Analytical study of second free harmonic induced by a submerged horizontal plate or a submerged rectangular step

The main objective of this work is to compare the analytical study results of a nonlinear wave, during its interaction with a submerged rectangular structure (horizontal plate or step), with the experimental results obtained by Brossard et al. (Coast Eng 56:11-22, 2009) and the numerical data of Lin et al. (J Waterw Port Coast Ocean Eng 140:04014016, 2014). The nonlinear problem leads to solving a system of nonhomogeneous linear partial differential equations, whose general solu-tion is decomposed into a homogeneous solution (called "free mode") and a particular solution (called "bound mode"). The latter has the same frequency (2w) as the free mode and at the same celerity as the fundamental mode. In this study, it is shown that the contribution of the incident-free mode is weak compared to that of the incident-bound modes. The comparisons related to the harmonic rate of the transmit-ted and reflected free mode, as well as the rate of energy flux of the transmitted free mode, show good agreement with the experimental results of Brossard and Chagdali (Coastal Eng 42:277-290, 2001), Brossard et al. (Coast Eng 56:11-22, 2009) and the numerical data of Lin et al. (J Waterw Port Coast Ocean Eng 140:04014016, 2014), and this in the case of a horizontal plate and that of a step.