A Harmonic Half-Space Fundamental Solution

The fundamental solution for a harmonic point force in a half-space of a linear elastic media is established. The fundamental solution consists of the displacements for both a vertical and a horizontal acting point force. The solution is given in cylindrical coordinates.

The solution is based on continuous wavesystems through a second order Fourier transform i.e. a Hankel transform of the horizontal geometrical coordinates.

The solution of all displacement components from the wave equation is established as infinite horizontal wavenumber integrals, here inverse Hankel transforms. The solution of each displacement component consists of both an analytical part describing the full-space solution and a part describing the influence of the free surface.

Accurate numerical schemes for integration of infinite wavenumber integrals are know to be difficult to establish. Therefore, a complex contour integration technique is used to set up an alternative solution method that for certain source-receiver configurations is more efficient than the direct integration method of the infinite wavenumber integral. It is shown that the two integration methods are efficient in each their complementary parameter domain.

An adaptive numerical integration scheme is set up and tested on an infinite wavenumber integral with known analytical solution. The established adaptive numerical integration scheme is used for computation of one fundamental solution displacement component as an example.

Possibilities of alternative solution methods and alternative numerical integration schemes and aspects by use of the established fundamental solution in a boundary element method code are finally discussed.