Green’s function for a transversely isotropic multi-layered half-space: an application of the precise integration method

Abstract

A numerical approach is presented for the evaluation of Green’s function for a multi-layered half-space. The formulation is unconditionally stable and has the computational simplicity with only the algebraic calculations involved. It imposes no limit to the thickness of the layered medium and the magnitude of frequency. In the analysis, the Fourier–Bessel transform and precise integration method (PIM) are employed. Here, the Fourier–Bessel transform is employed to convert the wave motion equation from the spatial domain to the wavenumber domain, which derives a second-order ordinary differential equation. Then, a dual vector representation of wave motion equation is introduced to reduce the second-order differential equation to first order. It is solved by PIM. Finally, the Green’s function in the wavenumber domain is obtained. In order to calculate the Green’s function in the spatial domain, the inverse Fourier–Bessel transform over the wavenumber is employed for deriving the solutions, which results in a one-dimensional infinite integral with Bessel functions involved. An adaptive Gauss quadrature is used for the evaluation of this integral. Numerical examples are provided to demonstrate the capability of the proposed method. Comparisons with other methods are made. Very promising results are obtained.