  Organisms that locomote by propagating waves of body bending can maintain
performance across heterogeneous environments by modifying their gait frequency
$\omega$ 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' $\omega\propto k^{\pm2}$. 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 $\omega$ dependence of the various underlying force terms and how
their relative importance changes with the external environment and the
neuronally commanded gait.
