This paper is a study of the vibration of an elastic plate under a time-harmonic point force. The plate is infinite in extent and rests on an elastic foundation. This study is made on the basis of the improved (Timoshenko) plate theory. The mathematical problem is to seek a fundamental solution of the time-reduced plate equation of the improved plate theory. Such a fundamental solution is constructed by the distributional Fourier transform method. From the explicit expressions of the fundamental solution is examined in detail the behavior of the fundamental singularity as a function of the vibration frequency and the foundation stiffness. Also, it is found that several types of plate resonance may occur depending upon the combination of the vibration frequency and the foundation stiffness. The paper is concluded with a discussion on how to construct integral representations for steady-state vibrations (Green's functions) for finite plates. [Research supported in part by the National Science Foundation.]