• M D Sharma

      Articles written in Journal of Earth System Science

    • Near-field ground motion during earthquake preparation process

      M D Sharma

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      Wave number discretization method is applied to study the near-field of seismic sources embedded in a cracked elastic solid. Near-field solutions are obtained for horizontal and vertical line forces. Effects of modifications in cracks of focal region on ground motion, in the near-field, are studied numerically for different

      ▪ values of crack density

      ▪ saturation of cracks

      ▪ width of cracks, and

      ▪ regimes of connection between cracks.

      An earthquake process is assumed to be going through five major stages. These stages represent continuous accumulation of stress, interconnections between cracks leading to eventual failure and drainage of fluid from cracks after the major shock. Variations in the velocity ratio of waves noted from accelerograms verify the process of preparation of an earthquake.P wave contribution to vertical acceleration is negligible when the source is a vertical line force andS wave contributes only a little to horizontal acceleration when the source is a horizontal line force.

    • Scattering of a spherical pulse from a small inhomogeneity: Dilatation and rotation

      M D Sharma

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      Perturbations in elastic constants and density distinguish a volume inhomogeneity from its homogeneous surroundings. The equation of motion for the first order scattering is studied in the perturbed medium. The scattered waves are generated by the interaction between the primary waves and the inhomogeneity. First order scattering theory is modified to include the source term generating the primary waves. The body force equivalent to the scattering source is presented in a convenient form involving the perturbations in wave velocities and gradient of density perturbation.

      A procedure is presented to study the scattering of a spherical pulse from a small inhomogeneity, in time domain. The size of inhomogeneity is assumed small as compared to its distance from source and receiver. No restrictions are placed on the positions of source, receiver and inhomogeneity. The dilatation and rotations are calculated for a pulse scattered from an arbitrary point in a spherical volume. The aggregate of the scattered phases from all the points of the inhomogeneity, reaching at a fixed receiver, gives the amount of scattering from the inhomogeneity. The interaction of bothP andS waves with inhomogeneity are considered. Dilatation and rotations for scattering are obtained as integral expressions over the solid angle of inhomogeneity. These expressions are computed numerically, for hypothetical models. The effects of source (unit force) orientations, velocity and density perturbations, and size of inhomogeneity, on the scattered phases, are discussed.

    • Strains of scattering of near-field of a point source

      M D Sharma

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      Three dimensional scattering of near-field is studied for dilatation and rotation in the time domain. The perturbation method is applied to solve the equation of motion for the first order scattering from a weak inhomogeneity in an otherwise homogeneous medium. The inhomogeneity is assumed close enough to the point source so that the near-field intermediate wave is dominating over the far-field sphericalP andS pulses. The integral expressions are derived to relate dilatation and rotation of scattering to the radial fluctuations of velocities and density in the inhomogeneity. These integrals are solved to calculate the strains of scattering from (a part of) an inhomogeneous spherical shell of arbitrary curvature. Variable curvature may allow the shape of inhomogeneity volume element to change uniformly from spherical to rectangular. Rotation of scattering from a spherical shell is independent ofP wave velocity inhomogeneity. Dilatation of scattering does not involveS wave velocity inhomogeneity but its gradient. The back scattering results are obtained as a special case. Strains are computed numerically, for hypothetical models to study the effects of various parameters viz., velocity inhomogeneity, distance of source from inhomogeneity and from receiver, and thickness of inhomogeneity. The curvature of the spherical shell is varied to study the effects of the shape of inhomogeneous volume element on scattering.

    • Acceleration of near-field scattering from an inhomogeneous spherical shell

      M D Sharma

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      The three dimensional scattering of near-field, from a point source, is studied for acceleration in the time domain. The perturbation method is applied to define the acceleration for the first order scattering from a weak inhomogeneity in a homogeneous surrounding. A body force, arising from the interaction between the primary waves and the inhomogeneity, acts as the source generating the scattered motion. The acceleration of scattered waves is related to the velocity and density fluctuations of the inhomogeneity. No restrictions are placed on the inhomogeneity size or locations of the source and receiver. Decoupling of scattered motion enables the identification of different phases. Integral expressions are derived for the scattering acceleration due to the incidence of near-field wave (from an impulsive point force) at a radially inhomogeneous volume element. These integrals are solved further for scattering from an inhomogeneous spherical shell. The accelerations for back scattering are obtained as a special case. These accelerations are simple analytically solvable expressions in closed form.

      Only spherical asymmetry ofP wave velocity inhomogeneity can affect the scatteredS acceleration. ScatteredP acceleration is affected by the gradient ofS wave velocity inhomogeneity. The back scattering of near-field from a spherical shell, is independent of radial inhomogeneity ofP wave velocity. Inhomogeneity with smoothly perturbedS wave velocity does not back-scatter any acceleration. Accelerations are computed numerically for scattering from a part of inhomogeneous spherical shell. Hypothetical models are considered to study the effects of the distances of spherical shell from source, receiver, its thickness and its position relative to the direction of impulsive force.

    • Polarisations of quasi-waves in a general anisotropic porous solid saturated with viscous fluid

      M D Sharma

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      Wave propagation is studied in a general anisotropic poroelastic solid saturated with a viscous fluid flowing through its pores of anisotropic permeability. The extended version of Biot’s theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in such media. The non-trivial solution of this system is ensured by a biquadratic equation whose roots represent the complex velocities of four attenuating quasi-waves in the medium. These complex velocities define phase velocity and attenuation of each quasi-wave propagating along a given phase direction in three-dimensional space. The solution itself defines the polarisations of the quasi-waves along with phase shift. The variations of polarisations of quasi-waves with their phase direction, are computed for a realistic numerical model.

    • Wave velocities in a pre-stressed anisotropic elastic medium

      M D Sharma Neetu Garg

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      Modified Christoffel equations are derived for three-dimensional wave propagation in a general anisotropic medium under initial stress. The three roots of a cubic equation define the phase velocities of three quasi-waves in the medium. Analytical expressions are used to calculate the directional derivatives of phase velocities. These derivatives are, further, used to calculate the group velocities and ray directions of the three quasi-waves in a pre-stressed anisotropic medium. Effect of initial stress on wave propagation is observed through the deviations in phase velocity, group velocity and ray direction for each of the quasi-waves. The variations of these deviations with the phase direction are plotted for a numerical model of general anisotropic medium with triclinic/ monoclinic/orthorhombic symmetry

    • Wave propagation in a general anisotropic poroelastic medium: Biot’s theories and homogenisation theory

      M D Sharma

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      Anisotropic wave propagation is studied in a fluid-saturated porous medium, using two different approaches. One is the dynamic approach of Biot’s theories. The other approach known as homogenisation theory, is based on the averaging process to derive macroscopic equations from the microscopic equations of motion. The medium considered is a general anisotropic poroelastic (APE) solid with a viscous fluid saturating its pores of anisotropic permeability. The wave propagation phenomenon in a saturated porous medium is explained through two relations. One defines modified Christoffel equations for the propagation of plane harmonic waves in the medium. The other defines a matrix to relate the relative displacement of fluid particles to the displacement of solid particles. The modified Christoffel equations are solved further to get a quartic equation whose roots represent complex velocities of the four attenuating quasi-waves in the medium. These complex velocities define the phase velocities of propagation and quality factors for attenuation of all the quasi-waves propagating along a given phase direction in three-dimensional space. The derivations in the mathematical models from different theories are compared in order to work out the equivalence between them. The variations of phase velocities and attenuation factors with the direction of phase propagation are computed, for a realistic numerical model. Differences between the velocities and attenuations of quasi-waves from the two approaches are exhibited numerically.

    • Effect of initial stress on reflection at the free surface of anisotropic elastic medium

      M D Sharma

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      The propagation of plane waves is considered in a general anisotropic elastic medium in the presence of initial stress. The Christoffel equations are solved into a polynomial of degree six. The roots of this polynomial represent the vertical slowness values for the six quasi-waves resulting from the presence of a discontinuity in the medium. Three of these six values are identified with the three quasi-waves traveling in the medium but away from its boundary. Reflection at the free plane surface is studied for partition of energy among the three reflected waves. For post-critical incidence, the reflected waves are inhomogeneous (evanescent) waves. Numerical examples are considered to exhibit the effects of initial stress on the phase direction, attenuation and reflection coefficients of the reflected waves. The phase velocities and energy shares of the reflected waves change significantly with initial stress as well as anisotropic symmetry. The presence of initial stress, however, has a negligible effect on the phase directions of reflected waves.

    • Wave propagation in thermoelastic saturated porous medium

      M D Sharma

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      Biot ’s theory for wave propagation in saturated porous solid is modified to study the propagation of thermoelastic waves in poroelastic medium. Propagation of plane harmonic waves is considered in isotropic poroelastic medium. Relations are derived among the wave-induced temperature in the medium and the displacements of fluid and solid particles. Christoffel equations obtained are modified with the thermal as well as thermoelastic coupling parameters. These equations explain the existence and propagation of four waves in the medium. Three of the waves are attenuating longitudinal waves and one is a non-attenuating transverse wave. Thermal properties of the medium have no effect on the transverse wave. The velocities and attenuation of the longitudinal waves are computed for a numerical model of liquid-saturated sandstone. Their variations with thermal as well as poroelastic parameters are exhibited through numerical examples.

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