Estimation of the Area of a Reverberant Plate Using ... · Reverberation Properties Hossep Achdjian ... This estimation requires an access to only a small portion of the plate and
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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