Harmonic Green’s functions for a thin semi-infinite plate with clamped or free edges are developed starting from either simply supported or roller supported solutions and applying corrections to account for boundary excitation. This is achieved by connecting the solutions in terms of polar coordinates with the solutions in Cartesian coordinates. The formal solutions in the form of improper wave-number integrals are numerically evaluated using adaptive Clenshaw–Curtis integration. Alternate solutions obtained for each boundary condition compare well. Only harmonic point loads are considered in this article but the methodology may be extended to moment excitation and distributed loads. The methodology developed here will form the basis for advancing the ray tracing technique for vibration analysis of finite plates.