Linearised water wave problems involving submerged horizontal plates

• A new Fourier transform approach is made to the solution of linearised water wave problems involving submerged thin horizontal barriers.
• A Galerkin method applied to integral equations result in simple systems of equations which are simple to compute and converge rapidly.
• An explicit low-frequency approximation is developed from a one-term series expansion.
• New results are shown for wave free oscillations and unidirectional wave radiation.

In this paper a number of related linearised water wave problems all involving thin submerged horizontal plates are considered. An integral transform approach is adopted and used to formulate integral equations in terms of unknown functions related to the jump in pressure across the plate. A Galerkin method is applied to the solution of these integral equations leading to elegant expressions for quantities of interest and a rapidly convergent numerical scheme. The focus of the paper is to demonstrate the application of this method in a number of settings including both two-dimensional problems applied to infinitely-long plates of constant width and three-dimensional problems involving circular discs. In the process we present new results including, for example, for wave-free forced oscillations of plates.