Thickness or phase velocity measurements using the

Green function comparison method

Nicolas Etaix, Alexandre Leblanc, Mathias Fink, and Ros-Kiri Ing∗,

Laboratoire Ondes et Acoustique, Institut Langevin

University Paris 7- ESPCI - CNRS UMR 7587

10 rue Vauquelin, 75231 Paris Cedex 05, France.

December 17, 2009

Abstract

The acoustic guided wave propagation for plate thickness measurement is generally

treated in free space in order to simplify the formal approach. In this paper, propaga-

tion in a closed environment is treated by dealing with plates of finite dimensions and

arbitrary geometries and linear boundary conditions. The present approach considers

the plate Green function to be composed of two terms. The first term corresponds to

the Green function of an infinite plate. The second term corresponds to a correction

term which in addition to the first term satisfies all equations. Assuming the boundary

∗e-mail : ros-kiri.ing@espci.fr

1

conditions to be linear it is found that the acoustic wave generated by a point source is

comparable to that of a circular array of source centred on it. By measuring difference

between the two signals, either the plate velocity or plate thickness should be deter-

mined. This determination has been achieved on isotropic or anisotropic homogeneous

plates of different geometries, on inhomogeneous plates, and also in a passive mode,

without theoretically any active emitter.

1 Introduction

Guided waves have been extensively used to estimate the thickness or the elastic constants of

structures. One advantage of the guided waves is to allow an averaged measurement over a

large area of the structures to be inspected as these waves propagate with low attenuation [1,

2]. These waves are also available in a broad range of frequencies. Selective and/or sensitive

measurements are then possible by properly selecting the mode and the frequency in order

to study the types, geometries or location of defects [3].

For plate-like structures, the guided waves are the Lamb waves. In the case of an isotropic

and homogeneous plate, only longitudinal and transverse waves can exist. The determination

of the materials parameters is thus mainly based on the analysis of the group or phase velocity

of these two parts of guided acoustic waves [4, 5, 6]. Basically, the measured velocities are

analyzed and compared to theoretical dispersion curves to determine the elastic constants of

materials and the thicknesses, as in [7, 8], or only one of these parameters [9, 10]. For other

type of materials, guided waves will depend on more constants, as for example in the case

2

of anisotropic materials [11], bilayered films [12] or coating layers on a substrate [13].

In general, existing methods [14] give the phase or group velocities of the excited modes

versus frequency. Considering this aspect, it should be noted that the frequency range

depends on the plate thickness, as Lamb modes are defined as a function of the frequency-

thickness product. Thus, in thin plates, high frequency ultrasound is generally used. To

obtain the plate constants from the experimental dispersion curves, least-squares fitting is

generally uses to fit the experimental data with the theoretical curves. For the cases of the

first antisymmetric and symmetric Lamb modes, approximations for the exact expression

for the Rayleigh-Lamb dispersion curves can also be used [7, 15] in order to simplify the

fitting procedures. However, the use of conventional dispersion curves requires a free space

approach. So, the finite dimension aspect of real structures are not really considered or

simply shortcut. Moreover, used frequencies are often higher than hundred kilohertz in order

to keep the wavelengths small versus the lateral dimensions of plates, leading to complex

measurements, based on expensive transducers.

In this paper, the plate velocity parameter as defined in [2] or thickness measurement is

performed through the Green function analysis of plates of finite dimensions and arbitrary

geometries. The used guided wave is the first antisymmetric Lamb mode A0, also known

as the flexural wave (from a mechanical point of view). As the plate velocity is specific to

the material, the proposed method gives the relative thickness with less computational effort

than existing methods and use only the audio frequency band (typically from 500 Hz to 18

kHz).

3

Γ

O

θ

r

x2

x1

x2x3

Figure 1: Homogeneous plate of constant thickness with arbitrary geometry andboundary conditions.

After introducing some theoretical considerations about the Lamb waves in Sec. 2, Sec. 3

shows experimental results and the application of the method proposed in this article to

isotropic plates. In Sec. 4, an extension of the method to orthotropic plates and plates

with inhomogeneities is demonstrated. Finally, the method is adapted a setup working in

passive mode is presented to perform a plate velocity - thickness parameter measurement

from ambiant acoustic noises.

2 Theoretical consideration

For a thin plate and for internal strains leading to curvatures of weak radius, the acoustic

wave displacement is represented by only one component directed along Ox3 as shown in

Fig. 2. Equation governing the flexural vibration for harmonic wave is expressed in terms of

transverse displacement w [16]:

4

D∇4w + ρhω2w = 0 (1)

with D = Eh3/[12 (1 − σ2)]. The coefficients ρ, h, E and σ are respectively the density,

thickness, Young modulus and Poisson ratio.

When the plate is submitted to an external surface load the right term of Eq. (1) is

replaced by the excitation function. For a unit concentrated load at a point rs the excitation

function is represented by the Dirac delta function and the solution of the inhomogeneous

equation corresponds to the free space Green function [17]:

Gfree (|~r − ~rs|) =

i

8k2D

[

H(1)0 (k|~r − ~rs|) − H

(1)0 (ik|~r − ~rs|)

]

(2)

with

k4 =ρhw2

D(3)

This solution satisfies the Sommerfeld condition that implies only waves diverging from

the point source are considered. The first term of Eq. (2) corresponds to diverging propa-

gating wave whereas the second term corresponds to evanescent wave.

For a plate of finite dimension, the initial wave emitted by a point source (from a temporal

point of view) is identical to that expressed by Eq. (2) (neglecting the amplitude factor and

5

before any reflections by the plate edges with continuous boundary Γ). Thereafter, the waves

are distorted due to multiple reflections and its amplitude expression diverges faster from

that given by Eq. (2). For an observation point located on the plate, the result observed

consists in waves interferences between the direct wave and multiple reflected waves. As it is

shown in [18, 19, 20], the free space Green function is still valid in a plate of finite dimension

but an additional correction term must be added. Thus, the sum of the two terms are

solution of the inhomogeneous plate equation and satisfy the boundary conditions. In the

absence of any external source to the plate, the Green function of the plate can be written

as follows:

Gplate (|~r − ~rs|) =

Gfree (|~r − ~rs|) + Crefl (Gfree (|~r − ~rs|)) (4)

Crefl represents only multiple reflected waves. Argument of Crefl is the free space Green

function because initial incident waves must be first defined before computing any reflected

waves.

The reflections at the edges are determined by the mechanical conditions of the coupling

between plate and support. These conditions are applied to the transverse displacement w

but also to its spatial derivatives up to order 3 (normal slope, bending moment, twisting

moment, shear force and Kelvin-Kirchhoff edge reaction). These conditions are assumed to

be linear. For example, and for a bending moment Mn:

6

Mn

(

∂2 (w1 + w2)

∂x2,∂2 (w1 + w2)

∂y2,∂2 (w1 + w2)

∂x∂y

)∣

∣

∣

∣

Γ

=

Mn

(

∂2w1

∂x2,∂2w1

∂y2,∂2w1

∂x∂y

)∣

∣

∣

∣

Γ

+

Mn

(

∂2w2

∂x2,∂2w2

∂y2,∂2w2

∂x∂y

)∣

∣

∣

∣

Γ

(5)

Without any further complex calculation Crefl should then satisfy the following linear

relation:

Crefl (x + y) = Crefl (x) + Crefl (y) . (6)

For a circular array of point sources of radius R and centred at ~rs the amplitude of the

total waves Wcircle(~r) is equal to the sum of the waves emitted by each source. Using the

linear property of Crefl and assuming each source of same strength ( 1N

) one obtains the

following relation of proportionality:

Wcircle (~r) =

1

N

N∑

i=1

Gfree (|~r − ~ri|) +1

N

N∑

i=1

Crefl (Gfree (|~r − ~ri|)) (7)

By increasing the source number N up to infinity the discrete sum of the free space

Green function is equivalent to a contour integral of Gfree along a curvilinear coordinate s

7

that follows a closed circular trajectory of radius R and centred at ~rs :

1

N

N∑

i=1

Gfree (|~r − ~ri|) →∮

circle (R,~rs)

Gfree (|~r − ~ri|)2πR

ds (8)

By using the theorem of sum of cylindrical harmonics (a property of the Hankel function)

it gives:

∮

circle (R,~rs)

H(1)0 (k |~r − ~ri|)ds = 2πRJ0 (kR) H

(1)0 (k |~r − ~rs|) (9)

According to Eq. (9), the contour integral of the free space Green function, is then equal

to:

∮

circle (R,~rs)

Gfree (|~r − ~ri|)2πR

ds =i

8k2D

(

J0 (kR) H(1)0 (k |~r − ~rs|) + J0 (ikR) H

(1)0 (ik |~r − ~rs|)

)

(10)

Far from the source (k|~r − ~rs| � 1), the left term that corresponds to evanescent term

is weak and can be neglected. Finally, the wave emitted by a circular array of point sources

of radius R (and when the number of sources is huge) is equal to the following term:

Wcircle (~r) = J0 (kR)Gplate (|~r − ~rs|) (11)

This result is obtained whatever the geometry of the plate and whatever the boundary

8

N

12 3

Plate

Laser vibrometer

Piezoelectric disc

Computer

DA converterand amplifier

DA converter

Figure 2: Experimental setup.

conditions as long as they stay linear. The frequency response of a circular array of point

sources is directly related to the frequency response of a unique point source - i.e. Green

function - located at the center of the circular array. This remarkable result is used to setup

a plate velocity - thickness measurement method that is named in the present paper the

Green function comparison (GFC) method.

3 Experimental validations on isotropic plates

3.1 Determination of the plate velocity - thickness product.

Figure 3.1 shows the experimental setup. A Duralumin plate of 180×180×1.03mm3 with

free edges is used. A circular piezoelectric transducer (PZT) is rigidly coupled to the plate.

The PZT is driven by an audio amplifier in the frequency band 500 Hz - 18 kHz using a chirp

excitation signal. Flexural waves are detected using a laser vibrometer. N measurements

are realized along a closed circular path of radius R using a constant angular pitch. The

value of N is chosen equal to 72. An additional measurement is achieved at the central point

9

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 104

−4

−2

0

2

4

frequency (Hz)

(rad

)

(b) Angular phase

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 104

0

0.2

0.4

0.6

0.8

1

frequency (Hz)

(nor

mal

ized

)

(a) Amplitude

ring−shaped sourcepoint source

Figure 3: Amplitude and phase of the signals emitted by a unique point sourceand a circular array of N point sources of radius 50 mm - N=72.

of the circular array. To determine the frequency response, each signal from the vibrometer

is deconvolved by the excitation signal. In the experimentation the reciprocity property is

largely exploited and its validity is experimentally verified to ensure its usability. Using the

reciprocity property a signal detected at point A and generated at point B is assumed to

be identical to the signal detected at point B and generated at point A. The reciprocity

property largely simplifies the measurements by allowing the use of only one actuator while

taking advantage of the ease of moving of the motorized vibrometer to measure the flexural

waves anywhere on the plate.

Figure 3.1 shows the frequency response of the circular array of sources. This result is

obtained by computing the mean value of the individual frequency responses of the N signals

measured along the circular path. It is compared to the frequency response of the signal

uniquely measured at the central point. The amplitude parameter of the two frequency

responses shown in Fig. 3.1(a) are different because according to Eq. (9) the frequency

10

response of the circular array of sources is proportional to the Green function of the plate but

modulated by the real Bessel function J0(kR). The phase parameters of the two frequency

responses shown in Fig. 3.1(b) are identical for some frequency bands. For other frequency

bands a phase shifting equal to π is observed. This result is due to the sign of the real Bessel

function J0(kR). The location of the phase shifting correspond to values of kR where the

Bessel function J0(kR) is equal to zero. From this information, it is possible to determine

either plate velocity or plate thickness. The wavenumber k is equal to ω/V where V is the

phase velocity that is according to Eq. (3) equals to

V 2 = hω

√

E

ρ [12 (1 − σ2)]= π

VP√3hf (12)

VP is defined as the plate velocity according to the Lamb wave approach [2].

By locating the frequencies where the phase shifting is encountered one can easily deter-

mine the value of the product VP h.

Figure 3.1 shows the phase difference between the frequency responses of the circular

array of point sources and unique centred point source. This phase difference is determined

for different values of radius of the circular array. As expected, the locations of the phase

shifting differ from one radius value to another one. By determining the frequency values

of the various phase shifting locations an averaged value of the plate velocity is (knowing

the thickness value of the plate) determined : VP = 5477 m.s−1. To validate this result,

another measurement is realized using the zero group velocity (ZGV) method developed by

Clorennec et al. [8]. In this case the plate velocity value is found equal to : VP (ZGV ) = 5498

11

Radius (mm)

Fre

quen

cy (

Hz)

10 15 20 25 30 35 40 45 50

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x 104

0

0.5

1

1.5

2

2.5

3

ϕ (rad)

Figure 4: Phase difference ϕ of the signals emitted by a unique point source anda circular array of N point sources of different radius. Solid lines represents thezero crossing curves of the Bessel function J0(kR).

m.s−1. The velocity difference between the two methods is less than 1%.

3.2 Determination of the relative thicknesses of plates.

For thickness determination purpose a second experiment is realized using two Duralumin

plates of different thicknesses (but supposed made by the same manufacturer) and different

and arbitrary geometries as shown in Fig. 3.2. The thicknesses of the two plates are measured

using a Vernier calliper : 3.98 mm and 2.95 mm. Assuming VP to be the same for the

two plates, the ratio of the product values VP h (determined with the GFC method for the

two plates) is computed. The ratio value obtained is equal to 1.346. The thickness ratio

calculated from the Vernier calliper measurement values is quite similar and equal to 1.349.

A second experiment was conducted in order to ascertain the variation in thickness of a

plate on its surface. For this, a plate of initial thickness 4 mm is used. An excavation of

0.5 mm in depth on an area of 100×100 mm2 is made in it. The initial experiment consists

12

Figure 5: Duralumin plates with different geometries and dimensions. Platethicknesses are equal to 2.95 mm (left plate) and 3.98 mm (right plate).

in achieving circular scans on the surface. The area of interest is 140×120 mm2 and the

circular scans, of radius 20 mm, are made every 5 mm pitch.

For this experiment, as the circular scans are not very large, there is only one phase

shifting in the used frequency bandwidth and then not enough sufficient to make a good

fitting with the zeros of the Bessel function. Instead, the amplitude of the frequency response

of the circular scan is directly compared to the amplitude of the frequency response of the

central signal multiplied by the Bessel function J0(kR) (see Eq. (11)).

Fig. 3.2 shows the relative thickness of the plate on this area. The excavation is not per-

fectly imaged, because the depth is averaged on the surface of the circular scans. Moreover,

for each point, the measurements are made on the central point and on the N points of the

circular array. Thus, this experimental method is heavy to implement and a new and more

efficient approach is then proposed as shown in the next paragraphs.

13

(mm)(m

m)

0 20 40 60 80 100 120 140

0

20

40

60

80

100

3.5

3.6

3.7

3.8

3.9

4

4.1

h (mm)

Figure 6: Mapping of the relative thickness of a plate with a 100×100 mm2

excavation. Measurements are made on circular arrays of N sources for eachpoint. N = 18.

d1

d2

Figure 7: Measurement of the plate velocity - thickness product from a rectan-gular scan.

3.3 Adaptation of the GFC method to a rectangular scan

The GFC method can also be computed from a rectangular matrix scan measurement (using

a laser vibrometer for example) by comparing the frequency response of a given point with

the eight points that surrounds it (Fig. 3.3). In this case, a thickness or plate velocity

mapping can be realized.

Instead of considering N points on a circular contour, the eight neighbour of a central

point are considered. Thus there are two different circles, each composed of 4 points. And

14

two different Bessel functions J0(kd1) and J0(kd2) (cf. Eq. (11)) have to be considered:

WCircle1(~r) = 2πJ0(kd1)Gplate (|~r − ~rs|) (13)

and

WCircle2(~r) = 2πJ0(kd2)Gplate (|~r − ~rs|) (14)

Instead of computing the amplitude difference between WCircle (~r) and J0(kR)Gplate (|~r − ~rs|),

the following comparison is performed between |WCircle1(~r) J0(kd2)+WCircle2

(~r) J0(kd1)| and

|Gplate (|~r − ~rs|)J0(kd1)J0(kd2)|.

The thickness mapping result shown in Fig. 3.3 demonstrates the accuracy of the new

approach. The square excavation is better resolved because the value of the scan pitch is

lower than the radius of the circular scans used in the previous section. The main advantage

of this matrix scan approach is to enhance the time duration to acquire all the signals from

the measurement area.

4 Plates with anisotropy or inhomogeneities.

4.1 Anisotropic plate

Subsequently, the same study is applied to anisotropic plates, especially orthotropic plates.

In this medium, there are two main directions of propagation. Such a type of materials

corresponds generally to plates manufactured by lamination or to unidirectional composites.

15

(mm)

(mm

)

0 20 40 60 80 100 120 140 160 180

0

20

40

60

80

100

120 3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

h (mm)

Figure 8: Mapping of the relative thickness of a plate with a 100×100 mm2

excavation. Measurement are made directly from a matrix scan (cf. Fig. 3.3.

Figure 9: Anisotropic plate studied (epoxy with stripes of copper).

Indeed, the plate is stretched in one direction and rigidity in this direction differs from the

stiffness in the transverse direction.

The experimental plate used is an epoxy plate 0.5 mm thick covered with 35 µm copper

(Fig. 4.1). After partially removing copper by applying a temporary mask and chemical

removing process, the residual copper is in parallel band to achieve a transversely isotropic

material.

Only the flexural mode is studied and its phase velocity is still proportional to√

f (as in

16

Eq. (12)). However, in this case, the product VPh depends on the direction of propagation.

For this orthotropic plate, the plate velocity is assumed to vary elliptically versus the

propagation direction in a first approach. As the phase velocity of the flexural mode is

proportionnal to the square root of the plate velocity, the scans are made on ellipse-like

contours rather than circular contours.

The first step is to determine the right ratio of the ellipse-like contour and its direction.

For this purpose, a two transducers T1 and T2 setup is used. The impulse responses of the

two transducers are recorded, and the product of their Fourier transforms is computed for

the elliptical scan:

Rcontour(f) =

(

∑

contour

Ui1(f)

)

·(

∑

contour

U∗

i2(f)

)

(15)

and its center:

Rcenter(f) = Ucenter.1(f) · U∗

center.2(f) (16)

Uij(f) is the Fourier transform of signal detected in position i along the ellipse-like contour

and emitted by the jth transducer Tj . Ucenter.j(f) is the Fourier transform of signal detected

in central position and emitted by the jth transducer Tj . The sign ∗ corresponds to a phase

conjugation operator.

The experimental method consists in changing the direction of the ellipse-like contour, θ,

and its form defined by the ratio R1/R2 (major axis/minor axis) in order to find the minimum

of phase difference for the product: Rcontour · R∗

center. For an isotropic plate and a circular

17

R1/R

2

|θ| (

°)

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

0

10

20

30

40

50

60

70

80

900.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

RMS phase difference

Figure 10: Evolution of the RMS of the phase difference of the product Rcontour ·R∗

center versus R1/R2 and θ.

contour, Rcontour is proportional to |J0(kR)|2Rcenter (using Eq. (11) for both receivers), the

result of the product Rcontour · R∗

center is then real leading to a minimum phase difference.

For an orthotropic plate and in a first approach, the product Rcontour is also assumed to be

proportional to the product Rcenter when the right ellipse-like contour is found. In this case,

the result of the product Rcontour · R∗

center is then also expected to be real. Figure 4.1 shows

the root mean square of the phase difference of the product for different values of the ratio

of the ellipse-like contour (i.e. ratio R1/R2) and different directions θ for the studied plate.

Once the right ellipse-like contour is determined, the plate velocity - thickness product

are obtained using the GFC method for radii R1 and R2 for the two main directions:

• VP1h = 1.99m2.s−1 in the direction of the stripes

• VP2h = 1.74m2.s−1 in the transverse direction.

These results have been succesfully compared to those obtained using a modal analysis

18

of the plate. In this last case, the flexural rigidities of the plate are the followings using the

method detailed in [21]

D11 = 0.57N.m

D22 = 0.33N.m

From these flexural rigidity values, it is shown [22] that the waves velocity follows the

equations:

cf = 4

√

D11/ρh√

ω

in the stripes direction

cf = 4

√

D22/ρh√

ω

in the transverse direction.

Finally, the product VP h is linked to flexural rigidities by

VPih = 2√

3

√

Dii

ρh(17)

The products VPih given by the modal analysis are

• VP1h = 2.01m2.s−1

• VP2h = 1.75m2.s−1

Both values obtained by the elliptical approach and the modal analysis are identical

19

Figure 11: Plate velocity - thickness product for a viscoelastic patch.

within 1% error. This result confirms the validity of the GFC method for materials with

weak anisotropy.

4.2 Inhomogeneous plate

In this section, a NDE approach is proposed, using the GFC method, for locating inhomo-

geneities or defects in plates. For this purpose, a viscoelastic patch of size 20×40 mm2 is

sticked under a plate of thickness 1 mm, and a rectangular matrix scan is conducted on

the plate with 2 mm pitch. The map of the plate velocity - thickness product is shown on

Fig. 4.2.

On this figure, the viscoelastic patch is obviously highlighted. It corresponds to the area

where the plate velocity - thickness product is lower than in the rest of the plate.

20

Figure 12: Circular and point sensors using PZT discs.

5 PZT sensors setup.

Equation (11) shows that radiation from a point source and a circular source are identical

except for the Bessel function. This result - involving the principle of reciprocity - is valid

whatever the point of emission of the acoustic wave. In the following setup, a Duralumin

plate of arbitrary geometry is equipped with a circular array of PZT discs connected in

parallel and a single PZT disc in the center of the array as shown in figure 5. The signals

received by the sensors, respectively s1(t) and s2(t) are processed using a Fourier transform

and their complex spectra are S1(f) and S2(f). According to Eq. (11) and the principle of

reciprocity, the result of the spectra ratio |S1(f)/S2(f)| is proportionnal to a Bessel function

J0(kR) with R the radius of the circular array of PZT.

Figure 5(a) shows the module of the spectra ratio S1(f)/S2(f) obtained for the config-

uration shown in figure 5 where the radius R is equal to 60 mm. In this experiment, the

source used is a loudspeaker that is manually moved in all the measurement room during the

21

acquisition of the signals from the two sensors. The excitation waveform is a chirp (band-

width 100 Hz - 20 kHz). The result shown in Fig. 5(a) represents the mean value of the

spectra ratio performed on 256 samples of signals. This result highlights the presence of the

J0(kR) factor between the spectra of signals from the two sensors. It should be compared

with the result of the measurement made with the direct method (see Section 3) shown in

Fig. 5(b). In this case, the laser vibrometer is used on 36 points of a circular contour of same

radius. Both results are comparable except for some frequency bands: at low frequencies

and at frequencies above 15kHz. For the first case, more experiments are needed to explain

the phenomenon. For the second case, the limit is due to the limited frequency bandwidth

of the loudspeaker. Product values VP h determined from both experiments through fitting

procedures with the function J0(kR) (dotted curves in 5(a) and 5(b)) are identical and equal

to 15.78 m2s−1.

It is shown from this experiment that the GFC method is usable in a passive mode by

simultaneously acquiring acoustic noise by two sensors. The acoustic noise should correspond

to acoustic waves emitted from air or contact source.

6 Conclusion

It has been shown that from the study of the Green function of the flexural A0 Lamb mode

the determination of the plate velocity - thickness product for plates of arbitrary geometries is

allowed. The GFC method, presented in this paper, takes into account the finite dimension of

the plate but demonstrates a local measurement. Furthermore, experiments showed that this

22

0 5 10 15 200

0.5

1

1.5

Frequency (kHz)

|S1 /

S 2|

(a)

0 5 10 15 200

0.5

1

1.5

Frequency (kHz)

|S1 /

S 2|

(b)

Figure 13: Ratio modulus of the Fourier transforms of signals from the PZT setup(a) and signals from the laser vibrometer measurement setup (b). Experimental(solid lines) and Bessel function fit (dash lines) curves.

method can be adapted to isotropic plates or orthotropic plates. Presence of inhomogeneities

in a plate is also demonstrated for NDE purpose. The GFC method works at low frequencies

(in the audio frequency band), allowing the use of inexpensive electronics and sensors. Finally

the GFC method should work in a passive mode, by simply exploiting the ambient acoustic

noise. From these results, a continuous thickness or plate velocity measurement should be

viewed in different application areas using low cost and green devices.

An alternative non contact measurement should also be realized using a unique vibrom-

eter with a powerful pulsed laser to successively generate under thermoelastic or ablation

regimes an impulse point source and impulse ring source.

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23

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