The thermoelastic excitation of air-solid interface waves using the pulsed laser

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1 Scence n Chna Ser. G Physcs, Mechancs and Astronomy 004 Vol.47 No The thermoelastc exctaton of ar-sold nterface waves usng the pulsed laser HU Wenxang & QIAN Menglu Insttute of Acoustcs, Tongj Unversty, Shangha 0009, Chna Correspondence should be addressed to Hu Wenxang (emal: wxhu@mal.tongj.edu.cn) Receved October 0, 003 Abstract Applyng the ntegral transform to the coupled problem of thermoelastc and heat conducton equatons, the ntegral representaton of the normal dsplacement feld for the ar-sold nterface waves excted by a pulsed laser lne source s obtaned. The pole resdues of the ntegrand are performed analytcally, and the transent dsplacement feld s calculated by usng FFT technque. The thermoelastc exctaton and detecton of the ar-sold nterface waves s carred out by a laser ultrasonc system, on whch the pulsed laser s focused nto a lne source onto the nterface to excte the ar-sold nterface waves: leaky Raylegh wave and Scholte wave, and the nterface waves dsplacement sgnal s detected successfully by a SH130 laser heterodyne nterferometer. The theoretc and expermental results are n good agreement. Keywords: ar-sold nterface, nterface waves, laser ultrasoncs. DOI: / 03yw0190 In the 190s, the sold-sold nterface wave, Stoneley wave, was frst studed by Stoneley. From the 1930s to 1940s, the flud-sold nterface waves, usually called Scholte wave or Scholte-Stoneley wave, were studed by Cagnard and Scholte respectvely [1]. The Scholte wave corresponds to the real root of the flud-sold nterface secular equaton, whch s usually called the Scholte equaton, and the velocty of Scholte wave s only slghtly lower than the longtudnal velocty of the flud. Whle the leaky Raylegh wave wth complex velocty corresponds to the complex root of the Scholte equaton, ts real part s close to the velocty of the Raylegh wave on the sold half-space, and ts magnary part s small. Leaky Raylegh wave rradates ts energy nto the flud when t propagates along the flud-sold nterface; therefore ts ampltude wll reduce gradually [1 3]. There are few research works on the exctaton and propagaton of the ar-sold nterface waves. Godnez-Azouage et al. [4] studed the nterface waves at the nterface between ar and the flud-saturated porous materal by usng analytc method n theory, and ther work showed that the soluton of ths acoustc feld has three sngulartes corre- Copyrght by Scence n Chna Press 004

2 00 Scence n Chna Ser. G Physcs, Mechancs and Astronomy 004 Vol.47 No spondng to the Ralegh wave, slow surface wave, and the arborne wave (ar-sold nterface wave), but they could not observe expermentally the arborne wave due to the mpedance msmatch between the ar and the transducer. The laser ultrasoncs s an all-optcal, non-contact and wde-band ultrasonc exctng and detectng method. Its unque exctng mechansm can excte varous wave modes at the same tme, but that s mpossble for the tradtonal pezoelectrc transducer. Therefore, laser ultrasoncs s an effectve means to excte the nterface waves. The theoretc soluton of flud-sold nterface waves, whch s generated by the thermoelastc exctaton of a pulsed laser, was frst presented by Gusev et al. [5]. They solved the coupled problem of thermoelastc and the heat conducton equatons by usng the tradtonal ntegral transform technque, and the asymptotcal approxmate soluton under varous medum condtons were gven. The laser ultrasonc expermental observaton for the lqud-sold nterface waves was frst done at the mercury-quartz and the water-copper nterfaces respectvely by Desmet et al. [6,7]. Rostyene et al. [8] attempted to dentfy the flud propertes by usng the lqud-sold nterface waves. But as far as we know, the numercal smulaton and the expermental observaton for the thermoelastc exctaton and propagaton of ar-sold nterface waves have not been reported up to now. In ths paper, we solve the problem of thermoelastc exctaton wth a method smlar to Gusev et al. s [5], then perform the analytc normal dsplacement soluton by usng resdue theory, and the transent acoustc feld of ar-sold nterface wave, whch s excted thermoelastcally by a pulsed laser lne source, s obtaned wth the FFT technque. The expermental thermoelastc exctaton and detecton are also done on the laser ultrasonc system, manly consstng of a Nd: YAG laser and a laser heterodyne nterferometer. The nterface waves: leaky Raylegh wave and Scholte wave at ar-ron nterface are excted and detected smultaneously. The numercal and expermental results are n good agreement. It s the frst tme to report what are to our knowledge the numercal smulaton and the dentcal expermental results of the thermoelastc generated nterface waves at the ar-sold nterface. 1 The dsplacement soluton of acoustc feld at flud-sold nterface excted thermoelastcally by a plused laser lne source Consder two homogeneous, elastcally sotropc half spaces, one of whch s the optcally transparent, separated by a flat nterface. The pulsed laser, whch s focused nto a lne source by a cylndrcal lens, s ncdent onto the nterface normally through the transparent medum. The geometry of ths problem s smlar to that n ref. [5]. The pulsed heatng of the nterface, at whch the laser energy s absorbed, leads to the exctaton of acoustc waves because of the thermoelastcty. It s a -D problem n xy plane. The correspondng thermoelastc equaton and thermal stresses are Copyrght by Scence n Chna Press 004

3 The thermoelastc exctaton of ar-sold nterface waves usng the pulsed laser 01 uy G c ( ) L c T u x cl c T =, t x y x y x ux G c ( ) T c L u y cl c T =, t x y x y y (1) () σ σ xy u u x y = ρct +, y x (3) u u y = c c + c G. (4) x y x yy ρ ( L T ) L Let the axes along and normal to the nterface be x and y, let the subscrpt 1 represent the opaque medum n regon y>0, and the subscrpt represent the transparent one n y<0. The heat conducton equatons are ρ = + + Q, (5) t x y T1 1c1 k1 T 1 ρ T c = k + T t x y, (6) where u y, u x are the normal and tangental dsplacements, σ yy, σ xy are the normal and tangental stresses, T 1, are the temperatures n medum 1 and, c L,T are the veloctes of longtudnal and transverse waves, ρ s the densty and c s the heat capacty of medum, respectvely. G = βkt/ρ, where β, K are the bulk thermal expanson coeffcent, bulk elastc modulus respectvely. Q s a heat source functon whch descrbes the release of the absorbed optcal energy [5]. Q = αie αy f(t)ψ(x), where α s the lght absorpton coeffcent, I s the laser ntensty rradated on the nterface, and functons f(t) and ψ(x) descrbe the shape of the lght pulse n tme and the dstrbuton of lght ntensty along the nterface respectvely. Applyng the double Fourer transforms on (t, x) and the Laplace transform on y to the equatons from (1) to (6), and usng the boundary condtons at y = 0, the dsplacement solutons n the transformed doman (ω, k x, y) for the sold-sold and flud-sold nterface can be derved. The followng s the normal dsplacement soluton at the flud-sold nterface [5] :

4 0 Scence n Chna Ser. G Physcs, Mechancs and Astronomy 004 Vol.47 No u ( ω, k,0) y x 1 k T1 ρ cl1 p L1 = ( k x + pt1) G1( ω, kx, pl1) kt1g( ω, kx, pl ), (7) sch cl1 ρ1 cl p L where ρ = + + L1 4 sch ( kx pt 1) 4kx pt 1pL1 kt1 ρ1 pl p s the Scholte functon. n whch k 1/ L1 = x L1 p ( k k ), L1 = ω / cl1, kl = ω / cl, kt1 = ω ct1 1/ L = ( x L) p k k /., p T1 ( kx = 1/ k T1 ), The transformed solutons of the temperature feld the expressons of G 1, G can be seen n ref. [5]. T 1( ω, k, p), T ( ω, k, p) x x and The normal dsplacement of acoustc feld at flud-sold nterface excted thermoelastcally by a pulsed laser lne source can be obtaned by transformng the uˆ( ω, k x,0) nto the tme-space doman (t, x, 0) as follows: 1 jωt+ jkxx uy(, t x,0) u (,,0). y ω kx e dk d 4π = x ω (8) Numercal calculatng on the transent response of the acoustc feld of ar-sold nterface waves excted by a pulsed laser lne source.1 The analytc soluton of the normal dsplacement for the acoustc feld of the flud-sold nterface waves The poles exst n the complex k x plane for the ntegrand n eq. (8). They are the roots of the Scholte equaton, correspondng to the nterface waves: Scholte wave, leaky Raylegh wave. Accordng to Jordan s Lemma, the path of the ntegraton can be changed nto a contour on the complex k x plane. Therefore, eq. (8), whch expresses the acoustc feld, can be converted nto the summaton of resdues for the poles n the contour and the branch cuts ntegratons by deformng the ntegraton contour along the branch cuts. The resdues of the poles correspond to the nterface modes, and the branch cuts represent the contrbutons of head waves. Thus, by the resdue theorem, the acoustc feld of flud-sold nterface waves can be represented by n u 1 jkxx y ( ω, x,0) =, e R (9) π = 1 where n s the numbers of the resdue n the contour, R s the resdue correspondng to the th pole kx = k x n that contour, whch can be performed as follows: Copyrght by Scence n Chna Press 004

5 The thermoelastc exctaton of ar-sold nterface waves usng the pulsed laser 03 M( kx ) R = π. (10) k k = k sch x x x From the transformed expresson uˆ ( ω, k,0), y x we can obtan 4 ω 1 ρ cl1 pl 1 1 M( k ) x = G1 G, ct1c L1 c c (11) T1 ρ1 cl pl c T1 3 sch ω 1 pl 1 pt 1 ρ c L pl pl 1 = 8 4 p L 1pT kx c c c T1 c pt1 p L1 ρ 1 c T1 pl 1 p, L where ω s angular frequency, c s the velocty of the nterface wave, p L1 = 1 c 1 c, L = 1 1, p 1 1 T 1 =. c c pl 1 c cl Substtutng eqs. (11), (1) and (10) nto eq. (9), we obtan the analytc expresson of acoustc feld dsplacement of flud-sold nterface waves, excted thermoelastcally by a pulsed laser lne source.. The numercal calculaton for the transent dsplacement feld of the ar-sold nterface waves Table 1 s the thermophyscal parameters [9,10] and acoustc parameters for ar and ron meda. Usng the parameters of longtudnal, transverse veloctes and denstes, the roots of Scholte equaton sch = 0 can be solved. For the ar-ron nterface, the veloctes of Scholte and leaky Raylegh waves are m/s and m/s respectvely. Bulk thermal expanson coeffcent β /K 1 Heat capacty c /J kg 1 K 1 T1 Table 1 Calculatng parameters (300K) Thermal conductvty k /W m 1 K 1 Lght absorpton coeffcent α /m 1 Densty ρ /kg m 3 Longtudnal velocty c L /m s 1 (1) Transverse velocty c T /m s 1 Ar Iron The pulse shape of Q swtch laser n tme can be usually represented by the followng functon [11] : whle ψ (x) can be consdered as [5] t t / ( ) e f() t =, τ τ

6 04 Scence n Chna Ser. G Physcs, Mechancs and Astronomy 004 Vol.47 No ψ ( x) =, 1 + ( x/ x ) where τ s the rase tme of the laser pulse, x 0 s the wdth of the lght beam. In our calculaton, τ =10 ns, x 0 = 00 µm. By usng functons f (t), ψ (x) and the parameters n table 1, we can calculate eqs. (11), (1), (10) and (9), then transform the u ( ω, x,0) nto the tme doman by usng the FFT technque, the dsplacement waveform u y (t, x, 0) s then obtaned. 3 The thermoelastc exctng and detectng experment by laser ultrasoncs The expermental system s shown n fg. 1. The system conssts of a Nd:YAG pulsed laser, the SH130 heterodyne laser nterferometer, the movable sample holder, the mcro-control platform for adjustng the focused laser beam, dgtal osclloscope and a mcrocomputer, and so on. y 0 Fg. 1. Schematc dagram of experment system. In order to mprove the reflectvty of the sample surface, enhance the detectve senstvty of the nterferometer, and ncrease the sgnal to nose rato, the metal surface must be polshed before experment. Adjustng the radated energy of the Nd: YAG pulsed laser, let t work n the thermoelastc regon. The pulsed laser beam, whch s focused nto a lne source by the cylndrcal lens L (M 1 and M n fg. 1 are reflectng mrrors), s normal ncdent onto the nterface between the ar and ron, whle the dsturbance caused by the nterface waves propagatng s detected by the nterferometer. The sgnals are acqured and stored by the dgtal osclloscope. Meanwhle, the scattered lght of the excted beam, whch s detected by a photodode, s used as a trggered sgnal of the dgtal osclloscope. The acqured sgnals can also be transported to the mcrocomputer for the further analyss and dsplay. The expermental parameters are: the laser pulse duraton s 10 ns at a wavelength Copyrght by Scence n Chna Press 004

7 The thermoelastc exctaton of ar-sold nterface waves usng the pulsed laser 05 Fg.. Ar-ron nterface waves excted thermoelastcally by a pulsed laser lne source. (a) Numercal result; (b) expermental result.

8 06 Scence n Chna Ser. G Physcs, Mechancs and Astronomy 004 Vol.47 No of 53 nm and the power of the Nd: YAG laser s 7 70 mj adjustable. The length and wdth of the focused lne source are 1 cm and 0. mm respectvely. The precson of the mcro-control platform s ± 1 µm. The thermoelastc exctng and detectng experment for the ar-sold nterface waves are done on the surface of an ron plate, wth the sze 10 mm 10 mm 5 mm. The energy of ncdent pulsed laser s 4 mj. g. (a) s the numercal dsplacement waveform of the ar-ron nterface waves excted by a pulsed laser lne source. The dstance between the lne source and detectng pont s 7 mm. The correspondng expermental result s dsplayed n fg. (b). The numercal and expermental results show that the nterface waves also exst at the ar-sold nterface just lke that at the lqud-sold nterface. But for the ar-sold nterface, the velocty of leaky Raylegh wave s almost the same wth that of the Raylegh wave on the surface of the sold half space, and the velocty of Scholte wave s almost the same wth that of the sound n ar, whle the characterstcs of the two nterface waves, such as the relatve ampltude, the pulse shapes, and so on, are also extremely smlar to those of the lqud-sold nterface waves. 4 Concluson In ths paper, we solved the coupled problem of thermoelastc and heat conducton equatons for the flud-sold nterface. The analytc soluton of dsplacement for nterface waves acoustc feld s presented by applyng the resdue theorem, and the nterface waves transent dsplacement feld excted by a pulsed laser lne source at the ar-sold nterface s calculated numercally by usng the FFT technque. The thermoelastc exctng and detectng experment was carred out at the ar-ron nterface on the laser ultrasonc system. Both the theoretc and the expermental results are n very good agreement. These results ndcate that the nterface waves also exst at the ar-sold nterface just lke that at the lqud-sold nterface. The thermoelastc exctng effect of the pulsed laser can not only excte the lqud-sold nterface waves, but also excte the ar-sold nterface waves effectvely. The theoretc and dentcal expermental results for the ar-sold nterface waves excted thermoelastcally by a pulsed laser are frst presented n ths paper. Acknowledgments Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grant Nos , ) and PRA MX00-. The authors are very grateful to Dr. W-J Xu, Valencennes Unversty n France for hs help wth the experment and numercal calculaton. References 1. Por e, B., Lupp, F., Evanescent plane waves and the Scholte-Stonelely nterface wave, J. Acoustque, 1991, 4(): Überall, H., Surface waves n acoustcs, n Physcal Acoustcs: Prncples and Methods (eds. Mason, W. P., Thurston, R. N.), New York: Academc Press, 1973, X: Nasr, S., Duelos, J., Ledue, M., Scholte wave characterzaton and ts decay for varous materals, J. Acoust. Soc. Am., 1990, 87(): Godnez-Azcuaga, V., Adler, L., Theoretcal smulaton of expermental observatons of surface wave propagaton on a flud-saturated porous materal, Revew of Progress n Quanttatve Nondestrusctve Evaluaton, Copyrght by Scence n Chna Press 004

9 The thermoelastc exctaton of ar-sold nterface waves usng the pulsed laser , 14: Gusev, V., Desmet, C., Laurks, W. et al., Theory of Scholte, Leaky Raylegh, and Lateral wave exctaton va the Laser-nduced thermoelastc effect, J. Acoust. Soc. Am., 1996, 100(3): Desmet, C., Gusev, V., Laurks, W. et al., Laser-nduced thermoelastc exctaton of Scholte waves, Appl. Phys. Lett., 1996, 69(1): Desmet, C., Gusev, V., Laurks, W. et al., All-optcal exctaton and detecton of Leaky Raylegh waves, Optcs Letters, 1997, (): Rostyene, K., Gloreux, C., Gao, W. et al., Laser ultrasonc measurements of acoustc waves generated at sold-lqud nterfaces, Acta Physca Snca, 1999, 8(supplement): s Rohsenow, W. M., Handbook of Heat Transfer Fundamentals, nd ed., New York: McGraw-Hll, 1985, 169, Holman, J. P., Thermodynamcs, 3rd ed., New York: McGraw-Hll, 1980, 59, McDonald, F. A., On the precursor n laser-generated ultrasound waveforms n metals, Appl. Phys. Lett., 1990, 56(3):