Theoretical Aspects and Numerical Computation of the Time-harmonic Green’s Function for an Isotropic Elastic Half-plane with an Impedance Boundary Condition

This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green’s function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Durán et al. (cf. [Numer. Math. 107 (2007) 295–314; IMA J. Appl. Math. 71 (2006) 853–876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important advantage because the obtention of explicit expressions for the surface waves. We show, in addition to the usual Rayleigh wave, another surface wave appearing in some special cases. Numerical results are given to illustrate that. This is an extended and detailed version of the previous article by Durán et al. [C. R. Acad. Sci. Paris, Ser. IIB 334 (2006) 725–731]. Mathematics Subject Classification. 31A10, 35E05, 65T50, 74B05, 74J15. Received June 10, 2008. Revised June 18, 2009. Published online February 23, 2010. Introduction The Green’s functions for elastic half-spaces have been studied by many authors, because of their applicability in important areas of science and engineering such as seismology, geophysics, and structural engineering. In particular, the main motivation of the present study comes from a difficulty that arises in mining engineering, during the rock blasting in an underground mine. When the blasting devices are set out, the engineers attempt to take full advantage of the energy irradiated, but at the same time they desire to prevent the expansive wave from concentrating an important amount of energy at the exploitation areas of the mine, as serious damage could be caused. As a preliminary approach to this problem, we consider a linear isotropic elastic model, where the ground is represented as a half-plane and the underground mine can be described as a local perturbation of the boundary. Additionally, an impedance boundary condition, also named in related literature non-absorbing or passive boundary condition, is imposed in order to have a more general description of the surface dynamics. We propose that the computation of resonant states associated with this domain can provide a basic knowledge about the most probable directions that waves could take to radiate the energy generated.