This paper considers the two-dimensional laminar boundary layer on an infinite flat plate normal to an oncoming stream, the plate making transverse oscillations in its own plane. The exact solution is shown to depend on a single ordinary differential equation, containing the frequency of oscillation as a parameter. Series methods are employed to evaluate the solution for small and large values of the frequency, and enough terms are calculated to give the solution with satisfactory accuracy over the whole frequency range. It is observed that the solution satisfies the full Navier-Stokes equations.

For a cylinder of arbitrary section making transverse or rotational oscillations, it is shown how the results for a flat plate can be used to describe the boundary layer in the neighbourhood of the front stagnation point. Difficulties which arise in extending the solution to cover the remainder of the cylinder are discussed, and an estimate is made of the fluctuating torque on a circular cylinder making transverse oscillations.