Scattering of elastic waves in cracked media using a finite difference method

This paper presents a simple, flexible way of introducing stress-free boundary conditions for including cracks and cavities in 2D elastic media by a finite difference method (FDM). The surfaces of cracks and cavities are discretized in a staircase on a rectangular grid scheme. When zero-stress is applied to free surfaces, the resulting finite difference schemes require a set of adjacent fictitious points. These points are classified based on the geometry of the free surface and their displacement is computed as a prior step to later calculation of motion on the crack surface. The use of this extra line of points does not involve a significant drain on computational resources. However, it does provide explicit finite difference schemes and the construction of displacement on the free surfaces by using the correct physical boundary conditions. An accuracy analysis compares the results to an analytical solution. This quantitative analysis uses envelope and phase misfits. It estimates the minimum number of points per wavelength necessary to achieve suitable results. Finally, the method is employed to compute displacement in various models with cavities in the P-SV formulation. The results show suitable construction of the reflected P and S waves from the free surface as well as diffraction produced by these cavities.