Abstract We investigate numerically the vibrational behavior of an elastic structure containing an inviscid fluid with a filling hole where the pressure is prescribed. The underlying mathematical model is detailed and its spectra is characterized. The finite element method relies upon the added‐mass formulation of Morand and Ohayon but avoids the explicit assembly of the dense added‐mass matrix. A parametric study allows to characterize the system's response to dimensionless parameters in terms of eigenfrequencies. The resulting mode shapes are then physically discussed, with emphasis on the presence and the behavior of singular modes caused by specific boundary conditions in the fluid.