Hydroelastic behavior of a floating ring-shaped plate

6 A.I. Andrianov & A.J. Hermanswhen ρ → ∞.The free-surface elevation ζ in the open water F equals to the sum of the incidentwave elevation ζ inc , and the additional wave elevation ζ pm , generated by the platemotionζ = ζ inc + ζ pm . (14)The total potential in F is also represented as the sum of the incident wave potentialand the potential of waves arising because of the presence of the plateφ F = φ inc + φ pm , (15)where φ pm is the diffraction potential plus the radiation potential. In fact, the potentialfunction φ pm must satisfy radiation condition (13). The governing equationsfor the regions F 0 and F 1 are the same; due to the fact that ρ > r 0 and ρ < r 1 ,respectively, differences in the analysis will appear, which will be discussed later.We introduce the Green’s function G(r, θ; ρ, ϕ), fullfilling the free-surface and theradiation conditions, and apply Green’s theorem to the potentials in the water andplate regions φ F and φ P , respectively. Analyzing the obtained relations and using thedynamic (10) and kinematic (2) conditions, we obtain the main integro-differentialequation in general form at the free surface z = 0{}D∆ 2 − µ + 1 w(ρ, ϕ)= K ∫{ }G(r, θ; ρ, ϕ) D∆ 2 − µ w(r, θ) r dr dθ + Ae ik 0ρ cos ϕ , (16)4πPwhere the last term represents the potential of the incoming waves. The presentedderivation of IDE can be done in polar and Cartesian coordinates, i.e., for plates ofany geometrical shape. Detailed information on the derivation of IDE is given in [6]for polar coordinates and in [7–8] for Cartesian coordinates.4. Green’s function and deflectionIn this section we describe the Green’s function, plate deflection and the operationson corresponding Bessel and Hankel functions for the ring-shaped plate. The generalexpressions for the Green’s function of a surface singularity for water of infinite andfinite depth can be found in [11]. The description of the modified Green’s function,which is used in this paper, was published by the authors in [6].We introduce the Green’s function for a source within the fluid that in Cartesiancoordinates fulfills ∆G = 4πδ(x − ξ), where δ is the Dirac δ-function, x is a sourcepoint and ξ is an observation point. The Green’s function obeys the boundary conditionsat the free surface, G z = KG, and at the bottom as well as the radiationcondition. The three-dimensional Green’s function at z = ζ = 0 takes the followingform [6]∫G(x, y; ξ, η) = −2 F (k)J 0 (kR) dk, (17)LAndrianovHermans.tex; 15/03/2005; 17:37; p.6