Hydroelastic behavior of a floating ring-shaped plate

Hydroelastic behavior of a floating ring-shaped plateAlexey I. Andrianov (a.i.andrianov@ewi.tudelft.nl) and Aad J. Hermans(a.j.hermans@ewi.tudelft.nl)Department of Applied Mathematics,Faculty of Electrical Engineering, Mathematics and Computer Science,Delft University of Technology,Mekelweg 4, 2628 CD Delft, The NetherlandsReceived 26 July 2004; accepted in revised form 3 March 2005Abstract. The interaction between incident surface water waves and floating elastic plates is studied.This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plateand its response to the incident waves. An analytic and numerical study of the hydroelastic behaviorof the plate is presented. An integro-differential equation is derived for the problem and an algorithmof its numerical solution is proposed. The representation of the solution as a series of Hankel functionsis the key ingredient of the approach. After an introduction, the problem is formulated. In Section 3the main integro-differential equation is on the basis of the Laplace equation and thin-plate theory.The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates asthe superpositions of Hankel and Bessel functions, respectively. These expressions, given in Section 4,are used in a further analysis of the integro-differential equation. The problem is solved for two casesof water depth: infinite and finite. For the coefficients in the case of infinite depth a set of algebraicequations is obtained, yielding an approximate solution in Section 6. Then a solution is obtained forthe general and most interesting case of finite water depth analogously in the seventh section. Theexact solution might be approximated by taking into account a finite number of the roots of theplate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the openwater field and within the gap of the ring is studied. Numerical results are presented for illustrativepurposes.Key words: diffraction, hydroelastic response, integro-differential equation, plate-water interaction,ring-shaped plate.Abbreviations: VLFP – very large floating platform, IWD – infinite water depth, FWD – finitewater depth, IDE – integro-differential equation.1. Introduction and backgroundThis paper considers the hydroelastic behavior of a floating elastic plate and itsresponse to the incident surface water waves. A thin plate with elastic propertiesserves as a model for a very large floating platform (VLFP) designed for the purpose ofan airport, ferry pier or other artificial construction, floating in offshore zone. VLFPshave a mat-like dynamic behavior which can be described by the plate equation. Asthe horizontal dimensions of VLFP are about several kilometers by several hundredmeters, while the thickness and draft are of order of several meters, it is possible toconsider the platform as a thin plate, using one of the standard theories of Kirchhoff,Timoshenko or Mindlin. Such a plate can also serve as a model of a huge ice field forstudying ice-water interaction. Water depth plays an important role; therefore, thetheory is divided into three cases: very deep water (depth assumed to be infinite),water of finite depth, and shallow water.There are many papers considering plate-water interaction. For this problem verydetailed literature survey for this problem has recently been published by Watanabec○ 2005 Kluwer Academic Publishers. Printed in the Netherlands.AndrianovHermans.tex; 15/03/2005; 17:37; p.1

2 A.I. Andrianov & A.J. Hermanset al. [1]. Mainly plates of rectangular planforms have been studied, while there arevery few papers considering arbitrarily shaped VLFPs. Several papers, studying thecircular plate, were published recently. The problem for the circular plate for shallowwater was solved by Zilman and Miloh [2] and Tsubogo [3]. For finite water depth theproblem was solved by Watanabe et al. [4] by use of the Galerkin method and Mindlinplate theory, and Peter et al. [5] who presented a solution based on decomposing thesolution into angular eigenfunctions. Andrianov and Hermans [6] solved the problemof the circular disk for water of finite and infinite depth, using an integro-differentialformulation.Here we extend the theory to the case of a ring-shaped plate (or, as in some ofthe literature, a circular plate with a circular hole). The floating structure of thisform may be either of large dimension, for purposes of VLFP, or small, for otherpurposes. The hydroelastic behavior of the ring-shaped floating elastic plate and itsresponse to incident surface waves are investigated. The ring floats on the surface ofideal, incompressible fluid. We consider two cases of water depth. First, we study thebehavior of the ring floating on the surface of water of infinite depth. Next the case ofwater of finite depth is considered, where the general analysis and set of equations aremore complicated as more roots of a water dispersion relation have to be taken intoaccount. An analytical study is carried out and presented for both cases. The problemfor water of shallow depth can be solved more easily by the use of well-known Stokerapproximation theory. In the latter case a set of equations is derived from free-edgeand transition conditions at the inner and outer edges of the ring.An integro-differential formulation, which allows us to solve the diffraction problemfor different geometrical configurations, was developed and applied in [6–9]. Thisformulation might be applied to a plate of finite or infinite dimensions and even tothe case of multiple plates. Also the thin plate theory, standard Laplace equation inthe fluid, supplemented by surface and boundary conditions, and Green’s theoremare used.We consider a plate having elastic properties of constant flexural rigidity andhomogeneous stiffness. The edges of the ring are free of shear forces, bending andtwisting moments. The plate deflection is generated by incoming plane surface waves.The solution for the plate deflection and free-surface elevation in the far field andring gap is derived. The plate deflection is represented as a superposition of Hankelfunctions with corresponding coefficients, containing amplitudes. In a similar way, werepresent the Green’s function, obeying the boundary conditions at the free surface,as a series of Bessel functions for both cases of water depth. Further, Graf’s additiontheorem is applied to the Green’s function.Next, we derive the governing integro-differential equation. The problem involvestwo dispersion relations: one in the plate region and the other in the open water.An analysis of integrals in the complex plane and use of Cauchy’s theorem and theWronskian lead us to the dispersion relation in the plate region. Furthermore, we derivefrom the integro-differential equation, supplemented by the free-edge conditions,a set of equations for the coefficients in the series expansion to determine the platedeflection. Also, we study the free-surface elevation in the ring gap and in the openwater field. To analyze the influence of the presence of the plate on the propagationof the incoming waves, results are obtained for the wave pattern generated by theAndrianovHermans.tex; 15/03/2005; 17:37; p.2

Hydroelastic Behavior of a Floating Ring-Shaped Plate 3plate’s motion. Numerical results are obtained for different values of the physicalparameters for the plate deflection and the free-surface elevation.2. Mathematical formulationA floating thin elastic ring-shaped plate of inner radius r 1 and outer radius r 0 coversa part of the surface of an ideal, incompressible expanse of water. The plate with zerothickness can serve as a model of the VLFP, as described above, due to the smallthickness and shallow draft of the platform. The water depth h is infinite for the caseof deep water and finite and constant for the other case. We assume that no spaceexists between the free surface and the plate. The flexural rigidity of the thin elasticisotropic plate is constant.The plate deflection is generated by incoming surface waves of length λ. It is alsoassumed that incoming waves are propagating in water that is still homogeneous inthe positive x-direction. The wave amplitude A is rather small in comparison withother length-parameters of the problem.The horizontal geometry sketch of the plate is shown in Figure 1. The radialcoordinate ρ is measured from the centre of the plate. We denote the surface of thefluid domain in the plate region (r 1 ≤ ρ ≤ r 0 ) as P and in the open water as F,where the open-water region outside the plate is F 0 (ρ > r 0 ) and the gap area F 1(ρ < r 1 ). Three-dimensional Laplacian is∆ = ∇ 2 = ∂2∂ρ 2 + 1 ∂ρ ∂ρ + 1 ∂ 2ρ 2 ∂ϕ 2 + ∂2∂z 2in polar coordinates, while their relations to Cartesian coordinates are the following:ρ 2 = x 2 + y 2 , ϕ = arctan y/x, z = z.yr 0r 1PSfrag replacementsϕρxz (up)Figure 1. Geometry and coordinate system of the problem.The velocity potential is introduced by ∇Φ(ρ, t) = V(ρ, t), where V(ρ, t) is thefluid velocity vector. The velocity potential Φ(ρ, t) is a solution of the governingLaplace equation in the fluid∆Φ = 0, (1)AndrianovHermans.tex; 15/03/2005; 17:37; p.3

4 A.I. Andrianov & A.J. Hermansfor z < 0 in the case of infinite water depth (IWD) or −h < z < 0 for finite waterdepth (FWD), supplemented by the boundary conditions at the free surface and atthe bottom for FWD. The linearized kinematic condition in the plate and waterregions, at the surface z = 0, has the form∂Φ∂z = ∂w∂t , (2)where w(ρ, ϕ, t) denotes either the free-surface elevation or the deflection of the plate;t is the time. Later, to distinguish the vertical displacements, we denote the freesurfaceelevation in the open water region F by ζ, while the plate deflection in Pis still denoted by w. The dynamic condition, derived from the linearized Bernoulliequation, has the formP − P atmρ w= − ∂Φ∂t− gw, (3)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 linearizedfree-surface condition in the water region F takes the form∂Φ∂z = −1 ∂ 2 Φg ∂t 2 (4)at z = 0. The other boundary condition at the bottom z = −h, applying for the caseof finite depth, is written in the following form∂Φ∂z= 0. (5)The edges of the ring are free of vertical shear forces, bending and twistingmoments. Therefore, the free-edge conditions at ρ = r 1 and ρ = r 0 are:{(∇ 2 (1 − ν) ∂−ρ ∂ρ + 1 )}∂ 2ρ ∂ϕ 2 w = 0, (6){ ( ∂ (1 − ν) ∂∂ρ ∇2 +ρ 2 ∂ρ − 1 ) }∂2ρ ∂ϕ 2 w = 0, (7)where ν is the Poisson ratio. Here and in the sequel the Laplacian is two-dimensional,except when the Green’s function is derived.3. Integro-differential equationIn this section the main integro-differential equation (IDE) is derived along the platecontour, consisting of the inner and outer edges of the ring. We use the approachdescribed and applied in [7–8] for plates of rectangular shape in Cartesian coordinates.The derivation of IDE in polar coordinates has recently been published by the authors[6].AndrianovHermans.tex; 15/03/2005; 17:37; p.4

Hydroelastic Behavior of a Floating Ring-Shaped Plate 5To describe the deflection of the plate w we apply the isotropic thin-plate theory,see, e.g., [10], which leads to a differential equation at z = 0 in the plate area P,known as the Gehring-Kirchhoff equationD∆ 2 w + m ∂2 w∂t 2 = P − P atm, (8)where m is the mass of an unit area of the plate, D is the flexural rigidity, expressedin terms of Young’s modulus E, Poisson ratio ν and the plate thickness h p :D =Eh 3 p12(1 − ν 2 ) .For both IWD and FWD cases we apply the operator ∂/∂t to (8) and use the surfaceconditions (2) and (3) to arrive at the following equation for the total potential Φ(Dρ w g ∆2 + mρ w g∂ 2∂t 2 + 1 )∂Φ∂z + 1 g∂ 2 Φ= 0. (9)∂t2 With the usual assumptions of an ideal fluid and small wave amplitude, thepotential can be written in the form Φ(ρ, ϕ, z, t) = φ(ρ, ϕ, z)e −iωt , where ω is thewave frequency; this means we consider harmonic waves. From (2) it follows thatthe surface elevation has the same harmonic behavior, whence we have the functionsw(ρ, ϕ) and ζ(ρ, ϕ) in P and F, respectively. Thus, we consider waves of the singlefrequency ω and obtain the following differential equation for the potential φ at z = 0() ∂φD∆ 2 − µ + 1 − Kφ = 0, (10)∂zwhere D = D/ρ w g, µ = mω 2 /ρ w g are additional physical parameters that areconstant as the plate considered is isotropic, and K = ω 2 /g.The potential of the undisturbed incident wave φ inc , in polar coordinates, is⎧gA⎪⎨ iω eik 0ρ cos ϕ+k 0 zfor IWD,φ inc (ρ, ϕ, z) =⎪⎩cosh k 0 (z + h)cosh k 0 hgAiω eik 0ρ cos ϕfor FWD,where A is the amplitude of the incident wave, and k 0 is the wave number. Thewave height is defined as twice the amplitude. For deep water the wave number isk 0 = K and for water of finite depth wave number k 0 is the positive real solution ofthe water-dispersion relation(11)k tanh kh = K. (12)The length of the incoming waves is λ = 2π/k 0 . Normally, in a practical situationthe outer diameter 2r 0 of the ring-shaped VLFP is longer than the wave length; thereverse case corresponds to extremely long waves.The potential must satisfy the Sommerfeld radiation condition√ ρ( ∂∂ρ − ik 0) (φ − φ inc) = 0 (13)AndrianovHermans.tex; 15/03/2005; 17:37; p.5

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

Hydroelastic Behavior of a Floating Ring-Shaped Plate 7where J 0 (kR) is the Bessel function, R measures horizontal distance, viz. R 2 = (x −ξ) 2 + (y − η) 2 in Cartesian coordinates. L is the integration contour in the complexk-plane from 0 to +∞, passing beneath the singularity k = k 0 , as shown in Figure 2,and chosen such that the radiation condition is satisfied. The function F (k) in (17)is given by⎧kfor IWD, (a)⎪⎨ k − k 0F (k) =⎪⎩k cosh khk sinh kh − K cosh khfor FWD,In the PSfrag FWD case replacements we have additional poles k = ±k i on the imaginary axis of thecomplex plane.0k 0k(b).(18)Figure 2. Contour of the integration.LThe horizontal distance R in polar coordinates is written as R 2 (ρ, ϕ; r, θ) = ρ 2 +r 2 − 2ρr cos (θ − ϕ). Here ρ and r are the distances from the center of the plate tothe source and observation points respectively, and θ − ϕ is the angle between r andρ.If we apply Graf’s addition theorem to the Bessel function J 0 (kR), which is alsocalled Neumann’s formula for the Bessel function of order zero, it can be replaced bythe series [12]∞∑J 0 (kR) = δ q J q (kr)J q (kρ) cos q(θ − ϕ), (19)q=0where δ 0 = 1 and δ q = 2 if q > 0. We truncate the series in (19) at q = N, chosensuch that convergence of the final results is guaranteed and taking into account thedecaying behavior of the Bessel functions with respect to the order. The value of N ischosen later; all terms of order higher than N are negligibly small. This is justified byinvoking a computational test. The Green’s function in polar coordinates now takesthe form∫G(r, θ; ρ, ϕ) = −2LN∑F (k) δ q J q (kr)J q (kρ) cos q(θ − ϕ) dk. (20)q=0In the sequel we use the Green’s function in the form (20).At the same time the deflection function has to be chosen for each shape of the plateseparately. For the circular plate it can be expressed by the superposition of Besselfunctions [6]. Analogously, for the ring-shaped plate the deflection can be representedas a series of Hankel functions with corresponding coefficients of the following formw(ρ, ϕ) =M∑N∑m=1 n=0[]a (1)mnH (1)n (κ m ρ) + a (2)mnH (2)n (κ m ρ) cos nϕ, (21)AndrianovHermans.tex; 15/03/2005; 17:37; p.7

8 A.I. Andrianov & A.J. Hermanswhere a (1)mn and a (2)mn are unknown amplitude functions, κ m are the roots of the platedispersion relation, which are the so-called reduced wave numbers, and M is thenumber of these roots taken into account in the computations.For the IWD case we use three roots of the dispersion relation in the plate area()Dκ 4 − µ + 1 κ = ±k 0 . (22)The roots of the plate dispersion relation (22), shown in Figure 3, are the following:two real roots ±κ 1 , and four complex roots ±κ 2 and ±κ 3 , which are symmetricallyplaced with respect to both the real and imaginary axes. Due to a property of Hankelfunctions, three roots are taken into account for the current situation: real positiveκ 1 and roots κ 2 and κ 3 , with equal imaginary parts and equal, but opposite-signedreal parts, all three being located in the upper half-plane. The real root κ 1 representsthe main traveling wave mode, the two complex roots represent damped waves.PSfrag replacementsκ 3κ 2κ 1−κ 1−κ 2 −κ 3Figure 3. Zeros of the dispersion relation for deep water.The plate dispersion relation for water of finite depth has the following form()Dκ 4 − µ + 1 κ tanh κh = K. (23)The dispersion relation (23) has two real roots ±κ 1 , and four complex roots ±κ 2 and±κ 3 symmetrically placed with respect to both the real and imaginary axes, thesesix being of the same order as in the IWD case, and infinitely many purely imaginaryroots. We take into account M roots of Equation (23): one real positive κ 1 , twocomplex roots κ 2 and κ 3 , having imaginary parts and equal but opposite-signed realparts, and M − 3 imaginary roots, all located in the upper half-plane.The positions of the roots κ m in the complex k-plane are similar to those of theroots of the water dispersion relation (12), which has one real root k 0 , correspondingto only one wave number of deep water, and a number of purely imaginary roots k i ,i = 1, ..., M −3, except for two complex roots κ 2 and κ 3 . Thus, in the FWD case M isthe truncation parameter, and we take into account M roots of the plate dispersionrelation (23) κ m , m = 1, ..., M, and M − 2 roots of the water dispersion relation (12)k i , i = 0, ..., M − 3.AndrianovHermans.tex; 15/03/2005; 17:37; p.8

Hydroelastic Behavior of a Floating Ring-Shaped Plate 95. Set of equationsAt this point we have only equations resulting from the free-edge conditions to determinethe unknown amplitudes a (1)mn and a (2)mn. Next we will derive relations from theintegro-differential equation to complete the set of equations.If we insert the relations for the deflection (21) and Green’s function (20) into(16), the integro-differential equation at the free surface z = 0 takes the expandedform{} M [D∆ 2 ∑ ∑ N− µ + 1 a (1)mn H(1) n (κ mρ) cos nϕ +m=1 n=0∫2π∫ r 0N∑a (2)mn H(2) n (κ mρ) cos nϕn=0+ K { } M [D∆ 2 ∑ N]∑N∑− µ a (1)2πmnH (1)n (κ m r) cos nθ + a (2)mnH (2)n (κ m r) cos nθ0 r 1m=1 n=0n=0∫ N∑N∑× F (k) δ q J q (kr)J q (kρ) cos q(θ − ϕ) dk r dr dθ = A ɛ n J n (k 0 ρ) cos nϕ, (24)Lq=0n=0where ɛ n = δ n i n . Due to the orthogonality relation for the cosine function, we onlyget a non-zero contribution in the integrand for n = q. Next we carry out the integrationwith respect to θ in (24). The integration over θ itself, when only componentsdependent on θ are considered, gives 2π for n = 0 and δ n π cos nϕ for n > 0, whichmeans 2π cos nϕ for all n. Then, using the orthogonality of the cosine function, weend up with the following set of N + 1 equations derived from IDE (24):M∑+Kr 1) [](Dκ 4 m − µ + 1 a (1)mn H(1) n (κ mρ) + a (2)mn H(2) n (κ mρ)m=1∫ r 0∑ Mm=1∫×L) [](Dκ 4 m − µ a (1)mn H(1) n (κ mr) + a (2)mn H(2) n (κ mr)F (k)J n (kr)J n (kρ) dk r dr = Aɛ n J n (k 0 ρ), (25)for n = 0, ..., N. In accordance with [13, Equation 21.8-22] the integration with respectto r leads us to the difference of two Lommel integrals at r 0 and r 1 . Equation (25)takes the form]∫+KLM∑m=1M∑m=1) [](Dκ 4 m − µ + 1 a (1)mn H(1) n (κ mρ) + a (2)mn H(2) n (κ mρ)(Dκ 4 m − µ ) F (k)(k 2 − κ 2 m)[]a (1)mn c(1) mn + a(2) mn c(2) mn J n (kρ) dk = Aɛ n J n (k 0 ρ),(26)AndrianovHermans.tex; 15/03/2005; 17:37; p.9

10 A.I. Andrianov & A.J. Hermanswhere we have introduced the function c (q)mn as a combination of Bessel and Hankelfunctions in the formc (q)mn = r 0[−r 1[kJ n+1 (kr 1 )H (q)kJ n+1 (kr 0 )H (q)nn(κ mr 0 ) − κ m J n (kr 0 )H (q) ]n+1 (κ mr 0 )], q = 1, 2. (27)(κ mr 1 ) − κ m J n (kr 1 )H (q)n+1 (κ mr 1 )From Equation (26) we can derive 2(M −2)(N +1) relations for the determinationof 2M(N +1) unknown amplitudes. The set of equations for both cases of water depthis completed by the free-edge conditions (6–7) when the deflection function is insertedthere. These conditions at the edges ρ = r 1 and ρ = r 0 give us 4(N + 1) equationsM∑m=1a (1)mn d(1) mn,ij + a(2) mn d(2) mn,ij = 0, (28)where n = 0, ..., N, i = 0, 1, j = 1, 2, and the functions d (q)[()d (q)mn,i1 = H (q)n (κ mr i ) −κ 2 (1 − ν)n(n − 1)m +r 2 imn,ij are[(d (q)mn,i2 = H (q)n (κ m r i ) − n κ 2 m + (1 − ν)n2 (1 − n)r i(+ H (q)n+1 (κ mr i ) κ 3 m +]+ H (q)n+1 (κ (1 − ν)mr i )κ m ,r i)r 3 i)](1 − ν)n2ri2 κ m , (29)where q = 1, 2 is the Hankel-function index and r i is either inner-ring contour r 1 orouter-ring contour r 0 .From now on we consider the IWD and FWD cases separately.6. Ring on water of infinite depthHere the analytic study is given for the case of infinite depth. In the IWD case we onlyconsider part of the solution, namely that related to the water dispersion relation.The analysis given in this section is a good basis from which to derive a solution forthe general and most complicated case of water of finite depth. To demonstrate themethod we wish to use for the FWD case, we consider the contribution of the polek = k 0 in the IWD case. Here we ignore the contribution of the integral along theimaginary axis after closing the contour in (20). This means that local disturbancesare disregarded.The deflection of the plate, the Green’s function and the function F (k) are representedby expressions (21), (20) and (18a), respectively. For deep-water wave numberk 0 equals K, so k 0 = ω 2 /g.Now we analyze the k-integral in IDE (26). The integrand has the poles k = ±κ m ,shown in Figure 3, and can be represented as a sum of four integrals∫∞k 00k(k − k 0 )(k 2 − κ 2 m)[]a (1)mn c(1) mn + a(2) mn c(2) mn J n (kρ) dk= a (1)mn (I 01 − I 11 ) + a (2)mn (I 02 − I 12 ) , (30)AndrianovHermans.tex; 15/03/2005; 17:37; p.10

Hydroelastic Behavior of a Floating Ring-Shaped Plate 11where the path of integration in (31) is below the singularities k = k 0 and k = κ 1 ,respectively, and the integrals I iq for i = 0, 1, q = 1, 2 are[×kJ n+1 (kr i )H (q)n∫∞I iq = k 0 r i0k(k − k 0 )(k 2 − κ 2 m)(κ m r i ) − κ m J n (kr i )H (q)n+1 (κ mr i )]J n (kρ) dk, (31)with r i as r 1 or r 0 , m = 1, ..., M and n = 0, ..., N. The contour of integration may beclosed in the complex k-plane. In the integral I iq we split up the Bessel function oflargest argument to be able to close the contour of integration and, furthermore, useJordan’s lemma.Let us consider the integral I 01 . For further manipulation we use the fact that ρ r 1 and, therefore, to close the contours of integration we represent the Besselfunctions J n (kρ) as the half-sums of corresponding Hankel functions of the first andsecond kind, as was done in (32).When the contours are closed for all integrals, we apply the Cauchy residue lemmaat the poles k = κ m for the integrals with the contour closed in the upper half-planeand at the poles k = −κ m for the integrals with the contour closed in the lowerhalf-plane. The Wronskian W {H (1)n (κ m r i ), H (2)n (κ m r i ) applies for the combination ofHankel functionsH (1)n+1 (κ mr i )H (2)n (κ mr i ) − H (1)n (κ mr i )H (2)n+1 (κ mr i ) = − 4i . (33)πκ m r iThen, adding up the resulting terms and considering the coefficients of J n (κr i ), wederive the plate dispersion relation for deep water (22), where the plus sign in theright-hand side corresponds to the integrals in the upper half-plane, and the minussign to the integrals in the lower half-plane. The derivation of the plate dispersionrelation from IDE is also a justification of our approach.In the upper half-plane we also obtain a contribution of the pole k = k 0 of theintegrand, as seen in Figure 2. This contribution has to cancel the term in the righthandside of IDE (26). If we neglect the integral along the imaginary axis, applicationAndrianovHermans.tex; 15/03/2005; 17:37; p.11

12 A.I. Andrianov & A.J. Hermansof Jordan’s lemma and the contribution of the pole k = k 0 lead us to 2(N +1) relationsandπir 0πir 1M ∑m=1M ∑m=1( )Dκ 4 k02m − µk 2 0 − κ2 m[k 0 H (1) ()n+1 (k 0r 0 ) a (1)mnH (1)n (κ m r 0 ) + a (2)mnH (2)n (κ m r 0 )()]−κ m H (1)n (k 0r 0 ) a (1)mn H(1) n+1 (κ mr 0 ) + a (2)mn H(2) n+1 (κ mr 0 )( )Dκ 4 k02m − µ−κ m J n (k 0 r 1 )k 2 0 − κ2 m= Aɛ n (34)[()k 0 J n+1 (k 0 r 1 ) a (1)mnH (1)n (κ m r 1 ) + a (2)mnH (2)n (κ m r 1 )()]a (1)mn H(1) n+1 (κ mr 1 ) + a (2)mn H(2) n+1 (κ mr 1 )= 0. (35)We have now derived a system of 2M(N + 1) equations (34–35) and (28) withM = 3 for the determination of 2M(N + 1) amplitudes a (1)mn and a (2)mn for ’infinitely’deep water. When the amplitudes are known, the plate deflection is calculated by(21).7. Ring on water of finite depthIn this section we consider a ring on water of finite depth. In general we follow theanalysis as presented in the previous section for the IWD case. For the FWD casethe deflection of the plate is represented by (21) and the Green’s function is givenby formulas (20) and (18b). Here we take into account M − 2 roots of the waterdispersion relation (12) and M roots of the plate dispersion relation (23).Let us consider the function F (k) given by (18b). For the FWD case this functionis meromorphic, as it is bounded for all poles. The poles are the roots of the dispersionrelation for the water region (12) k = ±k i , i = 0, ..., M − 3, where k 0 is the positivereal root and k i , for i ≠ 0, is the purely imaginary. So, the meromorphic functionF (k) for water of finite depth can be expressed by the following relationF (k) =M−3 ∑i=0ki2 ( 1ki 2h − + 1 ). (36)K2 h + K k + k i k − k iThe relation (36) is inserted into IDE (26). There are two integrals in the complexk-plane, those two can be combined into one integral from −∞ to +∞ with the polesk = k i . The contour of integration is defined as L ′ . Finally, we derive the governingintegro-differential equation for the case of finite water depthM∑m=1() [] ∫Dκ 4 m − µ + 1 a (1)mnH (1)n (κ m ρ) + a (2)mnH (2)n (κ m ρ) + K× J M−3n(kρ) ∑k 2 [i(k 2 − κ 2 m) (k 2 i=0 i h − a (1)K2 h + K)(k − k i )mnc (1)mn + a (2)M ∑L ′ m=1]mnc (2)mn( )Dκ 4 m − µdk = Aɛ n J n (k 0 ρ)(37)AndrianovHermans.tex; 15/03/2005; 17:37; p.12

Hydroelastic Behavior of a Floating Ring-Shaped Plate 13at z = 0, where n = 0, ..., N and the functions c (q)mn are given by (27) for q = 1, 2.To obtain the dispersion relation we repeat the procedure that was applied for theIWD case: the representation of k-integral as a sum of four integrals and, further,decomposition of each of those four as two integrals with contours closed in thedifferent half-planes of the complex plane. In the FWD case we have the poles, given inFigure 3, plus the poles at the imaginary axis ±κ m for m = 4, ..., M. The applicationof the residue lemma at the poles leads us to the standard plate dispersion relationfor water of finite depth (23).We consider the contributions of the poles of the water dispersion relation (12).The closure of the contour is shown in Figure 4. The contour for the integrals with(kr i ) might be closed in the upper half-plane, while the contour for the integralswith H (2)q (kr i ) in the lower half-plane. In the latter case we get a zero contributionbecause the poles k = k i are located as indicated in Figure 4.H (1)qPSfrag replacementsk i0kFigure 4. Closure of the contour of the integration in the upper half-plane.An application of the Cauchy theorem to the integrals with the contour closed inthe upper half-plane gives us equations to determine the amplitudes a (1)mn and a (2)mn.The poles k = k i , where i = 0, ..., M − 3, result in 2(M − 2)(N + 1) equations. First(M − 2)(N + 1) equations are derived along the outer ring edge ρ = r 0 and have thefollowing form[× k i H (1)−κ m H (1)n (k i r 0 )n+1 (k ir 0 )k 0M (∑ Dκ4πiKr m − µ ) ki2 0(k 2 m=1 i − κ2 m)(ki 2h − K2 h + K)(a (1)mn H(1) n (κ mr 0 ) + a (2)mn H(2)(a (1)mnH (1)n+1 (κ mr 0 ) + a (2)mnH (2)n+1 (κ mr 0 ))n (κ mr 0 ))]= A i , (38)whereA i ={ Aɛn , i = 0,0, i = 1, ..., M − 3,as the case i = 0 corresponds to the pole k = k 0 . The remaining (M − 2)(N + 1)equations are derived along the inner edge of the ring ρ = r 1[×M (∑ Dκ4πiKr m − µ ) ki2 1(k 2 m=1 i − κ2 m )(k2 i h − K2 h + K)(k i J n+1 (k i r 1 ) a (1)mnH (1)n (κ m r 1 ) + a (2)mnH (2)−κ m J n (k i r 1 )(a (1)mnH (1)n+1 (κ mr 1 ) + a (2)mnH (2)n+1 (κ mr 1 ))n (κ m r 1 ))]= 0. (39)AndrianovHermans.tex; 15/03/2005; 17:37; p.13

14 A.I. Andrianov & A.J. HermansWe have derived now 2(M −2)(N +1) relations (38–39) to determine the unknownamplitudes. The set of equations, like in the previous section, is completed by 4(N +1)edge conditions (28). The deflection can now be computed by (21).The finite water-depth model can be used to solve the problem of shallow orinfinitely deep water. Taking the limits we can derive the solutions for these situationsincluding transition from the dispersion relation for water of finite depth to thedispersion relations for deep and shallow water, respectively.8. Free-surface elevation and initiated wave patternWe continue the analysis of the plate-water interaction and consider the open waterregions, that is the main region F 0 and the gap inside the ring F 1 . The elevationζ(ρ, ϕ) of the free surface can be computed by (14), where the value of ζ inc may beobtained from the incident wave potential expression (11) by use of the kinematiccondition (2) and the value of ζ pm from the following analysis of IDE in the waterarea.The integro-differential equation has the form (37) for the FWD case. Therefore,we obtain the following expression for the free-surface elevation∫ζ(ρ, ϕ) = Ae ik 0ρ cos ϕ − KM−3 ∑×i=0L ′M ∑m=1k 2 i(k 2 i h − K2 h + K)(k − k i )( )Dκ 4 Jn (kρ)m − µ(k 2 − κ 2 m )[a (1)mnc (1)mn + a (2)]mnc (2)mndk. (40)We notice that in the open water region F 0 , where ρ > r 0 > r 1 , J n (kρ) is split upinto a half-sum of corresponding Hankel functions to close the contour of integration,and, to do so at the gap area F 1 , the Bessel functions J t (kr 1 ), t = n, n + 1, are splitup for ρ < r 1 < r 0 . Finally, after using of the residue lemma at the poles k = k i , weobtain for the free-surface elevationζ(ρ, ϕ) = Ae ik 0ρ cos ϕ − πiM∑m=1( )Dκ 4 m − µM−3 ∑ki 2 ×K[(k 2 i=0 i h − K2 h + K)(ki 2 − κ2 m ) a (1)mn f (1)mni + a(2) mn f (2) ]mni, (41)where the functions f (q)mni have the following expressions in the open water F 0 and inthe gap F 1f (q)mni = ⎧⎨⎩[r 0 H (1)n (k i ρ)r 1 J n (k i ρ)kJ n+1 (k i r 0 )H (q)n[kH (1)n+1 (k ir 1 )H (q)(κ m r 0 ) − κ m J n (k i r 0 )H (q) ]n+1 (κ mr 0 )n (κ mr 1 ) − κ m H (1)n (k ir 1 )H (q) ]n+1 (κ mr 1 )in F 0 ,in F 1 ,for q = 1, 2 and i = 0, ..., M − 3.For infinitely deep water the procedure, in general, is the same, but with the polek = k 0 of the water dispersion relation only. We derive the following equation for the(42)AndrianovHermans.tex; 15/03/2005; 17:37; p.14

free-surface elevationζ(ρ, ϕ) = Ae ik 0ρ cos ϕ − πiHydroelastic Behavior of a Floating Ring-Shaped Plate 15M∑m=1( )Dκ 4 k 2 [0m − µ(k0 2 − a (1)mnf (1)κ2 mn0m)+ a(2) mnf (2) ]mn0 , (43)where the functions f (q)mn0 , q = 1, 2, are defined by formula (42), for i = 0, for thewater regions F 0 and F 1 .The expressions (41) and (43) are for the total free-surface elevation for the FWDand IWD cases. We can subtract the incident field and use the second terms in theright-hand sides of these expressions, representing ζ pm , to study the resulting wavepattern as generated by the plate motion.9. Numerical resultsIn this section numerical results are given for the title problem for both the IWDand the FWD models. Results are presented for relevant and practically importantcases. We show results for the plate deflection w, free-surface elevation ζ and initiatedwave pattern elevation ζ pm , normalized by the wave amplitude A. Calculations arebased on the varied flexural rigidity D, while the plate radii r 1 and r 0 , Poisson ratioν = 0·25 and ratio m/ρ w = 0·25 m are constant. The wave amplitude is A = 1 m,water depth and incident wave length are varied, leading to different values of wavenumber k 0 and, respectively, of the frequency ω. In all given figures the values of theinner and outer radii are the following: r 1 = 100 m and r 0 = 500 m.The number of the roots of the plate dispersion relation that are taken into accountfor the FWD case is M = 10. The computation of the Bessel and Hankel functions ofcomplex argument must be done accurately. Problems may appear when the argumentis smaller than the order of the function and when the argument is too large or toosmall. Our algorithm was based on the Amos package [14], with some modifications toincrease the accuracy, and computations were performed using the Fortran compiler.Scaled functions were used, which remove the exponential behavior in both the upperand lower half-planes. Definitions and a description may be found in [15]. For zeroargument we have used table values of Bessel and Hankel functions. We take N = 30as the highest order of the modes of the Bessel functions, i.e., we consider Bessel andHankel functions of order 0 to 30. The choice of the truncation parameters M andN was also justified by computational testing such as to ensure sufficient accuracy.More details about the number of the roots can be found in [8] and about the orderof the Bessel functions in [6].The amplitude coefficients a (1)mn and a (2)mn, as well as the corresponding Hankel functions,decay rapidly. With the increase of the radii, and respectively of the argumentsof the functions, or the flexural rigidity, the decay becomes faster.In all figures the left subplot is denoted as (a), and the right as (b), if therefour subplots, then lower left is denoted as (c) and lower right as (d). All figuresare symmetric about the x-axis because incoming plane waves propagating in thex-direction and their crests are parallel to the y-axis.In Figures 5–6 numerical results are shown for the plate deflection for both casesof water depth. The wave length in the second figure is two times larger than in theAndrianovHermans.tex; 15/03/2005; 17:37; p.15

16 A.I. Andrianov & A.J. Hermansfirst, while the plate parameters are constant. The plate deflection for larger valuesof the rigidity is shown in Figure 7.Numerical results for the elevation of the free surface in the ring gap are givenin Figure 8 for different wave lengths. In Figure 9 we show results for the initiatedwave pattern, i.e., for the free-surface elevation ζ pm , generated by the motion of theplate, and total free-surface elevation. The subplots for the open water region F 0 aregiven for the surface of the fluid domain of the radius r f , for water of finite depth. InFigures 10–11 we show a complete set of results for the unknowns being studied inthe paper. Results are given for the plate deflection (a), free-surface elevation in F 1(b), initiated wave pattern (c) and the free-surface elevation in F 0 (d) for the IWDand FWD cases, respectively.The propagation of the wave through the plate area can be clearly seen, especiallyfor small values of plate rigidity. The wave propagates with a curved wave front, asis observing very well for these cases when the wave length is much smaller than theouter diameter of the ring. In a zone close to the plate edges the deflection can bequite different from that in the main zone.The plate deflection and its hydroelastic response to the wave field are highly dependenton the ratio between outer radius r 0 and the wave length λ. With decreasingwater depth the results for the plate deflection and free-surface elevation are changinggradually, upon which the water depth h itself has a growing influence on the results.For smaller values of the plate rigidity or stiffness the plate deflection increases, whichis demonstrated in Figures 5–7. If the wave length is decreasing, then the value ofthe deflection grows. The plate deflection is larger numerically also when the waterdepth increases. Also, we found that computational results for large values of depth,h > 100 m, are almost independent of water depth. For deep water the depth itselfdoes not have a strong influence on the results, and for this situation it is sufficientto take M = 10 also.We found that for a rigidity D > 10 7 m 4 the plate behaves as a rigid body withsignificant influence on the surface waves, whereas for D < 10 3 m 4 the plate hashardly any influence on the incident surface waves. Realistic values of the rigidityfor the VLFP can be of order of about 10 7 m 4 ; here many of the results shown forsmaller values serve to demonstrate well the nature of plate-water interaction effects,while the rigidity for the ice can be of order of about 10 5 m 4 .The influence of the ring gap on the results is also shown. The propagated wavechanges its behavior essentially after crossing the gap. The influence of the gapincreases for smaller lengths of the wave.The initiated wave pattern is highly dependent on water depth and the physicalproperties of the plate. For growing values of the plate flexural rigidity, stiffness orPoisson ratio, the influence of the plate motion on the total elevation of the watersurface is growing as well.10. ConclusionsThe problem of the interaction between a floating thin elastic plate of a ring-shapedplanform and incident surface water waves has been solved. An analytic and numericalstudy of the hydroelastic behavior of the plate has been presented. The integro-AndrianovHermans.tex; 15/03/2005; 17:37; p.16

Hydroelastic Behavior of a Floating Ring-Shaped Plate 17differential equation of the problem was derived and an algorithm of its numericalsolution proposed. For the case of finite water depth the system of equations for theexpansion coefficients has been obtained analytically. For infinitely deep water theproblem has been solved partly.The vertical displacements in the plate and water regions, the plate deflection andfree-surface elevation respectively, have been studied and obtained. The influence ofthe plate motion on the solution in the water regions was analyzed by the derivationof the initiated wave pattern. Also, the influence of the ring gap was analyzed anddemonstrated. For realistic values of the plate mass and rigidity, the plate behaviordoes not differ much from that of a circular plate, except for the area behind thering gap. In the inner free surface area F 1 the resonant situation, that is large freesurfaceelevation, is possible for some values of the wave length, see Figure 8, as wasshown by Hermans [9] for the channel between two plates. This phenomenon is alsodemonstrated by Molin in his study of sloshing modes in a moonpool [16].The solution for water of finite depth can be used for problems with shallow orvery deep (infinite) water. The floating platform should be located in offshore zonesof an ocean or sea. The water depth is rather small in such zones, but as wave lengthscan be both short and long, it is more natural to use the finite-water-depth model tostudy the hydroelastic response of the plate to water waves.The presented approach, particularly the analysis of the IDE and Green’s function,is valid for any rotational symmetric configuration of the plate. The solution presentedabove is a new application of the integro-differential formulation. This formulationallows us to solve many problems of interaction between water waves and floatingbodies. The main goal of the method is the derivation of an integro-differentialequation along the plate contour, followed by its analysis and solution. One of theadvantages of our approach is that we may find both vertical displacements, the platedeflection in P and the free-surface elevation in F, using a common set of equations.The presented approach is valid when the plate draft and thickness are assumedto be zero. So, the next task is to study the interaction of incoming water wavesinvolving plates of finite thickness.A possible application of the method is to use it for the hydroelastic analysis ofthe VLFP. The planform of the VLFP depends on the water depth, currents of thesea or ocean, where a floating airport could be located, distance from the coast etc.In some cases it can make sense to construct the VLFP of an arbitrary horizontalshape. One can also use the presented approach to study the motion of large ice fieldsin water waves, using the physical properties of the ice instead of those of the elasticplate. Most results are for the large plate, 2r 0 > λ, but the approach is also valid forsmall-sized floating plates 2r 0 < λ, which may be used for different purposes in thesea.References1. E. Watanabe, T. Utsunomiya and C.M. Wang, Hydroelastic analysis of pontoon-type VLFS: aliterature survey. Eng. Struct. 26 (2004) 245–256.2. G. Zilman and T. Miloh, Hydroelastic buoyant circular plate in shallow water: a closed formsolution. Appl. Ocean Res. 22 (2000) 191–198.AndrianovHermans.tex; 15/03/2005; 17:37; p.17

18 A.I. Andrianov & A.J. Hermans3. T. Tsubogo, The motion of an elastic disk on shallow water in waves. In: J.S. Chung, M.Sayeed, H. Saeki and T. Setoguchi (eds.), Proceedings of 11th International Offshore and PolarEngineering Conference. Vol.I. Stavanger, Norway: (2001) pp.229–233.4. E. Watanabe, T. Utsunomiya and C.M. Wang and Y. Xiang, Hydroelastic analysis of pontoontypecircular VLFS. In: J.S. Chung and S. Prinsenberg (eds.), Proceedings of 13th InternationalOffshore and Polar Engineering Conference. Vol.I. Honolulu, USA: (2003) pp.93–99.5. M.A. Peter, M.H. Meylan and H. Chung, Wave scattering by a circular plate in water of finitedepth: a closed form solution. In: J.S. Chung and S. Prinsenberg (eds.), Proceedings of 13thInternational Offshore and Polar Engineering Conference. Vol.I. Honolulu, USA: (2003) pp.180–185.6. A.I. Andrianov and A.J. Hermans, Hydroelasticity of a circular plate on water of finite or infinitedepth. J. Fluids Struct. 20 (2005) accepted.7. A.J. Hermans, The ray method for the deflection of a floating flexible platform in short waves.J. Fluids Struct. 17 (2003) 593–602.8. A.I. Andrianov and A.J. Hermans, The influence of water depth on the hydroelastic responseof a very large floating platform. Marine Struct. 16 (2003) 355–371.9. A.J. Hermans, Interaction of free-surface waves with floating flexible strips. J. Eng. Math. 49(2004) 133–147.10. S.P. Timoshenko, D.H. Young and W. Weaver, Vibration Problems in Engineering. New York:John Wiley & Sons (1974) 521pp.11. J.V. Wehausen and E.V. Laitone, Surface waves. In: W. Flügge (ed.), Encyclopedia of Physics.Vol.9. Berlin: Springer-Verlag (1960) pp.446–814.Also at http://www.coe.berkeley.edu/SurfaceWaves.12. C.J. Tranter, Bessel Functions with Some Physical Applications. London: The English UniversitiesPress (1968) 148pp.13. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions,Theorems and Formulas for Reference and Review. New York: McGraw-Hill, 1968, 831pp.14. D.E. Amos, A portable package for Bessel functions of a complex argument and nonnegativeorder. ACM Trans. Math. Software 12 (1986) pp.265–273.15. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables. Washington: U.S. Government Printing Office (1964) 1046pp.16. B. Molin, On the piston and sloshing modes in moonpools. J. Fluid Mech. 430 (2001) 27–50.PSfrag replacementsPSfrag replacements0.7w0-0.7-10.5010.60-0.6-1w0.501-0.5y/r 000.51 -1 -0.5x/r 0-0.5y/r 000.51 -1 -0.5x/r 0Figure 5. Deflection of the ring-shaped plate, for λ = 50 m, D = 10 5 m 4 : a) infinite depth, b) finitedepth, h = 20 m.AndrianovHermans.tex; 15/03/2005; 17:37; p.18

PSfrag replacementsPSfrag replacementsHydroelastic Behavior of a Floating Ring-Shaped Plate 191.4w0-1.4-100.511.40-1.4-1w00.51-0.5y/r 000.51 -1 -0.5x/r 0-0.5y/r 000.51 -1 -0.5x/r 0Figure 6. Plate deflection, as in Figure 5, for λ = 100 m.PSfrag replacementsPSfrag replacements0.6w0-0.6-100.510.50-0.5-1w00.51-0.5y/r 000.51 -1 -0.5x/r 0-0.5y/r 000.51 -1 -0.5x/r 0Figure 7. Plate deflection, for λ = 100 m, h = 100 m: a) D = 10 6 m 4 , b) D = 10 7 m 4 .AndrianovHermans.tex; 15/03/2005; 17:37; p.19

Sfrag replacementsSfrag replacements20 A.I. Andrianov & A.J. Hermans1.5ζ0-1.5-0.2y/r 000.2 -0.2 0x/r 00.22.50-2.5ζ-0.2y/r 000.2 -0.2 0x/r 00.2Figure 8. Free-surface elevation in the gap, for h = 100 m, D = 10 7 m 4 : a) λ = 50 m, b) λ = 100 m.PSfrag replacementsζ pmPSfrag replacements4ζ pm 31.42.2200-1.41 -2.20-5-5-4-1-3-2-2-10-31x/r2-4 0y/r 0345 -55ζ-4-3-2-1y/r 001234-2-3-45 -5543210-1x/r 0Figure 9. Initiated wave pattern (a) and free-surface elevation (b), for λ = 500 m, h = 100 m,r f = 2500 m, D = 10 7 m 4 .AndrianovHermans.tex; 15/03/2005; 17:37; p.20

Hydroelastic Behavior of a Floating Ring-Shaped Plate 21PSfrag replacementswPSfrag replacementswζ1w0-1-1PSfrag replacements-0.5y/r 000.51 -1 -0.50.50x/r 011.10-1.1ζ-0.2wζζ pmy/r 0PSfrag replacements00.2 -0.2 0x/r 00.24ζ pm 31.921.500-1.91 -1.50-5-5-4-1-3-2-2-10-31x/r2-4 0y/r 0345 -55ζ-4-3-2-1y/r 001234-2-3-45 -5543210-1x/r 0Figure 10. Deflection of the plate (a), free-surface elevation in F 1 (b), initiated wave pattern (c) andfree-surface elevation in F 0 (d), for λ = 500 m, h = 100 m, r f = 2500 m, D = 10 8 m 4 .AndrianovHermans.tex; 15/03/2005; 17:37; p.21

22 A.I. Andrianov & A.J. HermansPSfrag replacementswPSfrag replacementswζ0.4w0-0.4-1PSfrag replacements-0.5y/r 000.51 -1 -0.50.50x/r 011.40-1.4ζ-0.2wζζ pmy/r 0PSfrag replacements00.2 -0.2 0x/r 00.223ζ23ζ pm -31.20-1.2-3-2-1y/r 00123 -3 -2-101x/r 01.8-1.8 0-2-1y/r 00123 -3 -210-1x/r 0Figure 11. Deflection of the plate (a), free-surface elevation in F 1 (b), initiated wave pattern (c) andfree-surface elevation in F 0 (d), for λ = 200 m, r f = 1500 m, D = 10 8 m 4 .AndrianovHermans.tex; 15/03/2005; 17:37; p.22