Floquet scattering of shallow water waves by a vertically oscillating plate

Abstract: We report on the scattering of a plane wave from a vertically oscillating plate in the low frequency approximation by means of Floquet theory. In the case of a static plate, the scattering coefficients are evaluated via mode matching method for the full two-dimensional linearised water wave problem and are compared with the coefficients obtained from a reduced one-dimensional model in the shallow water approximation. The main part of the analysis is the extension of this 1D shallow water approximation to the case of a vertically oscillating plate, where time modulation is only encapsulated in the blockage coefficient. We show that the incident wave is scattered into Floquet sidebands and extract the scattering coefficients for each harmonic using a Floquet scattering formalism. Finally, considering a slowly oscillating plate, we propose a quasistatic approximation which appears to be particularly accurate.