Extension of 2D second-order irregular waves generation equations to non-continuous wavemaker shapes

Second-order Stokes-like solution of the wavemaker problem involves the participation of first order evanescent modes to the generation of spurious free harmonics. The unwanted free potential, and the paddle signal needed to suppress it, must be evaluated through series summation which, up to now, have been considered non-convergent for certain wavemaker configurations. This paper presents a demonstration of the convergence of these series summation for the historically discarded configurations. Following previous works, the free harmonics second order transfer function can obtained in two different ways. This operation reveals that the two alternative solutions match only if the lateral boundary condition is properly described. In particular, the domain restriction of non-continuous paddle shapes must be represented through left-continuous step functions. Conversely, if the support is limited only adjusting the integration limits, or the unit step function is non-zero at the origin, one of the two methods leads to an incomplete formulation.