  On surfaces with many motile cilia, beats of the individual cilia coordinate
to form metachronal waves. We present a theoretical framework that connects the
dynamics of an individual cilium to the collective dynamics of a ciliary carpet
via systematic coarse-graining. We uncover the criteria that control the
selection of frequency and wavevector of stable metchacronal waves of the cilia
and examine how they depend on the geometric and dynamical characteristics of
single cilia, as well as the geometric properties of the array. We perform
agent-based numerical simulations of arrays of cilia with hydrodynamic
interactions and find quantitative agreement with the predictions of the
analytical framework. Our work sheds light on the question of how the
collective properties of beating cilia can be determined using information
about the individual units, and as such exemplifies a bottom-up study of a rich
active matter system.
