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 liquid 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 solid 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 velocity 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 Anisotropy. 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 solids 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 scientists 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. stress 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.