We consider the two-dimensional problem of the free plate vibrations, affixed to the bottom of a water reservoir and partially protruding above the surface of the liquid. The bottom of the reservoir is assumed to be rigid and the surface of the liquid-free. The plate is rigidly fixed to the bottom and is free (or rigidly fixed, etc) on it's upper end. The exact analytical representation for the acoustical field in the liquid and for the displacements of the plate is obtained with account of their contact. In order to get this representation the full mathematical statement of this problem is given. It is considered to be a boundary-contact problem in a medium. On the basis of this solution we construct the dispersion equation for the eigen frequencies of the plate below the cutoff frequency of the wave guide. The influence of the liquid level on the eigen frequencies and on the eigen functions is analysed.