Second-order theory for multi-hinged directional wavemakers

The second-order directional wavemaker theory for regular and irregular waves is extended to multi-hinged wavemakers and combined piston--flap wavemaker systems. Derived expressions enable second-order signal correction, common in single-hinged wavemakers, to be applied to multi-hinged systems. Multi-hinged wavemakers offer additional degrees of freedom, with different combinations of paddle motion producing the same progressive wave. This is here exploited to better understand wavemaker behaviour. Single-harmonic signals are computed for double-hinged wavemakers that suppress spurious waves without introducing double-harmonic motions. Surprisingly, these flap motions are almost always in opposite phase, with the larger draft found underneath the water surface. %contradicting to assumptions from earlier studies. Due to the opposing paddle phase, the double-hinged wavemaker draft is usually smaller than the corresponding single-hinged draft. The ability of thsee systems to suppress spurious waves with single-harmonic motion is verified experimentally. The wavemaker theory further supports an arbitrary number of flap hinges, enabling the approximation of a fully flexible wavemaker through piecewise-linear segments. This is demonstrated with an exponential wavemaker profile that does not generate any evanescent waves at linear order. Such a wavemaker is likely to limit wave breaking and cross-modes, but is found to produce spurious second-order waves of a magnitude comparable to a single flap. The presented solution is complete, intrinsically including return flow through the second-order zero mode. This return flow is found to precisely match the Stokes drift.