Water wave interaction with an array of submerged circular plates: Hankel transform approach

In this paper, water wave interaction with an array of thin submerged horizontal circular plates is investigated within the framework of linear potential flow theory. To consider a more general case, the circular plates studied in this paper are not limited to be rigid and impermeable, and instead, they can be perforated or/and elastic. A Hankel transform approach is employed to formulate integral equations in terms of unknown functions related to the jump in velocity potential across each plate. A Galerkin method is adopted to the solution of these integral equations and the velocity potential jump across the plate is expressed in terms of Fourier-Gegenbauer series, incorporating the known square-root behavior at the edge of the plate in a rapidly convergent numerical scheme. For elastic plates, the plate motion is expanded in modes of free vibration with the edge constraint conditions accounted for intrinsically. The unknown coefficients of the plate motion are further expressed in terms of the unknown coefficients related to the velocity potential jump. The Hankel transform based model is found to be valid for multiple plates distributed arbitrarily, including the staggered arrangement, for which the traditional eigenfunction matching method would not work. In-depth discussions have been made to the hydrodynamic responses of staggered arrangement of plates. It is found that the staggered arrangement of plates can result in notable wave focusing, while less energy dissipation. The largest principal strain is observed on the front region of the plate submerged at a shallower depth.