  Biological systems can rely on collective formation of a metachronal wave in
an ensemble of oscillators for locomotion and for fluid transport. We consider
one-dimensional chains of phase oscillators with nearest neighbor interactions,
connected in a loop and with rotational symmetry, so each oscillator resembles
every other oscillator in the chain. Numerical integrations of the discrete
phase oscillator systems and a continuum approximation show that directional
models (those that do not obey reversal symmetry), can exhibit instability to
short wavelength perturbations but only in regions where the slope in phase has
a particular sign. This causes short wavelength perturbations to develop that
can vary the winding number that describes the sum of phase differences across
the loop and the resulting metachronal wave speed. Numerical integrations of
stochastic directional phase oscillator models show that even a weak level of
noise can seed instabilities that resolve into metachronal wave states.
