Dispersion relations for active undulators in overdamped environments

Organisms that locomote by propagating approximately sinusoidal waves of body bending maintain performance across different environmental substrates by modifying the frequency ω or wave number k of their gait. We identify a unifying relationship between these parameters for overdamped undulatory swimmers (including nematodes, spermatozoa, and mm-scale fish) moving in diverse environmental rheologies, in the form of an active "dispersion relation" ω∝k^{±2}. A model treating the organisms as actively driven viscoelastic beams in a surrounding fluid reproduces the experimentally observed scaling. The relative strength of rate-dependent dissipation in the body and in the environment determines whether k^{-2} or k^{2} scaling is observed. The existence of these scaling regimes reflects the k and ω dependence of the various underlying force terms and how their relative importance changes with the composition of the external environment and the parameters of neuronally commanded gait. In the regime where the body dissipation dominates, the application of boundary conditions does not introduce an explicit dependence on the body length, as would be expected in other wave systems in physics. Hence, mechanics constrains the relationship between the gait parameters but allows for their continuous variation along the dispersion curve.