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 Applied Mathematics, 2011, 2, 1129-1133 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 E-mail: 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 stress-free surface to obtain the expressions of the reflection coefficients of reflected quasi-P (qP) and quasi-SV (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 [1-12]. 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 stress-free surface of an initially stress-ed transversely isotropic solid half-space 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 x-z plane for an infinite, initially stressed me-dium, in the absence of external body forces are, 21311 2231 332ssuP,xz ztss wP,xz xt        (1) where  is the density, 1w u2x z  is rotational component, sij (i, j = 1, 3) are incremental stress compo- nents, u and w arethe displacement components. Following Biot [13], the stress-strainrelations are 11 111313 3144333313uwsCP CP,xzuww ussC,sC Czxzx ,    (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 RI111111 131313RI RI333333 444444CCiC,CCiCCCiC,CCiC  (3) where, i1, are real. Following Fung [15], the stress and strain components in dissipative medium are, RI RIRIR111113 13 33 33 44C,C,C,C,C,C,C,I44Cit itij ijiisse,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 RI11 11 1113 13RI31 134444RI RI33333313 13uwsC iC PC iCP,xzuwss CiC,zxwusCiCCiC,zx   (5) With the help of Equation (5), the Equation (1) be-comes, 222uRRR R1113 4444222222III I1113 444422uPwPCP CCC2xz 2xzuwuuiCC CC0,xzxz   (6) 22RRR331344222R44 2222IIII13 44443322wPuCCC2xzzPwCw2xuwwiC CCC0,xz xz   ,n (7) The displacement vector is given by, (n) nnu,O,wU nnninAeUd (8) where (n) assigns an arbitrary direction of propagation of waves, is the unit displacement vector nn13d,ddkct Xand is the phase factor, in nnnn. nnnwhich 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 , XnnThe displacement components un and wn are written kc . as (9) Making use of Equation (9) into the Equations (6) and (7nnn1 3nnnikxzc tn1nnn3Adue,Adw ), we obtain a system of two homogeneous equations, which as non-trivial solution if 222nnnccIP0, (10) where, 212n 132ID D,PDD D, 2222nnRR111 1443nnRR11 1443PDCR C2iCC , RRnnII nn21344 13134413PDCC iCC2,222 23nRnRn InI44133 344133 3DPCCiCC2,     22nn22nnII4P II4Pc,c22  The roots co si-P (qP) waves and quasi-SV (qSV) rrespond to quawaves 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 21c and 22c depend on initial stresses, damping and ction propagation ndire 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 Frabo3ee ean initially stressed dissipative half-space Surface consider Woccupying 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, 4jjj4iij1 j3j1 j1ux,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 12222 23333 34444 4kctsinexcosez,kct sinexcosez,kct sinexcosez,kct sinexcosez,     wave, incident qSV wave, reflected qP wave and reflected qSV wave, respecti vely. In the x-z plane, the displacements due to the incident qP wave ( 1111 ritten as ,311sin e,cose ) are w11i11de, 111i13uAwAde  1111i131411131siAQkdcosedsinee  (13) where, .In the x-z plane, the displacement and stress compo-ne itten as , 1111i33113312 11siAkQdcoseQdsinee RI RR11111213 13RI RI333 3344444QCiC,QC iC,QCiC,QC iC  nts due to the incident qSV wave ( 2212 2sin e,cose ) are wr22ide,32uA 22122i23wAde  2222i132421232siAQk[dcosedsine]e  2222i33223 32212siAkQdcoseQdsinee (14) In the x-z plane, the displacement and stress comne ritten as ,, (15) In the x-z plane, the displacement and stress compo-nents due to the reflected qSV wave 4are written as ,po-nts due to the reflected qP wave ( 3313 3sin e,cose ) are w33ide,33 33133i33uAwAde  3333i133431333siAQkdcosedsinee   3333i33333 33213siAkQdcoseQdsinee, ( 44143sin e,cose)   4444i4144i43uAde,dewA  4444i134441434siAQkdcosedsinee   4444i33443 342 14siAkQdcoseQdsinee , (16) The boundary conditions requ ired to be satisfied at the free surface z = 0, , nnz33fs eP0fs 0,x13 13n   (17) The above boundary cond itions are written as  1234 12313 131313131313131234ssssP(e ee 4e)0,  33 33 33 33ssss 0,  (18) The Equations (11) to (16) will satisfy the boundary conditions (18), if the following Snell, s law holds 312 4sinesinesinesineA11 + A22 + A33 + A437 + A48 = 0, (21) where 1234,cccc (19) and the following relations hold 4 = 0, (20) A15 + A26 + A 1113 1kL dcosedsine,      1112222123 233331 333444434141151331211226233221 233733332134842143kLd cosed sine,kLd cosed sine,kLd sined cose,kQdcose Qdsine,kQdcose Qdsine,kQdcose Qdsine,kQdsine Qd       434cose , (22) and 4PLQ 2 Copyright © 2011 SciRes. AM B. SINGH ET AL. 1132 1) For incident qP wave (A2 = 0), 3 45183517413847 13847AA,AA     , (23) 2) For incident qSV waves (A1 = 0), 34628 3624238472 384AA,AA    77. (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, 2From Equations (23) and (24), the reflectioncients of reflected qP and qSV waves are computed for thown 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, showthe 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 showR1C02R1011 3310213 44I 102I10211 33I 102I10213 442.62810N m,C1.56210N m,0.38510N m,C1.02510 Nm,C0.95010 NC0.425 10Nm,C0.325 10Nm,     R102RC0.50810N m,C  337.14 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 shFor P = 0, the reflection ave oscillates for the whole ranwdence of qP wave as shown by solid line in Figure 2. The variations of reflection coefficients of qP wave fos ven 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 stress-free 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 dor 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 ofqSV 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. 1-4. doi:10.4294/jpe1952.12.1 ” [2] R. N. Gupta, “Reflection of Plane Waves from a LinearTransition Layer in Liquid Media,” Geophysics, Vol. No. 1, 1965, pp. 122-131. doi:10.1190/1.1439528 Compressional Waves,” Geo-physics, Vol. 30, No. 4, 1965, pp. 552-570. 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. 511-526. doi:10.1190/1.1439622 [5] R. N. Gupta, “Propagation of SH-Waves in Inhomoge-neous Media,” Journal of the Acoustical Society of America, Vol. 41, No. 5, 1967, pp. 1328-1329. 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. 1101-1104. [7] V. Cerveny, “Reflection and Transmissionfor Transition Layers,” Studia Geophysica et Geodaetica, Vol. 18, No. 1, 1974, pp. 59-68. doi:10.1007/BF01613709 [8] B. M. Singh, S. J. Singh and S. D. Chopra, “Reflection atic Initial Stresses on Waves 7-0049-8and Refraction of SH-Waves and the Plane Boundary between Two Laterally and Vertically Heterogeneous Solids,” Acta Geophysica, Vol. 26, 1978, pp. 209-216. [9] B. Singh, “Effect of Hydrostin a Thermoelastic Solid Half-Space,” 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. 537-551. doi:10.1007/s12040-00 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. 1064-1074. [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 Mediumvances in Technologie s, Vol. 30, 2008, pp. 36-43. [15] Y. C. Fung, “Foundation of Solid Mechanics,” Prentice Hall of India, New Delhi, 1965.