Exact solutions for some oscillating motions of an Oldroyd-B fluid due to an oscillating circular cylinder are established as Fourier-Bessel series in terms of some suitable eigenfunctions. These solutions, presented as sum of steady-state and transient solutions, reduce to the similar solutions for Maxwell, second grade and Newtonian fluids as limiting cases. They describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the starting solutions tend to the steady-state solutions which are periodic in time and independent of initial conditions. Finally, the required times to attain the steady-state for cosine and sine oscillations of the boundary are obtained by graphical illustrations. These times decrease if the frequencies of the velocity of boundary increase. (c) 2008 Elsevier B.V. All rights reserved.