurate estimation of how properties are statistically distributed, for which different techniques exist, either based on quasi-static or dynamic measurements [13,17]. On the other hand, a deterministic response of the overall system is needed which may be readily measured using e.g. standard impedance tube techniques [18].
Depending on which parameters of the equivalent fluid model are affected, and thus on the type of uncertainty considered, the envelope generation process follows slightly different routes, essentially following Section 2.2.2. In cases when the uncertainty on q f is evaluated, the overall surface impedance and its computed deviation are independent. On the other hand, if either d or K f are affected by uncertainties, generating the frequency envelope requires the computation of Z B and thus the overall surface impedance Z s needs to be known. To obtain Z B , the impedance translation theorem is used anew, and the value of the overall surface impedance Z S has to be obtained implying a translation through the film thus its parameters has to be known. These may be obtained by averaging the corresponding experimentally characterised values, in order to obtain the average value of the impedance " Z B . Once " Z B is entirely determined, there only remains the computation of the deviation to the mean response from equation (7).

Numerical validation
As a validation of the proposed approach, numerical simulations are employed. In the current work, envelopes generated using the previously presented steps are compared to results obtained from a Monte Carlo based analysis.
For the numerical validation, three different configurations are used (see Fig. 3). In order to assess the limitations of the proposed method in relation to thin films of different design and manufacturing processes, a woven and a nonwoven film are investigated. The physical parameters of the media are taken from the literature [1,11,13] and gathered in Table 1 for reference. The following properties are used for air: q 0 ¼ 1:213 kg Á m À3 , Pr = 0.71, g 0 = 1.839 Â 10 À5 Pa Á s, P 0 = 1.01325 Â 10 5 Pa, and c 0 = 1.4. Note that in the following, a Johnson-Champoux-Allard (JCA) model [15,16] is used to determine the equivalent fluid quantities for the surface layer and only the corresponding parameters are used.
The two different films attached to a slab of foam and impinged by a plane wave at an angle h = 0°or h = 60°, give the results shown in Figure 4. These angles were selected in order to illustrate the resulting behaviour in both normal and oblique incidence. In this figure, the results from the proposed approach are compared with the Monte Carlo (MC) generated reference solutions. The MC references are generated by computing the response of the system corresponding to 300 samples of the varying parameter n drawn from a normal distribution N " n; Án À Á with Án ¼ 0:1 " n. Statistical indicators are then used to deduce the reference curves (standard deviation and extremal values for each frequency). The solutions from the proposed approach are generated for different deviations nÁn with n = 1, 2, 3 in order to reach different confidence levels.
Two different representations are proposed: the response envelope, plotted as a regular absorption coefficient and the centred envelope where the mean curve is subtracted from the envelope curves. This second representation (shown on the four bottom plots of Fig. 4) is especially useful when the comparison involves the generated envelope amplitudes over the entire frequency range under consideration.
Several points are worth noting about the results presented in Figure 4. The good agreement between the n = 1 envelopes predicted by the proposed approach and the MC standard deviation envelopes could perhaps have been anticipated given that this envelope correspond to a 68% confidence interval. Similarly, almost all the realisations are inside the n = 2 envelope (corresponding to a 95% confidence level). Over the considered frequency range, no realisation is completely outside the predicted envelopes) but some are partially outside of the envelope corresponding to a 99.7% confidence level (n = 3).
One must keep in mind that the differences between the predicted and empirical envelopes are of very low amplitudes and that a slight change in the model could cause such effects. It is also important to remember that the proposed approach is based on several approximations supported by the fact that the film is thin which might impair the overall accuracy. Furthermore, it is important to note that no major difference in accuracy is observed when increasing the angle of incidence h.
In order to demonstrate that the proposed approach works for different backings, two more cases are studied. Figure 5 shows the results obtained for a facing placed upstream a 200 mm air plenum which correspond to configuration in Figure 3c. This case triggers a large number of resonances but the predicted envelopes are very similar to the empirical ones. It is interesting to note, from a modelling point of view, that the envelopes constrict near resonances and expand in a very repeatable pattern. This feature was also visible in Figure 4 to a certain extent even though the resonances are not as sharp. An interesting phenomenon is observed on the h = 0°responses of Figure 4: constrictions of the envelope occur even though no resonance is visible for the absorption coefficient.
As a last validation test, Figure 6 shows the response and envelopes when the film facings are placed onto two  Table 1.    layers of PEM. This configuration is expected to be more damped meaning that the envelopes should be smoother. This effect is indeed observed and the standard deviation envelope of the experimental results is rather smooth and very well captured by the proposed method for n = 1. Note that the empirical envelopes might be asymmetrical (for h = 60°for instance) and that this effect is not quite captured by the proposed approach. Nevertheless, the above results support the validity of the proposed approach.

Experimental validation
Besides the numerical validation of the approach presented in Section 3, a number of experiments presented in the present section allow to assess the capabilities of the proposed method. These experiments use once again two different kinds of facing, one woven and the other nonwoven mimicking their numerical counterpart presented previously, and two different backing media. When needed for the proposed method, the properties of air were chosen the same as in Section 3. Photographs of samples used in the experiments are available in Figure 7.
As a first part of the experimental investigation, the screens and the different backing materials were characterised. The result of this first part is presented in Table 2 in terms of mean value over the measured samples. The characterisation procedure is based on three samples of melamine foam, two samples of glass wool and forty samples of each of the facings. The large number of samples in the last case was chosen in order to have statistically significant data.
A second part of the experiments consists in using an impedance tube to measure the response of twenty samples of each film set on one of the samples of melamine and one of the samples of glass wool. This process once again allows to obtain meaningful statistical indicators regarding the response of each system.
The remainder of this section comprises a note on the measurement rig and the post-processing followed by a section of results where the proposed approached is leveraged to predict envelope curves.
The measurements used in the present paper are released online and free to download for further examination or use [19].

Measurement rig & post-processing
Both for the characterisation of all samples and for the actual measurement of the response of the studied filmbacking configurations, an impedance tube of diameter 44.44 mm was used together with a dedicated commercial software [20]. All measurements use a three-microphones procedure [21,22] and span over the frequency band 10-4250 Hz. A schematic of the measurement rig is presented on Figure 8. The tube is split in two parts to allow for sample insertion and when needed, the film samples are clamped in the seam. Note that these are thin enough not to create leakage.
Note that the tube is split in two parts which overlap when assembled. If the tube's inner diameter is not impacted by this design, it provides a slight seam located at the beginning of the sample section which can be used for clamping the film samples.
Characterising the samples is entirely based on impedance tube measurements, following procedures described in the literature for facings [13] and backings [23,24]. In the first case, a 200 mm plenum is formed behind the screen sample by moving a piston located at the end of the tube. For characterising the backing samples, the piston is adjusted to their length such that the surface of each  The incident wave is a plane wave, travelling normal to the sample's surface. A three-microphones method is used, one being mounted through the rigid backing, the two others on the side wall between the source and the sample. Figure 9. Plastic grid positioned inside the tube to prevent the facings from bending. It adds a 0.89 mm shift due to its shoulder.
sample sits flush with the tube split line (i.e., exactly at the position of the seam). For the two films, the characterisation is performed using 40 samples; for the backings only 3 samples of melamine and 2 samples samples of glass wool were used. The results are gathered in Table 2.
When measuring the response of the multi-layered systems, in the second part of the experiments, a plastic grid shown in Figure 9 was inserted inside the tube to prevent the facing from bending. This grid has a known thickness of 0.89 mm which could be compensated for to achieve better accuracy although this was not considered to be necessary here. In an attempt to reduce most of the sources of variation except for these linked to the tested films, the same sample of backing material is used to measure the response with the 20 samples of film. To make this possible, the tested facings are not glued to the backings but simply clamped and mounted tight between the grid and the backing sample. This technique gives satisfying results while being very flexible. Figure 10 show the absorption coefficient for the two considered backing (melamine foam and glass wool) with the upstream grid as well as the absorption coefficients for each of the 20 samples of film and each backing. On this figure, the influence of the film is clearly visible. Analysing these results, it appears that the variability of the parameters of the film leads to a slight spreading of the responses with occasional features distributed over the spectrum. A glitch is seen around 600 Hz and is present on all measurements, it is linked to the measurement rig and will not be discussed any further.
To better understand the origin of the observed variations, Figure 11 show the range of variation of the physical parameters for both films in terms of percentage of deviation with respect to the average value. In this figure, it is seen that r is the most varying parameter for both screens, closely followed by the porosity / and the two characteristic lengths K and K 0 . It is known that the airflow resistivity, together with the thickness d, is of prime importance as it plays a prominent role in the airflow resistance. Even though the thickness does not vary much amongst the samples considered here, it has been observed to have high deviations for other facings and might sometimes have an important impact.

Generating envelopes for experimental data
In this subsection the envelope response is observed and two techniques to reconstruct these are compared. The proposed approach is compared to a simplified evaluation based on the mean values and standard deviations of the parameters of the thin facings as well as the backing parameters to compute upper-and lower bound responses. These responses are computed by using the average value of all parameters except the supposedly varying one (for instance r) for which n times the standard deviation is added/subtracted to mean value.
The other approach is the one described in Section 2 where statistical information on the facing parameter is embedded in an annotated envelope. When needed, the backing impedance Z B is computed using the average value of the parameters. All envelopes in this section are computed with the proposed method for n = 1 or n = 2. On Figure 12, the cases with a melamine backing is presented, on Figure 13 the one with a glass wool backing is shown.
Note that in these figures, both films are considered for both approaches.
From these figures, it is clear that the simplified approach is rarely a good approximation of the actual envelope, particularly at low frequencies. On the top rows of both Figures 12 and 13, one easily notices the discrepancies introduced by considering a JCA model both for the backing and for the film. Indeed, even though this hypothesis is reasonable for the film thanks to its very specific effect and thinness, it is a strong approximation for the poro-elastic backing in which solid-borne waves also propagate.
On the other hand the proposed approach closely follows the empirical envelope particularly the predicted envelope for n = 1 which almost exactly follows the standard deviation envelope. Some discrepancies are observed  around 1750 Hz on Figure 13 with a woven fabric, which might be linked to the fact that the surface of the glasswool sample is not perfectly flat and that the woven film is rather soft. Around this frequency, the proposed approach traces a plausible envelope but does not account for the existing resonance.

Conclusion
In the present contribution, a method to generate envelopes responses for multi-layered systems including a film with uncertain parameters is introduced. The proposed method uses a set of Taylor expansions to introduce a way to separate the contributions of uncertainties from the average trend. The said method is both suitable for standalone computations of envelope responses or as a tool to compute envelopes from a single response from the system and the parameters of the facing.
The method is tested both on simulated cases with materials and configurations sourced from the literature and on actual experiments. In both cases, the method is shown to give good results with predicted envelopes having similar trends and amplitudes as the ones based on measurements. The proposed solution is fitting the measured reference better than a simplified approach based on the bounds of a given variation interval of the parameters.