Physics Procedia 70 ( 2015 ) 139 – 142

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1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ICU 2015doi: 10.1016/j.phpro.2015.08.065

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2015 International Congress on Ultrasonics, 2015 ICU Metz

Estimation of the Area of a Reverberant Plate Using Average

Reverberation Properties

Hossep Achdjian∗, Emmanuel Moulin, Farouk Benmeddour, Jamal Assaad

IEMN UMR CNRS 8520, University of Valenciennes and Hainaut-Cambresis, Le Mont Houy, F-59313 Valenciennes, France

Abstract

This paper aims to present an original method for the estimation of the area of thin plates of arbitrary geometrical shapes. This

method relies on the acquisition and ensemble processing of reverberated elastic signals on few sensors. The acoustical Green’s

function in a reverberant solid medium is modeled by a nonstationary random process based on the image-sources method. In

that way, mathematical expectations of the signal envelopes can be analytically related to reverberation properties and structural

parameters such as plate area, group velocity, or source-receiver distance. Then, a simple curve fitting applied to an ensemble

average over N realizations of the late envelopes allows to estimate a global term involving the values of structural parameters.

From simple statistical modal arguments, it is shown that the obtained relation depends on the plate area and not on the plate

shape. Finally, by considering an additional relation obtained from the early characteristics (treated in a deterministic way) of

the reverberation signals, it is possible to deduce the area value. This estimation is performed without geometrical measurements

and requires an access to only a small portion of the plate. Furthermore, this method does not require any time measurement nor

trigger synchronization between the input channels of instrumentation (between measured signals), thus implying low hardware

constraints. Experimental results obtained on metallic plates with free boundary conditions and embedded window glasses will be

presented. Areas of up to several meter-squares are correctly estimated with a relative error of a few percents.c© 2015 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz.

Keywords: Physics of sound waves, reverberation, signal processing, characterization of a structure;

1. Introduction

The propagation of acoustic and elastic waves in a finite medium with low attenuation gives rise to signals of

long duration (reverberation). Such complex signals consist of three main parts. The first part is the wave packet

corresponding to direct path from the source to the receiver. The second one includes wave packets due to the early

reflections. These two parts depend directly on the positions of the source and receiver, as well as local properties of

the propagation medium. The third part of the signal is made up of high order reflections and named as coda. During

the acoustic propagation, these reflected signals accumulate information about the global environment of the medium

∗ Corresponding author. Tel.: +33-(0)3-27-51-12-34; fax: +33-(0)3-27-51-11-89.

E-mail address: hossep.achdjian@univ-valenciennes.fr

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ICU 2015

140 Hossep Achdjian et al. / Physics Procedia 70 ( 2015 ) 139 – 142

and gradually lose the influence of local properties. However, these coda waves are reproducible and contain useful

information on the traveled medium.

The first studies of reverberation phenomenon are indissociable from the history of architectural acoustics or room

acoustics (Warnock (1983)). Although the use of such complex acoustic signals is common in room acoustics, they

are not so commonly used for characterization of solid medium and for a non-destructive testing (NDT). Recently

researchers are focused on retrieving potentially useful information about the structural properties of solid medium.

Thus, powerful techniques have been developed on this purpose such as coda interferometry (Snieder (2006)), field

correlation (Chehami et al. (2014)) and time-reversal acoustics (Derode et al. (1999); Fink. (1997)). We have shown in

previous studies (Achdjian et al. (2014a, 2012); Moulin et al. (2012)), that the exploitation of the ensemble averaging

properties of codas offers another interesting way of estimating some structural properties with lower processing

complexity and even with limited number of sensors. The principle is to identify the fitted average data with theoretical

expressions of mathematical expectations delivered by a statistical reverberation model based on the image-source

method (Cuenca et al. (2009)). Then, the additional relations obtained from the first wave packets offer an elegant

way of extracting useful quantitative information on the structure such as material properties, attenuation, dimensions,

emitted acoustic energy and source-receiver distances.

In this paper, we propose an original way to estimate the value of the area of an arbitrary shaped solid plate with free

or embedded boundary conditions. First, we present a time-domain statistical description of reverberation modeled

for rectangular plates. This model allows to predict the expressions of the average characteristics of dispersive signals

received in a few points. Then, we show that the theoretical model is still valid regardless of the shape of the plate.

In section three, we present the principle of area estimation for thin plate based on these theoretical results. Finally

before a brief conclusion, we deal with the experimental validation of the method. This estimation requires an access

to only a small portion of the plate and without geometrical measurements.

2. Statistical Behaviour of Wave Packets in a Reverberant Signal

In order to simplify the theoretical developments, we first study the reverberations in a rectangular plate of area S,

assuming a single propagation mode.

Let us consider that a localized source emits an acoustic signal s0(t) of finite duration and narrow-band from a point

on the plate. The response h(t) obtained at the receiver consists of direct propagation hD(t) = s(r0, t) from source to

receiver and a reverberant part corresponding to a series of reflections at the plate boundaries,

hR(t) =∞∑

i=1

κi s(ri, t) , (1)

where s(r, t) is a dispersive signal received after propagation over a distance r and r0 is the distance between source-

receiver. κi is the number of wave packets arriving from ith image-sources located at distances between ri and ri + Δri

from the receiver. As shown in our previous works (Achdjian et al. (2014a); Moulin et al. (2012)), the mathematical

expectation of the squared envelope of hR(t) is characterized by a decreasing exponential function of the form:

E[∣∣∣HR(t)

∣∣∣2]� A e−2t/τ , (2)

where HR(t) the complex analytic representation of reverberant signal hR(t) and τ is a term related to the reverberation

time of the plate. A is the amplitude of the previous equation related both to structural characteristics of the medium

and to the acoustic source properties, it can be expressed as

A = βd vg0 Ds , (3)

with

βd =2 π

S , (4)

vg0 is the group velocity at frequency ω0 and Ds � 2 rm IDm is a term related to the acoustic power supplied by the

source to the medium, where rm is the distance from source to mth receiver and IDm =∫ +∞

0

∣∣∣hDm(t)∣∣∣2dt is the energy of

the first wave packet of the signal received at any sensor m.

Hossep Achdjian et al. / Physics Procedia 70 ( 2015 ) 139 – 142 141

a b

0 0.01 0.02 0.03 0.04−10

−8

−6

−4

−2

0

time (s)

log[envelope]

experimentaltheoreticalfitted

Fig. 1. (a) measurement device; (b) experimental result of an envelope average over 5 receiver positions and comparison to theoretical and fitted

envelopes.

The expression of βd given in Eq. (4) has been derived in the case of a rectangular plate. However, a statistical

study in the frequency domain allows to relate βd to the modal density of the plate in the following way (see Achdjian

(2014) for more details):

βd =D′s

vg0 Dsn(ω0), (5)

where D′s is a term related to the acoustic energy injected into the plate and n(ω0) is the modal density at ω0.

Weyl Balian and Bloch (1970) showed that the modal density depends only on the area of the plate and not its

shape. Thus, two plates of the same material, same thickness, with the same area but different geometrical shapes and

excited by sources of the same type and same amplitude, have the same values of βd, according to the Eq. (5).

3. Estimation of thin plate area

As shown in previous section, the amplitude A (Eq. 3) depends on structure parameters. We will use this prop-

erty to estimate the area of the plate in an original way. To achieve this goal, we consider a set of receivers dis-

tributed over a circle of radius r0 around an acoustic excitation source (Fig. 1-a). The received signals hm(t) at M(here, M = 5 & m = 1 . . . 5) receivers are acquired and processed. An example of an envelope averaged over five real-

izations is shown in Fig. 1-b (blue solid line) and compared to the theoretical expected envelope (red solid line) given

by Eq. (2). Good agreement is observed between these curves. Values A f it and τ f it of the parameters A and τ of

Eq. (2), respectively, are estimated through a linear curve fitting (Fig. 1-b, black dashed line) process applied to the

log envelope averaged. Once A f it and IDm are estimated from the measured signals, using the Eq. (3) and (4) it will

be possible to estimate one of the three parameters vg0, S or rm, if the other two are known. In the event that vg0 and

rm are known, we can deduced the plate area Sest � (2πvg0Ds)/A f it. For more precision on Ds, we can estimate the

energy of the first wave packet IDm received from an average over M receivers ID

M =∫ +∞

0

∣∣∣ 1M∑M

m=1 hDm(t)∣∣∣2dt, provided

that the receivers are at the same distance r0 from the source.

Experimental tests have validated the principle. The estimated values of area for aluminum plates with free bound-

ary conditions and for embedded window glasses of our laboratory (Fig. 2) are presented in Table 1.

Good agreement is shown between the actual and estimated values of S, as indicated in the first and the second

column of the table, respectively. These results are quite persuasive and validate the proposed method.

4. Conclusion

The studies presented in this paper illustrate an original method to measure the area of a thin plate using the acoustic

reverberations. Firstly, we have shown that theoretical expressions for the mathematical expectation of the envelope

of reverberant signals are related to the thin plate area. Then, encouraging experimental results of area estimation

are presented for metallic plates and window glasses, with relative error of a few percents. This technique does not

require geometrical measurements, needs an access to only a small portion of the plate and uses a low software and

142 Hossep Achdjian et al. / Physics Procedia 70 ( 2015 ) 139 – 142

Fig. 2. Description of the plates (3 mm of thickness) used for geometrical characterization testing. (a) aluminum plate; (b), (c) window glasses.

Table 1. Experimental results: estimation of area S for 2 aluminum plates and 4 window glasses of different geometrical shapes and different

boundary conditions, and comparison to actual area values. The last column shows the relative error (in %) between these two values.

Polygons actual S (m2) estim. S (m2) ΔS/S (%)

Rect.#1 (alu.) 2 1.975 1.25

Rect.#2 (alu.) 0.15 0.155 3.3Rect.#3 (glass) 4.02 4.21 4.7Rect.#4 (glass) 2.66 2.79 4.88

Rect.#5 (glass) 1.84 1.86 1.08

Poly.#1 (glass, Fig. 2-c) 1.25 1.28 2.4

hardware constraints. Thereafter, it could be interesting to apply this method to measure the sheet length of the roll,

follow the remaining surface after cutting, control the expansion of the surface, etc.

Acknowledgements

This work has been partially supported by the French National Research Agency (ANR): ANR2011 BS0903901,

PASNI project.

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