Computation of axisymmetric vibration transmission using a well-conditioned system for elastic layers over a half–space

In the context of range-independent solid media, we propose a well-conditioned dynamic stiffness matrix for an elastic layer sitting over an elastic half-space. This formulation overcomes the well-known problem of numerical ill-conditioning when solving the system of equations for deep-layered strata. The methodology involves the exact solutions of transformed ordinary differential equations in the wavenumber domain, namely a projection method based on the transformed equations with respect to the depth coordinate. By re-arranging the transformed equations, the solutions remain numerically well-conditioned for all layer depths. The inverse transforms are achieved with a numerical quadrature method and the results presented include actual displacement fields in the near-field of the load in plane-strain and three-dimensional axisymmetric cases. Verification against finite element method (FEM) calculations demonstrates the performance and complexity of the two approaches.