Wave effects on a floating, flexible and porous plate in 3D

Motivated by floating solar energy production, we derive by a variational approach the equation of motion of a floating, flexible, and porous plate exposed to incoming waves. The modal representation is orthogonal and complete. The wavenumber $K$ is up to $K{\ell }_x\le 30$ (${\ell }_x$ the plate length). The dimensionless stiffness $D/\left(\rho g{\ell }_x^2{\ell }_y^2\right)$ is in the range between $10^{-3}$ (small plate) and $10^{-7}$ (large plate) ($\rho$ the density of the fluid, $g$ the acceleration due to gravity, ${\ell }_y$ the lateral plate extension). The damping effect of the plate is modeled by a linear Darcy law. Even a small damping coefficient strongly reduces the plate responses. The porous dissipation depends on the flexibility of the plate. The plate-fluid system is connected with a set of integral equations for the radiation and diffraction problems using suitable Green functions in 3D and 2D. The integral equations are developed in two different versions. A set of generalized Haskind relations for the modal exciting force is developed. The damping coefficients are predicted by three different theoretical formulas obtaining convergent results. The overall energy equation is evaluated. The horizontal drift force on a damped plate is essential.