Improvements to the Mathematical Model of Acoustic Wave Scattering from Transversely Isotropic Cylinders

Transcription

1 Transation B: Mehanial Engineering Vol. 17, No. 3, pp. 157{166 Sharif University of Tehnology, June 21 Improvements to the Mathematial Model of Aousti Wave Sattering from Transversely Isotropi ylinders Abstrat. S. Sodagar 1 and F. Honarvar 1; This paper onsiders the sattering of an innite plane aousti wave from a long immersed, solid, transversely isotropi ylinder. The mathematial model whih has already been developed for this problem does not work in the ase of a normally inident wave. Modiations to the mathematial model are proposed in order to make it appliable to all inidene angles, inluding. Numerial results are used to demonstrate the orretness of the modied equations. Moreover, using a mathematial disussion, it is shown that at normal inidene, the whole displaement eld is onstrained within the isotropi plane of the ylinder (ylinder ross setion) and only the two elasti onstants haraterizing this plane appear in the remaining equations. A perturbation study on the ve elasti onstants of the transversely isotropi ylinder onrms this result. Keywords: Aousti wave; Sattering; Transversely isotropi; ylinder. INTRODUTION irular omponents suh as ylinders, rods, pipes, and tubes are widely used in oil, gas, petrohemial, transportation and power industries. The nondestrutive evaluation of ylindrial omponents has reeived muh attention in reent years. Among various tehniques used for haraterizing material properties and deteting defets, ultrasoni tehniques are the most widely used. When the immersion ultrasoni tehnique is employed for material haraterization, a theoretial model of aousti sattering (or reetion) from the sample is needed for a quantitative evaluation [1]. Most previous studies are onerned with isotropi ylinders. However, many engineering omponents have anisotropially embedded reinforements or unintended anisotropy produed during manufaturing proesses. Examples are axially ber-reinfored omposite rods manufatured by the extrusion proesses. Resonane Aousti Spetrosopy (RAS) is the study of resonane eets present in reeted aousti ehoes from an elasti target. These resonane eets 1. Faulty of Mehanial Engineering, K. N. University of Tehnology, Tehran, P.O. Box , Iran. *. orresponding author. honarvarkntu.a.ir Reeived 12 Deember 28; reeived in revised form 2 Deember 29; aepted 2 February 21 are aused by the exitation of eigenvibrations of the target by an inident aousti wave. RAS and other aousti sattering tehniques have been used for nondestrutive evaluation of materials, material haraterization and remote lassiation of submerged targets [2-4]. The interest in aousti wave sattering from solid obstales dates bak to the time of Lord Rayleigh [5]. Early studies of wave sattering from solid elasti ylinders, onduted by Faran [6], dealt with normally inident ompression waves inident on a submerged in- nite homogeneous elasti isotropi rod. The more general problem of the sattering of an obliquely inident plane wave from an innite elasti ylinder was studied by Flax et al. [7]. Similar problems for a ylindrial shell were studied by Leon et al. [8] and Veksler [9]. Until a deade ago, all mathematial models developed for aousti wave sattering from elasti targets only dealt with isotropi materials. The rst mathematial model for the sattering of plane aousti waves from an anisotropi ylinder was developed by Honarvar and Sinlair in 1996 [1]. They used a normal mode expansion method for solving the aousti wave sattering problem. The anisotropy symmetry onsidered was hexagonal (transverse isotropy). An alternative formulation, based on the same mathematial method for this problem, was later presented by Ahmad and Rahman [11]. These works were omplemented by a

2 158 S. Sodagar and F. Honarvar number of other papers disussing various aspets of this problem [12-14]. The mathematial models developed in [1,11] have been used by many researhers in studying different problems dealing with the sattering of aousti waves from transversely isotropi ylinders. Following the formulation presented in [1], Qian et al. [15,16] studied the problem of the sattering of P-waves from 1-3 piezo-omposite ylinders. They onsidered the sattering of a longitudinal plane elasti wave inident at arbitrary angles on a transversely isotropi piezoeletri ylinder surrounded by a polymeri matrix medium. Pan et al. [17] studied the aousti eld of a transversely isotropi ylinder generated by a laser line pulse in either the ablation or thermo-elasti regime. In another study, this group onsidered waves generated by a laser point soure in an isotropi ylinder [18] and used it for measurement of the elasti onstants of the ylinder [19]. They also modeled bulk and surfae aousti waves generated in a transversely isotropi ylinder by a laser point soure [2] and used it for stiness tensor measurements [21]. All the above mathematial models use analytial or semi analytial methods for solving aousti wave sattering problems. There are also a number of numerial methods for modeling aousti wave sattering problems inluding nite element [22-24], innite element [22], boundary element, and oupled nite/boundary element methods [25]. The mathematial models presented for the sattering of plane aousti waves from immersed transversely isotropi ylinders in [1,11], although orret, both suer from a deieny; they annot be used for normally inident waves. In this paper, we show how these mathematial models an be modied to overome this deieny. Moreover, using the derived equations, we show that in the ase of a normally inident wave, the ylinder behaves exatly as an isotropi material and the wave only has displaement omponents in the isotropi plane (ross setion) of the ylinder. The normal mode expansion method will be used throughout this paper for solving the aousti wave sattering problem. OVERVIEW OF URRENT FORMULATION OF THE PROBLEM To present the proposed modiations, we rst need to review the mathematial model developed for the sattering of a plane aousti wave from an immersed innite transversely isotropi solid elasti ylinder following [1]. In formulating the problem, an innite plane aousti wave of irular frequeny,!, inident at an angle,, on an innite submerged transversely isotropi ylinder, is onsidered (Figure 1). A ylindrial oordinate system (r,, z) is hosen suh that Figure 1. Innite plane aousti wave obliquely inident on a submerged innite transversely isotropi ylinder. the z diretion oinides with the axis of the ylinder. The inident wave pressure, p i, at an arbitrary point, M(r; ; z), is: p i p " n i n J n (k?r) os(n)e i(k zz!t) ; (1) where k z k sin, k? k os, and k!, is the ompression wave veloity in the liquid medium surrounding the ylinder, " n is the Neumann fator (" 1 and " n 2 for n > ), p is the inident pressure wave amplitude, and J n are the Bessel funtions of the rst kind of order n. The sattered wave pressure, p s, at an arbitrary point, M, is symmetrial about and of the following form: p s p " n i n A n H n (1) (k?r) os(n)e i(kzz!t) ; (2) where H n (1) are the Hankel funtions of the rst kind of order n, and A n are unknown sattering oeients. A transversely isotropi material is haraterized by ve independent elasti onstants, 11, 12, 13, 33, and 44. The general Hooke's law for a transversely isotropi material is: 8 >< >: rr zz z rz r 9 8 > >< >; >: 8 >< >: ( )2 " rr " " zz " z " rz " r 9 > >; ; 9 > >; (3) where ij are stress omponents and " ij are strain omponents. In a ylindrial oordinate system, the

3 Wave Sattering from Transversely Isotropi ylinders 159 equations of motion of a ontinuum in the absene of body fores an be written as: 2 U z 44 zr + 2 U r 2 U r z r U r r r 3 U 2r U r 2r r 2 12 U r 2 U z + 13 zr 2 U r 44 z U z + 1 r + 1 U 2 1 r r 2 2 U r 2 U r r 44 1 r 2 2 U z U r zr U z z 2 U r r U 2r r 1 U r r + U r t 2 ; (4) U z 1 U r r 2 U r 2 1 2r 2 U r + 1 2r + 1 2r 2 U r r U r U 2 r U r 2r 2 1 U 2 r U z r z 2 U t 2 ; (5) + 2 U z r r 2 U z + 1 U z r r + 1 U r r z Ur + 13 z r + 1 U r + U r r 2 U z t 2 ; (6) where is the ylinder density and U r, U, U z are displaement omponents in the r, and z diretions, respetively. The potential funtion method is used for solving this problem and the displaement vetor is written in terms of three salar potential funtions,, and, as follows: ~u r + r (^e z ) + ar r ( ^e z ); (7) where a is the radius of the ylinder; a onstant with dimensions of length. The substitution of Equation 7 into Equations 4-6 gives: r 2 2 z 2 11 r 2 + ( ) 2 z 2 2 t 2 + a ( ) r 2 z + ( ) 2 2 z 2 t 2 ; (8) ( )r 2 + ( ) 2 z z t 2 + a z 2 r 2 44 r 2 + ( ) 2 z 2 2 t 2 ; (9) r (11 12 ) z ( ) 2 r 2 2 z 2 2 t 2 : (1) Equations 8 and 9 represent the L (longitudinal) and SV (vertially polarized shear) waves. The longitudinal wave represented by, and the SV wave represented by are oupled. Aording to Equation 1, the SH (horizontally polarized shear) wave, represented by, is deoupled from the other two wave types. To solve Equations 8-1 for, and, the normal mode expansion method is used and solutions of the following forms are assumed: B n J n (sr) os ne i(kzz!t) ; (11) n J n (sr) os ne i(kzz!t) ; (12) D n J n (sr) sin ne i(kzz!t) : (13) Substituting Equations 11 and 12 into 8 and 9 gives: a11 a 12 a 21 where: a 22 Bn n ; (14) a 11 [ 11 s 2 ( )k 2 z +! 2 ]s 2 ; a 12 aik z [ ( )s 2 44 k 2 z +! 2 ]s 2 ; a 21 ik z [ ( )s 2 33 k 2 z +! 2 ]; a 22 as 2 [ ( 44 s 2 ( )k 2 z +! 2 ]: (15) For a nontrivial solution, the oeient determinant of Equation 14 must vanish. This yields the following harateristi equation: s 4 s 2 + ; (16)

4 16 S. Sodagar and F. Honarvar where: ( ) 2 k 2 z + 11 (! 2 33 k 2 z) + 44 (! 2 44 k 2 z); (! 2 44 k 2 z)(! 2 33 k 2 z): (17) Solving Equation 16 yields two roots, s 1 and s 2, as follows: s 2 1 p ; (18) s p : (19) Therefore, the potential funtions, and, should be of the form: [B n J n (s 1 r) + q 2 n J n (s 2 r)] os ne i(kzz!t) ; (2) where: q 1 [q 1 B n J n (s 1 r) + n J n (s 2 r)] os ne i(k zz!t) ; (21) 11 s 2 1 ( )k 2 z +! 2 aik z [ ( )s k 2 z +! 2 ] ; (22) q 2 aik z[ ( )s k 2 z +! 2 ] 11 s 2 2 ( )k 2 z +! 2 : (23) Moreover, for the salar potential funtion,, orresponding to the SH wave: s 2 3 2(! 2 44 k 2 z) ; (24) and therefore: D n J n (s 3 r) sin ne i(kzz!t) : (25) The boundary onditions at r a are: 1 r (p i + p s ) 2 U r t 2 ; rr (p i + p s ); r ; rz ; (26) where is the density of the uid surrounding the ylinder. Inserting the potential funtions from Equations 2, 21 and 25 in Equation 26, a system of four linear algebrai equations is obtained for eah value of n: B a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 32 a 33 a 34 a 42 a 43 a 44 1 A B A n B n n D n 1 A B b 1 b 2 1 A : (27) Elements a ij and b ij of the matries appearing in Equation 27 are given in the Appendix. Equation 27 an be solved for A n for any desired frequeny and position angle,. The usual approah is to solve the problem in the far eld (r >> a) at a spei angle,, for a range of frequenies. The resulting far-eld amplitude spetrum, whih is alled the `form funtion', is obtained from the following equation [26]: jf1j 2r a 12 ps p e ikr : (28) The above formulation annot be used if the wave inidene angle is zero. In the following setion, we present the required modiations in order to avoid this deieny. PROPOSED MODIFIATIONS TO THE FORMULATION Although the above mathematial formulation is derived for any arbitrary angle of inidene, it turns out to be singular when the wave is normally inident ( ) on the ylinder. In this ase, the axial wave vetor, k z, equals zero (k z k sin ) and, onsequently, Equation 22 tends to innity. We suggest the following modiations to Equations 22 and 23 in order to avoid this situation and make the formulation appliable to all values of. For having a nontrivial solution to Equation 14, the determinant of the oeient matrix should vanish, i.e: a 11 a 12 a 21 a 22 : (29) From Equations 15, 22 and 23, we note that: q 1 11 s 2 1 ( )k 2 z +! 2 aik z [ ( )s 2 44 k 2 z +! 2 ] a 11 a 12 ; q 1 aik z[ ( )s k 2 z +! 2 ] 11 s 2 2 ( )k 2 z +! 2 (3) a 12 a 11 : (31)

5 Wave Sattering from Transversely Isotropi ylinders 161 The terms on the right hand side of Equations 3 and 31 an be replaed with their equivalent values based on Equation 29. This replaement gives: q 1 a 11 a 12 a 21 a 22 q 2 q 2 ik z [ ( )s k 2 z +! 2 ] as 2 1 [ 44s 2 1 ( )k 2 z +! 2 ; (32) aik z[ ( )s k 2 z +! 2 ] 11 s 2 2 ( )k 2 z +! 2 : (33) The above transformation does not aet the nal results and makes the equations appliable to the ase of normally inident waves. The transformation has moved the k z term from the denominator to the numerator of Equation 32. The substitution of Equations 32 and 33 into Equations 2 and 21 gives: [B n J n (s 1 r)+q2 n J n (s 2 r)] os ne i(kzz!t) ; [q1b n J n (s 1 r)+ n J n (s 2 r)] os ne i(k zz!t) : (34) (35) The same modiation an be applied to the mathematial model presented in [11] and would lead to similar results. THE ASE OF A NORMALLY INIDENT WAVE From the physis of the problem it is known that for a normally inident wave, the surfae waves travel along the irumferene of the ylinder and, therefore, do not depend on the axial properties of the ylinder. However, onsidering that the elasti onstants, 13 and 44, relate the elasti properties of the isotropi plane (ylinder ross setion) to those in the axial diretion, it annot readily be reognized whether or not, for normally inident waves, the sattering eld depends on these onstants. In the following, we try to answer this question. For the ase of a normally inident wave k z ( ) and Equations 32 and 33 redue to: q 1 q 2 : (36) onsequently, the potential funtions will be of the following forms: B n J n (s 1 r) os ne i!t ; (37) n J n (s 2 r) os ne i!t ; (38) D n J n (s 3 r) sin ne i!t ; (39) where: s 2 1! 2 11 s 2 2! 2 44 s 2 3! 2 ( 11 ) ; (4)! 2 ( 44 ) ; (41)! 2 [( )2] : (42) Although Equations are similar to the potential funtions of an isotropi ylinder, as given in [6], in the new formulation, these potential funtions are funtions of three independent elasti onstants, 11, 12 and 44, while, for an isotropi material, there are only two independent elasti onstants. Equations 4-42 indiate that, in the ase of a normally inident wave, is a funtion of 11, is a funtion of 44, and is a funtion of both 11 and 12. Moreover, orresponds to the longitudinal wave, and and orrespond to shear waves. The substitution of k z into the oeients matrix of Equation 27 gives (see Appendix): a 13 a 23 a 33 a 42 a 44 ; (43) and therefore: B a 11 a 12 a 14 a 21 a 22 a 24 a 32 a 34 a 43 This requires that: 1 A B A n B n n D n 1 A B b 1 b 2 1 A : (44) n : (45) Aording to Equation 38, if n is equal to zero, then,, whih means that when the wave is normally inident on the ylinder, only the longitudinal wave and one type of shear wave are present. Moreover, these two waves depend only on the elasti onstants haraterizing the ross-setional plane (isotropi plane) of the ylinder, i.e. 11 and 12. By taking, the

6 162 S. Sodagar and F. Honarvar displaement deomposition introdued in Equation 7 will redue to the familiar Helmholtz deomposition for potential funtions, and, whih is ommonly used for solving problems dealing with isotropi materials. We observe that in the ase of a normally inident wave, s 1 and s 3 are funtions of only two elasti onstants ( 11 and 12 ). Therefore, it an be onluded that, in this ase, the ylinder is indeed behaving as an isotropi material. ylinder with transversely isotropi elasti properties is alulated at inidene angles of and 5. The orresponding funtions are shown in Figures 7 and 8, respetively. The physial properties of obalt are given in Table 1. In these numerial alulations, and in those whih follow, the number of normal modes, N, used in evaluating the series is N ka max + 5 where ka max is the maximum value of the NUMERIAL RESULTS To verify the modied mathematial model, form funtions of both isotropi and transversely isotropi ylinders are alulated at dierent inidene angles. First, the sattered eld from an immersed isotropi aluminum ylinder is alulated at, and 3 degrees. The physial properties of aluminum are given in Table 1. The orresponding form funtions are shown in Figures 2 and 3 for the frequeny range of < ka < 2. These results are idential to earlier results, whih are reported in [27] and shown in Figure 4. In Figures 5 and 6, the sattered eld of the immersed isotropi aluminum ylinder alulated at higher inident angles of 2 and 5 is shown. Next, the sattered pressure eld from a obalt Figure 3. Form funtion of an aluminum ylinder at 3. Figure 2. Form funtion of an aluminum ylinder at. Figure 4. Form funtion of an aluminum ylinder at and 3 [27]. Table 1. Material properties. Material Type Stiness 1 11 (N/m 2 ) Density (kg/m 3 ) Aluminum Isotropi obalt Trans. iso

7 Wave Sattering from Transversely Isotropi ylinders 163 interior displaement eld of the isotropi aluminum ylinder are plotted in Figure 9. In this gure, the displaements shown on the top are alulated by the modied model at and those on the bottom are alulated based on the formulation of Ref. [1], at :1. It is observed that although the displaement elds in the r and diretions are almost idential, the two displaement elds in the z diretion are ompletely dierent. For the ase of the normally inident wave, the displaement omponent in the z diretion are zero over the entire ross setion of the ylinder, while, at :1 degrees, the displaement eld along the ylinder axis is not zero. The nullity of the displaement eld in the axial diretion onrms our earlier disussion regarding the independeny of the Figure 5. Form funtion of an aluminum ylinder at 2. Figure 7. Form funtion of a obalt ylinder at. Figure 6. Form funtion of an aluminum ylinder at 5. normalized frequeny, ka, on the graphs. This number of normal modes guarantees the orretness of the form funtion for the onsidered range of frequenies ([9], p. 245). Also, the runtime of the omputations for the modied model is less than one seond, whih is similar to that of the original model disussed in [1]. As mentioned earlier, the mathematial models given in [1,11] annot be used in the ase of normally inident waves. If using these models, one would have to hoose a very small inidene angle that would approximate normal inidene. However, no matter how small this angle is, it would produe extra resonanes in the ylinder. To show the eet of small inidene angles, the three omponents of the Figure 8. Form funtion of a obalt ylinder at 5.

8 164 S. Sodagar and F. Honarvar a transversely isotropi obalt ylinder are perturbed, and the eet of this perturbation is studied on the form funtion. Figures 1a to 1e show the eet of perturbing eah of the ve elasti onstants of a obalt ylinder on its form funtion at for the frequeny range < ka < 1. These gures show that only 11 and 12 aet the resonane frequenies at normal inidene and the other three elasti onstants have no eet. This omplies with our previous disussion, where it was shown that 13 and 44 have no eet on the sattered eld when. ONLUSIONS Figure 9. Displaement elds of an aluminum ylinder in r, and z diretions. Top: alulated based on the modied formulation suggested in this paper at. Bottom: alulated based on the formulation of [1] at :1. surfae waves from the axial properties of the ylinder when the wave angle is normal to the ylinder axis. To demonstrate that the elasti onstants, 13 and 44, do not aet the sattered eld in the ase of a normally inident wave, all ve elasti onstants of The existing mathematial model for the sattering of a plane aousti wave from an immersed, innite solid, transversely isotropi ylinder fails in the ase of normally inident waves. In this paper, modiations were suggested to make it work in the ase of normally inident waves. It was also shown that in the ase of a normally inident wave, the orresponding equations redue to those of the isotropi ylinder and only elasti onstants haraterizing the isotropi plane of the material aet the resonane frequenies. The modi- ed model was numerially veried for both isotropi Figure 1. Eet of the perturbation of the ve elasti onstants of a obalt ylinder on resonane frequenies when. a) 1% inrease in 11; b) 1% inrease in 12, ) 1% inrease in 33, d) 1% inrease in 13, e) 1% inrease in 44.

9 Wave Sattering from Transversely Isotropi ylinders 165 and transversely isotropi ylinders. A perturbation study showed that, in the ase of a normally inident wave, the only elasti onstants aeting the resonane frequenies are those whih haraterize the isotropi plane of the transversely isotropi ylinder. This modied model an be employed in an examination of transversely isotropi ylinders by Resonane Aousti Spetrosopy (RAS). REFERENES 1. Kim, J.Y. and Ih, J.G. \Sattering of plane aousti waves by a transversely isotropi ylindrial shell - Appliation to material haraterization", Applied Aoustis, 64, pp (23). 2. Fan, Y., Tysoe, B., Sim, J., Mirkhani, K., Sinlair, A.N., Honarvar, F., Sildva, H., Szeket, A. and Hardwik, R. \Nondestrutive evaluation of explosively welded lad rods by resonane aousti spetrosopy", Ultrasonis, 41, pp (23). 3. Ogi, H., Nakamura, N., Hirao, M. and Ledbetter, H. \Determination of elasti, anelasti, and piezoeletri oeients of piezoeletri materials from a single speimen by aousti resonane spetrosopy", Ultrasonis, 42, pp (24). 4. Datta, S.K. and Shah, A.H., Elasti Waves in omposite Media and Strutures with Appliations to Ultrasoni Nondestrutive Evaluation, R Press (29). 5. Strutt, J.W., The Theory of Sound, I, (Third Baron Rayleigh) 2nd Ed., London, Mamillan & o., Reprinted by Dover, New York (1945). 6. Faran Jr., J.J. \Sound sattering by solid ylinders and spheres", J. Aoust. So. Am., 23, pp (1951). 7. Flax, L., Varadan, V.K. and Varadan, V.V.\ Sattering of an obliquely inident aousti wave by an innite ylinder", J. Aoust. So. Am., 68, pp (198). 8. Leon, F., Leroq, F., Deultot, D. and Maze, G. \Sattering of an obliquely inident aousti wave by an innite hollow ylindrial shell", J. Aoust. So. Am., 91, pp (1992). 9. Veksler, N.D., Resonane Aousti Spetrosopy, Springer (1993). 1. Honarvar, F. and Sinlair, A.N. \Aousti wave sattering from transversely isotropi ylinders", J. Aoust. So. Am., 1, pp (1996). 11. Ahmad, F. and Rahman, A. \Aousti sattering by transversely isotropi ylinders", Int. J. Eng. Si., 38, pp (2). 12. Fan, Y., Honarvar, F., Sinlair, A.N., Jaari, M. \irumferential resonane modes of solid elasti ylinders exited by obliquely inident aousti wave", J. Aoust. So. Am., 113, pp (23). 13. Sodagar, S., Honarvar, F. and Sinlair, A.N. \Aousti wave sattering from immersed transversely isotropi ylinders overed by isotropi ladding", Proeeding of Thirteenth International ongress on Sound and Vibration, Austria (26). 14. Sodagar, S., Honarvar, F. and Sinlair, A.N. \Resonane modes of immersed type I and type II transversely isotropi lad rods", Proeeding of 19th International ongress on Aoustis, Madrid (27). 15. Qian, Z., Jin, F., Wang, Z. and Kishimoto, K. \Investigation of sattering of elasti waves by ylinders in 1-3 piezoomposites", Ultrasonis, 43, pp (25). 16. Jin, F., Kishimoto, K., Qian, Z. and Wang, Z. \Study on the sattering of elasti waves for the nondestrutive evaluation of piezoeletri omposites", Key Engineering Materials, 1139, pp (26). 17. Pan, Y., Rossignol,. and Audoin, B. \Aousti waves generated by a laser line pulse in a transversely isotropi ylinder", Appl. Phys. Lett., 82, pp (23). 18. Pan, Y., Rossignol,. and Audoin, B. \Aousti waves generated by a laser point soure in an isotropi ylinder", J. Aoust. So. Am., 116, pp (24). 19. Pan, Y., Rossignol,. and Audoin, B. \Aousti waves generated by a laser line pulse in ylinder; appliation to the elasti onstants measurement", J. Aoust. So. Am., 115, pp (25). 2. Pan, Y., Petron, M., Audoin, B., and Rossignol,., \Aousti waves generated by laser point pulse in a transversely isotropi ylinder", J. Aoust. So. Am., 119, pp (26). 21. Petron, M., Audoin, B., Pan, Y. and Rossignol,. \Bulk onial and surfae helial aousti waves in transversely isotropi ylinders; appliation to the stiness tensor measurement", J. Aoust. So. Am., 119, pp (26). 22. Bettess, P. and Zienkiewiz, O.. \Diration and refration of surfae waves using nite and innite elements", Int. J. Num. Methods Eng., 11(8), pp (1977). 23. Eaton, J.A. and Regan, B.A. \Appliation of the - nite element method to aousti sattering problems", AIAA J., 34(1), pp (1996). 24. Ihlenburg, F., Finite Element Analysis of Aousti Sattering, Springer (1998). 25. Dubus, B., Lavie, A., Deultot, D. and Maze, G. \oupled nite element/boundary element method for the analysis of the aousti sattering from elasti strutures", ASME Des. Eng. Div. Publ. DE., 84, USA (1995). 26. Flax, L., Gaunaurd, G.. and Uberall, H. \Theory of resonane sattering", in Physial Aoustis, W.P. Mason and R.N. Thurston, Eds., Aademi, New York, 15, p. 191 (1981). 27. Li, T. and Ueda, M. \Sound sattering of a plane wave obliquely inident on a ylinder", J. Aoust. So. Am., 86, pp (1989).

10 166 S. Sodagar and F. Honarvar APPENDIX Elements of the matries given in Equation 29 are as follows: a 11 i p i n " n hnh n (1) (k?a) (k?a)h n+1 (1) (k?a)!! 2 ; (A1) a 12 (1 + iq 1 ak z )[nj n (s 1 a) s 1 aj n+1 (s 1 a)]; (A2) a 13 (q 2 + iak z )[nj n (s 2 a) s 2 aj n+1 (s 2 a)]; (A3) a 14 nj n (s 3 a); (A4) a 21 p i n " n a 2 H (1) n (k?a); (A5) a 22 [ 11 + i( )q 1 ak z ] b(n 2 n s 2 1a 2 )J n (s 1 a) + s 1 aj n+1 (s 1 a) + [ 12 + ( )iq 1 ak z ] [nj n (s 1 a) s 1 aj n+1 (s 1 a)] + [ 13 a 2 k 2 z 12 n 2 + ( )in 2 q 1 ak z ]J n (s 1 a); a 23 [ 11 q 2 + i( )ak z ] b(n 2 n s 2 2a 2 )J n (s 2 a) + s 2 aj n+1 (s 2 a)] + [ 12 q 2 + ( )iak z ] [nj n (s 2 a) s 2 aj n+1 (s 2 a)] [( 13 a 2 k 2 z 12 n 2 )q 2 (A6) + ( )in 2 ak z ]J n (s 2 a); (A7) a 24 ( )n[(n 1)J n (s 3 a) s 3 aj n+1 (s 3 a)]; (A8) a 31 ; a 32 2n(1+iq 1 ak z )[(1 a 33 2n(q 2 +iak z )[(1 (A9) n)j n (s 1 a)+s 1 aj n+1 (s 1 a)]; (A1) n)j n (s 2 a)+s 2 aj n+1 (s 2 a)]; (A11) a 34 bs 2 3a 2 2n(n 1)J n (s 3 a) 2s 3 aj n+1 (s 3 a); (A12) a 41 ; a 42 [q 1 (s 2 1a 2 a 2 k 2 z) + 2iak z ] (A13) [nj n (s 1 a) s 1 aj n+1 (s 1 a)]; (A14) a 43 (s 2 2a 2 a 2 k 2 z + 2iak z q 2 ) [nj n (s 2 a) s 2 aj n+1 (s 2 a)]; (A15) a 44 inak z J n (s 3 a); (A16) b 1 p i n " n [nj n (k?a) (k?a)j n+1 (k?a)]!! 2 ; (A17) b 2 p i n " n a 2 J n (k?a): (A18) BIOGRAPHIES Farhang Honarvar is an assoiate professor of Mehanial Engineering at K. N. Toosi University of Tehnology (Tehran, Iran) and an adjunt assoiate professor of Mehanial Engineering at the University of Toronto (Toronto, anada). He reeived his PhD from University of Toronto (1997), MAS from University of Waterloo (1993), and BS from University of Tehran (1989), all in mehanial engineering. Dr. Honarvar is the diretor of the Nondestrutive Evaluation Laboratory (NDE Lab) at K. N. Toosi University of Tehnology. His sienti interests inlude advaned nondestrutive testing tehniques, theoretial and experimental studies of aousti and elasti waves, signal proessing of ultrasoni waves, and modeling of measurement systems. Dr. Honarvar has several inventions and patents in the area of ultrasoni testing of materials. He has published more than one hundred and fty papers in national and international journals and onferenes. Sina Sodagar reeived a BS degree from hamran University, Ahvaz, Iran, in 2 and a MS degree in mehanial engineering from K. N. Toosi University of Tehnology, Tehran, Iran, in 23, where he is urrently pursuing his PhD degree in mehanial engineering. From 28-29, he was with the Ultrasoni Nondestrutive Laboratory (UNDEL) at the University of Toronto in anada, working on Resonane Aousti Sattering as a Guest Researher. He has been with the Petroleum University of Tehnology in Abadan, Iran, as a Guest Leturer sine 29. His researh interests inlude ultrasoni testing of materials, biomedial ultrasonis, miroeletromehanial ultrasoni transduers and ultrasoni wave propagation.