  Over a sufficiently long period of time, or from an appropriate distance, the
motion of many swimmers can appear smooth, with their trajectories appearing
almost ballistic in nature and slowly varying in character. These long-time
behaviours, however, often mask more complex dynamics, such as the side-to-side
snakelike motion exhibited by spermatozoa as they swim, propelled by the
frequent and periodic beating of their flagellum. Many models of motion neglect
these effects in favour of smoother long-term behaviours, which are often of
greater practical interest than the small-scale oscillatory motion. Whilst it
may be tempting to ignore any yawing motion, simply assuming that any effects
of rapid oscillations cancel out over a period, a precise quantification of the
impacts of high-frequency yawing is lacking. In this study, we systematically
evaluate the long-term effects of general high-frequency oscillations on
translational and angular motion, cast in the context of microswimmers but
applicable more generally. Via a multiple-scales asymptotic analysis, we show
that rapid oscillations can cause a long-term bias in the average direction of
progression. We identify sufficient conditions for an unbiased long-term effect
of yawing, and we quantify how yawing modifies the speed of propulsion and the
effective hydrodynamic shape when in shear flow. Furthermore, we investigate
and justify the long-time validity of the derived leading-order solutions and,
by direct computational simulation, we evidence the relevance of the presented
results to a canonical microswimmer.
