Half-Space Green’s Function for Lamb’s Problem as Applied in Seismic Geotechnical Engineering

Abstract

The propagation of waves in a homogeneous, isotropic linear elastic half-space is investigated herein. A buried point pulse generates a vibration in an arbitrary direction. Both source and receiver points are located in the interior of the three-dimensional domain. To solve the elastodynamic problem, also known as Lamb's problem, the source image and the superposition principle are both used to derive a numerical transient solution. Accordingly, the transient response of the problem in the time domain can be considered as the superposition of the responses to the real and imaginary sources in the full-space and some additional vertical loads on the surface of the half-space. The additional vertical loads are distributed over a rectangular area on the surface of the half-space and are space- and time-dependent functions that vary with time as the Heaviside step, the Dirac delta, and derivatives of the Dirac delta functions. The motion at depth due to a point source applied on the surface is obtained using some well-known approaches reported in the literature. To achieve the Laplace transform displacement, Helmholtz potentials have been employed for the displacement field and the Laplace transform wave equation as well as the Hankel transform of boundary conditions have been satisfied. The inverse Laplace transform (time-domain solutions) is found via a modified version of some other methods reported in the literature. The solutions obtained in this way automatically satisfy the traction-free boundary conditions over the surface of the half-space and can be implemented in the three-dimensional time-domain boundary element method (BEM), and no discretization of the ground surface is needed.