Ground vibrations due to seismic detonation of a buried source

The problem of vibrations at the free surface of an elastic half-space due to detonation of a buried source is studied here from the viewpoint of the geophysical seismic technique employed in oil exploration. The fundamental assumption of the theory, therefore, is that the pressure at the source is impulsive and a Dirac delta function of time. The depth of the source below the surface of the medium is considered in the present theory as an additional parameter which has hitherto made the buried source problem formidable and, therefore, has limited almost all previous works to the relatively easy problem of surface blasting. An exact formulation is presented by dividing the half-space into a stratum above the level of the buried source and a half-space below it. For boundary conditions at the interface, it is assumed that the vertical displacement and shear stress at the level of the source are continuous while the direct stress is discontinuous. A numerical evaluation of the contour integration in the analysis is presented for the components of vibrations at any point of the free surface for a Poissonian medium. Finally, the analysis provides a theoretical justification for the interesting results obtained from field experiments recorded in an earlier work.     

On the disturbance due to a point source in an inhomogeneous medium

Summary The problem of a point source in an isotropic, inhomogeneous fluid medium is discussed. It is assumed that the density of the fluid is constant and the acoustic velocity varies with depth asc=c ₀(1 +m z) wherem is a constant andc ₀ is, the velocity at the level of the origin. An approximate expression for the field due to a point source in such a medium is obtained when the medium is infinite as well as when it is semi-infinite. It is found that the results obtained agree with the WKB solution of the problem.     

Three-dimensional elastic wave scattering and diffraction due to a rigid cylinder embedded in an elastic medium by a point source

The formal solutions of displacement field to the problem of elastic wave scattering and diffraction due to an infinitely long rigid cylinder embedded in an infinite elastic medium by an impulsive point source have been obtained in the integral form. The integrals for the reflected and the diffracted waves both in the shadow zone and in the illuminated zone are evaluated asymptotically for the early time motion by the Reisdue-Cagniard method and the Saddle-point-Cagniard method.Numerical results of the diffractedP, S andPS waves at a fixed circum-distance from the surface of the rigid cylinder show noticeably that (1) the energy partition for the diffractedS wave is small in comparison with that for the diffractedP wave, (2) the wave form of the diffractedS wave is broader and more diffused than that of the diffractedP wave, (3) the direction of the radial motions of the diffractedP andS waves varies as a function of the observational point, and (4) the energy partition for the diffractedP wave is much smaller than that for the direct or the reflectedP waves.This paper has been presented at the 46th Annual International Meeting of Society of Exploration Geophysicists in Houston, Texas, Oct. 28, 1976.     

Disturbance due to a point source in a layered half-space

Summary In this paper the displacement components due to a compressional point source in a layered half-space consisting of a liquid layer of finite thickness overlying a semi-infinite solid homogeneous medium have been deduced. Two different cases have been considered. In the first case the source is taken in the liquid layer and in the second case the source is taken in the solid layer. The displacement components in cylindrical coordinates have been obtained in closed form in detail at the interface for all values of distance from the source for the second case.     

Numerical simulation on seismic liquefaction by adaptive mesh refinement due to two recovered fields in error estimation

The finite element method is widely employed in numerical analysis of seismic liquefaction at present. However, due to the mesh distortion in strong dynamic action, the results are generally inaccurate and can even stop the calculation. Therefore, an adaptive mesh refinement (AMR) scheme is introduced to improve the numerical results based on a finite element and finite difference coupled dynamic method (FEM–FDM). The changing law about the errors due to the linear and the bilinear recovered fields of coarse meshes, AMR meshes and fine meshes is discussed. The seismic response on soil is compared between different AMR meshes. The AMR strategy can better simulate seismic liquefaction and offer more accurate results than the normal finite element simulation.     

Seismic ground motion in a layered half-space due to a Haskell-type source. I: Theory

In this paper, closed-form analytic expressions for the frequency-wave number domain Fourier amplitudes of the displacement field at the free surface of a layered, anelastic half-space are established. The displacement field is caused by a seismic source described by a shear dislocation propagating with constant velocity over a rectangular fault (Haskell's model). Three-dimensional plane wave propagation is considered in the layered half-space using a propagator-based formalism. The wave radiation from the source is decoupled into P-SV and SH motions and the two problems are treated separately. First, analytic expressions are calculated for the displacement field at the free surface due to unidirectional unit impulses. Then, these expressions are used to compute solutions for the displacement field due to effective point sources associated with a pure strike slip and a pure dip slip. Finally, these solutions are combined and integrated over the rectangular fault area to establish closed-form analytic expressions of the total displacement field at the free surface.     

Transient ULF Electric and magnetic fields of an electrical current source in a medium with continuously varying conductivity

Recent analytical studies of ULF electromagnetic fields in the atmosphere are reviewed. These fields have their origin in the discharge of thunderclouds. The problem for a vertical electrical dipole source played in an atmosphere where the conductivity increases exponentially with altitude is described. The analytical expressions for the electric and magnetic field, which vary in time and space, are approximately obtained by a vector potential. It is seen that the amplitudes of the pulse decrease with both increases of the horizontal distance and the gradient of the conductivity. However, the shapes of the pulse are almost constant.On leave from the Department of Electronic Engineering, Gumma University Kiryu, 376, Japan     

The disturbances in an infinite inhomogeneous medium due to transient forces and twists on the surface of a buried spherical source

Summary In this paper the disturbances in an inhomogeneous medium due to two types of forces (i) transient normal forces, (ii) transient twists in presence of a buried spherical source has been considered. The material of the inhomogeneous medium is assumed to be transversely isotropic. The results are in terms of modified Bessel functions. In a particular case, the equation reduces to an ordinary differential equation of second order.     

Disturbances due to time-dependent body forces in a non-isotropic elastic medium

Summary In this paper, the disturbances produced due to time-dependent body forces in an anisotropic elastic medium resting on a semi-infinite non-isotropic layer have been discussed.     

On the disturbances in an inhomogenous medium due to transient forces and twists on the surface of a burried spherical source

Summary In this paper the displacements and stresses are obtained due to two types of forces: (i) transient normal forces, (ii) transient twists on the surface of a burried spherical source. The material is assumed to be inhomogeneous and spherically aeolotropic. Laplace transform technique has been used here.     

Waves produced in an infinite elastic medium due to normal forces and twists on the surface of a burried spherical source

Summary In this paper, the waves propagated in an infinite medium has been considered by disturbances on the surface of a burried spherical source. The disturbances are produced by two types of forces viz. i) Transient normal force, ii) Transient twist applied on the surface of the source. The solutions are obtained in a very simple form. Graphical representations of some numerical solutions has been added at the end of each problem.     

OnSH type of motion due to body forces and due to stress-discontinuity in a semi-infinite viscoelastic medium

Summary In this paper, surface displacements of theSH type of motion due to time-dependent body forces and due to sudden introduction of a discontinuity in the shearing stress in a semi-infinite, homogeneous, isotropic, viscoelastic medium have been worked out by the use of Laplace and Fourier transforms. The displacements due to a point force and due to a stress discontinuity over a fixed region have been calculated numerically.     

Disturbances in a viscoelastic medium due to impulsive forces on a spherical cavity

Summary The disturbances produced in a visco-elastic medium by impulsive forces on the surface of a spherical cavity inside the medium have been studied.     

Generation of rayleigh waves due to stress-discontinuity in a semi-infinite elastic medium

Summary In this paper, the displacements produced at the surface of a semi-infinite, clastic medium by the sudden creation and subsequent motion of discontinuities in the normal and tangential stresses have been obtained in the form of definite integrals byCagniard's method. Numerical values are given in a particular case.Published with the kind permission of the Director General, Geological Survey of India, Calcutta.     

On the disturbances in a viscoelastic medium due to blast inside a spherical cavity

Summary In this paper we have considered the disturbances in a viscoelastic medium due to a blast of intensityP and durationT sec inside a spherical cavity.     

On the disturbance due to a spherical distortional pulse in an elastic medium

Summary Theoretical expressions are derived for the displacement, velocity and stress in the time domain induced by an axially symmetric shearing stress applied at the inner surface of a spherical cavity in a hornogeneous, isotropic, elastic medium of infinite extent. Theoretical seismograms are computed for a step source and for three sources with exponential decay in time. A satisfactory time-dependence of the source can be obtained by combining the step source with one or more exponentially decaying sources.     

Thermo-elastic interactions in an infinite elastic solid due to a concentrated transient heat source

Summary The present paper is concerned with the determination of thermo-elastic stress and temperature distribution in an infinite elastic solid when it is subjected to a concentrated transient heat source taking into account the effect of coupling. The fundamental partial differential equation for the thermo-elastic potential is solved by means of the operational method. The solutions are obtained for small values of time in terms of known functions such as complementary error functions and associated complementary error functions and have the form of power series with respect to the parameter of coupling . Since this parameter is generally very small for most of the metals, the expressions for stress and temperature distribution have been obtained by considering terms upto linear .     

Axisymmetric wave propagation in multilayered poroelastic grounds due to a transient acoustic point source

This paper deals with the study of axisymmetric wave propagation in various acoustic/porous stratified media coupling configurations. It presents the theoretical developments of a semi-analytical method, its validation for a limit test-case half-space ground, and an extension to a realistic multilayered seabed, when spherical waves are emitted from a transient point source in water.     

The disturbance due to a point source in a homogeneous liquid layer over a heterogeneous liquid half-space

Summary The problem of a periodic point source in a homogeneous liquid layer overlying a heterogeneous liquid half-space is discussed. After obtaining the formal solution, the path of integration for the displacement potential of the layer is transformed from the positive real axis to the positive imaginary axis and the Sommerfeld contour and the latter is further distorted to the modified Sommerfeld contour. The residues of the integrand at the poles contained within the Sommerfeld loop constitute the normal mode solution to the problem. The integrands in the expressions for the integrals along the imaginary axis are expanded in a series of negative powers of exponentials and then some of the terms in these expansions are evaluated approximately. This gives various waves reflected from the interface and the integral along the Sommerfeld loop vanishes. The frequency equation is obtained, also by the principle of constructive interference. An expression for the reflection coefficient at an interface of two liquid media, the upper medium being homogeneous and the lower one inhomogeneous, is obtained.