Hydroelastic buoyant circular plate in shallow water: a closed form solution

Hydroelastic Behavior of a Floating Ring-Shaped Plate

“…Later, to distinguish the vertical displacements, we denote the free-surface elevation in the open water region F by ζ , while the plate deflection in P is still denoted by w. The dynamic condition, derived from the linearized Bernoulli equation, has the form at z = 0, where ρ w is the density of the water, g is the gravitational acceleration, P (ρ, ϕ, t) is the pressure in the fluid, and P atm is the atmospheric pressure. Relations (2)(3) are the kinematic and dynamic conditions at the free surface. The linearized free-surface condition in the water region F takes the form…”

“…For both IWD and FWD cases, we apply the operator ∂/∂t to (8) and use the surface conditions (2) and (3) to arrive at the following equation for the total potential…”

“…From (2) it follows that the surface elevation has the same harmonic behavior, whence we have the functions w(ρ, ϕ) and ζ(ρ, ϕ) in P and F, respectively. Thus, we consider waves of the single frequency ω and obtain the following differential equation for the potential φ at z = 0…”

“…where a (1) mn and a (2) mn are unknown amplitude functions, κ m are the roots of the plate dispersion relation, which are the so-called reduced wave numbers, and M is the number of these roots taken into account in the computations.…”

“…Several papers, studying the circular plate, have been published recently. The problem for the circular plate for shallow water was solved by Zilman and Miloh [2] and Tsubogo [3]. For finite water depth the problem was solved by Watanabe et al [4] by use of the Galerkin method and Mindlin plate theory, and Peter et al [5] who presented a solution based on decomposing the deflection into angular eigenfunctions.…”