A note on the dynamic and static displacements from a point source in multilayered media

A simple and unified approach is presented to solve both the elasto-dynamic and elasto-static problems of point sources in a multi-layered half-space by using the Thompson-Haskell propagator matrix technique. It is shown that the apparent incompatibility between the two is associated with the degeneracy of the dynamic problem when ω = 0 and both can be handled uniformly using the Jordan canonical forms of matrices. We re-derive the propagator matrices for both the dynamic and static cases. We then show that the dynamic propagator matrix and the solution converge to their static counterparts as ω → 0. Satisfactory static deformation can be obtained numerically using the dynamic solution at near-zero frequency.