Coupled vibrations of a membrane in contact with fluid are found in many engineering applications. This paper studies the interactions of a compressible and inviscid fluid on a circular elastic membrane at the top of a rigid cylindrical container fully filled with fluid. The fluid velocity potential is calculated, assuming harmonic motion, from the Helmholtz wave equation by a method of separations of variables. Next, the fluid pressure is derived from Bernoulli’s linearized momentum equation. Considering the dynamic equation of the membrane, and after an integration procedure of the squared integrands, considering the different terms expressed in series expansions and taking into account the orthogonality properties of the functions involved, an analytical expression is deduced for the calculus of the coupled natural frequencies. The influence of different parameters on these frequencies is studied, such as drum radius, drum height, membrane thickness, membrane tension, and fluid density. Also, the effects of the sound speed or fluid compressibility on these frequencies are analyzed. Furthermore, applying the generalized work equation for the membrane dynamic equation and after integration of the different terms expanded in series and considering the orthogonality properties of the integrand functions, a system is obtained for the calculus again of the natural frequencies that coincides with those calculated with the previous method, and also the fluid mass parameters of the different modes are obtained analytically in a simple way as function of the excitation frequency. Validation of this method is given by comparing the results with other authors and theories.