Ultrasonics 54 (2014) 860–866

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Ultrasonics

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Propagation of Lamb waves in an immersed periodically grooved plate:Experimental detection of the scattered converted backward waves

0041-624X/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ultras.2013.10.012

⇑ Corresponding author. Postal address: CSC, BP 64, Cheraga, Algiers, Algeria. Tel.:+213 21 36 18 50; fax: +213 21 36 18 54, mobile: +213 556 81 58 67.

E-mail addresses: nharhad@yahoo.fr (N. Harhad), elkettani@univ-lehavre.fr(M.Ech-Cherif El-Kettani), hdjelouah@usthb.dz (H. Djelouah), jean-louis.izbicki@univ-lehavre.fr (J.-L. Izbicki), predoi@gmail.com (M.V. Predoi).

Nadia Harhad a,⇑, Mounsif Ech-Cherif El-Kettani b, Hakim Djelouah c, Jean-Louis Izbicki b,Mihai Valentin Predoi d

a Scientific Center of Research on Welding and Control, CSC, BP 64, Cheraga, Algiers, Algeriab Laboratoire Ondes, Milieux Complexes (LOMC), Groupe Ondes Acoustiques (GOA), UMR CNRS 6294, University of Le Havre, Le Havre, Francec U.S.T.H.B, Material Physics Laboratory, BP 32 El Allia, Bab-Ezzouar, Alger, Algeriad Department of Mechanics, University Politehnica of Bucharest, Bucharest, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 June 2013Received in revised form 15 October 2013Accepted 16 October 2013Available online 29 October 2013

Keywords:Liquid–solid interfacesRough platePeriodic gratingLamb wavesConverted modes

Guided waves propagation in immersed plates with irregular surfaces has potential application todetection and assessment of the extent, depth and pattern of the irregularity. The complexity of theproblem, due to the large number of involved parameters, has limited the number of existing studies.The simplest case of irregularities of practical interest is the two-dimensional corrosion profile. Even thiscase is in general so complex, that one can extract several amplitude dominant periodic surfaces only byusing a Fourier spectrum of the surface. Guided waves in plates, with one or both free surfaces havingperiodic perturbations of different shapes, have been presented in specialized literature.

In this paper is studied the propagation of Lamb waves in an aluminum plate with a periodic groovedsurface on only one side and immersed in water. The interaction between an incident Lamb wave and thegrating gives rise to retro-converted waves. Preliminary numerical simulation by the finite elementmethod is performed in order to obtain key parameters for the experiments. It is shown that retro-converted waves radiating into the water are detectable although their amplitudes are small. The phononrelation is verified for the leaky Lamb modes. The damping coefficients of the leaky Lamb modes in thegrooved immersed plate are evaluated.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Rough liquid–solid interfaces have been the object of severaltheoretical and experimental studies especially in the field of non-destructive testing (NDT) and evaluation (NDE) [1] and in optics[4]. The potential method constitutes one of the theoretical ap-proaches to formulate mathematically the scattering of ultrasoundwaves by rough liquid–solid interfaces [1–3]. In the past, Rayleighsurface waves have been used to investigate periodically roughliquid–solid interfaces. Attenuation of surface waves propagationalong a rough liquid–solid interface results from its scattering bythe roughness. The equations governing this phenomenon aresolved by the method of small perturbations when the roughnessis small in comparison with the wavelength [5,6]. In the case of arough plate, Lamb waves have been used since they are sensitive

to the roughness [7–9]. Their interaction with the rough surfacegives rise to converted modes. In the case of a periodic grating,the incident, the retro-converted wavenumbers and the gratingperiod verify the phonon relation [10]. For some particular fre-quencies, the incident mode is reflected as the same mode or asa different mode at the edge or inside the first Brillouin zone [11].

This work is an experimental study of the propagation of leakyLamb waves in a water immersed plate which is corrugated on oneside only. The corrugation is a grating with a periodic geometry.Two main objectives are pursued: (a) the possibility to detectexperimentally the existence of roughness on an immersed struc-ture and (b) the sensitivity of the proposed method. These areimportant issues for the underwater detection of radiated wavesby a rough surface, as for example in the case of an immersed cor-roded structure. Even the two-dimensional roughness case is ingeneral so complexes, one can extract several amplitude dominantperiodic surfaces only by using a Fourier spectrum of the surface.

In this paper, after a brief description of the studied sample,results from numerical simulations are given in order to guide inselecting the experimental parameters such as the incidence angleand the modal damping of a free corrugated plate. The phonon

N. Harhad et al. / Ultrasonics 54 (2014) 860–866 861

relation is verified for the leaky Lamb modes. The damping coeffi-cient of the incident leaky Lamb wave is measured. The attenuationdue to the periodic corrugation is obtained and a comparison ismade between experimental and theoretical results.

2. Numerical simulation

2.1. Description of the studied sample

The studied sample is an aluminum plate with a thicknessh = 5 mm and a total length of 500 mm. Longitudinal and shearvelocities in aluminum are respectively CL = 6320 m/s andCT = 3115 m/s and its density is 2700 kg/m3. Only the middle partof one side of the plate is grooved with 20 identical rectangulargrooves. The grooves have a spatial periodicity K = 6 mm and adepth p = 0.1 mm as shown in Fig. 1. The depth of the periodic cor-rugation is small as compared to the thickness of the plate so thesmall perturbation approximation can be used. In a such approxi-mation, the phonon relation for a 1D periodic grating can be used.

2.2. Numerical processing

The interaction of an incident Lamb wave with the grating in-volves different Lamb modes in the reflected Lamb waves. In orderto study the guided waves scattering, a numerical solution is ob-tained by using the commercially available finite elements simula-tion software COMSOL [12] in transient analysis. In order to avoid along computation time, the plate described in Section 2.1 is consid-ered in vacuum, knowing that for metallic plates immersed inwater, the real part of a modal wavenumber is with a very goodapproximation, the same as the corresponding wavenumber inthe free plate. An analytical expression is used for the displace-ments applied on the left edge of the plate in order to generate asingle Lamb wave at a given frequency. To represent the time-space image of the propagating waves in the plate, the normal sur-face displacements are recorded on the smooth plane surface of theplate, on the opposite side of the corrugated one. Three simulationsare made:

(a) The fundamental antisymmetric A0 mode, propagating at250 kHz towards the grating.

(b) The fundamental symmetric S0 mode propagating towardsthe grating at 320 kHz.

(c) The fundamental symmetric S0 mode propagating towardsthe grating at 450 kHz.

In Fig. 2a a time-space (x, t) representation is given for S0 inci-dent wave at 320 kHz. The part (1) of the signal corresponds tothe incident wave and part (2) to the reflection from the grating[13].

Fig. 1. Geometry of the studied sample with 20 grooves, h is the plate thickness,p the groove depth and K the geometric period.

Fig. 2. (a) Normal displacements in (x, t) space: (1) the incident signal and (2) thereflected signal. (b) Dual space (f,k) representation of the S0 incident mode at320 kHz. (c) (f,k) Representation of the A0 retro-converted mode at 320 kHz.Frequency shift according to the phonon relation. Amplitudes according to colorlegend. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

2.3. Results

In order to separate and to identify the incident and thereflected waves, a two-dimensional Fast Fourier Transform

Table 1Numerical results: verification of phonon relation for each incident propagating wave and its corresponding retro-converted wave from the grating.

Incident wave Converted wave f (kHz) kinc (m-1) kconv (m-1)P

k (m-1) Relative error (%)

A0 S0 290 715 337 1052 0.5S0 A0 290 306 735 1041 0.6S0 A1 450 644 430 1074 2

f, Frequency; kinc, incident wavenumber; and krefl, reflected wavenumber.

Table 2Numerical results: wavenumbers values of the incident and retro-converted waves when the modes are identical.

Incident wave Reflected wave f (kHz) kinc (m-1) krefl (m-1) p/K (m-1) Relative error (%)

A0 A0 200 490 550 523 5S0 S0 400 528 551 523 5

f, Frequency; kinc, incident wavenumber; krefl, reflected wavenumbe; and K, grating period.

Fig. 3. Representation of the Poynting vector on the normal oriented cross-section.

862 N. Harhad et al. / Ultrasonics 54 (2014) 860–866

(2D FFT) is performed. In the dual space (f,k) the results are super-imposed on the theoretical dispersion curves of a free infinite planeplate. The wavenumbers of the incident and the reflected wavesare respectively positive and negative. The S0 incident mode at320 kHz is identified as shown in Fig. 2b. While a fraction of theincident S0 mode energy is converted into reflected A0 mode(Fig. 2c), named in the following the retro-converted mode dueto the inversion of the direction of propagation. Dual space (f,k)representation of the Lamb mode permit to reach for each fre-quency its corresponding wavenumber and amplitude. At the fre-quency for which the retro-converted A0 mode amplitude ismaximum, the phonon relation will be verified.

The phonon relation detailed in Ref. [10] which can be writtenas:

kinc þ kconv ¼ 2p=K ð1Þ

represents a resonance phenomenon, relating the wavenumbers ofthe incident wave kinc, the retro-converted wave kconv and the pho-non related to the grating, which has an equivalent wavenumber2p/K = 1047 m�1. At f = 290 kHz, the wavenumbers of respectively,S0 incident and A0 retro-converted waves (kS0inc = 306 m�1 andkA0conv = 735 m�1) satisfy the phonon relation: kS0inc + kA0conv =2p/K with a relative error of 0.5%.

Other simulations have been made for A0 incident mode at250 kHz and for S0 incident mode at 450 kHz. In Table 1 are re-ported the values of the wavenumbers of each incident modeand of the corresponding retro-converted mode. The values ofthe frequencies correspond to a maximum of the retro-convertedmode amplitude. Eq. (1)shows that the incident mode and theretro-converted mode play symmetric roles: the simulationconfirms within a numerical error that whichever of A0 or S0 isthe incident mode, the retro-converted mode S0 or A0 is obtainedat a specific frequency.

In Table 2 are reported the values of the incident mode wave-numbers for which the reflected mode is the same as the incidentmode. In this case, the values of the wavenumbers correspond tothose of the edge of the Brillouin zone.

From the values of the normal displacements along the free sur-face of the plate, it is possible to evaluate the energy carried out byeach propagating incident and retro-converted mode as shown in[14]. The power flux U carried out by a mode along 0x directionis the flux of Poynting vector P

![15] through a normal oriented sur-

face dS!

cross-section located in Oyz plane with 1 m length alongthe 0y axis and thickness of which along 0z axis is h, as shown inFig. 3:

ZS

P!:dS!¼Z þh=2

�h=2

Z þ1=2

�1=2Pxdzdy ð2Þ

with

Px ¼ �12

v�xrxx �12

v�zrzx ð3Þ

The particle velocity ~v is the time derivative of the acoustic dis-placement ~UðUx;Uy;UzÞ; v�x;v�z are the complex conjugate of thevelocity components vx, vz and rxx, rzx represent the correspondingnormal and shear stress components. Since the experimentalaccessible values are the normal displacements Uz on the surfaces+h/2 or �h/2, a relation between the normal displacements and theenergy carried by a propagating Lamb wave has been establishedin Ref. [16] by using a parameter f , defined by:

f ¼jUzjtheory

surfaceffiffiffiffiffiffiffiffiffiffiffiffiffiUtheory

p ð4Þ

where jUzjtheorysurface and Utheory are respectively the theoretical normal

displacements and the energy obtained analytically. From the val-ues of the normal displacements obtained either by the finite ele-ments method or by experiments, the energy values of theincident and retro-converted modes are calculated from therelation:

Ufem;exp ¼jUzjfem;exp

surface

f

" #2

ð5Þ

where jUzjfem;expsurface and Ufem;exp are respectively the normal displace-

ments and the energy obtained by simulation and/or by experi-ment. The conversion coefficient is defined by the ratio betweenthe energy of the retro-converted mode Uconv and the energy ofthe incident mode Uinc:R = Uconv/Uinc. In the same way, the trans-mission coefficient is defined by: T = Utrans/Uinc, where Utrans is

Table 3Numerical results: Energy balance.

Incidentwave

Converted/reflected wave

f(kHz)

T R T + R Relativeerror (%)

A0 S0 271 0.9948 0.0037 0.9986 0.1A0 210 0.9989 0.0002 0.9991 <0.1

S0 A0 290 0.9747 0.0077 0.9892 1.7

S0 A1 434 0.9930 0.0014 0.9944 0.56S0 390 0.9854 0.0012 0.9866 1.34

f, Frequency; T, transmission coefficient; and R, reflection coefficient.

Table 4Computed angles for different incidents and retro-converted waves.

Incident wave f (kHz) hi Converted wave f (kHz) ht

S0 320 17� A0 290 35�

S0 450 19.3� A1 450 11�S0 400 18�

A0 250 S0 290 16�.437� A0 200 40�

f, Frequency; hi, incidence angle; and ht, receiving angle.

N. Harhad et al. / Ultrasonics 54 (2014) 860–866 863

the transmited mode energy. Each retro-converted mode from thegrating has a small energy ratio which is lower than 1% of the inci-dent mode energy. In Table 3 are reported the conversion and trans-mission coefficients values of each incident mode and thecorresponding retro-converted and transmitted mode.

3. Experiments

3.1. Experimental setup

The experimental setup is shown in Fig. 4. All the measure-ments are performed in a water immersion tank. The generationand the reception of Lamb waves are done using two identical pie-zoelectric transducers of 500 kHz central frequency. The transduc-ers are used one as the emitter and the other as the receiver, withthe acoustic axes in a plane perpendicular to the plate. The dis-tance between the plate and the emitter or the receiver is approx-imately 15 cm which satisfies the far field condition. The incidenceand reception angles can be determined from the wavenumbers ofrespectively the generated and received Lamb mode. For a givenLamb mode the value of the angle h is deduced from the Snell–Descartes formula [15]:

h ¼ sin�1ðk0Lamb=kLiqÞ ð6Þ

in which kLiq is the wavenumber of the ultrasonic wave in water andk0Lamb is the real part of the complex valued wavenumber of the

Fig. 4. Experimental setup for (a) the incident mode (b) the retro-converted mode.

selected leaky Lamb mode obtained from a numerical solution ofthe dispersion equation of Lamb waves. A waveform generatordelivers a 10 V amplitude burst with eight periods to the emittingtransducer. For a selected Lamb mode, the emission and the recep-tion are performed at the same angle, denoted hi. Then, the receiv-ing transducer is moved parallel to the plate surface by steps of0.5 mm along the O1x1 axis as shown in Fig. 4a. The received signalis recorded for several positions, located by their abscissa x1. Thedata acquisition starts when the receiving transducer is out of the

Fig. 5. Representation (x, t) of the incident wave (a). Dual space (f,k) representation ofS0 incident mode at 320 kHz frequency (amplitudes according to legend) superposedon the dispersion curves (white lines) (b). (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Representation (x, t) of the retro-converted mode (a). Dual space (f,k)representation of A0 rectro-converted mode. Amplitudes according to legend (b).

Fig. 7. Experimental amplitude of S0 incident mode versus propagation distanceand its corresponding fitting curve.

864 N. Harhad et al. / Ultrasonics 54 (2014) 860–866

direct emitted field and ends when the amplitude of the retro-converted Lamb mode is too weak to be detected by the receiver.To detect the retro-converted modes predicted by the numericalsimulation, the corresponding receiving angle ht is obtained fromEq. (6). The receiving transducer Fig. 4b is turned at this angleand then displaced in opposite direction of the incident Lamb modealong the O2x2 axis. In this case, the scan starts at the beginning ofthe first groove and finishes when the retro-converted Lamb modebecomes undetectable.

Three main experiments are reported. The first one is based onthe incident antisymmetric A0 Lamb mode at a central frequency of250 kHz. The second one uses the incident symmetric S0 Lambmode at a central frequency of 350 kHz and the same incidentmode but at a central frequency of 450 kHz is used in the thirdexperiment. The same values were used in the numerical simula-tions by finite element method.

3.2. Experimental results

For an immersed plate, Lamb waves are called leaky waves dueto energy loss resulting from the radiation in the fluid and thewavenumbers are complex numbers, which can be written as:

kx ¼ k0x þ ik00x ð7Þ

The real part k0x is governing the wave propagation being associatedto the phase velocity and the imaginary part k00x can be associated tothe wave attenuation, mainly due to the reemission in the sur-rounding medium, but also to the retro-conversion. The Lamb wavenumbers values kx deduced from the simulation in the case of aplate in vacuum are nearly equal to the real parts of the wavenum-bers values k0x [7,17]. The variation of the imaginary part k00x for thecorrugated plate is studied in the following. In Table 4 are indicatedthe angles computed from Eq. (6), which are necessary to generateand receive the different Lamb modes.

Generally speaking, the recorded signals provided by the recei-ver are represented on time-space diagrams. A two-dimensionalFast Fourier Transform (2D FFT) is then applied to identify theLamb modes by superimposing the results on the theoretical dis-persion curves of the Lamb waves.

For hi = 17�, the S0 mode is generated at 320 kHz central fre-quency. To observe the A0 retro-converted mode, the receivingtransducer is adjusted around the computed angle (35�) and is dis-placed in the opposite direction to the incident wave direction ofpropagation. In Fig. 5a is represented the incident signal amplitudeand the S0 mode is identified in Fig. 5b and in Fig. 6a is representedthe amplitude of A0 retro-converted mode and the A0 mode is iden-tified in Fig. 6b.

The wavenumbers of S0 incident mode and A0 retro-convertedmode are respectively kinc = 379 m�1 and kconv = 706 m�1 atf = 300 kHz. With these experimental values, the phonon relationis satisfied with a relative error of 3.6%, for the water immersedgrating.

The damping coefficient of the leaky Lamb modes of a planeplate corresponds to the reemission of the wave in the surroundingfluid. Solving the dispersion equations for water immersed platewithout roughness permits to determine the theoretical value ofthe damping coefficient which is 8.90 m�1 for S0 incident modeat 300 kHz. Experimental maximum value of the signal corre-sponding to this mode is represented versus the propagation dis-tance x1 in Fig. 7. To determine the values of the dampingcoefficient before, under and after the grating, exponential fittingof the raw data is performed. Fig. 7 represents the experimentalamplitude and the corresponding fitting curve.

Therefore, the damping coefficients values before and after thegrating, are respectively 9.05 m�1 and 9.27 m�1, which are veryclose. These values are slightly larger than the theoretical value:this fact can be explained by the spreading of the acoustic beam.This three dimensional effect is not taken into account in the

Table 5Experimental results: verification of the phonon relation.

Incident wave Converted wave f (kHz) kinc (m-1) kconv (m-1)P

k (m-1) Relative error (%)

A0 S0 260 675 282 959 8.5S0 A0 300 379 706 1085 3.6S0 A1 460 676 380 1056 0.8

f, Frequency; kinc, incident wavenumber; and krefl, reflected wavenumber.

Table 6Experimental results: wavenumbers values of the incident and the retro-concerted waves when the modes are identical.

Incident wave Reflected wave f (kHz) kinc (m-1) krefl (m-1) p/K (m-1) Relative error

A0 A0 200 558 498 523 4.7%S0 S0 400 518 503 523 3.8%

f, Frequency; kinc, incident wavenumber; krefl, reflected wavenumber; and K, grating period.

Table 7Experimental and theoretical values of the damping coefficients.

Incident wave a (m-1) Before the grating a (m-1) Under the grating a (m-1) After the grating Theoretical value (plane plate)

A0 (250 kHz) 30.74 32.22 * 30.10S0 (320 kHz) 9.05 10.47 9.27 8.90S0 (450 kHz) 50.71 * * 50

a, Damping coefficient and *, not measured damping value.

N. Harhad et al. / Ultrasonics 54 (2014) 860–866 865

theoretical approach. The measured damping coefficient under thegrating is 10.46 m�1, which is greater than the values before andafter the grating and this is the consequence of the interaction ofLamb modes with the corrugation. Despite of the weak surface per-turbation, the retro-converted mode is easily detected because theangle of reemission is far different from that of the direct transmit-ted signal.

The second experiment consists in the generation of the inci-dent A0 mode at 250 kHz. The computed incident angle ishi = 37�. The numerical simulation predicts a reflected A0 modeand a retro-converted S0 mode. These modes are detected whenthe receiving transducer is oriented at ht = 40� and ht = 16.4�respectively.

The last experiment is dedicated to the S0 incident mode at450 kHz, generated for an incident angle hi = 19.3�. To detect thetwo reflected modes predicted by the numerical simulations, thereceiving transducer is oriented at ht = 18� to detect the S0 reflectedmode and at ht = 11� for A1 retro-converted mode.

Tables 5–7 summarize the experimental results. In Table 5 arereported the experimental wavenumbers values of each incidentmode and its retro-converted mode which always satisfies thephonon relation with a good agreement. In Table 6 are reportedthe experimental wavenumbers values for which there is noretro-conversion and the only reflected mode is the same as theincident one. Overall there is a good agreement between thetheoretical values (Eqs. (1) and (6)) and the experimental ones.

In Table 7 are reported the damping coefficient values for A0

incident mode at 250 kHz, S0 incident mode at a frequency of320 kHz and S0 incident mode at a frequency of 450 kHz. Whenthe plate is corrugated, the damping of the incident Lamb modeis larger because converted modes are generated. Moreover, onecan see from Table 7 that the S0 mode at 320 kHz is more sensitiveto the presence of the corrugation than the A0 mode at 250 kHz, asthe relative increase of its damping coefficient between the planeand corrugated domains is around 15% (5% for the A0 mode).

Conversely the measurement of the damping coefficientcould bring information about the presence of roughness. There-fore the corrugation which was investigated on the opposite sideof the plate, can be detected either by the damping measurement

of the incident guided mode, or by the angular scanning in the sur-rounding fluid, by searching the optimal position corresponding tothe reemission of the retro-converted mode.

4. Conclusion

Numerical simulations followed by an experimental validationof the retro-converted leaky Lamb modes by an immersed corru-gated plate have been presented. The retro-converted mode inwater is well detectable, even if its energy predicted by numericalsimulations is weak. Its angular direction of reemission is analyti-cally determined. Moreover, we show that the damping coefficientof the incident mode is larger under the grating than along the im-mersed plane plate, as predicted by numerical simulations. Thesupplementary attenuation introduced by the grating is physicallydue to the generation of converted Lamb modes. The phononrelation linking the incident mode wavenumber and the retro-converted mode has been verified for the immersed grating,explaining the frequency shift of the retro-converted mode. Apotential application of this study could be the detection and theassessment of an internal or unattainable periodic corrugation bythe damping measurement of the incident guided mode, or bythe angular scanning of the radiated ultrasound wave on the reach-able face of the plate to evaluate. The future of this work is to studydifferent types of grating with various periods and depths in orderto compare their influence on the propagating modes. A numericalmodel for an immersed grating will also be investigated.

References

[1] M. De Billy, G. Quentin, Backscattering of acoustic waves by randomly roughsurfaces of elastic solids immersed in water, J. Acoust. Soc. Am. 72 (2) (1982)591–601.

[2] J.M. Claeys, O. Leroy, A. Jungman, L. Adler, Diffraction of ultrasonic waves fromperiodically rough liquid–solid surface, J. Appl. Phys. 54 (10) (1983) 5657–5662.

[3] S.W. Herbison, N.F. Declercq, Angular and frequency spectral analysis of theultrasonic backward beam displacement on a periodically grooved solid, J.Acoust. Soc. Am. 126 (6) (2009) 2939–2948.

[4] T. Tamir, H.L. Bertoni, Lateral displacement of optical beams at multilayeredand periodic structures, J. Acoust. Soc. Am. 61 (10) (1971) 1397–1413.

866 N. Harhad et al. / Ultrasonics 54 (2014) 860–866

[5] A.D. Lapin, Scattering of surface waves propagating over an uneven liquid–solid interface, Sov. Phys. Acoust. 15 (3) (1967) 336–339.

[6] A.D. Lapin, Reflection of a surface wave from periodic irregularities of a liquid–solid interface, Sov. Phys. Acoust. 24 (3) (1978) 209–212.

[7] M.F.M. Osborne, S.D. Hart, Transmission, reflection, and guiding of anexponential pulse by a steel plate in water. I. Theory, J. Acoust. Soc. Am. 17(1) (1945) 1–18.

[8] M.F.M. Osborne, S.D. Hart, Transmission, reflection, and guiding of anexponential pulse by a steel plate in water. II. Experiment, J. Acoust. Soc.Am. 18 (1) (1946) 170–184.

[9] I.A. Viktorov, Rayleigh and Lamb waves, Plenum Press, New York, 1967.[10] D. Leduc, A.-C. Hladky, B. Morvan, J.-L. Izbicki, P. Pareige, Propagation of lamb

waves in a plate with a periodic grating: interpretation by phonon, J. Acoust.Soc. Am. 118 (4) (2005) 2234–2239.

[11] M. Bavencoffe, A.-C. Hladky-Hennion, B. Morvan, J.L. Izbicki, Attenuation oflamb waves in the vicinity of a forbidden band in a phononic crystal, IEEETrans. Ultrason. Ferroelectr. Freq. Control 56 (9) (2009) 1960–1967.

[12] COMSOL Multiphysics – User Manual, 2013. <www.comsol.com>.[13] N. Harhad, M. Ech-Cherif El-Kettani, H. Djelouah, J.L. Izbicki, M.V. Predoi,

Propagation of Lamb waves on an immersed plate containing a periodicgrating: experimental study, in: IEEE Int. Ultrason. Symp., Dresden, Germany,October 7–10, 2012, pp. 1762–1765.

[14] B.A. Auld, Acoustic Fields and Waves in Solids, Wiley ed., New York, 1973.[15] D. Royer, E. Dieulesaint, Elastic Waves in Solids, Springer ed., New York, 2000.[16] B. Morvan, N. Wilkie-Chancellier, H. Duflo, A. Tinel, J. Duclos, Lamb wave

reflection at the free edge of a plate, J. Acoust. Soc. Am. 113 (3) (2003) 1417–1425.[17] F. Ahmed, N. Kiyani, F. Yousaf, M. Shams, Guided waves in a fluid-loaded

transversely isotropic plate, Math. Prob. Eng. 8 (2) (2002) 151–159.