  In addition to conventional planar and helical flagellar waves, insect sperm
flagella have also been observed to display a double-wave structure
characterized by the presence of two superimposed helical waves. In this paper,
we present a hydrodynamic investigation of the locomotion of insect spermatozoa
exhibiting the double-wave structure, idealized here as superhelical waves.
Resolving the hydrodynamic interactions with a non-local slender body theory,
we predict the swimming kinematics of these superhelical swimmers based on
experimentally collected geometric and kinematic data. Our consideration
provides insight into the relative contributions of the major and minor helical
waves to swimming; namely, propulsion is due primarily to the minor wave, with
negligible contribution from the major wave. We also explore the dependence of
the propulsion speed on geometric and kinematic parameters, revealing
counter-intuitive results, particularly for the case when the minor and major
helical structures are of opposite chirality.
