SUMMARYNumerical simulations of a heaving airfoil undergoing non-sinusoidal motions in an incompressible viscous ow 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 coe cients, and propulsive e ciency 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 e ciencies 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.