Trapping of surface gravity waves by a vertical flexible porous plate near a wall

Abstract

The present study deals with the trapping of oblique surface gravity waves by a vertical submerged flexible porous plate located near a rigid wall in water of finite as well as infinite depths. The physical problem is based on the assumption of small amplitude water wave theory and structural response. The flexible plate is assumed to be thin and is modeled based on the Euler–Bernoulli beam equation. Using the Green’s function technique to the plate equation and associated boundary conditions, an integral equation is derived which relates the normal velocity on the plate to the difference in velocity potentials across the plate involving the porous-effect parameter and structural rigidity. Further, applying Green’s second identity to the free-surface Green’s function and the scattered velocity potentials on the two sides of the plate, a system of three more integral equations is derived involving the velocity potentials and their normal derivatives across the plate boundary along with the velocity potential on the rigid wall. The system of integral equations is converted into a set of algebraic equations using appropriate Gauss quadrature formula which in turn solved to obtain various quantities of physical interest. Utilizing Green’s identity, explicit expressions for the reflection coefficients are derived in terms of the velocity potentials and their normal derivatives across the plate. Energy balance relations are derived and used to check the accuracy of the computational results. As special cases of the submerged plate, wave trapping by the bottom-standing as well as surface-piercing plates is analyzed. Effects of various wave and structural parameters in trapping of surface waves are studied from the computational results by analyzing the reflection coefficients, wave forces exerted on the plate and the rigid wall, flow velocity, plate deflections and surface elevations. It is observed that surface-piercing plate is more effective for trapping of water waves compared to the bottom-standing and submerged plates. Further, irrespective of plate configurations, full reflection occurs for the same values of the distance between the plate and the rigid wall. Similar phenomenon is observed in case of angle of incidence. Irrespective of plate configurations, in the very long wave regime, full reflection occurs in case of partial plate of any length due to the occurrence of the wave diffraction through the gap region while zero reflection occurs in case of fully extended plate.