Computation of the Green’s function associated with the elasticity system in a half-plane with impedance boundary condition

Abstract

An effective numerical analytical method for calculating the Green’s function associated with the time-harmonic
elasticity system in a half-plane is presented, where an impedance boundary condition has been imposed. This need of calculation arises when studying resonant states in underground mining and seismological engineering. The proposed method is based on the work done by Durán, Muga and Nédélec for the Helmholtz’s equation. The desired Green’s function is expressed as sum of two terms, namely, the first one associated with the whole plane and the second taking into account the half-plane and the boundary conditions. Special attention requires the study of the poles of this last term, which are associated with surface waves. We have found that in addition to the well-known Rayleigh pole, under certain conditions on the impedance and the pulsation, an additional pole appears, which produce another surface wave.