The flow of a viscous fluid produced by a plane boundary moving in its own plane with a sinusoidal variation of velocity is considered. An analytical solution describing the flow at small and large times after the start of the boundary is obtained by the Laplace transform method. The solution gives not only the steady solution but also the transient solution. The time required to attain steady flows for the cosine oscillation of the boundary is one-half cycle and it is a full cycle for the sine oscillation of the boundary. (C) 1999 Elsevier Science Ltd. All rights reserved.