A Neuromechanical Dispersion Relation for Undulatory Swimming

Many swimmers operate in environments where inertia is unimportant, from low-Reynolds number (Re) flagellates like spermatozoa, to macroscopic vertebrates in overdamped frictional environments, like sand. In these regimes, the kinematic efficiency of the gait (the distance traveled in a single undulatory cycle) is purely geometry dependent - depending on parameters such as the wavenumber k=1/λ, but not the temporal frequency of undulation ω. However, the gait frequency is biologically significant, partly because executing a particular gait more frequently increases the speed of travel. Using a simple neuromechanical model where the organism is treated as a viscoelastic beam actively driven along the body, we identify a unifying relationship between k and ω for organisms ranging from microscopic spermatozoa to nematodes, polychaetes, and macroscopic fish larvae. This 'dispersion relation' scales as ω ∝ k^±2. Quadratic scaling is observed for organisms dominated by external fluid dissipation and inverse-quadratic scaling is observed for organisms dominated by internal dissipation within the body. Thus, frequency and wavenumber can not be set independently, due to fundamental neuromechanical constraints on body movement. Surprisingly, these scaling relations hold for inertial swimming at Re up to 2,000 and worms in viscoelastic gel environments. This suggests that linear rate-dependent drag imposes a universal constraint on gaits in low-inertia undulatory swimming.