Decomposition of measured ground vibrations into basic soil waves

Man-made vibrations from different types of sources are usually measured on the surface of the ground or building. The measured signal is always the superposition of all travelling basic waves. For a homogeneous half space there are three basic waves – the Compressional (P-wave), Shear (S-wave) and Rayleigh wave (R-wave). Depending on the measuring equipment, only the accelerations or velocities in time of the superposed wave can be measured, but not the distribution of the individual basic waves. Additional problems are that each of the basic waves has its own velocity, besides the body and surface waves have different attenuation laws. By using the rules of superposition of harmonic waves and also the propagation laws of the P-, S- and R-waves, it should be theoretically possible to split the measured superposed signal into the basic waves, because mathematically a system of equations can be assembled which describes the displacements at multiple measuring points in time. In this paper this problem has been solved for a homogenous, elastic and isotropic soil, which is disturbed by a harmonically oscillating disc on the surface. A numerical simulation was performed using a finite element method. The displacements in time were recorded in 10 points on the surface and a system of superposed equations was assembled and solved. The findings prove that each of the three basic waves has its own phase shift with the source, something which was not known before