Off-line envelope estimation for acoustic screens with uncertain properties
Mathieu GABORIT(1,2) , Olivier DAZEL(1) , Peter GÖRANSSON(2)(1) LAUM UMR CNRS 6613, Le Mans Université, Le Mans, France(2) MWL, KTH Royal Institute of Technology, Stockholm, Sweden

Abstract
The development of new solutions to reduce impact of noise receives constant attention both from the researchand engineering communities. Even though the proposed systems tend to diversify, multi-layered absorbers
based on foams remain the most common. Such structures feature one or several thick slabs of foam and arecompleted by protective or tuning layers of small thickness inserted between slabs or on the free surface. Theselayers (acoustics screens or films) bring a number of challenges. Firstly, their thickness makes them hard tocharacterise and some parameters are uncertain. Secondly, the bonding to the surrounding system alter theeffective properties. Given the strong influence the films have on the system’s response, it appears important toaccount for these uncertainties from the design phase on. The present contribution introduces a new techniqueto efficiently account for uncertainty linked to the screens. More precisely, the proposed approach is basedon separating the average film influence from the deviation due to uncertainties. This allows to estimate theresponse envelope using a pre-computed average response and the properties. The results computed for a rangeof typical and challenging configurations are promising and validate the robustness of the method at any angle
of incidence.
Keywords: poroelastic media, uncertainties, envelope curves

1

INTRODUCTION

The requirements in terms of noise reduction keep being raised so to reduce the impact on health and wellness [1, 2]. This trend calls for a better understanding of the most common noise absorbers and for new ways
way towards optimised systems. The context of the present study is composite absorbers usually constructed byassembling several layers of poroelastic media [3]. This is a widely used strategy to create absorbing panel andit proved to be efficient on a wide frequency band at a reasonable cost. Many contributions from the literature
study this kind of absorbers [3, 4] or their modelling [5, 6, 7] and thus, for conciseness, this part will not be
considered in the following.
The films often added on the absorber’s free surface are one of the easiest ways to to tune these absorbers [8].
Despite their usual thinness (sub-millimetre), these components alter the coupling between the surrounding air
and the absorbing layers and tend to improve the performances if chosen carefully. Nevertheless, their natureand geometry imply a difficult characterisation [9, 10] and thus, reported values for the different parameters ofthese media are often inaccurate. In the present contribution, the choice is made to account for these uncertainties and represent the systems’ response as an envelope controlled by the film’s most uncertain parameters.
The remainder of this contribution is as follows. After this introduction, a second section briefly presents how
to separate the contribution of the uncertainties that apply to one of the parameters and to derive its influenceon the classical quantities of interest. It is shown how these developments allow to quickly generate envelopesaround an average response so to represent the system’s variability. A third section gathers examples of using
the method both for simulating a response and for annotating an envelope on measured responses. A last section
discusses the results and concludes.
√
In the following, a positive time convention ejωt is used where j = −1 is the imaginary number, t the time
variable and ω the angular frequency.

1992

2

TheoryThis first part studies systems such as the one shown in figure Figure 1 and their responses to an excitation
by a PW. Particularly, the aim is to identify how the uncertainty on one of the film’s parameters impacts theoverall behaviour.
Backing system

Film

Zs

ZB
z

θ

L

d

Figure 1. Model under study. A laminate which surface layer is a thin film (d  1) is impinged by an air-borne
wave. Despite the backing system being represented as a single layer on a rigid backing, the proposed approach
can handle more intricate backings without requiring adaptation.
The film is thin (d  1) and it is proposed to neglect the elastic effects, focusing mostly on the impact on
air-borne waves and considering the film as an equivalent fluid [3]. Regarding the mathematical modelling, thisassumption allows to represent the film through an equivalent density ρ f and compressibility K f which in turncan be combined to access the associated characteristic impedance Z f and the wavenumber k f :p
Zf = ρf Kf ,r
k f = ω cos θ

ρf
Kf

(1)

where θ is the incidence angle of the wave impinging the laminate.
A key point of the development is to identify that one can derive the surface impedance Zs for any backing
impedance ZB through the knowledge of the film’s properties and geometry. Indeed the impedance translation
theorem yields
Zs = Z f

ZB + jZ f tan k f d.
Z f + jZB tan k f d(2)Assuming the film is thin with respect to the wavelength such that k f d  1, it comes that the tangents in (2)can safely be approximated by their first order Taylor expansion which, after simplification, leads to
Zs ≈ZB + jZ f k f d.
1 + jZB Z −1
f kf d(3)

In order to further simplify the expression, a stronger assumption must be considered. In cases where ZB ' Z f(which turns out to be true in many cases due to the highly resistive nature of the films [3, 11]) the denominatorcan be approximated by a second Taylor expansion ((1 + ε)−1 = 1 − ε + O (ε 2 ) with ε  1).
Substituting in (3), distributing the product, replacing Z f and k f by their expressions from (1) and neglecting
the second order terms yields
Zs = ZB − jωd cos θ ZB2 K −1
f + jωd cos θ ρ f .

(4)

1993

This expression describes the surface impedance of systems where a thin, resistive film is placed atop a socalled backing system represented through its acoustic impedance ZB in cases where the film is considered as
an equivalent fluid.
In the context of this contribution, one of the film’s parameters is imprecisely known which in turns affectthe different geometrical and physical descriptors K f , ρ f and d. Mathematically, the uncertainty is accounted
for by representing each descriptor ξ by the sum of a nominal value ξ¯ and the associated deviation ∆ξ . In
the remainder of this paper, only uncertainties on parameters ρ f alone are discussed. Extending the approachto other parameters and descriptors is straightforward. As a last note, the equivalent fluid model used in thepresent document is a motionless skeleton model commonly known as the Johnson-Champoux-Allard (JCA)
model [12, 13, 3]. In the scope of said model, a modification of ρ f alone occurs if the airflow resistivity σ isuncertain.
Replacing ρ f = ρ̄ f + ∆ρ f into 4 leads to
Z̄s

}| 
{
z 
+
jωd
cos
θ
ρ̄
Zs = ZB 1 − jωd cos θ ZB K −1
f
f

(5)

+ jωd cos θ ∆ρ f .
|
{z
}
∆Zs

where the first part Z̄s is entirely determined from the nominal value of ρ f and other deterministic quantities(such as the backing impedance ZB ) and the second ∆Zs is directly linked to the uncertainties.
From the overall surface impedance Zs and the surrounding air characteristic impedance Z0 , it is straightforward
to derive the reflection coefficient R in a purely deterministic case:R=

Zs − Z0.
Zs + Z0(6)

Then, to obtain the so-called nominal and deviated components R̄ and ∆R, (5) is substituted into (6). FactorizingZ̄s + Z0 at the numerator and denominator, the latter takes the form of 1 + ε with the small parameter being∆Zs (Z̄s + Z0 )−1 . This last operation allows to use once more a first order Taylor expansion such thatZ̄s − Z0
2Z0 ∆ZsR=
1+ 2
Z̄s + Z0Z̄s − Z02
| {z }| {z }
R̄

!


+O

∆Zs2
Z̄s2 − Z02


(7)

η

Equation (7) introduces the parameter η which turns out to control both the deviation of R and of the absorption
coefficient α = 1 − |R|2 = 1 − |R̄|2 |1 + η|2 . To further detail the expression it is important to note that η iscomplex (due to both its denominator and numerator being complex) and formally writes η = η 0 + jη 00 , which
after substitution into the expression of α and simplification leads to:
α = 1 − |R̄|2 + |R̄|2 2η 0 + |η|2
| {z } |
{z
}
ᾱ

(8)

∆α

From equations (7) and (8), it is clear that evaluating η for all frequencies allows to compute the deviation of
α and hence generate an envelope from an average response. This process, hereby called envelope annotation, iscompletely independent of the determination of the average response itself. Note that in the case of uncertainties
impacting the other descriptors (d for instance) if might be needed to computed the average backing surface
impedance ZB using the translation theorem. Indeed, introducing d = d¯ + ∆d in (4) shows that ZB appears in
the computation of η.

1994

Table 1. Properties of media used in tests, the ones for simulations are taken from the literature [14]. CL
stands for characteristic length and only the Johnson-Champoux-Allard parameters are available for the second
experimental media.

3

Simulations
Film

Property [unit]

Foam

Open-Porosity φ
Airflow resistivity σ [N · s · m−4 ]Dynamic Tortuosity α∞
Thermal CL Λ0 [µm]Viscous CL Λ [µm]Density ρ1 [kg · m−3 ]Poisson ratio νYoung’s modulus E [Pa]Loss factor

0.994
9045
1.02
197
103
8.43
0.42
194.9·103
0.05

0.99
65·103
1.98
120
37
16
0.3
46.8·106
0.1

Foam
0.97
23·103
1.06
149
54

Experiments
Film
0.07±0.01(1120.2±161.6)·10−3
1.14
42±1
42±1

APPLICATIONSIn order to demonstrate the technique, two examples are provided in this section. The first one uses the proposed
approach to generate envelope curves from scratch and compare the results with the results of Monte Carlosimulations. The second, uses actual measurements of a two-layers system’s response to compute an averagereponse and annotate an envelope using the films parameters.
In both cases, the system considered is similar to the one in Figure 1. The dimensions used are d = 0.5 mm
and L = 20 mm for the first test case and d = 0.3 mm and L = 17.2 ± 0.3 mm for the second (the uncertainty
on L is linked to small differences in the porous backing samples geometry). The films and backing media
properties for both cases are presented in Table 1. The properties of air are as follows: ρ0 = 1.213 kg · m−3 ,Pr = 0.71, η0 = 1.839 · 10−5 Pa · s, P0 = 1.01325 · 105 Pa and γ0 = 1.4.
In order to render the results analysis easier, not only the absorption coefficient of the laminates is presented
but also the so-called centred envelope for which the nominal response is subtracted to all plotted quantities.
Note that the nominal response is the one obtained when all the parameters have their nominal value and that,
because of the inherent non-linearity of the models and sampling concerns, this response might be different
from the actual average one.
3.1 Computing a response envelope
In the first test case, the envelopes generated by the proposed method are compared to Monte Carlo simulations
with a set of 300 samples drawn from a normal deviation. On this basis, a so-called observed envelope iscomputed by adding (or subtracting) the sample-wise standard deviation to the sample-wise average. These
quantities are computed both at normal incidence θ = 0 rad and at θ = π3 rad and shown respectively in Figures 2
and 3. The Monte Carlo results are generated using a in-house solver [15] that implements a method from theliterature [7]. For this test, the backing PEM is modelled as a Biot-JCA material (hence including mechanicaleffects) and the film as a limp medium [3, 16, 17].
The empirical envelopes are computed from the Monte Carlo simulations by adding (or subtracting) the samplewise deviation to the sample-wise average. For both figures, these envelopes are closely following these computed using the proposed approach and a deviation of 1β (with β the standard deviation of the uncertain
parameter, set to 20% of the nominal value here). The method gives reliable results both at normal incidence(Figure 2) and at oblique incidence (Figure 3).
It is observed that the 1β curves fits correctly the empirical standard deviation envelope, which is an expected

1995

a)

1.0

Abs. Coef.

0.8
0.6
0.4

MC standard deviation envelopes
Proposed method (1 )Proposed method (2 )

0.2
0.0

1000

2000

4000

5000

6000

4000

5000

6000

b)

0.3
Centred Envelope of Abs. Coef.

3000
Frequency (Hz)

0.2
0.1
0.0

1000

2000

0.1

3000
Frequency (Hz)

0.2
0.3

Figure 2. Comparison of the Monte Carlo envelopes (solid line) and the results of the proposed approach(dashed line for one standard deviation of the uncertain parameter and dotted line for two) at normal incidence.
Light lines are Monte Carlo realisations. a) Absorption coefficient. b) Centred envelope.
a)1.0

Abs. Coef.

0.8
0.6
0.4

MC standard deviation envelopes
Proposed method (1 )Proposed method (2 )

0.2
0.0

1000

2000

3000
Frequency (Hz)

4000

5000

6000

4000

5000

6000

b)
Centred Envelope of Abs. Coef.

0.15
0.10
0.05
0.00
0.05

1000

2000

3000
Frequency (Hz)

0.10
0.15

Figure 3. Comparison of the Monte Carlo envelopes (solid line) and the results of the proposed approach(dashed line for one standard deviation of the uncertain parameter and dotted line for two) at θ = π/3 rad. Lightlines are Monte Carlo realisations. a) Absorption coefficient. b) Centred envelope.

1996

result. Note though that a number of realisations remain out of the envelope. The uncertain parameter (σ in thepresent case) is drawn from a normal law, centred on its nominal value. Then, the 1β envelope then correspondto the 95% confidence interval and one could reach higher confidence values by computing the envelopes for
2β (as presented on the figures) or even 3β .
Similar tests were performed for other media and backing structures, leading to similar results, not reported here
for conciseness. Note that the tests are not performed at high frequency and that the film parameters ensurethat none of the assumptions are violated. A more thorough testing is require to fully validate the approach
3.2 Envelope annotation on experimental results
In the second test, three experimental measurements of the absorption coefficient of three samples of a multilayer system are used as an input. Using these, an average response is computed and an envelope annotatedfrom the statistical characterisation of the film. The parameters are recalled in Table 1 and from the associateduncertainties, the envelope is generated for a variation of the airflow resistivity σ . Note that some discrepancies
are expected as only 3 traces of the full system and of the films are released (which clearly is not enough forstatistical meaningfulness).
As seen in the previous test, envelopes generated for 1β tend to let many traces out of the envelope. In thissecond test and, which results are presented on Figure 4, the envelopes are generated for deviations of both 1β
and 2β .
a)

1.0Abs. Coef.

0.8
0.6
0.4

Experimental envelopes
Proposed method (1 )Proposed method (2 )

0.2
0.0

500

1000

1500

2000

2500
Frequency (Hz)

3000

3500

4000

3000

3500

4000

b)
Centred Envelope of Abs. Coef.

0.10
0.05
0.00

500

1000

1500

2000

2500
Frequency (Hz)

0.05
0.10

Figure 4. Comparison between the envelopes annotated on experimental results (dotted and dashed lines), experimental traces (light lines) and empirical envelope (solid lines, average plus/minus standard deviation) undernormal incidence. a) Absorption coefficient. b) Centred envelope.
Several phenomena are seen on the figure. First, it is observed that, as for simulations, the 1β envelope fit
reasonably well the empirical envelope except around the resonances observed at 1500 Hz. The said resonancebeing due to mechanical effects which are not accounted for in the equivalent fluid model used both for thefoam and film, it is not accurately represented in the envelope. Note though that, despite not formally accounting for mechanical effects in the film, the annotation is based on the experimental traces and thus partly

1997accounts for the resonance. On this side, using 2β as a deviation allows to better fit with the experimental
envelopes, as seen on Figure 4. This last remark illustrate the fact the the datasets are very scarce which makes
all statistical usage quite subject to deviation.

4

CONCLUSIONS

The method proposed in this contribution is a fast solution for generating envelope curves straight from theparameters of the considered system or over experimental data. This so-called envelope annotation process
is fast and requires only little information about the system. The developments come at the cost of using aequivalent fluid model for film placed upstream. This implies little to no information about mechanical effectwill be rendered. Extending the method this way would be of great interest.

ACKNOWLEDGEMENTS
The authors thank the Matelys Research Lab and particularly L. Jaouen for the support and providing theexperimental traces used in section

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