Bounded passive schooling of in-line self-propelled swimmers with different undulatory wavelengths

Collective swimming is often attributed, at least in part, to hydrodynamic interactions between neighboring individuals. However, previous studies of passive hydrodynamic schooling have focused primarily on swimmers with identical kinematics. Here, we investigate whether bounded collective motion can emerge between swimmers with different undulatory wavelengths, frequencies, and amplitudes. We identify bounded schooling for several heterogeneous kinematic combinations within the sampled parameter space. Frequency matching is the predominant condition, although bounded states also occur for a short-wavelength, higher-frequency, and larger-amplitude follower. The bounded simulations exhibit several well-separated long-time spacing ranges. The close-spacing state promotes collective acceleration through a coherent unified wake, whereas the intermediate- and large-spacing states allow the follower to maintain nearly its isolated swimming speed while reducing its mechanical power expenditure. Flow-field analysis links these outcomes to near-field coupling, fragmented vortex-body interactions, and more coherent periodic wake encounters. Within the present two-dimensional, streamwise-constrained framework, these results demonstrate that passive hydrodynamic interactions can organize the streamwise relative motion of a heterogeneous in-line swimmer pair and produce distinct propulsive and energetic outcomes.