The natural frequencies of annular plates on an aperture of an infinite rigid wall and in contact with a fluid on one side are theoretically obtained by using the added mass approach. The fluid is assumed to be incompressible and inviscid and the velocity potential describes its irrotational motion. The Hankel transform is applied to solve the fluid-plate coupled system; boundary conditions are expressed by integral equations. Mode shapes are first assumed not to be modified by the fluid. Accurate numerical results are given for different plate boundary conditions; they are suitable for engineering applications. The accuracy of the assumed-modes approach is theoretically studied by using the Rayleigh-Ritz method that removes the simplifying hypothesis that dry and wet mode shapes are the same. Eigenfunctions of the plate vibrating in vacuum are assumed as admissible functions and the Rayleigh quotient for coupled vibration is used to obtain a Galerkin equation. It was found that the fundamental mode and frequency, for all the plate boundary conditions considered, is well estimated by the assumed-modes approach; higher modes are computed with less accuracy by this formulas and for some enhanced applications the Rayleigh-Ritz approach is necessary.