The initial value problem related to axisymmetric forced oscillations of a rigidly rotating inviscid fluid enclosed in a finite circular cylinder is examined in linear approximation with the aid of the Laplace transform technique. An impulsive starting motion is considered. The solution consists of a ‘periodic’ motion which oscillates with the forcing frequency, together with a doubly infinite set of inertial modes whose presence is determined by the initial conditions and whose frequencies form a dense set in the range (−2ω, 2ω), where ω is the angular velocity. The nature of the periodic or ‘steady-state’ part of the solution is strongly dependent on the precise value of the forcing frequency α (α > 0) when α ≤ 2ω. In particular the system will resonate if α equals any one value of the dense set of resonant frequencies. It is shown that no internal sets of discontinuities in velocity or velocity gradient are present in the inviscid flow for finite times. Effects of viscosity on the inviscid solution are also discussed, and it is argued that when the inertial modes decay the steady-state flow will contain pseudo-random patterns of internal shear layers for some values of α < 3ω. It seems possible that these shear layers may be interpreted as owing their existence indirectly to viscosity.