Abstract
For sustained swimming and flights, vertebrates and insects oscillate their propulsors periodically within a narrow range of Strouhal number (St), a dimensionless quantity describing the rate and density of the motion, suggesting a close relationship between the range and cruising optimality. The persistence of this range across size and fluids has puzzled biologists and engineers, resulting in multiple interpretations of its cause. Here, we propose that the optimal St range is largely constrained by power output efficiency of the trailing edge of the caudal fin. A mathematical model of the periodic wake of the trailing edge, which defines the proportion of power lost without contributing to propulsion, predicts that such energy loss is minimal in the observed range of St preferred by fish. The constraints apply across a range of Reynolds number in cruising fish. The same constraints dictate the optimal speed across a wide range of swimmers, in combination with morphological characteristics. Other factors such as drag properties also affect the optimal swimming speed, but probably to a smaller extent. The result that the geometry of periodic waveforms is key to cruising optimality provides an additional angle to study animal locomotion in fluids and related bioinspired robotics.