  Spermatozoa flagella are known to synchronize when swimming in close
proximity. We use a model consisting of two-dimensional sheets propagating
transverse waves of displacement to demonstrate that fluid forces lead to such
synchronization passively. Using two distinct asymptotic descriptions (small
amplitude and long wavelength), we derive the synchronizing dynamics
analytically for ar- bitrarily shaped waveforms in Newtonian fluids, and show
that phase locking will always occur for sufficiently asymmetric shapes. We
characterize the effect of the geometry of the waveforms and the separation
between the swimmers on the synchronizing dynamics, the final stable
conformations, and the energy dissipated by the cells. For two closely-swimming
cells, synchronization always oc- curs at the in-phase or opposite-phase
conformation, depending solely on the geometry of the cells. In contrast, the
work done by the swimmers is always minimized at the in-phase conformation. As
the swimmers get further apart, additional fixed points arise at intermediate
values of the relative phase. In addition, computations for large-amplitude
waves using the boundary integral method reveal that the two asymptotic limits
capture all the relevant physics of the problem. Our results provide a
theoretical framework to address other hydrodynamic interactions phenomena
relevant to populations of self-propelled organisms.
