Abstract Free vibrations of circular cylindrical shells and tubes completely filled with a dense fluid and partially immersed in a different fluid (liquid) having a free surface are studied. Elastic shell constraints, varying from simply supported to clamped ends are assumed. Fluids are assumed to be stationary, inviscid and incompressible. The fluid outside the shell is assumed to be unlimited in the radial direction and limited in the vertical direction by a rigid bottom and a free surface. The effect of free surface waves is considered, so that both sloshing and bulging modes of the system are investigated. A velocity potential is used to describe the fluid oscillations, and the Rayleigh–Ritz method has been extended to the case of fluid–structure interaction to obtain the solution of the coupled system.