Scientia Iranica A (2016) 23(6), 2469{2477

Sharif University of Technology Scientia Iranica Transactions A: Civil Engineering www.scientiairanica.com

Dispersion of surface waves in a transversely isotropic half-space underlying a layer

A. Bagheria, A. Khojastehb, M. Rahimiana; and R. Attarnejada a. School of Civil Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran. b. School of Engineering Science, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran.

Received 23 November 2014; received in revised form 23 August 2015; accepted 28 November 2015

KEYWORDS Abstract. Surface waves dispersion is studied in a two-layer half-space consisting of a nite liquid layer overlying a transversely isotropic half-space. A couple of complete Surface wave; potential functions are utilized to uncouple the equation of motion of the transversely Dispersion; isotropic solid along with a displacement potential for the liquid. The frequency equation Liquid; and dispersion curves are developed. Several solid materials are considered for Transversely isotropic; the bed, and both phase and group velocity curves are calculated. Higher modes are also Rayleigh wave; discussed and the respective curves are plotted. Various special cases are considered by Stoneley wave; letting the liquid layer depth take zero value, to form a solid half-space, or very large values Scholte wave; to form a full-space bi-material. Moreover, an isotropic bed material can be obtained as a Frequency equation; special case by appropriately setting the respective elastic constants. Reduced frequency . equations and numerical results are derived for each case to con rm the results with existing ones. © 2016 Sharif University of Technology. All rights reserved.

1. Introduction isotropy of the geologic materials [4]. This anisotropy can considerably a ect the dispersive properties of Propagation of elastic surface waves in various media surface waves. Studying wave propagation in trans- has been of importance and interest in di erent elds of versely isotropic is the research subject of many science such as seismology, earthquake engineering, ma- investigators due to its importance in di erent elds of terial engineering, geophysical studies, etc. The media science and technology. Stoneley [5] was the rst to consisting of a liquid layer lying over an isotropic solid point that transverse isotropy signi cantly in uences half-space has been studied by interested in the wave propagation in a medium in comparison with elds such as ocean engineering, etc. [1-3]. isotropic one. Later, elastodynamics of transversely However, in a wide range of cases, there are isotropic media was subsequently studied by Synge [6], observations revealing the existence of anisotropy in Buchwald [7], and Payton [8]. Earth's crust. It is believed that the fact of following Rahimian et al. [9] presented an ecient ana- a preferred pattern instead of random orientations by lytical formulation to obtain the response of a three- materials depositing in the water and also existence of dimensional transversely isotropic half-space to a time- vertically aligned microcracks result in the transverse harmonic surface loading. Khojasteh et al. [10] extended the previous work to present explicit ex- pressions for the three-dimensional dynamic Green's *. Corresponding author. Tel.: +98 21 61112256; functions in a transversely isotropic half-space and Fax: +98 21 88078263 E-mail addresses: [email protected] (A. Bagheri); investigated the wave propagation. They continued the [email protected] (A. Khojasteh); [email protected] (M. work for a two-layer half-space [11] as well. Also, Kho- Rahimian); [email protected] (R. Attarnejad) jasteh et al. subsequently studied wave propagation in 2470 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477

  @2u 1 @u u 1 @2u 1 @u transversely isotropic bi-materials, tri-materials, and a c r + r r +c r 2c  multilayered half-space [12-14]. 11 @r2 r @r r2 66 r2 @2 11 r2 @ Surface waves dispersion in a model of a liquid   layer overlying a transversely isotropic half-space has @2u (c + c ) 1 @2u 1 @u + c r + 11 12 r +  been investigated in some papers. Abubakar and 44 @z2 2 r @r@ r2 @ Hudson [15] studied the dispersive properties of sur- face waves in such a system in the two-dimensional @2u + (c + c ) z +  !2u = 0; case. Sharma et al. [16,17] obtained the frequency 13 44 @r@z s r equation of a three-layer model consisting of a liquid   2 2 2 layer lying over poroelastic and anisotropic layers. @ u 1 @u u 1 @ u @ u c66 + + c11 + c44 Sharma [18] also tackled the dispersion of Stoneley @r2 r @r r2 r2 @2 @z2 waves in an oceanic crust model. In addition to   2 Kumar and Miglani [19], Kumar and Kumar [20] 1 @ur (c11 + c12) 1 @ ur 1 @ur + 2c11 + solved the problem for the proposed oceanic crust r2 @ 2 r @r@ r2 @ models. In this paper, the dispersive characteristics 2 of a three-dimensional earth crust model, consisting 1 @ uz 2 + (c13 + c44) + s! u = 0; of a homogeneous transversely isotropic elastic me- r @@z dia overlaid by a homogeneous compressible liquid   @2u 1 @u 1 @2u @2u layer, are investigated utilizing the potential method c z + z + z + c z 44 2 2 2 33 2 presented by Rahimian et al. [9] and extended by @r r @r r @ @z   Khojasteh et al. [10]. The velocity dispersion curves of 2 2 @ ur 1 @ur 1 @ u surface waves are plotted. Arrival times of the waves + (c13 + c44)  + + are distinctly dependent on the elastic properties of @r@z r @z r @@z the bed. Analyzing this phenomenon to realize the 2 earth crust's characteristics by reverse calculation is + s! uz = 0; (1) of signi cant importance in marine seismology and where the time factor exp(i!t) is suppressed. In order underground geophysical explorations. The formula- to uncouple these equations, a couple of potential func- tion can also be utilized to derive Green's functions of tions introduced by Eskandari-Ghadi [21] are applied. and displacement elds for various load types These two potential functions, F and , are related to in such a system which is applicable to earthquake displacement components u , u , and u as: engineering. r  z @2F (r; ; z) 1 @ (r; ; z) ur(r; ; z) = 3 ; 2. Statement of the problem and governing @r@z r @ equations 1 @2F (r; ; z) @ (r; ; z) u(r; ; z) = 3 + ; The geometry of the system and the coordinate axes r @@z @r   selected for the problem are shown in Figure 1. For @2  !2 the homogeneous transversely isotropic half-space, the u (r; ; z)= (1+ )r2 + 2 + s F (r; ; z); z 1 r @z2 c time-harmonic equations of motion are expressed as [9]: 66 (2) where: @2 1 @ 1 @2 r2 = + + ; (3) r @r2 r @r r2 @2 and also:

c12 + c66 c44 c13 + c44 1 = ; 2 = ; 3 = : (4) c66 c66 c66 Rewriting the equation of motion (Eq. (1)) in terms of the potential functions yields to the two separate partial di erential equations as:   @2 r2r2 + !2 F = 0; 1 2 @z2

2 Figure 1. Geometry of the system. r0 = 0; (5) A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2471

  d2 de ning: r~ 2 r~ 2 + !2 F~m(z) = 0; 1m 2m dz2 m 1 @2 1  !2 r2 = r2 + + s ; i = 0; 1; 2; (6) i r 2 2 s @z i c66 ~ 2 m i r0m ~m = 0; (13) and: in which:  = 1 0  !2 1 d2 r~ 2 = s 2 + ; i = 0; 1; 2: (14) c c im 2 2 44 11 ic66 s dz 1 = 2 = ; 2 = 1 + 1 = ; i c66 c66    The general solutions for equations in Eq. (13) are: delta 1 1 1 c = + 1 + 33 ; (7) 2 2 F~m(; z) =A ()e1z + B ()e1z + C ()e2z s c44s2 c11s1 c11 c44 m m m m where: 2z + Dm()e ; 1 s = p : (8) 0 m 3z 3z 2 ~m(; z) = Em()e + Fm()e ; (15)

Note that s1 and s2 are not zero or pure imaginary introducing: numbers with respect to the positive de niteness of the r strain energy function [22]: 1p 2 4 2  1 = a + b + c + d + e; 4 2 2 2 c33c44s + c13 + 2c13c44 c11c33 s + c11c44 = 0: (9) r 1p  = a2 + b c4 + d2 + e; The two real roots of this equation can be easily 2 2 obtained since no unknowns of odd order exist in s Eq. (9). Hence s1 and s2 are obtained by Eq. (10) as  !2 2 s shown in Box I. With respect to the angular direction, 3 = s0  : (16) c66 Fourier series expansion is now applied as follows: X1 In the above terms: im [F (r; ; z); (r; ; z)] = [Fm(r; z); m(r; z)] e : 1 2 2 1 (11) a = (s + s ); 2 1 2 Also, with respect to the radial direction, mth order   1 2 1 1 Hankel transform may be expressed as: b = s! + ; h i 2 c33 c44 ~m m F (; z); ~ (; z) 2 2 2 c = (s2 s1) ; Z 1   = [F (r; z); (r; z)] rJ (r)dr; 1 1 m d = 2 !2 + (s2 + s2) 0 s 1 2 c33 c44 [F (r; z); (r; z)]   c11 1 1 Z 2 + ; 1 h i c33 c11 c44 ~m m = F (; z); ~ (; z) Jm(r)d: (12) 0  2 2 4 1 1 Applying Hankel transform and Fourier series expan- e = s! : (17) c33 c44 sion, one may arrive at the two following ordinary di erential equations for transformed components of The unknown functions Am, Bm, Cm, Dm, Em, and and F and as [9]: Fm will be determined from the boundary conditions.

s p 2 2 2 2 (c13 + c13c44 c11c33)  (c13 + c13c44 c11c33) 4c33c11c44 s1; s2 = (10) 2c33c44

Box I 2472 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477

 ,  , and  are multi-valued functions and are made X1 1 2 3 '(r; ; z) = ' (r; z)eim; (21) single-valued by specifying the branch cuts emanating m p 1 from the branch points  = ! s=c11,  = p p1 2 Z !  =c , and  = !  =c on the complex - 1 s 44 3 s 66 m '~ (; z) = '(r; z)rJm(r)dr; (22) plane, such that the real parts of 1, 2, and 3 remain 0 non-negative for all values of  [10]. The positive sign of these branch points denotes the wave numbers of where the inverse transform can be used to return to the body waves, P, SV, and SH, respectively. Unlike the real domain as: an isotropic media, transverse isotropy of the medium Z 1 m causes di erent for SV and SH waves in this '(r; z) = '~ (; z)Jm(r)d: (23) kind of solids [5]. The Fourier components of the 0 transformed stress and displacement can be expressed The same is followed for stress and displacement com- as [10]: ponents. The respective ordinary di erential equation m   of motion in terms of' ~m can be obtained as: 2 2 m d s! 2   u~ = +  (1 + ) F~m; 2 2 zm 2 2 1 m  ! d dz c66 l 2 + '~m = 0: (24) K dz2 m dF~m m+1 m+1 m m A general solution to this equation is: u~r + iu~ = 3 i ~m; m m dz '~m = S ()e4z + R ()e4z; (25) ~m m m m m1 m1 dFm m u~r iu~ = 3 i ~m; m m dz where:  s d2 !2 m+1 m+1 2 2 ~zr +i~ = c44 ( 3 2) +  (1 + 1) 4 =  : (26) m zm dz2 2 cl  p  !2 d ~m s ~m m Here, cl = K=l is the velocity of the compressional Fm c44i ; c66 dz wave in the liquid, and Sm and Rm will be calculated  from the boundary conditions, where Rm is associated d2 with the amplitude of the transmitting wave from ~m1i~m1 = c  ( ) + 2(1 + ) zrm z 44 3 2 2 1 m dz the bottom boundary, and Sm is associated with the  re ecting wave amplitude from the free surface of the 2 m s! ~m d ~m liquid. Fm c44i : (18) c66 dz Fourier factors of stress and displacement compo- nent in the transformed mode are written as: For the compressible liquid, the governing equation of ~ 2 m time-harmonic motion in cylindrical coordinate system ~zz = P = l! '~m; is [1]:   m + 1 m 1 u~m =  '~m1 '~m+1 ; K rm 2m m 2m m r2'(r; ; z) = !2'; (19) l m m @'~m u~z = ; where ', K, and l are the displacement potential, m @z bulk modulus, and density of the liquid, respectively. i  The displacement components and can now be u~m = '~m1 +' ~m+1 : (27) expressed in terms of ' as: m 2 m m

@' @' 1 @' 3. Boundary conditions and frequency u = ; u = ; u = ; r @r z @z  r @ equation

2 2 P = Kr ' = l! ': (20) The boundary conditions for the problem under con- sideration are given below. The expressions provided Similar to the solid substratum, the procedure of in Eqs. (18) along with Eqs. (27) are utilized. The Fourier series expansion is followed with respect to coecients Am, Cm, and Em are omitted according to the angular direction, and applying mth-order Hankel the radiation condition. transform is followed with respect to the radial direc- The free surface condition requires the vanishing tion: of the liquid pressure, P : A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2473

+ zz(z = 0) = 0: (28) zz(z = h ) = zz(z = h ); (36)     2 This imposes: 2 s! 2 2 3c13 + c33  (1 + 1) + c33 21 c66 Sm() = Rm(): (29) 

1h 2 At the interface of the solid and liquid, i.e. at z = h, (1e )Bm() + 3c13 the boundary conditions imply continuity of the normal    component of the displacements of the two media: 2 s! 2 2 + c33  (1 + 1) + c33 22 C66 u (z = h) = u (z = h+); (30) z z 

2h 2 4h which means: (2e )Dm() (s! )(Sme    2  s! (e1h) 2 + 2(1 + ) B () 2 1 1 m + R e4h) S () = 0: (37) C66 m m     !2 Two separate sets of equations for the ve remaining 2h 2 s 2 + (e ) 22 +  (1 + 1) c66 unknown functions, Bm, Dm, Fm, Sm, and Rm, are   now provided. These equations constitute the matrix given by Eq. (38) as shown in Box II, where: 4h 4h Dm()+4 Sm()e Rm()e =0: (31)  !2  = ( )2 + 2(1 + ) s ; i 3 2 1 1 c Meanwhile, since the liquid does not support shear 66 stresses, the two tangential stress components,  and 2 z #i = 3i i;  , vanish at the interface: rz   c13 2 2  (z = h) = 0; (32) i = i 3  3i i; i = 1; 2: (39) z c33 The nontrivial solution is obtained when the determi- rz(z = h) = 0: (33) nant of the matrix vanishes, that is: These two conditions yield to the following equations, det[A] = 0: (40) respectively:   The expression for m is separated from the other 3h 2C44i3e Fm() = 0; (34) unknowns and reduces to 3 = 0. According to the assumed non-viscos nature of the liquid, no Love    2 s! wave is expected to appear at the interface which 2C ( )2 +2(1+ ) e1h B () 44 3 2 1 1 m is mathematically observed here, since no roots are c66 obtained in the expression given for , except one  m determining the SH wave-number. + 2c ( )2 + 2(1 + ) 44 3 2 2 1 Eq. (40) provides the frequency equation with re- spect to the two-layer half-space; it relates , the wave-    !2 number, to ! which is the angular frequency; it presents s 2h e Dm() = 0: (35) the dispersive properties of the system. For any value c66

of !, there is at least one root, R > max(1 ; 2 ), as- The normal stress continuity is preserved at the solid- sociated with the Rayleigh wave-number which satis es liquid interface: the frequency equation. Dependent on the liquid layer

2 3  h  h  h  h e 1 #1 e 2 #2 4e 4 4e 4 6 e1hc  e2hc   !2e4h  !2e4h7 A = 6 33 1 33 2 s s 7 : (38) 4 1h 2h 5 2c441e 2c442e 0 0 0 0 1 1

Box II 2474 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 depth, frequency of the excitations, and properties Table 1. Material properties. of the liquid and bed material, there might be other Material Ev=Eh h c11 c12 c13 c33 "(%) roots which are associated with the higher modes. 1a 1 0.25 3.00 1.00 1.00 3.00 0 Notwithstanding, only a single root may exceed the 2b 2 0.33 3.00 1.12 1.03 5.51 22.78 value !=cl which will be associated with the Stoneley- or Scholte-wave-number. If the compressional wave 3b 3 0.17 2.65 0.52 0.79 7.91 33.25 velocity of the liquid is higher than the SV wave 4b 4 0.25 2.72 0.72 0.86 12.93 39.48 velocity, the only possible root will be related to the a b 3 Isotropic; Transversely isotropic, s = 2500 kg/m , Stoneley wave and no higher modes will appear. cij = cij =c44, and c44 = 10 GPa.

4. Special cases which provides the surface waves velocity. By means of a computer program written in Mathematica, the roots To draw a comparison between the results obtained of the equation are calculated and the dispersion curves here with some existing ones, several special cases of are obtained consequently. The non-dimensional phase the problem are now considered: and group velocities of the surface wave are plotted - Case 1. If h tends to in nity or equivalently the versus another dimensionless variable, = ncnh=2cl, liquid layer depth takes very large values compared in semi-logarithmic gures. Four solid materials, whose to the wavelength, the velocity of the surface waves elastic properties are presented in Table 1, are consid- tends to the Stoneley - or Scholte-wave velocity, ered as the underlying medium. The elastic constants propagating in a bi-material full space consisting of are normalized with respect to c44. In this table, the liquid and transversely isotropic solid half-spaces. materials are sorted with respect to the anisotropy This results in the wave-number, st, of Stoneley parameter, ", which is de ned as follows [23]: wave showing exact agreement with the frequency c c equation presented by Abubakar and Hudson [15]: " = 11 33  100: (42) 2 2c33 l! (2#1 1#2) + c334(21 12) = 0: (41) Material 1, shown in Table 1, is an isotropic material; - Case 2. If h tends to zero, meaning that the hence, its anisotropy parameter is equal to zero. In all wave-length becomes very large in comparison to con gurations, the liquid is taken to be water with the the depth of the liquid layer, the frequency equation properties:  = 1000 kg/m3 and K = 2:2 GPa. reduces to that of a Rayleigh wave for a transversely l The positive de niteness of the strain energy is isotropic solid half-space developed by Khojasteh et considered when de ning the material. The al. [10], i.e.     = 0. 1 2 2 1 constants should comply with the requirements be- - Case 3. If the solid material is considered to low [21]: be isotropic, the wave-numbersp associated with the body waves become  = !  =(2 + ) and  = 2 p P s SH c11 > jc12j; (c11 + c12)c33 > 2c13; c44 > 0: (43) SV = ! s=, and the frequency equation reduces to the equation developed by Ewing [3] for an The phase velocity associated with the rst mode, and isotropic bed material underlying a liquid layer. the next three higher modes for systems consisting of half-space of kind Materials 2 and 4 (Table 1) - Case 4. If the solid material is reduced to an iso- underlying a water layer are provided in Figures 2 tropic medium and h tends to zero, the Rayleigh wave velocity for isotropic half-space is obtained. As a more speci c case, where  = , the well-known function for Rayleighq wave velocity is deduced [10] 2  as: cR = p p : Also, if the solid is simpli ed 3+ 3 s to an isotropic case and h becomes very large, the interface Stoneley wave is obtained for a full-space bi-materials system consisting of liquid and isotropic solid half-spaces. These special cases are numerically veri ed in the following section.

5. Numerical results and discussion

The derived frequency equation in the previous section Figure 2. Phase and group velocities for the rst four implicitly expresses the wave-number, , in terms of !, modes of Material 4 underlying a water. A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2475

Figure 3. Phase and group velocities for the rst four Figure 5. Group velocities of the rst modes for four bed modes of Material 2 underlying a water layer. materials as the bed underlying a layer of water.

waves can always arise in the systems of contacting liquid and solid media. Nevertheless, it should be noted that in contacting solid layers, the possibility of this type of surface waves depends on the properties of the layers [3]. The Stoneley branch of the curve arises in a higher value of h as the elastic constants of the material increase. Normal mode group velocities for the de ned materials as the solid half-space are plotted in Figure 5 for comparison. Unlike the phase velocity, the group velocity takes a minimum value (Airy phase) which occurs at a higher h value in the higher modes. All the gures indicate that the velocities of SV wave in the solid domain and the in the liquid mainly Figure 4. Phase velocity of the rst modes for four bed control the range of the values that the dispersion materials as the bed. curves covers. and 3. The results show exact agreement with those 6. Summary and conclusion presented by Abubakar and Hudson [15]. It can be observed that the higher modes take a maximum value In this paper, dispersion of generalized Rayleigh waves of cuto phase velocity equal to the velocity of SV is studied in a model of a homogeneous compressible pwave propagation in the solid substratum, i.e. cSV = nite liquid layer lying over a transversely isotropic c44=s. Nevertheless, for the rst mode, it takes the solid half-space. Hankel transform and Fourier series Rayleigh wave velocity for the respective vacuum-solid expansion along with the method of potential functions half-space system. are used to deal with the equations of motion. The dis- The dimensionless phase velocities of the rst persive nature of the surface waves is described through modes of three transversely isotropic materials along the derived frequency equation. Various special cases with one isotropic material underlying a nite water are studied to con rm the formulation and numerical layer are shown in Figure 4. It can be observed results with the existing solutions. In this regard, both that the curve for the isotropic material is in exact the mathematical formulation and numerical results agreement with the results presented by Ewing et are derived for several simpler half- and full spaces al. [3]. An isotropic solid can be degenerated from degenerated from the most general case. The results a transversely isotropic one appropriately setting the are compared and con rmed with the existing ones. elasticity constants as c11 = c33 =  + 2, c12 = The gures clearly show that the material anisotropy c13 =p, and c44 = c66 = . Inp an isotropic material, can play an important role in the dispersive characteris- 2 2 2 2 1 =  P , and 2 = 3 =  S. tics of surface waves. The considerable in uence of the As it is apparent in Figures 2, 3, and 5, all values liquid layer depth relative to the wave-length, as well of the phase velocities for higher modes fall above the as the bed material properties, on the characteristics value cl, whereas, for the normal mode, there are values of the phenomenon as well as the elastic properties of of the velocity less than c1 which tend to the asymptotic the bed can be observed from the gures presented for value of Stoneley or Scholte wave velocity. Stoneley di erent bed materials. The velocity dispersion curves 2476 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477

reveal the fact that the di erence between the velocities h Poisson ratio characterizing the e ects of SV and SH waves in a VTI solid is important for of horizontal strain on complementary the lower limit of the velocities of higher modes. As horizontal strain; in this case, the lower limit is the SV wave velocity  Hankel's parameter and wave-number; which is di erent from the SH wave velocity. This  Rayleigh wave-number; indicates the importance of the anisotropy, as the SV R  Stoneley wave-number; and SH wave velocities may signi cantly di er in the st highly anisotropic materials, and hence in the velocity n nth Mode wave-number; dispersion curves. Consequently, neglecting the e ect l Liquid density; of the anisotropy may clearly results in considerable s Solid density; inaccuracies. ij Stress components; The obtained results can be eciently extended Potential function; to calculate the elastodynamic response of described system subjected to dynamic excitations which is being ! Frequency. processed by the authors and can be e ectively applied to solve bed- uid-structure problems. References

1. Stoneley, R. \The e ect of ocean on Rayleigh waves", Nomenclature Mon. Not. R. Astro. Soc. Geophys. Suppl., 1, pp. 349- 356 (1926). c Dispersive wave velocity; 2. Haskell, N.A. \The dispersion of surface waves in cij Elasticity constants; multilayered media", Bull. Sesmol. Soc. Am., 43, pp. 17-34 (1953). cl Velocity of compressional wave in the liquid; 3. Ewing, W.M., Jardetzky, W.S. and Press, F., Elastic c nth mode interface wave velocity; Waves in Layered Media, McGraw-Hill Book Com- n pany, INC., New York, Toronto, London (1957). c Rayleigh wave velocity; R 4. Crampin, S. \Review of wave motion in anisotropic cSV SV wave velocity; and cracked elastic media", Wave Motion, 3, pp. 343- Eh Young's moduli in the direction normal 391 (1981). to the plane of transverse isotropy; 5. Stoneley, R. \The seismological implications of ae- Ev Young's moduli in the plane of olotropy in continental structures", Roy. Astron. Soc. transverse isotropy; Mon. Notices, Geophysical Supplement, 5, pp. 343-353 (1949). F Potential function; 6. Synge, J.L. \Elastic waves in anisotropic media", J. h Liquid layer depth; Math. Phys., 35, pp. 323-334 (1957). J Bessel function of the rst kind and m 7. Buchwald, V.T. \Rayleigh waves in transversely mth order; isotropic media", Q. J. Mech. Appl. Math., 14(4), pp. K Liquid bulk modulus; 293-317 (1961). kp Dilatational wave-number; 8. Payton, R.G., Elastic Wave Propagation in Trans- versely Isotropic Media, Martinus, Nijho , the Nether- ks Distortional wave-number; lands (1983). m Hankel integral transform order; 9. Rahimian, M., Eskandari-Ghadi, M., Pak, R.Y.S. and P Liquid dynamic pressure; Khojasteh, A. \Elastodynamic potential method for r Radial coordinate; transversely isotropic solid", J. Eng. Mech, ASCE, 133(10), pp. 1134-1145 (2007). s1; s2 Roots of strain energy function; t Time variable; 10. Khojasteh, A., Rahimian, M., Eskandari, M. and Pak, R.Y.S. \Asymmetric wave propagation in a ur Displacement component in r-direction; transversely isotropic half-space in displacement po- uz Displacement component in z- tentials", Int. J. Eng. Sci., 46, pp. 690-710 (2008). direction; 11. Khojasteh, A., Rahimian, M., Pak, R.Y.S. and Eskan- u Displacement component in - dari, M. \Asymmetric dynamic Green's functions in a direction; two-layered transversely isotropic half-space", J. Eng. " Anisotropy parameter; Mech. ASCE, 134(9), pp. 777-787 (2008).  Angular coordinate; 12. Khojasteh, A., Rahimian, M. and Pak, R.Y.S. \Three- dimensional dynamic Green's functions in transversely  Lame's constant; isotropic bi-materials", Int. J. Solids. Struct., 45, pp.  Lame's constant; 4952-4972 (2008b). A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2477

13. Khojasteh, A., Rahimian, M. and Eskandari, M. Three contributions have been published by him in \Three-dimensional dynamic Green's functions in international journals and conferences. He was also transversely isotropic tri-materials", J. Math. Model., ranked 383rd among more than 800,000 applicants 37, pp. 3164-3180 (2013). participated in the nationwide undergraduate program 14. Khojasteh, A., Rahimian, M., Eskandari, M. and Pak, entrance exam. He has published ve papers in R.Y.S. \Three-dimensional dynamic Green's functions scienti c journals and also several conference papers. for a multilayered transversely isotropic half-space", Int. J. Solids. Struct., 48, pp. 1349-1361 (2011). Ali Khojasteh, currently an Assistant Professor in 15. Abubakar, I. and Hudson, J.A. \Dispersive properties the University of Tehran, received his BSc in Civil of liquid overlying an aelotropic half-space", Geophys. Engineering from the University of Shiraz, Iran in J. Roy. Astr. Soc., 5, pp. 217-229 (1961). 2003, and MSc and PhD degrees from the University 16. Sharma, M.D., Kumar, R. and Gogna, M.L. \Surface of Tehran, Iran, in 2005 and 2009, respectively, both wave propagation in a liquid saturated porous solid ranked 1st among graduates. layer overlying a homogeneous transversely isotropic His research interests are computational mechan- half-space and lying under a uniform layer of liquid", ics, wave propagation in solids/ uids, mechanics of Int. J. Solids. Struct., 127, pp. 1255-1267 (1991). composites, BEM, soil-structure Interaction. He was 17. Sharma, M.D., Kumar, R. and Gogna, M.L. \Surface ranked 262nd among more than 800,000, 7th among wave propagation in a transversely isotropic elastic more than 15000, and 9th among more than 15000 layer overlying a liquid saturated porous solid half- participating in the nationwide undergraduate, MSc space and lying under the uniform layer of liquid", program entrance exams and the Civil Engineering Pure. Appl. Geophys., 133(3), pp. 523-539 (1990). Olympiad, respectively. More than 15 contributions 18. Sharma, M.D. \Dispersion in oceanic crust during have been published in the international journals by earthquake preparation", Int. J. Solids. Struct., 36, him. pp. 3469-3482 (1999). 19. Kumar, R. and Miglani, A. \Surface wave propagation Mohammad Rahimian, currently the Full Professor in an oceanic crust model", Acta Geophys. Polonica, of Civil Engineering Department at the University of 52(4), pp. 443-456 (2004). Tehran, received his BSc degree in Civil Engineering 20. Kumar, R. and Kumar, R. \Analysis of wave motion in from the University of Tehran. He received his PhD transversely isotropic elastic material with voids under degree in Ecole National Des Ponts et Chaussees, an inviscid liquid layer", Can. J. Phys., 87(4), pp. 377- France, in 1981. His research elds are earthquake 388 (2009). engineering, wave propagation in anisotropic materials, 21. Eskandari-Ghadi, M. \A complete solutions of the soil-structure- uid interaction, etc. He is the former wave equations for transversely isotropic media", J. chair and assistant to chair of the University of Tehran Elast., 81, pp. 1-19 (2005). and the former chair of the Faculty of Engineering of 22. Lekhnitskii, S.G., Theory of Anisotropic Elastic Bod- the University of Tehran. Near 50 papers published in ies, Holden-Day, San Francisco (1963). ISI indexed journals and 6 written or translated books 23. Ben-Menahem, A. and Sena, A.G. \Seismic source are of his scienti c works. theory in strati ed Anisotropic Media", J. Geophyis. Res., 95, pp. 15, 395-15,427 (1990). Reza Attarnejad received his PhD degree in 1990 from UPC in Spain focused on computational mechan- ics on continuous media. He is currently the (full) Biographies Professor of Structural Engineering Division of Civil Amirhossein Bagheri received his BSc degree in Engineering Department at University of Tehran and Civil Engineering from Sharif University of Technology also was the former chairman of the division. He in 2008, MSc degree in Hydraulic Structures Engineer- has extensive publications on his specialties which are ing from University of Tehran in 2011. He is currently uid-structure interaction, structural dynamics, FEM, a PhD candidate of Hydraulic Structures Engineering applied mathematics, and semi-analytical methods. in University of Tehran, working in elds such as He has punlished near 70 papers in the ISI indexed wave propagation and soil-water-structure interactions. international journals.