An approximate solution for the displacement and the asymptotic far pressure field of a thin, elastic, finite cylindrical shell immersed in a perfect fluid at rest is proposed. Two different problems are considered: the baffled shell which is free to vibrate within a finite region and is rigidly bonded at its ends to two semi-infinite cylindrical rigid screens and the constrained shell, which is free to vibrate both within and outside the finite region but is constrained on the boundary between them. In each case, the shell is excited by a point unit force located in the finite region. A boundary element method is proposed for the solution of the exact equations. A semi-analytical method for the constrained shell is developed. The displacement of the baffled shell is approximated by that of the constrained shell, restricted to the domain limited by the constraints. It is easy to obtain the sound pressure field radiated by the shell, because the sound pressure radiated at infinity is calculated from the Fourier transform of the normal component of the displacement. Numerical examples show that the exact and approximated solutions agree well.