Numerical simulations of a heaving airfoil undergoing non-sinusoidal motions in an incompressible viscous flow is presented. In particular, asymmetric sinusoidal motions, constant heave rate oscillations, and sinusoidal motions with a quiescent gap, are considered. The wake patterns, thrust force coefficients, and propulsive efficiency at various values of non-dimensional heave velocity are computed. These have been compared with those of corresponding sinusoidal heaving motions of the airfoil. It is shown that for a given non-dimensional heave velocity and reduced frequency of oscillation, asymmetric sinusoidal motions give better thrust and propulsive efficiencies in comparison to pure harmonic motion. On the other hand, constant rate heave motion do not compare favourably with harmonic motion. A train of sinusoidal pulses separated by a quiescent gap compares favourably with a pure sinusoidal motion, but with the notable exception that the quiescent gap induces a discontinuity that induces large impulses to the wake pattern. Copyright © 2005 John Wiley & Sons, Ltd.