The swimming of a micro-organism by the propagation of helical waves on a long slender flagellum is analysed. The model developed by Higdon (1979) is used to study the motion of an organism with a spherical cell body (radius A) propelled by a cylindrical flagellum (radius a, length L).

The average swimming speed and power consumption are calculated for helical waves (amplitude α, wavenumber k). A wide range of parameter values is considered to determine the optimal swimming motion. The optimal helical wave has ak ≈ 1, corresponding to a pitch angle of 45°. The optimum number of waves on the flagellum increases as the flagellar length L/A increases, such that the optimum wavelength decreases as L/A increases. The efficiency is relatively insensitive to the flagellar radius a/A. The optimum flagellar length is L/A ≈ 10.

The results are compared to calculations using two different forms of resistance coefficients. Gray-Hancock coefficients overestimate the swimming speed by approximately 20% and underestimate the power consumption by 50%. The coefficients suggested by Lighthill (1976) overestimate the swimming speed for large cell bodies (L/A < 15) by 20% and underestimate for small cell bodies (L/A > 15) by 10%. The Lighthill coefficients underestimate the power consumption up to 50% for L/A < 10, and overestimate up to 25% for L/A > 10. Overall, the Lighthill coefficients are superior to the Gray-Hancock coefficients in modelling swimming by helical waves.