Dispersion Relations for Active Undulators in Overdamped Environments
Christopher J. Pierce, Daniel Irvine, Lucinda Peng, Xuefei Lu, Hang Lu, Daniel I. Goldman
TL;DR
This work identifies a unifying dispersion relation between undulation frequency $\omega$ and wavenumber $k$ for overdamped locomotors, observed across nematodes and diverse environments. An active viscoelastic beam model, with internal and external dissipation and a phase-shifted neuromuscular input, yields $\operatorname{Re}\{\omega(k)\}$ and $\operatorname{Im}\{\omega(k)\}$ and reveals two scaling regimes: $\omega \propto k^{-2}$ in internal-dissipation-dominated conditions and $\omega \propto k^{2}$ when external dissipation dominates; stability imposes $k<1$. The model introduces regime constants $\alpha=(2\pi t_c s_c^2)^{-1}$ and $\beta=2\pi s_\eta^2/t_c$, linking frequency scales to wavelength $\lambda$ via $\omega \approx \alpha \lambda^2$ or $\omega \approx \beta/\lambda^2$, respectively. Experimental data on $\omega(k)$ from C. elegans, together with cross-species evidence from spermatozoa and fish larvae, support this unified framework, suggesting that environmental rheology and gait control co-determine how organisms adjust locomotor parameters in overdamped regimes.
Abstract
Organisms that locomote by propagating waves of body bending can maintain performance across heterogeneous environments by modifying their gait frequency $ω$ or wavenumber $k$. 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' $ω\propto k^{\pm2}$. A model treating the organisms as actively driven viscoelastic beams reproduces the experimentally observed scaling. The relative strength of rate-dependent dissipation in the body and 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 external environment and the neuronally commanded gait.