A theoretical and experimental study is presented of the viscoplastic version of the Stokes problem, in which a oscillating wall sets an overlying fluid layer into one-dimensional motion. For the theory, the fluid is taken to be described by the Herschel-Bulkley constitutive law, and the flow problem is analogous to an unusual type of Stefan problem. In the theory, when the driving oscillations are relatively weak, the overlying viscoplastic layer moves rigidly with the plate. For sufficiently strong oscillations, the fluid yields and numerical solutions illustrate how localized plug regions coexist with sheared regions and migrate vertically through the fluid layer. For the experiments, a layer of kaolin slurry in a rectangular tank is driven sinusoidally back and forth. The experiments confirm the threshold for shearing flow, equivalent to a balance between inertia and yield-stress. However, although kaolin is well described by a Herschel-Bulkley rheology, the layer dynamics notably differs between theory and experiments, revealing theological behaviour not captured by the steady flow rule. (C) 2008 Elsevier B.V. All rights reserved.