Applied Mathematics, 2011, 2, 11291133 doi:10.4236/am.2011.29156 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Reflection of Plane Waves from a Free S u r f a ce o f a n I ni t i a l l y Stresse d Transversely Isotropic Dissipative Medium Baljeet Singh1, Jyoti Arora2 1Department of Mat hem at i cs, Post Graduate Government College, Chandigarh, India 2B.S.A.I.T.M, Alampur, Faridabad, India Email: bsinghgc11@gmail.com Received June 23, 2011; revised July 15, 2011; accepted July 23, 2011 Abstract The governing equations of a transversely isotropic dissipative medium are solved analytically to obtain the speeds of plane waves. The appropriate solutions satisfy the required boundary conditions at the stressfree surface to obtain the expressions of the reflection coefficients of reflected quasiP (qP) and quasiSV (qSV) waves in closed form for the incidence of qP and qSV waves. A particular model is chosen for numerical computation of these reflection coefficients for a certain range of the angle of incidence. The numerical val ues of these reflection coefficients are shown graphically against the angle of incidence for different values of initial stress parameter. The impact of initial stress parameter on the reflection coefficients is observed significantly. Keywords: Transversely Isotropic, Dissipative Medium, Initial Stress, Plane Waves, Reflection, Reflection Coefficients 1. Introduction We can not know the Earth completely by assuming mere an elastic body. If we consider various additional parameters, e.g. porosity, initial stress, viscosity, dissipa tion, temperature, voids, diffusion, etc., then we can un derstand better the interior of the Earth. Initial stresses in a medium are caused by various reasons such as creep, gravity, external forces, difference in temperatures, etc. The reflection of plane waves at free surface, interface and layers is important in estimating the correct arrival times of plane waves from the source. Various research ers studied the reflection and transmission problems at free surface, interfaces and in layered media [112]. The study of reflection of plane waves in the presence of ini tial stresses as well as dissipation is interesting. With the help of Biot [13] theory of incremental deformation, Selim [14] studied the reflection of plane waves at a free surface of an initially stressed dissipative medium. In the present paper, we studied the problem on reflection of plane waves at a stressfree surface of an initially stress ed transversely isotropic solid halfspace with dissipation. The reflection coefficients of reflected waves are com puted numerically to observe the effect of initial stress. 2 . Formulation of the Problem and Solution Following Biot [13], the basic dynamical equations of motion in xz plane for an infinite, initially stressed me dium, in the absence of external body forces are, 2 13 11 2 2 31 332 s su P, xz zt ss w P, xz xt (1) where is the density, 1w u 2x z is rotational component, sij (i, j = 1, 3) are incremental stress compo nents, u and w arethe displacement components. Following Biot [13], the stressstrainrelations are 11 1113 13 3144333313 uw sCP CP, xz uww u ssC,sC C zxzx , (2) where Cij are the incremental elastic coefficients.
B. SINGH ET AL. 1130 , , For dissipative medium, elastic coefficients are re placed by the complex constants: RI RI 111111 131313 RI RI 333333 444444 CCiC,CCiC CCiC,CCiC (3) where, i1, are real. Following Fung [15], the stress and strain components in dissipative medium are, RI RIRIR 111113 13 33 33 44 C,C,C,C,C,C,C, I 44 C it it ij ijii sse,uue , (4) where (i, j = 1, 3) and being the angular frequency. With the help of Equations (3) and (4), the Equation (2) becomes, RI RI 11 11 1113 13 RI 31 134444 RI RI 33333313 13 uw sC iC PC iCP, xz uw ss CiC, zx wu sCiCCiC, zx (5) With the help of Equation (5), the Equation (1) be comes, 222 u RR R R 1113 4444 22 222 2III I 1113 4444 22 uPwP CP CCC 2xz 2 xz uwu uiCC CC0, xz xz (6) 22 RRR 331344 2 22 R 44 2 222 IIII 13 444433 22 wPu CCC 2xz z Pw Cw 2x uww iC CCC0, xz xz , n (7) The displacement vector is given by, (n) nn u,O,wU n nn i n Ae Ud (8) where (n) assigns an arbitrary direction of propagation of waves, is the unit displacement vector nn 13 d,dd kct X and is the phase factor, in n nnn. nn n which 13 is the unit propagation vector, cn is the velocity of propagation = (x, z), and kn is corresponding wave number, which is related to the an gular frequency by , X nn The displacement components un and wn are written kc . as (9) Making use of Equation (9) into the Equations (6) and (7 nn n1 3n n nikxzc t n1 n nn3 Ad ue, Ad w ), we obtain a system of two homogeneous equations, which as nontrivial solution if 2 22 nnn ccIP 0, (10) where, 2 12n 132 ID D,PDD D, 22 22 nn RR 111 1443 nn RR 11 1443 P DCR C 2 iCC , RRnnII nn 21344 13134413 P DCC iCC 2, 2 22 2 3 n RnRn InI 44133 344133 3 D P CCiCC 2, 22 nn 22 nn II4P II4P c,c 22 The roots co siP (qP) waves and quasiSV (qSV) rrespond to qua waves respectively. The above two roots give the square of velocities of propagation as well as damping. Real parts of the right hand sides correspond to phase velocities and the respec tive imginary parts correspond to damping velocities of qP and qSV waves, respectively. It is observed that both 2 1 c and 2 2 c depend on initial stresses, damping and ction propagation n dire of . In the absence of initial stresses and damping, the ve analysis corresponds to the case of transversely isotropic elastic solid. . Reflection of Plane Waves from Fr abo 3ee ean initially stressed dissipative halfspace Surface consider W occupying the region z > 0 (Figure 1). In this section, we shall drive the closed form expressions for the reflection coefficients for incident qP or qSV w ave s. The displacement components of incident and re flected waves are as, 4 j jj 4 ii j1 j3 j1 j1 ux,z,tAde,wx,z,t Ade, j (11) where, Copyright © 2011 SciRes. AM
B. SINGH ET AL.1131 Figure 1. Geometry of the problem. (12) Here, subscripts 1, 2, 3 and 4 correspond to incident qP nt and stress compo ne 1111 1 2222 2 3333 3 4444 4 kctsinexcosez, kct sinexcosez, kct sinexcosez, kct sinexcosez, wave, incident qSV wave, reflected qP wave and reflected qSV wave, respecti vely. In the xz plane, the displaceme nts due to the incident qP wave ( 11 11 ritten as , 3 1 1 sin e,cose ) are w 1 1i 11 de, 1 11 i 13 uA wAde 1 111 i 131411131 siAQkdcosedsinee (13) where, . In the xz plane, the displacement and stress compo ne itten as , 1 111 i 33113312 11 siAkQdcoseQdsinee RI RR 11111213 13 RI RI 333 3344444 QCiC,QC iC, QCiC,QC iC nts due to the incident qSV wave ( 22 12 2 sin e,cose ) are wr 2 2i de, 3 2 uA 2 21 22 i 23 wA de 2 222 i 132421232 siAQk[dcosedsine]e 2 222 i 33223 32212 siAkQdcoseQdsinee (14) In the xz plane, the displacement and stress com ne ritten as , , (15) In the xz plane, the displacement and stress compo nents due to the reflected qSV wave 4 are written as , po nts due to the reflected qP wave ( 33 13 3 sin e,cose ) are w 3 3i de, 3 3 3 31 33 i 33 uA wAde 3 333 i 133431333 siAQkdcosedsinee 3 333 i 33333 33213 siAkQdcoseQdsinee, ( 44 143 sin e,cose) 4 4 44i 41 44 i 43 uAde, de wA 4 444 i 134441434 siAQkdcosedsinee 4 444 i 33443 342 14 siAkQdcoseQdsinee , (16) The boundary conditions requ ired to be satisfied at the free surface z = 0, , nn z33 fs eP0 fs 0, x1 3 13 n (17) The above boundary cond itions are written as 1234 123 13 13131313131313 1234 ssssP(e ee 4 e)0, 33 33 33 33 ssss 0, (18) The Equations (11) to (16) will satisfy the boundary conditions (18), if the following Snell, s law holds 3 12 4 sine sinesinesine A11 + A22 + A33 + A4 37 + A48 = 0, (21) where 1234 , cccc (19) and the following relations hold 4 = 0, (20) A15 + A26 + A 1113 1 kL dcosedsine, 11 1 22 22123 2 33 331 333 44 443414 11 51331211 22 6233221 2 33 73333213 4 842143 kLd cosed sine, kLd cosed sine, kLd sined cose, kQdcose Qdsine, kQdcose Qdsine, kQdcose Qdsine, kQdsine Qd 4 34 cose , (22) and 4P LQ 2 Copyright © 2011 SciRes. AM
B. SINGH ET AL. 1132 1) For incident qP wave (A2 = 0), 3 45183517 4 13847 13847 AA , AA , (23) 2) For incident qSV waves (A1 = 0), 34628 362 4 238472 384 AA , AA 7 7 . (24) For isotropic case, C11 = + 2 + P, C13= P = –S11, then, the above theoretical derivations reduce to Selim [14]. 4. Numerical Example For numerical purpose, a particular example of the mate rial is chosen with the followin g physical constants, 2 From Equations (23) and (24), the reflection cients of reflected qP and qSV waves are computed for th own graphically in Figure 2 for incident qP wave and in Figure 3 for incident qSV wave. coefficient of reflected qP ge of the angle of inci r P = 1 and P = 2 also change at each angle of incidence as shown by solid line with asters and solid lines with trian gles, respectively. The comparison of the variations of reflection coefficients for P = 0, P = 1 and P = 2, show the significant effect of initial stress on reflected qP wa for incident qP wave. Similarly, reflected qSV is also affected significantly due to the presence of initial stress as show R1 C02R10 11 33 102 13 44 I 102I102 11 33 I 102I102 13 44 2.62810N m,C1.56210N m, 0.38510N m, C1.02510 Nm,C0.95010 N C0.425 10Nm,C0.325 10Nm, R102R C0.50810N m,C 33 7.14 10 kg m. m , coeffi e incident qP and qSV waves. The numerical values of the reflection coefficients of reflected qP and qSV waves are sh For P = 0, the reflection ave oscillates for the whole ranw dence of qP wave as shown by solid line in Figure 2. The variations of reflection coefficients of qP wave fo s ve n in Figure 2. For incident qSV wave, the reflection coefficient of reflected qP wave first increases to its maximum value and then decreases to its minimum value at angle e2 = 45˚ when P = 0. Thereafter, it oscillates as shown by solid line in Figure 3. The variations of reflection coef ficients of qP wave for P = 1 and P = 2 are similar to that for P = 0. The comparison of solid line, solid line with asters and solid line with triangles shows the significant effect of initial stress on reflected qP wave for incident qSV wave. Similarly, reflected qSV is also affected sig nificantly due to the presence of initial stress as sh own in Figure 3. Figure 2. Variation of the reflection coefficients of qP an qSV waves against the angle of incidence for incidence of qP wave. 5. Conclusions The reflection from the stressfree surface of a trans versely isotropic dissipative medium is considered. The expressions for the reflection coefficients of reflected qP and qSV waves are obtained in closed form for the inci dence of qP and qSV waves. For a particular material, these coefficients are computed and depicted graphically against the angle of incidence for different values of ini tial stress parameter. From the figures, it observed that 1) the initial stresses affect significantly the reflection coef ficients of all reflected waves. 2) For incident qP wave, the critical angle for reflected qSV wave is observed at e1 = 45˚ and for incident qSV wave, the critical angle f d or reflected qP wave is observed also at e2 = 45˚. 3) The effect of initial stresses on the reflection coefficients is Copyright © 2011 SciRes. AM
B. SINGH ET AL. Copyright © 2011 SciRes. AM 1133 Figure 3. Variation of the reflection coefficients of qP and qSV waves against the angle of incidence for incidence of qSV wave. minimum at e1 = 45˚ for incidence qP wave and at e2 = 45˚ for incidence qSV wave. 6. References [1] S. B. Sinha, “Transmission of Elastic Waves through a Homogenous Layer Sandwiched in Homogenous Media, Journal of Physics of the Earth, Vol. 12, No. 1, 1999, pp. 14. doi:10.4294/jpe1952.12.1 ” [2] R. N. Gupta, “Reflection of Plane Waves from a Linear Transition Layer in Liquid Media,” Geophysics, Vol. No. 1, 1965, pp. 122131. doi:10.1190/1.1439528 Compressional Waves,” Geo physics, Vol. 30, No. 4, 1965, pp. 552570. Reflection of Elastic Waves from a Linear Transition Layer,” Bulletin of the Seismological Society [3] R. D. Tooly, T. W. Spencer and H. F. Sagoci, “Reflection and Transmission of Plane [4] R. N. Gupta, “ of America, Vol. 56, 1966, No. 2, pp. 511526. doi:10.1190/1.1439622 [5] R. N. Gupta, “Propagation of SHWaves in Inhomoge neous Media,” Journal of the Acoustical Society of America, Vol. 41, No. 5, 1967, pp. 13281329. doi:10.1121/1.1910477 [6] H. K. Acharya, “Reflection from the Free Surface of In Coefficients homogeneous Media,” Bulletin of the Seismological So ciety of America, Vol. 60, No. 4, 1970, pp. 11011104. [7] V. Cerveny, “Reflection and Transmission for Transition Layers,” Studia Geophysica et Geodaetica, Vol. 18, No. 1, 1974, pp. 5968. doi:10.1007/BF01613709 [8] B. M. Singh, S. J. Singh and S. D. Chopra, “Reflection atic Initial Stresses on Waves 700498 and Refraction of SHWaves and the Plane Boundary between Two Laterally and Vertically Heterogeneous Solids,” Acta Geophysica, Vol. 26, 1978, pp. 209216. [9] B. Singh, “Effect of Hydrost in a Thermoelastic Solid HalfSpace,” Applied Mathe matics and Computation, Vol. 198, No. 2, 2008, pp. 498 505. [10] M. D. Sharma, “Effect of Initial Stress on Reflection at the Free Surfaces of Anisotropic Elastic Medium,” Jour nal of Earth System Science, Vol. 116, No. 6, 2007, pp. 537551. doi:10.1007/s1204000 30, ,” Recent Ad [11] S. Dey and D. Dutta, “Propagation and Attenuation of Seismic Body Waves in Initially Stressed Dissipative Medium,” Acta Geophysica, Vol. XLV1, 1998, pp. 351 365. [12] M. M. Selim and M. K. Ahmed, “Propagation and Atte Nuation of Seismic Body Waves in Dissipative Medium under Initial and Couple Stresses,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 10641074. [13] M. A. Biot, “Mechanics of Incremental Deformation,” John Wiley and Sons Inc., New York, 1965. [14] M. M. Selim, “Reflection of Plane Waves at Free Surface of an Initially Stressed Dissipative Medium vances in Technologie s, Vol. 30, 2008, pp. 3643. [15] Y. C. Fung, “Foundation of Solid Mechanics,” Prentice Hall of India, New Delhi, 1965.
