This work deals with the linear two-dimensional free vibrations of an inflated cylindrical membrane. An air-inflated membrane is considered first. Vibration frequencies and modes are determined for various cases. The lowest mode shape is anti-symmetric. In the rest of the work, the membrane is inflated with water. In some cases there is a reservoir of water on one side of the membrane. The membrane equation of motion is solved using a finite difference method, and the hydrodynamic pressures on the membrane, caused by the motion of the internal and external water domains, are treated by the boundary element method. The effects of the membrane parameters, internal and external water head, and density of the membrane on the lowest four frequencies are illustrated. For the membrane without the outside water, the first two natural frequencies agree well with experimental values. The existence of the upstream head has a significant influence on the frequencies, and the mode shapes are shown to be tilted toward the downstream side of the membrane. / Ph. D.