We employ a model system, called rowers, as a generic physical framework to define the problem of the coordinated motion of cilia (the metachronal wave) as a far from equilibrium process. Rowers are active (two-state) oscillators in a low Reynolds number fluid, and interact solely through the forces of hydrodynamic origin. In this work, we consider the case of fully deterministic dynamics, find analytical solutions of the equation of motion in the long-wavelength (continuum) limit, and investigate numerically the short-wavelength limit. We prove the existence of metachronal waves below a characteristic wavelength. Such waves are unstable and become stable only if the sign of the coupling is reversed. We also find that with normal hydrodynamic interaction, the metachronal pattern has the form of stable trains of traveling wave packets sustained by the onset of anti-coordinated beating of consecutive rowers.

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Journal Phys. Rev. E
Cosentino Lagomarsino, M, Jona, P, & Bassetti, B. (2003). Metachronal waves for deterministic switching two-state oscillators with hydrodynamic interaction. Phys. Rev. E, 68(Article number: 21908), 1–9.