The wave induced hydrodynamics of rigid and flexible foils are investigated by means of experimentation, including Particle Image Velocimetry (PIV), and analytically through Theodorsen’s unsteady loading theory. High-amplitude high-frequency fluctuations were investigated in order to explore extreme flow fluctuations, which cannot be matched by a pitch control system, and to assess the limits of Theodorsen’s linear approach. We found a strong non-linear interaction between the separated boundary layer and the trailing edge vortex, with circulation counter to the bounded case present. This resulted in low lift-to-drag ratio (i.e. low efficiency) and poor agreement between Theodorsen and experiments for the rigid foil. Conversely, the flexible foil was very promising: preventing large flow separation, it resulted in 25% higher lift-to-drag ratio, 30% lower flow fluctuations and reasonable agreement with Theodorsen’s prediction.