Aerodynamic focusing of particles in spatially periodic flows is investigated numerically in model potential two-dimensional flows with axial harmonic modulations of the stream function. A distinction is made between cases where the two walls oscillate approximately in phase (antisymmetric) or with a phase difference of 180 degrees (symmetric). The latter case has been most widely used in the past. It focuses all particles smaller than a critical size into a unique terminal streamline, the axis or plane of symmetry. It also alternates regions of high and low speed, with strong adverse pressure gradients along the walls, and tendency towards early flow separation. Antisymmetric walls focus particles along oscillating terminal trajectories (as in earlier work by Maxey), with amplitudes and phase shifts that depend on particle size. Because the flow cross-section is almost constant, pressure variations along the walls are greatly reduced with respect to the symmetric case, with an expected tendency to delay flow separation and turbulent transition. Analysis in the limit of a small particle inertia parameter S reveals that the mean drift velocity towards the terminal trajectory is proportional to S epsilon(2), where epsilon is a corrugation parameter measuring the amplitude of the wall oscillation in wavelength units. Surprisingly, focusing is as effective in symmetric and antisymmetric walls, even though acceleration is predominantly along streamlines in the first case, while it is dominantly centrifugal in the latter. However, because the maximum expected epsilon value attainable without flow separation is larger in antisymmetric than in symmetric walls, the latter would focus considerably better at fixed Re, or could presumably be run at considerably higher Re at fixed epsilon. The symmetric flows studied have high particle transmission efficiency even at very high S (similar to 100), and can focus particles with apparently no upper size limit. Antisymmetric walls also focus arbitrarily large particles, but good transmission efficiency requires using special particle introduction schemes. (C) 2005 Elsevier Ltd. All rights reserved.