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Numerical Modeling of Love Waves in Dry Sandy Layer Under Initial Stress Using Different Order Finite Difference Methods

  • Jayantika PalEmail author
  • Anjana P. Ghorai
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

This stated manuscript is concerned with the propagation of surface waves in a dry sandy layer under initial stress. The analysis is based on Biot’s theory. The dispersion equation of phase velocity of this proposed layer has been derived using convenient second-order finite difference scheme, staggered-grid finite difference scheme, and higher order finite difference scheme where, in each case, second-order central difference operator has been used for temporal derivatives, but second, fourth, and higher order finite difference scheme are used for spatial derivatives, respectively. A comparison study using these three methods has been done and presented in graphs. It has been shown that staggered-grid finite difference scheme is more accurate than second-order finite difference scheme and higher order finite difference scheme is more accurate than second-order finite difference scheme and staggered-grid finite difference scheme both.

Keywords

Sandy layer Surface waves Initial stress Phase velocity Finite difference scheme 

References

  1. 1.
    Weiskopf WH (1945) Stresses in solids under foundation. J Franklin Inst 239:445MathSciNetCrossRefGoogle Scholar
  2. 2.
    Biot MA (1956a) Theory of propagation of elastic waves in a fluid -saturated porous solid. I. low frequency range. J Acoust Soc Am 28(2):168–178MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biot MA (1956b) Theory of propagation of elastic waves in a fluid-saturated porous solid. II. higher frequency range. J Acoust Soc Am 28(2):179–191MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biot MA (1955) Theory of elasticity and consolidation for a porous Anisotropic solid. J Appl Phys 26:182–185MathSciNetCrossRefGoogle Scholar
  5. 5.
    Biot MA (1956) General solution of the equation of elasticity and consolidation for a porous material. J Appl Mech 32:91–95MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ewing M, Jardetzky W, Press F (1957) Elastic waves in layered media. McGraw-Hill, New YorkCrossRefGoogle Scholar
  7. 7.
    Achenbach JD (1999) Wave propagation in elastic solids. North- Holland Publishing Co., AmsterdamzbMATHGoogle Scholar
  8. 8.
    Abd-Alla AM, Mahmoud SR, Helmi MIR (2009) Effect of initial stress and magnetic field on propagation of shear wave in non-homogeneous anisotropic medium under gravity field. Open Appl Math J 3:58–65Google Scholar
  9. 9.
    Gupta S, Chattopadhyay A, Majhi DK (2010) Effect of initial stress on propagation of Love waves in an anisotropic porous layer. Solid Mech 2:50–62Google Scholar
  10. 10.
    Gupta S, Chattopadhyay A, Majhi DK (2011) Effect of irregularity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer. Appl Math Mech Engl Ed 31(4):481–492MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pal J, Ghorai AP (2015) Propagation of love wave in sandy layer under initial stress above anisotropic porous half-space under gravity. Transp Porous Media 109(2):297–316MathSciNetCrossRefGoogle Scholar
  12. 12.
    Alterman Z, Karal FC (1968) Propagation of elastic waves in layered media by finite difference method. Bull Seismol Soc Am 58(1):367–398Google Scholar
  13. 13.
    Virieux J (1984) SH-wave propagation in heterogeneous media: velocity stress finite difference method. Geophysics 49:1933–1957CrossRefGoogle Scholar
  14. 14.
    Virieux J (1986) P-SV wave propagation in heterogeneous media: velocity stress finite difference method. Geophysics 51:889–901(1986)CrossRefGoogle Scholar
  15. 15.
    Levander AR (1988) Fourth order finite difference P-SV Seismograms. Geophysics 53:14251436CrossRefGoogle Scholar
  16. 16.
    Graves RW (1996) Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bull Seismol Soc Am 86:1091–1106Google Scholar
  17. 17.
    Hayashi K, Burns DR (1999) Variable grid finite difference modeling including surface topography. In: 69th annual international meeting, SEG, Exp. Abstracts, pp 523–527Google Scholar
  18. 18.
    Saenger EH, Gold N, Shapiro SA (2000) Modeling the propagation of elastic waves using a modified finite difference grid. Wave Motion 77–92MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tessmer E (2000) Seismic finite difference modeling with spatially variable time steps. Geophysics 65:1290–1293CrossRefGoogle Scholar
  20. 20.
    Kristek J, Moczo P (2003) Seismic wave propagation in visco-elastic media with material discontinuities: a 3D forth-order staggered-grid finite difference modeling. Bull Seismol Soc Am 93(5):2273–2280CrossRefGoogle Scholar
  21. 21.
    Saenger EH, Bohlen T (2004) Finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid. Geophysics 69:583–591CrossRefGoogle Scholar
  22. 22.
    Kristek J, Moczo P (2006) On the accuracy of the finite difference schemes: The 1D elastic problem. Bull Seismol Soc Am 96:2398–2414CrossRefGoogle Scholar
  23. 23.
    Finkelstein B, Kastner R (2007) Finite difference time domain dispersion reduction schemes. J Comput Phys 221:422–438MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu Y, Sen MK (2009a) A practical implicit finite difference method: examples from seismic modeling. J Geophys Eng 6:231–249CrossRefGoogle Scholar
  25. 25.
    Liu Y, Sen MK (2009b) A new time-space domain high order finite difference method for the acoustic wave equation. J Comput Phys 228:8779–8806MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liu Y, Sen MK (2009c) Advanced finite-difference method for seismic modeling. Geohorizons 516Google Scholar
  27. 27.
    Dublain MA (1986) The application of high-order differencing to the scalar wave equation. Geophysics 51:54–56CrossRefGoogle Scholar
  28. 28.
    Liu Y, Sen MK (2010) Acoustic VTI modeling with a time-space domain dispersion-relation based on finite-difference scheme. Geophysics 75(3):A11–A17CrossRefGoogle Scholar
  29. 29.
    Zhu X, Mc Mechan GA (1991) Finite difference modeling of the seismic response of fluid saturated, porous, elastic solid using Biot theory. Geophysics 56:424–435CrossRefGoogle Scholar
  30. 30.
    Ghorai AP, Tiwary R (2013) Modeling of surface waves in a fluid saturated poro-elastic medium under initial stress using time-space domain higher order finite difference method. Appl Math 4:469–476CrossRefGoogle Scholar
  31. 31.
    Liu Y, Sen MK, (2011) Scalar wave equation modeling with time-space domain dispersion-relation based staggered-grid finite-difference schemes. Bull Seismol Soc Am 101(1):141–159CrossRefGoogle Scholar
  32. 32.
    Liu Y, Xiucheng W (2008) Finite difference numerical modeling with even order accuracy in two phase anisotropic media. Appl Geophys 5(2):107–114Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Engineering and Applied SciencesUsha Martin UniversityRanchiIndia
  2. 2.Department of MathematicsBIT, MesraRanchiIndia

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