Hydrofoil pumping — the technique competitive swimmers and foil surfers use to sustain speed by oscillating a submerged wing — involves a coupled system of heave (vertical motion) and pitch (rotational motion). The dynamics are complex: fluid forces on the foil depend on its angle of attack, which depends on both the pitch and the heave-induced velocity, which depends on the fluid forces.
Jourdain, Godoy-Diana, and Hecht reduce this to a minimal model: a coupled second-order system with two degrees of freedom. The model captures the essential dynamics — the foil pitches about its mount point, the body heaves vertically, and the two motions are coupled through the lift and moment generated by the foil.
The counterintuitive finding: pitch stability is controlled by the rear wing's moment arm, not its lift contribution. The rear wing generates less lift than the front wing, but its moment arm — the distance from the center of mass to the rear wing's attachment point — determines whether pitch oscillations grow or decay. A small rear wing at a large moment arm stabilizes better than a large rear wing close in.
The stability criterion reduces to a single dimensionless parameter: the ratio of the rear wing's moment arm to the system's pitch inertia. Above a critical value, the system is stable. Below, the foil porpoises — undergoes growing pitch oscillations that destroy the pumping rhythm.
A minimal model of a complex fluid-structure interaction. Two degrees of freedom, one stability parameter, and the answer is in the lever arm, not the force.