The transient response of floating elastic plates to wavemaker forcing in two dimensions

Transcription

1 The transient response of foating eastic pates to wavemaker forcing in two dimensions F. Montie, L.G. Bennetts 1, V.A. Squire Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 954, New Zeaand Abstract The time-dependent inear motion in a two-dimensiona fuid domain containing a group of foating thin eastic pates is considered. Forcing is provided by a wavemaker and the waves that transmit through the group of pates are partiay refected by a beach. Fourier Transforms are used to reate the soutions in the time and frequency-domain, where a soution is found using a combination of eigenfunction matching and transfer matrices. This aows refections from the wavemaker and the beach to be incuded or excuded in a simpe manner. Numerica resuts show that the frequency response of a singe pate is significanty affected by the resonances introduced by the presence of the atera boundaries. For a two-pate system the reative fexura response is found to be strongy dependent on pate spacing. Key words: Linear water waves, Eastic pates, Time-domain, Wavemaker 1. Introduction Interactions between surface gravity waves and foating compiant bodies have been studied extensivey, particuary in the ast two decades. Linear theory is often used aong with thin eastic pate theory, and this has enabed a wide range of mathematica techniques to be deveoped. On the other hand, few wave tank experiments have taken pace to vaidate these modes. In support of research on Very Large Foating Structures (VLFSs), some experiments were conducted in the ate 199 s in Japan (see, e.g., Yago and Endo, 1996; Ohta et a., 1997; Kagemoto et a., 1998), ooking at the hydroeastic response of rectanguar eastic pates for norma and obique wave incidence. The aim of the experiments was to monitor the wave oads on a particuar structure, scaed from the origina fu-size VLFS. More genera studies are needed to understand how eastic structures of various thicknesses and shapes respond to waves across a wide frequency range. In this context, the most reevant series of experiments was conducted by S. Sakai and K. Hanai in Japan in a two-dimensiona wave fume. These experiments were focused on determining the dispersion reation that characterises the propagation of fexura gravity waves in ice-covered seas (see Sakai and Hanai, 22). 1 Present address: Schoo of Mathematica Sciences, University of Adeaide, Adeaide, South Austraia, 55, Austraia Corresponding author. Te.: Emai address: fmontie@maths.otago.ac.nz (F. Montie) Preprint submitted to Journa of Fuids and Structures October 24, 211

2 The current authors, in coaboration with members of the Laboratoire de Mécanique des Fuides of Écoe Centrae de Nantes (see, e.g., Roux de Reihac et a., 211), are invoved in a series of wave tank experiments to extract the fexura response of a sma group of circuar compiant pates under inear wave conditions. It is expected that these experiments wi produce benchmark resuts to vaidate a fuy three-dimensiona inear hydroeastic mode. A description of the experimenta campaign and preiminary resuts may be found in Montie et a. (211). Whie these experiments motivate the present work, modeing the hydroeastic interactions between a three-dimensiona wave tank and an eastic scatterer presents a significant chaenge. Athough methods exist to cacuate the circuar wave fied produced by a foe or foes (see Meyan, 22; Peter and Meyan, 24; Bennetts and Wiiams, 21), it is not obvious how to coupe this to the refected wave fied induced by the rectanguar boundaries of the domain. The probem is much easier to sove in a two-dimensiona setting, and this approximation deineates the scope of this study where the aim is to provide anaytica insights concerning the fexura response of foating eastic pates due to a transient reguar wave forcing produced by a wavemaker. In the present paper, we propose a two-dimensiona wave tank mode, which incudes a wavemaker and a beach, to characterise the transient hydroeastic response of a group of foating thin eastic pates. Athough we introduce atera boundaries to the fuid domain, no direct impications to the wave tank experiments mentioned previousy is intended. The effects of the side was and the wave propagation over the directiona spectrum are not considered in the present two-dimensiona mode, and wi be incuded in a future three-dimensiona study. Water wave scattering by foating eastic pates has been of interest in two main areas. The first appication concerns the propagation of ocean waves into arge fieds of sea-ice foes in the poar seas. Research in this area has been summarised by Squire et a. (1995) and, more recenty, by Squire (27). The second area of appication reates to the design of pontoon-type VLFSs mentioned above, such as offshore foating runways. Literature surveys on the hydroeastic response of VLFSs are due to Kashiwagi (2a) and Watanabe et a. (24). Our mode is based on potentia fow theory. The equations are inearised, assuming that water surface osciations are of sma ampitude compared to the waveength. The foating eastic pates are modeed as Euer-Bernoui thin eastic beams, assuming the vertica deformations in a pate are sma compared to its thickness. The vertica dispacement of a pate then competey characterises its motion. Unike most hydroeastic modes, the fuid domain is bounded ateray. At one end, a wavemaker exists; its motion transmitted to the fuid through a kinematic condition. At the opposite end, an absorbing beach refects a fraction of the ampitude of the waves that reach it. Finding a soution of the proposed mode is chaenging, as the equations are soved in the time-domain. A number of time-domain methods are avaiabe in the iterature. The memory-effect and reated Lapace transform methods have been widey used to sove the radiation probem (fuid initiay at rest) of a thin 2

3 eastic pate in two dimensions (see, e.g., Kashiwagi, 2b; Korobkin, 2; Sturova, 26; Meyan and Sturova, 29). Those methods require a time stepping evauation of the soution. The spectra generaised eigenfunction method, deveoped for a foating eastic pate by Hazard and Meyan (27), aows for the transient response to be obtained for arbitrary initia conditions. The spectra method provides a mapping from the time-domain to the frequency-domain through a sef-adjoint operator, so that the tempora evoution of the system may be obtained by cacuating an integra over the corresponding harmonic excitations. This is simiar to the approach adopted in the present work. However, because the system is initiay at rest we may use a simpe Fourier Transform as a map between the time and frequency domains. The transient response is then obtained after soving the governing equations for a sufficient number of frequencies through the inverse Fourier integra. In the frequency domain, the probem resembes a number of existing modes of inear wave scattering by ice foes. In particuar, the scattering occurring at the transition between an open water region and an ice-covered sea (ice edge) was originay soved by Fox and Squire (1994) using an eigenfunction matching method (EMM). An EMM invoves the decomposition of the fuid motion in each homogeneous region of the overa domain into an infinite sum of propagating and decaying waves. Truncated representations are then matched at their common interface and combined with fuid continuities in some manner to generate equations that determine the ampitudes (at east approximatey). In Fox and Squire (1994) a minimisation of a residua error was appied to derive the fina equations but different approaches have been deveoped subsequenty. Usuay inner-products of the continuity reations are taken with vertica moda functions over the fuid depth (see, e.g., Kohout et a., 27). A three-dimensiona appication of this method was proposed by Peter et a. (23) to sove for the singe circuar foe. Other soution methods are possibe and may give superior convergence and may aso aow for semi-infinite fuid depth. For exampe, Chung and Fox (22) and Linton and Chung (23) soved the semi-infinite ice sheet probem, using a Wiener-Hopf technique and residua cacuus, respectivey. However, EMMs remain popuar due to their speed for reasonabe accuracy and their versatiity. In particuar, they are the basis for the current modes of wave attenuation through a vast array of ice foes in 2-D (Kohout and Meyan, 28) and 3-D (Bennetts et a., 21). The current mode extends these previous approaches by incuding atera boundaries and a non-zero draught of the pates. The soution method is based on eigenfunction matching to cacuate the scattering produced by a singe pate s edge. It accounts for the singuarity induced by the draught of the pate by use of appropriatey weighted Gegenbauer poynomias (see Wiiams and Porter, 29). From this canonica probem a transfer matrix is obtained that maps ampitudes of the waves on one side of a pate s edge to the other, which aows us to deveop an iterative technique to extend the method to mutipe pates. This approach is simiar to the one used by Bennetts et a. (29) to generate the soution to the scattering by periodic variations embedded in a continuous ice sheet. The governing equations of the probem are given in 2, incuding the conditions at the wavemaker and 3

4 the beach. The transformation to the frequency-domain is defined in 3. The frequency-domain soution method, briefy outined above, is deveoped in 4 and numerica resuts are presented in 5. The transient response of the system is anaysed in terms of the strain energy in the pate. The infuence of the atera boundaries and mutipe pates on this quantity is investigated. In particuar, the steady-state response, as we as the maximum strain energy are anaysed over a frequency range that is reevant to the dimensions of a prototype wave tank. 2. Mathematica mode We investigate the scattering of sma-ampitude waves by a group of foating eastic pates in a twodimensiona fuid domain which is bounded ateray. Let x and z be the horizonta and vertica coordinates, and assume that the z-axis points upwards. A wavemaker, ocated at the eft-hand boundary of the domain (x = ), generates waves that wi set the system in motion. At the right-hand end (x = x b ) a beach exists that party attenuates the waves that reach it. We consider that the set of pates is incuded in a sub-domain bounded at its eft-hand end at x = x, and at its right-hand end at x = x r. The equiibrium fuid surface coincides with z = and the depth of the fuid domain is assumed to be constant, H say. We denote Ω = {x, z : x < x, H z } and Ω r = {x, z : x r < x x b, H z } the two exterior open water regions, and Λ and Λ r their respective free surfaces. It is reasonabe to begin the anaysis with a singe pate before extending to mutipe pates. We suppose that this singe pate has a constant thickness h, which is sma compared to its ength L. Let d = h (ρ/ρ ) be the Archimedean draught of the pate, where ρ is the density of the pate and ρ is the density of the fuid. We denote Ω = {x, z : x x x r, H z d} the water domain upon which the pate is foating, and Λ the pate s underside surface. In addition, we define the vertica interfaces between the fuid regions as Υ = {x, z : x = x, H z d} and Υ r = {x, z : x = x r, H z d}. The pate is aowed to have heave and pitch motions but surge motion is not considered. The pate aso experiences bending under wave forcing. Fig. 1 shows a schematic of the probem modeed Boundary vaue probem in the time-domain We assume the Reynods number is arge enough that viscous effects in the fuid can be negected. It foows that the fow is irrotationa, so the veocity fied can be written as the gradient of a scaar fied, known as the veocity potentia and denoted Φ(x, z, t). We aso suppose the fuid is incompressibe, so that Φ satisfies Lapace s equation 2 Φ =, in Ω Ω Ω r, (1) where = ( x, z ). Moreover, the foor of the fuid domain is impermeabe, so that the no-fow condition z Φ =, at z = H, 4 (2)

5 appies, and we express the inearised free-surface condition 2 t Φ + g z Φ =, on Λ Λ r, (3) where g 9.81 ms 2 is the acceeration due to gravity. The fexura behaviour of the pate is modeed by Euer-Bernoui beam theory. In doing so, we assume that the thickness of the pate and its deformation are sufficienty sma compared to the pate s ength. It then foows that the vertica dispacement of the pate s midde pane, η(x, t), fuy describes its behaviour. With reference to Timoshenko and Woinowsky-Krieger (1959), D 4 xη + ρh 2 t η = P P ρ w gd, on Λ, (4) where D and ρ are respectivey the fexura rigidity and density of the pate. The quantities P = P(x, t) and P are, respectivey, the pressure on Λ and the atmospheric pressure, whie ρ w gd is the buoyancy term. The pressure term in Eq. (4) is derived from the inearised version of Bernoui s equation appied on Λ, assuming no cavitation between the pate and water surface. After substitution, we obtain ( D 4 x + ρh t 2 + ρ wg ) z Φ + ρ w t 2 Φ =, on Λ, (5) where we have used the inearised kinematic surface condition t η = z Φ on Λ. Because free bending in the pate is assumed, the bending moment and shearing stress must vanish at its ends. In terms of veocity potentia this is expressed as 2 x zφ = 3 x zφ = at z = d, x = x, x r. (6) The motion of the wavemaker, ocated at x = (see Fig. 1), is described by the function X(z, t), which represents the ampitude of the padde inearised about x =. Fig. 2 shows the geometry of the hinged-fap wavemaker that we use in our mode, athough other shapes coud be considered. Foowing Schäffer (1996), the motion of the padde is expressed as X(z, t) = f(z)x (t), where f(z) is the shape function representing the padde geometry and X (t) is the time-dependent evoution of the wavemaker s ampitude at z =. In the present paper, we have 1 + z/(h ), (H ) z, f(z) = H z < (H ). The energy is transmitted to the fuid through the inearised kinematic condition (7) t X = x Φ, at x =. (8) Most wave tank experiments are designed to generate accuratey controed waves by a wavemaker over a period of time ong enough to reach a steady-state. For this reason, wave tanks are a equipped with an 5

6 absorbing zone, known as a beach, ocated at the opposite end from the wavemaker so that returning waves traveing back through the wave tank cause minima interference with the generated wave fied. Ideay a beach woud absorb a of the wave energy incident on it, but in practice even the most efficient beaches refect part of the energy. To account for this refection in our mode, the beach is simuated by a piston-ike wave absorber that can extract part of the incoming wave energy depending on the frequency. This approach was introduced by Cément (1996), who incuded such a condition as part of a non-inear numerica wave tank BEM sover. Let X b (t) be the horizonta dispacement of the wave absorber. A kinematic condition is prescribed, which is t X b = x Φ at x = x b. (9) In addition, the system is initiay at rest. Therefore the initia conditions Φ(x, z, ) = η(x, ) = X(z, ) = X b () = are appied. This competes the set of governing equations Non-dimensionaisation From here on we scae the spatia variabes by the depth H and the time variabes by H/g. Therefore we can express the non-dimensiona variabes (denoted by an over bar) as x = x H, z = z H, t = t H/g, X = X H, Xb = X b H, and Φ = Φ H Hg. For carity, the overbars are dropped for the remainder of the text, with the understanding that the variabes are non-dimensiona. Note that Eqs. (1), (2), (6), (8) and (9) remain unchanged, whie Eqs. (3) and (5) become 2 t Φ + z Φ =, on Λ Λ r, (1) and ( β 4 x + γ 2 t + 1 ) z Φ + 2 t Φ =, on Λ, (11) where β = D/ρ w gh 4 and γ = d/h. 3. Transformation to the frequency-domain Whist a direct soution of the time-dependent governing equations, as set out in the previous section, presents a chaenge, the corresponding time-harmonic probem may be tacked using standard methods. Accordingy, a Fourier transformation of the time variabes is used to recast the probem in the frequencydomain. For the present probem, we assume that the wavemaker forcing, characterised by X (t), is a continuous rea function of the time parameter and is causa, that is X (t) = for t <. A typica profie 6

7 is X (t) = R(t)cosω t, where ω is the radian frequency and R(t) is the enveope function, which is defined by the finite time signa ( th(t) (t t1 )H(t t 1 ) R(t) = A w + (t T)H(t T) (t t ) 2) H(t t 2 ), t 1 T t 2 where the function H(t) is the Heaviside step function and the parameters ω, t 1, t 2, T and A w are nondimensiona. The initia inear ramp settes after t 1 at A w and the decaying inear ramp starts at t 2 unti rest at T. The wavemaker ampitude time-evoution profie is potted in Fig. 3(a) for A w = 1, t 1 = 5, t 2 = 3, T = 4 and ω = π. The decaying ramp has the effect of smoothing the system s return to its equiibrium position. The time-dependent evoution of the function X(z, t) determines the transient response of the system, once the geometry is set. As we intend to convert the equations to the frequency-domain (equivaent to the harmonic excitation probem), we must Fourier transform the wavemaker signa. Let X define the one-sided Fourier transform of X, so that X(z, α) = F[X(z, t)] = X(z, t)e i αt dt = f(z)f[x (t)], (12) where α = Hω 2 /g is a non-dimensiona frequency parameter. For the present wavemaker forcing, the Fourier transform can be cacuated anayticay, the signa being non-zero during a finite duration. It can be shown that F[X (t)] = A w 2 where α = Hω 2 /g and [ F ( α α ) + F ( α + α )], F(θ) = e iθt1 1 + iθt 1 e iθt t 1 θ 2 + e iθt2 (1 + iθ (T t 2 )) e iθt (T t 2 )θ 2, θ R. For the set of parameters defined above, the rea and imaginary parts of F[X (t)], which are reated through the Kramers-Kronig reations, are given in Fig. 3(b). We recover the origina time-dependent signa by an inverse Fourier transformation X(z, t) = F 1 [ X] = 1 X(z, α)e i αt d α. 2π Likewise, the pair of transforms for the veocity potentia is written Φ(x, z, α) = Φ(x, z, t)e i αt dt and Φ(x, z, t) = 1 Φ(x, z, α)e i αt d α, 2π and, for the wave absorber s dispacement, it is X b (α) = X b (t)e i αt dt and X b (t) = 1 X b (α)e i αt d α. 2π 7

8 Taking the Fourier transform of Eqs. (1), (2), (6), (8), (9), (1) and (11), we obtain the foowing set of PDEs that describes the probem in the frequency-domain 2 Φ =, (x, z) Ω Ω Ω r, (13a) z Φ =, z = H, (13b) z Φ = α Φ, (x, z) Λ Λ r, (13c) ( β 4 x αγ + 1 ) z Φ = α Φ, (x, z) Λ, (13d) 2 x z Φ = 3 x z Φ =, z = d, x = x and x = x r, (13e) i α X = x Φ, x =, (13f) i α X b = x Φ, x = xb. (13g) The frequency-domain boundary vaue probem is then competey determined by the frequency parameter α, and the properties of the pate β and γ. To obtain the transient response of the system, this set of frequency-dependent equations must be soved over a range of frequencies that is determined by the spectra distribution of the input variabe, X (t) (see Fig. 3(b)). A soution method for these equations wi be outined in the subsequent section. In practice, we take advantage of the particuar spectra distribution, which is narrowband around the exciting frequency ω. Once the system of PDEs is soved for a sufficient number of frequencies, the timedomain soution is reconstructed by evauating the inverse Fourier integra of the potentia Φ. Severa integration schemes have been compared for this purpose: the Fast Fourier Transform (FFT) agorithm, a direct integration rue and a numerica quadrature method based on a doube exponentia formua (see Ooura, 25). We performed tests on the wavemaker input signa X (t) and it was found that the FFT was superior in both accuracy and speed. Such performances are expained as the FFT ony requires O(N s og N s ) operations to compute the transform, where N s is the number of frequency sampes, whie direct integration requires an inversion at each time sampe so that O(N s N t ) operations are needed, where N t is the number of time sampes. In impementing the FFT inversion, an idea ow-pass fiter (sinc fiter) is appied to the frequency-domain variabes by truncating the spectrum at an appropriate maximum frequency to imit the number of frequency sampes. This induces distortion in the reconstructed time signa in the form of sma rippes, and is characteristic of the truncation of Fourier series (here the Discrete Fourier expansion used by the FFT), commony caed the Gibbs phenomenon. Athough we can observe these osciations on the time responses, they are sma enough not to affect the resuts. 8

9 4. Frequency-domain soution method 4.1. Expansion of the veocity potentia We partition the fuid domain into 3 subdomains, as shown in Fig. 4. In each region, the potentia is expressed as the sum of two components Φ (±) i (i =, r), for the exterior regions, and Φ (±), for the interior region, where the superscript denotes the direction in which the wave propagates or decays. In open water regions (Ω and Ω r ), we approximate the potentia as foows Φ (±) i (x, z) N [ n= A (±) i ] (±) n e kn(x x i ) ϕ n (z) (i =, r), (14) where the vertica modes are ϕ n (z) = cosk n (z +H)/ cosk n H and we have truncated the infinite sums to N terms. The quantities k n are the roots K of the free-surface dispersion reation K tan KH = α, with k on the positive imaginary axis, and k n (n = 1,...,N) rea, positive and in ascending order. The exponentia functions, used to describe the horizonta motions, are associated with incident, refected and transmitted traveing waves (n = ) and aso decaying waves that are ony excited in the vicinity of the pate s edges (n = 1,...,N). In Eq. (14), the abscissa x (±) i the particuar wave is generated, so that x (+) expansion (14) using matrix and vector notations as where A (±) i (i =, r) corresponds to the ocation where =, x ( ) = x, x (+) r = x r and x r ( ) = x b. We can rewrite Φ (±) i (x, z) C (z)e (±) (x x (±) i )A (±) i (i =, r), (15) is the (N + 1)-ength coumn vector of the unknown ampitudes defined in Eq. (14). The row vector C (z) = (ϕ (z),..., ϕ N (z)) and the diagona matrix E (±) (x) = diag { e kx,..., e knx} have aso been introduced. In region Ω, we approximate the potentia as Φ (±) (x, z) N n= 2 [ A (±)] n e κn(x x(±)) ψ n (z), (16) where the vertica modes are ψ n (z) = cosκ n (z+h)/ cosκ n (H d), defined for a n 2, and the truncation parameter N is used again. The quantities κ n are the roots K of the pate-covered dispersion reation ( βk αγ ) K tan K(H d) = α, where we denote the unique root that ies on the positive imaginary axis as κ, and the positive rea roots κ n (n 1), which are ordered in ascending magnitude. In this case there aso exists a pair of compex conjugate roots with positive rea part, denoted κ 2 and κ 1, that support damped modes (traveing waves 9

10 that decay with distance from the scattering source). Bennetts et a. (27) proved that the vertica modes ψ n (z) (n 2) are not ineary independent and the degree by which this set of vertica modes can be reduced is two. It foows that ψ j (z) = v n,j ψ n (z), j = 1, 2. (17) n= The coefficients v n,j are expressed, foowing Bennetts (27), as v n,j = κ nr (κ j )(κ 2 n κ 2 j )cosκ n(h d) κ j R (κ n )(κ 2 j κ2 j )cosκ j(h d), n, where j = (3 ( 1) j )/2 and the prime superscript signas the derivative of the singe variabe function it is appied to. We have aso defined the foowing compex function R(ξ) = (1 αγ + βξ 4 )ξ sinξ(h d) + α cosξ(h d), ξ C. Appying the free edge conditions (13e) to Φ (±) aows us to express the ampitudes of the damped modes in terms of the ampitudes (A (±) ) n (n =,...,N), so that where [ A (±)] j = T(±) 1j A(+) + T (±) 2j A( ), j = 1, 2, (18) A (±) = ([A (±)],..., [ A (±)] N) T. Expressions for the row vectors T (±) ij truncated to N. We therefore write (i, j = 1, 2) can be found in A, and use Eq. (17), with the infinite sum Φ (±) (x, z) = C(z)D (±) (x)a (+) + C(z)E ( ) (x)a ( ), (19) where C(z) = (ψ (z),..., ψ N (z)). The (N + 1)-size square matrices D (±) (x) and E ( ) (x) are obtained by substituting Eqs. (18) into Eq. (16). The coefficients of the matrices D (±) (x) and E ( ) (x) are, as foows [ D (±)] n,p = δ n,p e κn(x x(±)) + [ E (±)] n,p = δ n,p e κn(x x(±)) + 2 [ j=1 2 [ j=1 T (±) 1j T (±) 2j ] ] p v n,j e κ j(x x(±)), p v n,j e κ j(x x where δ n,p = 1 if n = p and otherwise. We aso introduce the two matrices D(x) = D (+) (x) + D ( ) (x) and E(x) = E (+) (x) + E ( ) (x), for convenience. (±)), 1

11 4.2. Scattering by a pate s edge A version of eigenfunction matching method (EMM) is used to cacuate the scattering at a pate s edges. The method is demonstrated for the eft-hand edge, which is ocated at x = x. On the fuid coumn at this ocation we enforce continuity of fuid pressure and veocity. To do this the function u (z) is introduced, which is defined as the fuid norma veocity aong Υ. The continuities are then expressed as Φ (x, z) = Φ(x, z) and x Φ (x, z) = x Φ(x, z) = u (z), for z Υ, where Φ = as x Φ1 (x, z) =, d z. (+) ( ) Φ + Φ and Φ = Φ (+) + Φ ( ). Additionay, the no surge assumption is appied The submerged corner creates a singuarity in the fuid veocity. Numerica evidence suggests that this singuarity is of the same order as that in the case of a rigid pate of the same shape, i.e. it is of order 1/3 (Wiiams and Porter, 29). Foowing Bennetts and Wiiams (21), we capture this singuarity by approximating the function u as a sum of, appropriatey weighted, even Gegenbauer poynomias. We define the set { } of poynomias as V g = (z ) : m N, where we have introduced z = (z + H)/(H d) (, 1). Therefore et u (z) M m= C (1/6) 2m U () m C (1/6) 2m (z ), where the infinite sum has been truncated to M + 1 modes, chosen such that convergence towards u (z) is satisfactory. We express the eements of V g as C (δ) m (y) = 2 m!(m + δ)γ 2 (δ) (H d) π2 1 2δ Γ(m + 2δ) ( 1 y 2 ) δ 1/2 G (δ) m (y), y 1, with Γ the Gamma function and G (δ) m (y) a Gegenbauer poynomia (see, Abramowitz and Stegun, 197, 22). For convenience in what foows, we re-express the previous expansion as (2) u (z) = C g (z )U, (21) where U = (U (),..., U() M )T and C g (z ) = (C (z ),..., C 2M (z )). The subscript g denotes a quantity associated with the Gegenbauer projection basis, and the subscript denotes a quantity associated with the pate s eft edge. Using the matrix expansions of the potentia defined in the previous section, we appy the EMM as ( ) B E (+) (x )A (+) + E ( ) ()A ( ) = B g U, (22) ( B D (x )A (+) + E (x )A ( )) = B g U, to approximate the continuity of fuid veocity, and B g T ( E (+) (x )A (+) + A ( ) 1 ) = B g T ( D(x )A (+) + E(x )A ( )), (23) 11

12 for the continuity of fuid pressure, noting that E ( ) () is the identity matrix of dimension N + 1. In the above, the (N + 1)-size square matrices B and B respectivey contain the inner-products of the vertica modes of V = {ϕ n (z) : n N} and the eements of V = {ψ n (z) : n N}, and are defined as [B ] i+1,j+1 = H ϕ i ϕ j dz and [B] i+1,j+1 = d H ψ i ψ j dz (i, j =,...,N). Simiary, we introduced B g, the matrix of inner-products between the eements of V and V g, and B g, the matrix of inner-products between the eements V and V g. These matrices are non-square and have dimension (N +1) (M +1). The entries are cacuated using Gegenbauer s addition theorem (see, Wiiams and Porter, 29) and the orthogonaity of the Gegenbauer poynomias. They are given by d ( ) δ [B g ] i+1,j+1 = ϕ i C j (z 2 (z))dz = Γ (δ) (2j + δ)( 1) j J 2j+δ (k i (H d)), k i (H d) cosk i H and [B g ] i+1,j+1 = H d H ( ψ i C j (z (z))dz = Γ (δ) 2 κ i (H d) where J p is the Besse function of the first kind of order p. ) δ (2j + δ)( 1) j J 2j+δ (κ i (H d)), cosκ i (H d) Aowing N and M to be independent provides more numerica freedom, as their effect on the accuracy and convergence rate can be studied separatey. Consequenty, non-square matrices appear and care must be taken to manipuate them. The soution is found by expressing the vectors of ampitude of the scattered waves, i.e. A ( ) and A (+), in terms of the vectors of ampitude of the incident waves and the interface vector U. We eiminate this vector using Eq. (23), in which an (M + 1)-size square matrix appears that can be inverted. We may cacuate the scattering matrix for the pate s edge. It is denoted S, and maps the ampitudes of the waves incident on the eft edge of the pate to the ampitudes of the scattered waves induced by it. Aternativey, we may express the ampitudes of the waves in the pate-covered fuid region in terms of the ampitudes in the open fuid region, through transfer matrix T. The matrices S and T are defined by A( ) = S A(+) and A(+) = T A(+), (24) A (+) A ( ) A ( ) A ( ) and their entries may be reated in a straightforward manner (see Bennetts and Squire, 29). Simiar expressions can be obtained for the pate s right-hand edge, where we denote the respective scattering and transfer matrices by S r and T r. This means that we may write the transfer matrix for the pate, T say, as the simpe expression T = T r T. The matrices T and T r are cosey reated due to the symmetry of the pate, as one maps from the open water region to the pate-covered region and the other, vice-versa. However, they are not direct inverses of one another, as the phases of the incident and refected waves are different at each edge. Note that T depends on the pate s ength and thickness. 12

13 4.3. Scattering by mutipe pates We now consider the case in which there are two pates, separated by a free-surface region, and which may have different properties. We re-partition the fuid domain into 5 fuid regions; the two exterior open water regions Ω p (p =, r), the two pate regions Ω q (q = 1, 3) and the interior open water region Ω 2. They are contained in the respective intervas (x (+) i, x ( ) i ), such that x (+) =, x ( ) = x (+) 1 = x, x ( ) 1 = x (+) 2, x ( ) 2 = x (+) 3, x ( ) 3 = x (+) r = x r and x r ( ) = x b. We denote the potentias in the 5 regions Φ i (x, z) accordingy. In regions Ω i, i =, 2, r, they are given by Eq. (15), whie in pate-covered fuid regions Ω i, i = 1, 3, they are expressed as Φ i (x, z) = C i (z) ( D i (x)a (+) i ) + E i (x)a ( ) i, i = 1, 3. (25) The vector C i (z) and the matrices D i (x) and E i (x) denote the corresponding arrays C(z), D(x) and E(x) when evauated using the properties of the pate in region Ω i. Using the soution method described in 4.2, we define T 1 and T 3 as the transfer matrices associated with pates in regions Ω 1 and Ω 3, respectivey. The use of transfer matrices provides an efficient means of combining the interactions of mutipe pates. In particuar, we can map the vectors of ampitudes in Ω to the ones in Ω r as foows A(+) r A r ( ) = T A(+) A ( ), (26) where T = T 3 T 1 is the transfer matrix associated with the group of pates. This procedure can be generaised to an arbitrary number of pates in a straightforward manner. To do this, the transfer matrix, T, associated with the corresponding group of pates is obtained by mutipying the transfer matrices associated with each individua pate. The scattering matrix S associated with the group of pates is then obtained from the entries of T, so we have A( ) A (+) r = S A(+) A r ( ) 4.4. Wavemaker forcing and beach refection. (27) The soution method for the first-order wavemaker is we-known (see, e.g., Linton and McIver, 21) and we ony rewrite the method (EMM) here so the reation with our previous notations is apparent. The scattering condition is deveoped by integrating Eq. (13f) mutipied by C T (z) over the vertica domain, so we obtain the foowing matrix equation i α X ( ) F = B E (+) ()A (+) + E ( ) ( x )A ( ), (28) where we have introduced the (N + 1)-ength vector F = H f(z)c T (z)dz, 13

14 which has entries that may be cacuated using standard techniques. We may re-arrange Eq. (28) to obtain the vector of ampitudes of the waves generated in terms of the incoming wave ampitudes A ( ), and the externa forcing vector F, as A (+) = S A ( ) + F. (29) The (N + 1)-size square matrix S and (N + 1)-ength vector F are easiy cacuated by re-arranging (28). On the far end of the fuid domain (x = x b ), we may expand the refective beach condition. We consider a refection coefficient of magnitude R b, between and 1, which depends on the frequency α as ( ) R b (α) =.9 e 5 α 1 + 1, (3) which quaitativey characterises the refection spectrum of a typica beach in a wave tank (see, e.g., Bonnefoy et a., 26b). We aso assume the beach does not refect evanescent waves, assuming the pates are far enough from this boundary. This approach is equivaent to ignoring the vertica dependence and expressing the condition at z =. As a consequence, the contro aw of the wave absorber s dispacement X b is defined such that a singe traveing mode is generated at x = x b as foows C b Φ( ) r (x b, ) = R b C b Φ(+) r (x b, ), (31) where C b = (1,,..., ) T is a (N + 1)-ength vector. Using Eq. (14) and re-arranging aows us to express the beach condition as a matrix equation A ( ) r = S b A (+) r, (32) where S b = R b C b C ()E (+) (x b x r ) is the scattering matrix associated with the beach. The soution of the probem can now be obtained, for any frequency α, by soving the couped system of matrix equations of size N + 1 defined by Eqs. (27), (29) and (32), for an arbitrary number of pates. The soution method is based on mutipe substitutions and is straightforward to impement. The soution method for the frequency-domain probem, which is described above, is highy efficient. In particuar, we note that, in most cases, ony 5 Gegenbauer poynomias are required to capture the fuid veocity beneath the pate s edge to four decima pace accuracy. It is therefore possibe to sove for the O( ) frequencies required, depending on the samping frequency and the signa duration, to produce the time-dependent response. 5. Numerica resuts The numerica resuts presented in this section concentrate on the fexure of the pate (or pates) under wavemaker forcing. Our intention is that these resuts may give some insights into issues arising from typica 14

15 aboratory experiments in two-dimensiona wave fumes, such as the infuence of the transient response and the boundaries of the fuid domain. Athough the experimenta campaign mentioned in the Introduction is not intended to be directy reated to the resuts and concusions drawn in the present section, we have chosen vaues for the parameters in accordance with the experimenta setup described in Montie et a. (211). In the subsequent computations, we have set H = 2 m and x b = 15 m as the dimensions of the fuid domain, and the wavemaker padde is hinged such that =.3 m. The eastic properties of the pates, made of expanded PVC, are described by E = 5 MPa and ρ = 5 kgm 3, which are vaues that are simiar to typica sea-ice when scaed. Additionay, it is appropriate to set the pate s ength to be constant, L = 1.5 m say, so the pate ength to waveength ratio ony varies with the frequency, and we set x = 1 m. The initia ramp of the wavemaker time evoution is chosen foowing typica experimenta vaues to be t 1 = 2 s. We generate resuts for three pate thicknesses, namey h = 3, 5 and 1 mm, as they are common vaues for the materia utiised here Strain energy The strain energy of a pate is an integrated vaue of the bending over the ength of the pate. Foowing Timoshenko and Woinowsky-Krieger (1959), the strain energy of a two-dimensiona pate, ocated between x = and x = r, is given by V = 1 r 2 D ( 2 x η ) 2 dx, (33) where xη 2 is the curvature. We scae the strain energy by the incident wave energy, i.e. ρ gi/2, 2 where I is the steady-state ampitude of the wave generated Frequency response of a singe pate We begin by anaysing the frequency spectrum of the strain energy for a singe pate. It is noted that inear modes are characterised by resonances, typicay due to phase interactions. However, in a wave tank we expect that such effects woud be tempered by auxiiary physica processes and therefore woud not be repicated exacty, if at a. To understand how the wavemaker and the beach infuence the fexura response of the pate, we study four cases. In case 1, we set the beach refection coefficient to be (perfect beach) and we aso assume that the wavemaker is invisibe to waves traveing from right to eft, so we have no refection on both atera boundaries. This case is equivaent to the frequency response of a singe pate in an unbounded domain (studied by a number of previous authors, e.g., Meyan and Squire, 1994). In case 2, the wavemaker refects waves but the beach is sti perfect (R b = ), whie in case 3, the wavemaker is invisibe and the beach induces refection dependent on the frequency, as described by Eq. (3). In case 4, both wavemaker and beach refect waves. 15

16 In Fig. 5, the scaed strain energy is potted against frequency for the three thicknesses mentioned above. We choose a frequency range, f = Hz, corresponding to waveengths λ.7 6 m. We compare the infuence of the domain atera boundaries on the response using cases 1 (dashed ine) and 4 (soid ine). Whie the fexura response varies smoothy when no atera refections are considered, the presence of the wavemaker and the beach ceary disturbs the spectra. Primariy the incident wave passes through the pate but atera refections add a structure over this. The resonances correspond to natura modes of the system {Tank+Pate}. Fig. 6 dispays resuts simiar to those in Fig. 5(b) (i.e. for the 5 mm pate ony) but for an extended frequency spectrum. Even though it is unreaistic to generate such waves in rea wave tanks, the arger bandwidth is used to highight the infuence of the beach and the wavemaker. Their effects are anaysed separatey. In Fig. 6(a), we ook at the infuence of the wavemaker. The soid ine represents the fexura response spectrum when the beach is not present (case 2). Case 1 is given for reference as a dashed ine. We note that, for case 2, the magnitudes of the resonance peaks vary smoothy with frequency and a trend enveope drives the evoution of those peaks. This enveope coincides with the case 1 curve, when the atter reaches a maximum. Those maxima correspond to the pate s natura modes. Meyan and Squire (1994) noticed that the resonances in fexure for case 1 occur when the refection coefficient of the pate is zero. As the waves generated by the wavemaker are perfecty transmitted and no beach is present, case 1 and case 2 are equivaent at those frequencies. Fig. 6(c) shows the refection coefficient spectrum and confirms these remarks. In Fig. 6(b), we anayse the effects of the beach on the fexura response of the pate (case 3 as a soid ine and case 1 as a thick dashed ine for reference). The beach acts ike a wa which attenuates most of the incident wave s energy, so the magnitudes of the resonance peaks are significanty smaer than those induced by the wavemaker (wa with no attenuation). We notice that the succession of peaks aso fits in a smoothy varying enveope. The main difference with case 2 is that the resonance enveope reaches its maxima at the same frequencies as in case 1. Perfect transmission (see Fig. 6(c)) aows a the wave energy to reach the beach, inducing waves of arger ampitude there, so the natura modes of the system {Pate+Beach} have higher peaks. Athough difficut to see, it is aso noted that the resonance enveope and the response of case 1 match at f = 1.9 Hz and f = 1.99 Hz, corresponding to the cases in which the beach has no effect on the response. As the draught of the pate is taken into account in the present mode, we can anayse the effect it has on the resuts compared to the commony used shaow-draught approximation. In Fig. 6(b) and (c), the strain energy and refection coefficient spectra are given as dotted ines to aow comparisons to be made. For this pate s thickness, the frequency responses seem to stretch towards high frequencies compared to the ones when draught is incuded, suggesting that high frequencies are more affected than ow frequencies. A simiar infuence was found for other thicknesses, athough the resuts are not shown in the present paper. 16

17 5.3. Transient response of a singe pate In this section, we anayse the transient evoution of the scaed strain energy, V. We are particuary interested in the infuence of the fuid domain s atera boundaries on generating strain energy of high magnitude after the wavefront has passed, due to mutipe refections, as we as the corresponding response time, defined as the time it takes for the system to reach its steady-state. Fig. 7 shows the time-dependent evoution of a singe pate s strain energy unti it reaches steady-state. Fig. 7(a) (c) dispays the fexura response of 3, 5 and 1 mm-thick pates at f =.87 Hz, respectivey. Simiary, Fig. 7(d) (f) give the respective responses at f = 1.4 Hz. We note that, for case 1 (thick dashed ines), the responses tend to their steady-state reativey rapidy after the wavefront has passed. This behaviour is of interest, as the strain imposed on an eastic pate due to a transient periodic excitation can be arger than the frequency-domain anaysis suggests. For case 4 (soid ines), the infuence of the atera boundaries becomes apparent through the onger response-time. Whie it takes 25 s to reach the steady-state for a frequency f =.87 Hz in case 1 (see Fig. 7(a) (c)), the response-time is more than 1 s in case 4. The mutipe refections on the wavemaker and the beach expain this behaviour and simiar observations are made for f = 1.4 Hz (see Fig. 7(d) (f)), where the response-time is even arger than at ower frequencies. At this frequency the pate generates more scattering, as can be seen in Fig. 6(c), and the waves trave back and forth between the wavemaker and the pate, and the pate and the beach, before setting to a steady-state. Note that the case 4 responses reach case 1 steady-state for a short time interva, before the refected waves re-disturb the system. This means that, whie performing experimenta tests in a wave tank, i.e. case 4, it is possibe to capture the steady-state of case 1 during a time window that depends on the frequency of the incident wave, assuming the pate is ocated far enough from the boundaries of the tank. This is particuary usefu for comparing experimenta data to frequency-domain modes and this approach has been used by Montie et a. (211) to compare their experimenta and numerica data. In practice, the efficiency of this experimenta approach argey depends on the frequency. In particuar, at ow frequency (f <.5 Hz), the incident wave traves quicky through the tank, so the time window is very short or nonexistent, but in this case there is itte scattering produced by the pate, so very itte interest is given to this regime as part of the arge body approximation. We aso note that two major disturbances occur in the response of case 4 before the curves sette down. Those events can be identified as the wavefront of the wave generated by the wavemaker, as in case 1, and a wave that has traveed from the pate to the wavemaker and back to the pate after experiencing refection. This impies that in most cases the effects of the boundaries are overcome after two refections ony. In Fig. 7(e), this is not true and the steady-state is reached graduay at a time significanty different from the one reached after the second disturbance. Athough the response to the second major disturbance may give an idea of the steady-state vaue, other minor disturbances sti occur afterwards. This is particuary true for thick pates, as we can see in Fig. 7(c) and (f), in which cases the mutipe refections affect the response 17

18 for a ong time, which is reated to the fact that thicker pates induce more wave scattering. The atera boundaries aso affect the time at which the maximum scaed strain energy, V max, say, is obtained. Athough the first wavefront aways induces a maximum, the mutipe refections may cause the time-response to reach a higher peak subsequenty (see Fig. 7(c)). In what foows, we ony consider the pate of thickness h = 5 mm. From the transient response, we can extract characteristic properties that may be anaysed over a reevant frequency range. We are particuary interested in the maximum strain energy V max and the time at which this maximum is reached t max. Fig. 8 dispays V max and t max over the frequency range f = Hz. Case 1 (unbounded domain) is anaysed in Fig. 8(a) and (b). The maximum strain energy (soid ine) behaves simiary to the frequency-domain response (dashed ine), as we can see in Fig. 8(a), and the difference in magnitude seems to increase with the frequency. We note from Fig. 8(b) that t max increases monotonicay over the spectrum as the frequency rises. In this case the maximum is aways reached as the incident wavefront starts exciting the pate, as previousy seen in Fig. 7. Simiar resuts for case 4 (ateray bounded fuid domain) are dispayed in Fig. 8(c) and (d). We note that at ow frequencies, i.e. f <.9 Hz, V max and t max are the same as for case 1, so the initia wave front generates the maximum bending in the pate, as we see in Fig. 7(b). For higher frequencies, there is a succession of peaks in Fig. 8(c), emerging from the curve representing case 1. Reating these peaks to the time at which they occur (see Fig. 8(d)), we note that they correspond to maxima reached ater than the time the initia wavefront hits the pate, after refections on the atera boundaries Mutipe pates anaysis In this section, we investigate the response of mutipe identica pates. A first arrangement of four pates of thickness h = 1 mm is studied in the time-domain for case 4 (bounded fuid domain). The abscissae of the pates eft edges are 4, 6, 1 and 12 m and the frequency of the wave generated is f = 1 Hz. Fig. 9 shows the evoution of the pates dispacements and fuid-surface eevations for t = 1, 5, 1, 15, 2, 25 s. A decrease in the ampitude of fexure of the pates from eft to right is ceary visibe (for 2 and 25s). This is a resut of the attenuation of wave energy produced by scattering (see, e.g., Kohout and Meyan, 28). As the system {Pates+Fuid} is conservative, the attenuated energy is refected back towards the wavemaker (athough it is distributed amongst waves with different phases). A more detaied anaysis on this mutipe pate arrangement woud be compex to perform, so that we focus our attention for the remainder of this section on a more fundamenta study case. Consider two identica pates, with properties the same as the singe pate case, and focus the present anaysis on a singe thickness, h = 5 mm, as the infuence of this parameter was studied earier. We denote the spacing between the two pates as s. Additionay, we set x = 6 m as the ocation of the eft edge of pate 1; it foows that the right edge of pate 2 is ocated at x r = 9 + s. It is of use for the anaysis of two pates to define the reative strain energy of the system as V r = V 2 /V 1, where V 1 and V 2 respectivey are the 18

19 strain energy of pates 1 and 2. We choose to remove the refections from the wavemaker and beach (i.e. case 1). This aows us to determine the interactions of the two pates without interference from the fuid domain s atera boundaries. In Fig. 1(a) and (c) we show the reative strain energy potted against frequency for two spacings that are s = L/3 =.5 m and s = 3L = 4.5 m, respectivey. Fig. 1(b) and (d) dispay the corresponding refection coefficient spectra. Simiary to Fig. 6, we ook at the reative response of the system over an extended spectrum f =.5 3 Hz. For the smaer spacing (Fig. 1(a) and (b)), the reative response osciates over the spectrum, such that when V r > 1 the fexura motion is dominated by pate 2, whie when V r < 1, pate 1 s fexure dominates. In the atter case, Fig. 1(b) indicates that the cause is a high refection coefficient. For the arger spacing (Fig. 1(c) (d)), the number of resonances discussed in the previous paragraph is significanty increased, but we observe that the same features are present. The resonances are reated to the natura frequencies of the system {Pate 1+Pate 2}. When the spacing increases, harmonics of ower frequencies are aowed to resonate and this is what is demonstrated here. Simiar to the smaer spacing, minima of reative strain energy occur for arge refection coefficients. It is aso interesting to note that a the curves shown in Fig. 1 fit into an enveope that is the same for both spacings. The important features of this enveope are driven by the spectrum of the singe pate. In particuar, when comparing with the dashed curve from Fig. 6, we note that the enveope coincides with V r = 1 when the response of a singe pate is maxima and when the refection coefficient of a singe pate is zero. The enveopes of the refection coefficient curves in Fig. 1 are aso simiar and it is true for any spacing. The behaviour of the refection coefficient is in agreement with the resuts given by Wiiams and Squire (28) for an open ead. It is aso interesting to note that for the frequencies at which we have perfect transmission, we aso have V r = 1, i.e. equipartition of the energy in the pates. The reciproca proposition is, however, not true, i.e. V r = 1 does not aways correspond to perfect transmission. The transient response of the system {Pate 1+Pate 2} is studied for case 1 in the foowing. Let us define the ratio of the maxima strain energy as Ṽmax = Vmax 2 /V max 1, where V max i is the scaed strain energy in pate i (i = 1, 2). We show Ṽmax for both spacings in Fig. 11(a) and (b). For comparison, we aso define Ṽmax, to be the ratio of maximum strain energy experienced by the pates separatey, in the absence of the other pate. This quantity is potted in Fig. 11(a) and (b) as sma circes. We note that this quantity is not 1 in genera, as the ampitude of the wavefront varies with the distance from the wavemaker. The transient wave generated by the wavemaker is composed of superposed components of different frequencies, which therefore trave at different group veocities, resuting in a time evoution of the wavefront ampitude. We can see that Ṽmax tends to vaues ess than 1 as f decreases. This ow frequency behaviour is ceary due to the transient evoution of the wavefront. At ow frequencies very itte scattering is occurring and the incident wave traves through the fuid domain unperturbed. This is confirmed by the ow frequency spectrum of Ṽ max, as we can see in Fig. 11(a) and (b). 19