Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium

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 values 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.

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 values 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

Transversely Isotropic, Dissipative Medium, Initial Stress, Plane Waves, Reflection, Reflection Coefficients

Cite this paper

nullB. Singh and J. Arora, "Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1129-1133. doi: 10.4236/am.2011.29156.

nullB. Singh and J. Arora, "Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium,"

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 Linear Transition Layer in Liquid Media,” Geophysics, Vol. 30, No. 1, 1965, pp. 122-131. doi:10.1190/1.1439528

[3] R. D. Tooly, T. W. Spencer and H. F. Sagoci, “Reflection and Transmission of Plane Compressional Waves,” Geophysics, Vol. 30, No. 4, 1965, pp. 552-570.

[4] R. N. Gupta, “Reflection of Elastic Waves from a Linear Transition Layer,” Bulletin of the Seismological Society of America, Vol. 56, 1966, pp. 511-526. doi:10.1190/1.1439622

[5] R. N. Gupta, “Propagation of SH-Waves in Inhomogeneous 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 Inhomogeneous Media,” Bulletin of the Seismological Society of America, Vol. 60, No. 4, 1970, pp. 1101-1104.

[7] V. Cerveny, “Reflection and Transmission Coefficients for 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 and Refraction of SH-Waves and the Plane Boundary between Two Laterally and Vertically Heterogeneous Solids,” Acta Geophy-sica, Vol. 26, 1978, pp. 209-216.

[9] B. Singh, “Effect of Hydrostatic Initial Stresses on Waves in a Thermoelastic Solid Half-Space,” Applied Mathematics 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,” Journal of Earth System Science, Vol. 116, No. 6, 2007, pp. 537-551. doi:10.1007/s12040-007-0049-8

[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 Medium,” Recent Advances in Technologies, Vol. 30, 2008, pp. 36-43.

[15] Y. C. Fung, “Foundation of Solid Mechanics,” Prentice Hall of India, New Delhi, 1965.

[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 Linear Transition Layer in Liquid Media,” Geophysics, Vol. 30, No. 1, 1965, pp. 122-131. doi:10.1190/1.1439528

[3] R. D. Tooly, T. W. Spencer and H. F. Sagoci, “Reflection and Transmission of Plane Compressional Waves,” Geophysics, Vol. 30, No. 4, 1965, pp. 552-570.

[4] R. N. Gupta, “Reflection of Elastic Waves from a Linear Transition Layer,” Bulletin of the Seismological Society of America, Vol. 56, 1966, pp. 511-526. doi:10.1190/1.1439622

[5] R. N. Gupta, “Propagation of SH-Waves in Inhomogeneous 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 Inhomogeneous Media,” Bulletin of the Seismological Society of America, Vol. 60, No. 4, 1970, pp. 1101-1104.

[7] V. Cerveny, “Reflection and Transmission Coefficients for 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 and Refraction of SH-Waves and the Plane Boundary between Two Laterally and Vertically Heterogeneous Solids,” Acta Geophy-sica, Vol. 26, 1978, pp. 209-216.

[9] B. Singh, “Effect of Hydrostatic Initial Stresses on Waves in a Thermoelastic Solid Half-Space,” Applied Mathematics 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,” Journal of Earth System Science, Vol. 116, No. 6, 2007, pp. 537-551. doi:10.1007/s12040-007-0049-8

[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 Medium,” Recent Advances in Technologies, Vol. 30, 2008, pp. 36-43.

[15] Y. C. Fung, “Foundation of Solid Mechanics,” Prentice Hall of India, New Delhi, 1965.