Displacement produced in an elastic half-space by the impulsive torsional motion of a circular ring source

Summary In this paper the problem of disturbance in an elastic semi-infinite medium due to the torsional motion of a circular ring source on the free surface of a medium are studied. Two cases, when the medium is either homogeneous or inhomogeneous, are treated. In order to solve the problem, the Laplace transform and the Hankel transform and the Laplace inversion by Cagniard's method as modified byDe Hoop (1959) are applied. Finally, the integrals for displacement are evaluated numerically. The displacement on the free surface as a function of time is shown by means of graphs, in the case of both a homogeneous and an inhomogeneous medium, indicating clearly the variation in displacement due to the presence of an inhomogeneity.     

A note on the approximate determination of the dispersion of Love waves in a laterally nonhomogeneous layer lying over a homogeneous half-space

Rayleigh's principle and the concept of the local wave number have been utilised for the approximate determination of the dispersion of Love waves propagating in a laterally heterogeneous layer lying over a homogeneous half-space. The shear wave velocity and the rigidity in the surface layer have been assumed to decrease with the increase of the lateral distance from the origin. The range of validity of the dispersion equation obtained by this method has been examined critically. It was found that: (a) for existence of Love waves the minimum value of shear wave velocity in the layer must be less than that in the matter below, and (b) the phase velocity of Love waves decreases with the increase of the lateral distance from the origin.     

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.     

On the displacements produced in a porous elastic half-space by an impulsive line load. (Non-dissipative case)

Summary On the basis of Biot's dynamical theory of poroelasticity the disturbance produced by an impulsive line load in a porous elastic half-space is studied. Using the Laplace-Fourier transform we have solved for the displacement potentials in terms of which the displacements and stresses in the transformed space are expressible. The expressions for the solid displacements in the interior as well as on the surface of the half-space are obtained by Cagniard's technique. The displacements are expressed in terms of six algebraic terms — three of which are identified as the disturbance due to specific wave fronts and the others represent the head wave contributions. When specialized our results agree with those for the common elastic half-space.     

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.     

Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function

A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly.     

The transient disturbance produced in an elastic layer by a buried spherical source

Summary In this paper approximate expressions for the surface displacements due to a buried spherical source inside an isotropic elastic layer with one surface stress-free and the other (i) rigid, (ii) stress-free, have been obtained. Graphical representations for the surface displacements at a large distance have also been found.     

Disturbance produced in a semi-infinite anisotropic elastic medium by an impulsive twist applied on the plane surface

Summary In this paper the displacement due to an impulsive twist produced by impulsive shearing stresses acting on a finite circular area on the plane surface of a semi-infinite solid of homogeneous anisotropic material has been found.Paper presented by Dr. A. K. Mitra of Jadavpur University.     

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.     

Weighted residual method for diffraction of plane P-waves in a 2-D elastic half-space III: on an almost circular arbitrary-shaped alluvial valley

Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV- scattered waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by Lee and Liu. This paper uses their technique of defining these stress-free scattered waves, which Brandow and Lee previously used to solve the problem of the scattering and diffraction of these in-plane waves on an almost-circular surface canyon that is arbitrary in shape, to the study of the scattering and diffraction of these in-plane waves on an almost circular arbitrary-shaped alluvial valley.     

3D scattering of obliquely incident plane SV waves by an alluvial valley embedded in a fluid-saturated,poroelastic layered half-space

The indirect boundary element method is used to study the 3D dynamic response of an infinitely long alluvial valley embedded in a saturated layered half-space for obliquely incident SV waves. A wave-number transform is first applied along the valley’s axis to reduce a 3D problem to a 2D plane strain problem. The problem is then solved in the section perpendicular to the axis of the valley. Finally, the 3D dynamic responses of the valley are obtained by an inverse wave-number transform. The validity of the method is confirmed by comparison with relevant results. The differences between the responses around the valley embedded in dry and in saturated poroelastic medium are studied, and the effects of drainage conditions, porosity, soil layer stiffness, and soil layer thickness on the dynamic response are discussed in detail resulting in some conclusions.