The dynamic response and wave propagation behavior resulting from anti-plane diffraction around an elastic wedge-shaped medium with an arbitrary-shaped cylindrical rigid foundation at its vertex has been studied. Numerical computation of the wave displacement field is carried out on and near the rigid foundation surfaces using the method of weighted-residuals, also known as the moment method. The wave displacement fields for the cases of elliptic, circular, rounded-rectangular and flat-elliptic rigid foundations are presented. As in previous works, the analysis demonstrates that the resulting surface displacement depends on many factors, including 1) the angle of the wedge-shaped half space, 2) the density (measure of rigidity) of the foundation, its geometry or shape at the vertex, 3) the frequencies of the incident waves, 4) the angle of incidence, and 5) the material properties of the media. The analysis provides results that help to explain geophysical observations regarding the amplification of seismic energy as a function of site conditions.