The aim of this thesis is to investigate and develop alternative methods of analyzing problems in dynamic soil-structure-interaction. The main focus is the major difficulty posed by such an analysis - the phenomenon of waves which radiate outward from the excited structures towards infinity. In numerical calculations, only a finite region of the foundation metium is analyzed and something is done to prevent the outgoing radiating waves to reflect from the regions's boundary. The prosent work concerns itself with the study of such effects, using the finite element method, and artificial transmitting boundary at the edges of the computational mesh. To start with, an investigation of the main effects of the interaction phenomena is carried out employing a widely used model, considering dynamic stiffness of the unbounded soil as frequency independent. Then a complete description, with the comments put forth, follows for different physical and mathematical formulations of transmitting boundary schemes. Both formulations, applicable for time and frequency domain analysis are considered emphasizing more the temporally and spatially local boundaries. From this class Doubly Asymptotic (DA) and Multi-Directional (MD) transmitting boundary are found attractive.An attempt is made here to give a different formulation and implementation of the two components of DAMB boundary. After an investigation of physical models in foundation vibration analysis, the DA boundary for three-dimentional analysis is formulated based on the one-dimensional wave propagation in a cone model resulting in the amplitude decay of inversely proportion to the distance travelled. So the transmitting boundary for body waves is constructed in analogy to springs and dashpots connecting the boundary nodes to a rigid base. For absorbing surface waves the one-dimensional model based on the amplitude decay of inversely proportion to the square-root of the distance travelled is formulated which also results in springs and dashpots at the lateral boundary. Based on the radiation criterion, for two-dimensional analysis body waves dacay in the same way as R-waves for three-dimensional analysis . Concerning surface waves, they propagate with constant amplitude and the lateral boundary has zero stiffness and can be modelled with normal impedance or standard viscous boundary. This boundary condition. This boundary condition represents an attempt to construct a local stiffness for the unbounded soil domain.