Floquet scattering of shallow water waves by a vertically oscillating plate

TL;DR

This paper investigates how shallow water waves scatter off a vertically oscillating plate using Floquet theory, extending static models to include time modulation and proposing a quasistatic approximation for slow oscillations.

Contribution

It extends the shallow water wave scattering model to account for a vertically oscillating plate using Floquet theory, introducing a quasistatic approximation for slow oscillations.

Findings

Wave is scattered into Floquet sidebands.

Scattering coefficients are derived for each harmonic.

Quasistatic approximation is highly accurate for slow oscillations.

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.