  Swimming eukaryotic microorganisms such as spermatozoa, algae and ciliates
self-propel in viscous fluids using travelling wave-like deformations of
slender appendages called flagella. Waves are predominant because Purcell's
scallop theorem precludes time-reversible kinematics for locomotion. Using the
calculus of variations on a periodic long-wavelength model of flagellar
swimming, we show that the planar flagellar kinematics maximising the
time-averaged propulsive force for a fixed amount of energy dissipated in the
surrounding fluid correspond for all times to waves travelling with constant
speed, potentially on a curved centreline, with propulsion always in the
direction opposite to the wave.
